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4151(complexity)X
4578(via)X
4715(loga-)X
2612 3384(rithmic)N
2915([)X
2991(Paterson)X
3341(Ruzzo)X
3613(Snyder)X
3909(])X
3985([)X
4061(Thompson)X
4485(Fourier)X
4791(])X
4867([)X
2612 3496(Thompson)N
3031(Raghavan)X
3425(and)X
3593(linear)X
3835(delay)X
4067([)X
4137(Bilardi)X
4424(Preparata)X
4797(])X
4867([)X
2612 3608(Chazelle)N
2943(Monier)X
3224(to)X
3315(signal)X
3548(propagation)X
3992(delay)X
4206(that)X
4362(is)X
4444(square)X
4696(in)X
4788(the)X
2612 3720(length)N
2861(of)X
2962(wires)X
3180([)X
3237(Bilardi)X
3511(Preparata)X
3871(])X
3928([)X
3985(Chazelle)X
4321(Monier)X
4607(])X
4663([)X
4719(Seitz)X
2612 3832(Ensemble)N
3002(In)X
3117(all)X
3248(of)X
3363(these)X
3586(models)X
3883(the)X
4034(width)X
4278(\(or)X
4423(thickness\))X
4823(of)X
2612 3944(wires)N
2828(is)X
2913(assumed)X
3242(to)X
3336(be)X
3444(a)X
3508(unit,)X
3693(depending)X
4085(on)X
4198(the)X
4331(minimal)X
4652(feature)X
2612 4056(width)N
2856(of)X
2973(the)X
3125(underlying)X
3551(technology)X
3987(as)X
4104(seems)X
4363(suggested)X
4754(in)X
4867([)X
2612 4168(Mead)N
2849(Conway)X
3179(])X
3244([)X
3309(Mead)X
3545(Rem)X
3748(performance)X
4228(However,)X
4607(for)X
4745(sub-)X
2612 4280(linear)N
2860(signal)X
3118(propagation)X
3586(delay)X
3824(this)X
3999(is)X
4105(a)X
4192(misunderstanding)X
4867([)X
2612 4392(Mead)N
2842(Rem)X
3039(delays)X
3294(])X
3353(of)X
3456(which)X
3701(the)X
3839(consequences)X
4352(are)X
4489(explored)X
4827(in)X
2612 4504(the)N
2742(present)X
3018(paper.)X
3 f
2612 4728(2.)N
2722(Electronic)X
3131(Basics)X
1 f
2812 4868(The)N
2978(time)X
3165(it)X
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3900(transistor)X
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4357(drive)X
4568(a)X
4637(wire)X
4823(of)X
2612 4980(length)N
2 f
2860(L)X
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2909(,)X
2958(width)X
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3186(W)X
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3286(and)X
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3795(H)X
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3885(can)X
4034(be)X
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3978(D)X
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4041 5109(l)N
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4093 5092(to)N
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2771(D)X
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2834 5221(w)N
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2916 5204(to)N
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4218(W)X
1 f
8 s
4301 5221(0)N
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4371 5204(is)N
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3359(the)X
3509(current)X
3800(technology,)X
4256(then)X
4450(the)X
4600(minimal)X
2612 5428(transistor,)N
2992(consisting)X
3380(of)X
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3550(wire)X
3735(crossing,)X
4079(occupies)X
4416(area)X
2 f
4591(W)X
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8 s
4669 5445(0)N
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4708 5428(.)N
4759(The)X
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2972(T)X
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3043(to)X
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3337(a)X
3398(wire)X
3576(is)X
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4164(by:)X
2 f
3452 5708(T)N
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9 f
3530 5691(~)N
3530 5713(~)N
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3600 5709(\()N
2 f
3629(R)X
8 s
3683 5726(t)N
11 s
9 f
3708 5709(+)N
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3756(R)X
8 s
3810 5726(w)N
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11 s
3860 5709(\))N
2 f
3903(C)X
8 s
3962 5726(w)N
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11 s
4794 5709(\(1\))N
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2864(R)X
8 s
2918 5893(t)N
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11 s
2982 5876(is)N
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3227(resistance)X
3613(of)X
3725(the)X
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4645(R)X
8 s
4699 5893(w)N
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11 s
4788 5876(the)N
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3618(C)X
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3677 6005(w)N
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3766 5988(its)N
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2980(T)X
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3055(can)X
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3312(thought)X
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4532(T)X
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4581 6117(d)N
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11 s
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4282(C)X
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4391 6212(,)N
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2612 6324(R)N
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2789 6341(w)N
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2874 6324(needed)N
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2995(source.)X
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3642(T)X
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3691 6453(d)N
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11 s
3765 6436(is)N
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3911(R)X
8 s
3965 6677(w)N
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4074 6677(w)N
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4150 6660(is)N
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4419(w)X
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4478(.)X
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2994(of)X
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3176(wire)X
3367(is)X
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4036(its)X
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3741(we)X
3865(have:)X
2 f
3484 7199(R)N
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3538 7216(w)N
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2 f
9 f
3610 7199(=)N
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3680(r)X
2 f
8 s
3728 7216(w)N
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2 f
3822 7260(WH)N
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10 f
3809 7173(h)N
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1 f
4794 7199(\(2\))N
ep
%%Page: 3 3
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11 s 0 xH 0 xS 1 f
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0 32(--)N
4323(--)X
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240(r)X
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8 s
288 993(w)N
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11 s
364 976(is)N
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960(of)X
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436(of)X
534(a)X
598(wire)X
779(is)X
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959(layers,)X
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536(that)X
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767 1532(w)N
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839 1515(=)N
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992 1532(w)N
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1126(\()X
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1191 1576(D)N
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1216 1463(H)N
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1339 1515(+)N
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1423 1576(D)N
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1486 1593(l)N
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1431 1463(W)N
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1410 1489(h)N
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1547 1515(\))N
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241(e)X
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8 s
280 1761(w)N
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11 s
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1126 1985(2)N
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1165 1968(\).)N
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813(2)X
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1273 2182(\()N
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1633 2260(l)N
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1767(.)X
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2182(\(4\))X
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2026(.)X
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727(R)X
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781 2540(t)N
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11 s
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961(L)X
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1044(under)X
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924 4444(2)N
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247(t)X
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319(is)X
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1326(C)X
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1325(r)X
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1381(drivers:)X
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9 f
628(=)X
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698(log)X
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821 5505(a)N
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924 5549(C)N
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898 5462(h)N
902(hhh)X
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1086 5488(&)N
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1206(a)X
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1283(=)X
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1367 5400(I)N
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1585 5339(1)N
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1569 5358(hh)N
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1 f
1690 5488(,)N
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248(T)X
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297 5752(d)N
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358 5735(=)N
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428(r)X
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483(t)X
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619(time)X
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898 6079(C)N
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1004 6079(=)N
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1074(e)X
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8 s
1113 6096(t)N
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10 f
1160 6053(h)N
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238(D)X
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311 6325(0)N
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1346(insulator)X
1680(and)X
9 f
1832(e)X
2 f
8 s
1871 6325(t)N
1 f
11 s
1921 6308(is)N
2005(the)X
2138(pro-)X
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789(y)X
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1345(and)X
1499(the)X
1634(dielectric)X
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417(insulator.)X
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1501(a)X
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2 f
536(C)X
8 s
595 6661(w)N
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11 s
671 6644(through)N
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1479(r)X
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1538(drivers)X
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2135(a)X
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2215(in)X
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9 f
901 6976(=)N
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971(a)X
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1101(log)X
8 s
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1214 6993(a)N
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1309 7037(C)N
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1356 6923(w)N
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10 f
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1287(hhh)X
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2143(\(8a\))X
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667(a)X
747(ramp)X
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1085(r)X
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1161(drivers,)X
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1841(\256xed)X
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2059(r)X
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2093(,)X
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2878(a)X
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2955(accordingly:)X
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3554 1252(=)N
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3624(r)X
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3665(t)X
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3774 1313(C)N
8 s
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8 s
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1 f
11 s
10 f
3748 1226(h)N
3752(hhh)X
3892 1164(M)N
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2 f
8 s
3939 1185(r)N
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3936 1103(1)N
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4041 1252(.)N
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4750(\(8b\))X
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3920(an)X
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4423(for)X
4547(T.)X
2 f
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2989 1702(~)N
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3211(log)X
8 s
9 f
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3559 1720(+)N
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3629(2)X
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(r)S
2 f
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3721 1737(w)N
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4575(.)X
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4755(\(9a\))X
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2 f
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3005 2207(~)N
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3152(t)X
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3227 2137(I)N
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3462 2158(r)N
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3542 2225(+)N
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3612(2)X
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(r)S
2 f
8 s
3704 2242(w)N
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3768 2225(e)N
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3971 2138(2)N
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3910(hhh)X
1 f
4064 2225(\()N
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11 s
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2 f
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4277 2225(+)N
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1 f
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4558(.)X
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4750(\(9b\))X
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3645(,)X
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3273(a)X
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3489 3280(w)N
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4071 3263(+)N
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2612 3491(T)N
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2683(grows)X
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3440(L)X
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3489(.)X
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9 f
3052 3767(t)N
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3119 3679(I)N
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3273 3839(e)N
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8 s
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8 s
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3177 3697(e)N
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8 s
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(-)S
2 f
3532(r)X
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3163 3741(h)N
3181(hhhhhhhhh)X
1 f
3613 3767(\()N
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3703 3715(H)N
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3664 3741(h)N
3672(hhh)X
2 f
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3826 3767(+)N
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3910 3828(D)N
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3918 3715(W)N
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3897 3741(h)N
3924(hh)X
1 f
4034 3767(\))N
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4063 3679(M)N
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8 s
4110 3700(r)N
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4107 3618(1)N
10 f
4091 3637(hh)N
11 s
1 f
4706 3767(\(10b\))N
9 f
3585 4105(~)N
3585 4127(~)N
3655 4123(r)N
2 f
8 s
3703 4140(w)N
1 f
11 s
9 f
3767 4123(e)N
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8 s
3806 4140(w)N
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3892 4184(WH)N
3936 4071(L)N
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3879 4097(h)N
3910(hhh)X
1 f
4072 4123(\()N
2 f
4137 4184(D)N
8 s
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11 s
4162 4071(H)N
1 f
10 f
4123 4097(h)N
4131(hhh)X
2 f
9 f
4285 4123(+)N
2 f
4369 4184(D)N
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4432 4201(l)N
11 s
4377 4071(W)N
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10 f
4356 4097(h)N
4383(hh)X
1 f
4493 4123(\))N
4566(,)X
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2612 4369(T)N
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2702(grows)X
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3710(.)X
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2968 4710(T)N
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3046 4693(~)N
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3221(a)X
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4058 4711(+)N
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1 f
4266 4711(\))N
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4345 4711(+)N
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4407(1)X
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4473 4623(M)N
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4711 4711(\(11a\))N
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2 f
3020 5233(T)N
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9 f
3098 5216(~)N
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3168 5234(\()N
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3197(r)X
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3252(+)X
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3314(1)X
3372(\))X
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3415(t)X
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3482 5146(I)N
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8 s
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9 f
4066 5234(+)N
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4274 5234(\))N
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8 s
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4706(\(11b\))X
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4316(in)X
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9 f
3130 6663(~)N
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3273 6742(C)N
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3320 6628(w)N
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11 s
10 f
3247 6655(h)N
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9 f
3405 6681(+)N
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3467(C)X
8 s
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11 s
3590 6681(R)N
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3644 6698(w)N
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11 s
4750 6681(\(12\))N
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9 f
3130 6961(=)N
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3200(\()X
9 f
3273 7033(e)N
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8 s
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8 s
3415 7050(0)N
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3288 6891(t)N
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3327(D)X
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8 s
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3489 6961(+)N
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3551(r)X
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8 s
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3685 7022(WH)N
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3672 6935(h)N
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3865 6961(\))N
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9 f
(e)S
2 f
8 s
3999 6978(w)N
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4063 6961(L)N
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4141(\()X
2 f
4214 7022(D)N
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4200 6935(h)N
4208(hhh)X
2 f
9 f
4362 6961(+)N
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4446 7022(D)N
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4509 7039(l)N
11 s
4454 6909(W)N
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10 f
4433 6935(h)N
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4578 6961(\))N
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4651(.)X
1 f
2812 7229(The)N
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11 s 0 xH 0 xS 1 f
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0 32(--)N
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668(Delay)X
905(and)X
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962 2551(+)N
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985(L)X
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8 s
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230(W)X
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1834(L)X
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2025(/)X
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2057(\()X
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2086(a)X
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2141(-)X
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2189(1\).)X
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815(hhh)X
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5682(1)Y
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(-)S
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1779(.)X
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2094(\(13b\))X
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1870(\))X
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1494(L)X
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636(\271)X
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1392(X)X
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1467(=)X
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1529(a)X
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6617(X)Y
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2125(a)X
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6617(X)Y
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2245 6600(a)N
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1085(X)X
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1139(.)X
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947(a)X
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1024(by:)X
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959(=)X
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1168(-)X
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1216(a)X
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1271(\))X
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3608(\()X
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3637(L)X
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3696 1476(2)N
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3785 1520(2)N
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2806(=)X
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2854 MX
(12)130 767 oc
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4740(\()X
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4867 2984(\))N
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4427(\()X
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(-)S
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4582(1)X
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4674 3096(\))N
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2812 3572(The)N
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3873(-)Y
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2801(6)X
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3146(6)X
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3716(\264)X
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3447(root,)X
3638(then)X
3817(either)X
2 f
4044(r)X
9 f
4092(-)X
2 f
4140(p)X
1 f
4210(or)X
2 f
4309(r)X
9 f
4357(-)X
2 f
4405(q)X
1 f
4475(is)X
4560(a)X
9 f
4625(W)X
1 f
4693(\()X
9 f
4722 1883 MXY
(sr)214 549 oc
10 f
4770(d)X
4794(d)X
2 f
4770 1900(A)N
1 f
4838(\)-)X
2612 2012(length)N
2855(subpath)X
3151(of)X
3246(a)X
3307(path)X
3481(from)X
3674(the)X
3804(root)X
3968(to)X
4059(a)X
4120(leaf.)X
10 f
4296(`)X
2 f
2812 2152(De\256nition.)N
1 f
3244(We)X
3401(denote)X
3671(by)X
2 f
3794(A)X
1 f
3862(\()X
2 f
3891(N)X
1 f
3957(\))X
4021(the)X
2 f
4164(minimal)X
4488(area)X
1 f
4684(to)X
4788(lay)X
2612 2264(out)N
2766(a)X
2846(complete)X
3211(binary)X
3477(tree)X
3649(with)X
2 f
3846(N)X
1 f
3945(leaves)X
4205(under)X
4445(the)X
4593(different)X
2612 2376(assumptions)N
3070(on)X
3180(wire)X
3358(area)X
3526(and)X
3675(cross)X
3877(over.)X
3 f
2612 2600(5.1.)N
2788(Logarithmic)X
3280(Signal)X
3538(Propagation)X
4025(Delay.)X
1 f
2812 2740(Let)N
2966(the)X
3110(area)X
3292(occupied)X
3647(by)X
3772(a)X
3848(wire)X
4041(of)X
4151(length)X
2 f
4409(L)X
1 f
4495(be)X
2 f
4615(aL)X
1 f
8 s
4718 2705(2)N
11 s
4794 2740(for)N
2612 2852(some)N
2820(constant)X
3136(0)X
2 f
()X
1 f
3322(0,)X
3195 5116(0)N
2 f
3253(<)X
1 f
9 f
3326(a)X
2 f
3395(<)X
1 f
3521 5177(2)N
9 f
3490 5047 MXY
(sr)214 549 oc
10 f
3538(d)X
3552(d)X
1 f
3538 5064(2)N
10 f
3476 5090(h)N
3477(hhh)X
2 f
3661 5116(&)N
1 f
3781(0)X
2 f
3839(<)X
3912(a)X
1 f
9 f
3977(\243)X
1 f
4061 5195(3)N
9 f
4119 5178 MXY
(sr)214 549 oc
10 f
4167(d)X
4181(d)X
1 f
4167 5195(6)N
4121 5064(2)N
10 f
4047 5090(h)N
4062(hhhh)X
2 f
4290 5116(.)N
1 f
2612 5373(Note)N
2819(that)X
2988(in)X
3093(the)X
3237(limit)X
3442(for)X
9 f
3580(a)X
3649 MX
(->)174 987 oc
3750 5356 MXY
(sr)214 549 oc
10 f
3798(d)X
3812(d)X
1 f
3798 5373(2)N
2 f
3863(/)X
1 f
3895(2)X
3975(we)X
4113(have)X
4315(that)X
2 f
4484(a)X
1 f
9 f
4549 MX
(->)174 987 oc
1 f
4650(0.)X
4774(For)X
2612 5485(given)N
9 f
2836(a)X
1 f
2919(and)X
2 f
3074(a)X
1 f
3146(in)X
3243(the)X
3378(appropriate)X
3806(ranges,)X
4084(an)X
2 f
4194(upper)X
4426(bound)X
1 f
4673(on)X
4788(the)X
2612 5597(total)N
2797(wire)X
2980(area)X
3153(plus)X
3327(node)X
3525(area)X
3698(for)X
3827(the)X
3962(H-tree)X
4213(layout)X
4461(is)X
4548(computed)X
2612 5709(by:)N
2 f
3052 5919(A)N
1 f
3113(\()X
2 f
3142(N)X
1 f
3208(\))X
9 f
3259(\243)X
2 f
3329(a)X
8 s
6009(k)Y
9 f
3411(=)X
1 f
3446(0)X
16 s
9 f
3387 5945(S)N
8 s
3402 5836(\245)N
11 s
1 f
3477 5919(2)N
2 f
8 s
5884(k)Y
1 f
11 s
9 f
3556 5919(a)N
8 s
1 f
3611 5884(2)N
2 f
(k)S
9 f
3681(-)X
1 f
3716(2)X
2 f
(m)S
1 f
11 s
2 f
9 f
3823 5919(+)N
1 f
3893(2)X
2 f
8 s
5884(m)Y
9 f
3993(+)X
1 f
4028(1)X
2 f
11 s
9 f
3259 6205(=)N
1 f
3351 6277(1)N
2 f
9 f
(-)S
1 f
3443(2)X
9 f
(a)S
8 s
1 f
3542 6242(2)N
2 f
11 s
3353 6153(a)N
1 f
9 f
3404(a)X
2 f
8 s
9 f
3459 6118(-)N
1 f
3494(2)X
2 f
(m)S
1 f
11 s
10 f
3337 6179(h)N
3374(hhhhh)X
2 f
9 f
3624 6205(+)N
1 f
3694(2)X
2 f
8 s
6170(m)Y
9 f
3794(+)X
1 f
3829(1)X
11 s
2 f
3926 6205(.)N
1 f
2612 6445(Therefore,)N
2 f
3293 6619(A)N
1 f
3354(\()X
2 f
3383(N)X
1 f
3449(\))X
9 f
3500(\243)X
2 f
3570(CN)X
8 s
9 f
3698 6571(-)N
1 f
3733(2)X
3775(log)X
5 s
3857 6583(2)N
8 s
9 f
3892 6571(a)N
11 s
2 f
9 f
3961 6619(+)N
1 f
4031(2)X
2 f
(N)S
1 f
4185(,)X
2612 6787(with)N
2 f
2799(C)X
1 f
2 f
9 f
2887(=)X
1 f
2 f
2957(a)X
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1 f
3047(\(1)X
2 f
9 f
3120(-)X
1 f
3168(2)X
9 f
(a)S
8 s
1 f
3267 6752(2)N
11 s
3306 6787(\).)N
3387(From)X
3608(the)X
3746(relation)X
4046(between)X
2 f
4369(a)X
1 f
4444(and)X
9 f
4602(a)X
1 f
4688(it)X
4769(fol-)X
2612 6899(lows)N
2840(that)X
2 f
3035(C)X
9 f
3108(\243)X
1 f
9 f
3156(a)X
1 f
3273(for)X
3436(0)X
2 f
(<)S
1 f
9 f
3539(a)X
2 f
3594(<)X
1 f
9 f
3653 6882 MXY
(sr)214 549 oc
10 f
3701(d)X
3715(d)X
1 f
3701 6899(2)N
2 f
3766(/)X
1 f
3798(2.)X
3925(Setting)X
2 f
4237(a)X
9 f
4302(=)X
1 f
4364(\(1)X
2 f
9 f
4437(-)X
1 f
4485(2)X
9 f
(a)S
8 s
1 f
4584 6864(2)N
11 s
4623 6899(\))N
2 f
4659(/)X
1 f
4691(2,)X
4818(so)X
2612 7011(1)N
2 f
2663(/)X
1 f
2695(2)X
9 f
2753(\243)X
2815(a)X
2 f
2884(<)X
1 f
9 f
2957 6994 MXY
(sr)214 549 oc
10 f
3005(d)X
3019(d)X
1 f
3005 7011(2)N
2 f
3070(/)X
1 f
3102(2)X
3168(and)X
3317(therefore)X
3656(0)X
2 f
3714(<)X
3787(a)X
1 f
9 f
3852(\243)X
1 f
3914(1)X
2 f
3965(/)X
1 f
3997(4)X
4063(and)X
2 f
4212(C)X
9 f
4292(=)X
1 f
4354(1)X
2 f
4405(/)X
1 f
4437(2,)X
4525(yields)X
2 f
3230 7185(A)N
1 f
3298(\()X
2 f
3327(N)X
1 f
3393(\))X
9 f
3444 7168(~)N
3444 7190(~)N
2 f
3514 7186 MXY
(12)130 767 oc
3581(N)X
1 f
8 s
3650 7138(1)N
2 f
9 f
(-)S
1 f
3717(log)X
5 s
3799 7150(2)N
8 s
3824 7138(\(1)N
2 f
9 f
3877(-)X
1 f
3912(2)X
2 f
(a)S
1 f
3981(\))X
11 s
2 f
9 f
4031 7186(+)N
1 f
4101(2)X
2 f
(N)S
1 f
2 f
4255(.)X
ep
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bp
11 s 0 xH 0 xS 2 f
1 f
10 s
0 32(--)N
4323(--)X
11 s
2375 568(-)N
2426(6)X
2492(-)X
0 976(Therefore,)N
429(for)X
592(each)X
9 f
814(e)X
2 f
853(>)X
1 f
912(0)X
1017(there)X
1254(is)X
1374(an)X
9 f
1519(a)X
2 f
1596(<)X
1 f
9 f
1677 959 MXY
(sr)214 549 oc
10 f
1725(d)X
1739(d)X
1 f
1725 976(2)N
2 f
1790(/)X
1 f
1822(2)X
1928(such)X
2151(that)X
2 f
0 1088(A)N
1 f
68(\()X
2 f
97(N)X
1 f
163(\))X
2 f
9 f
214(\316)X
1 f
2 f
9 f
299(O)X
1 f
362(\()X
2 f
391(N)X
1 f
8 s
460 1053(1)N
2 f
9 f
(+)S
1 f
9 f
527(e)X
11 s
1 f
562 1088(\).)N
642(Since)X
2 f
867(a)X
1 f
940(must)X
1141(be)X
1253(greater)X
1526(than)X
1707(0)X
1780(this)X
9 f
1937(e)X
1 f
2005(remains)X
0 1200(greater)N
266(than)X
440(0)X
506(as)X
601(well.)X
0 1340(For)N
9 f
158(a)X
235 MX
(->)174 987 oc
344 1323 MXY
(sr)214 549 oc
10 f
392(d)X
406(d)X
1 f
392 1340(2)N
2 f
457(/)X
1 f
489(2)X
569(\(so)X
712(the)X
856(aspect)X
1112(ratio)X
2 f
1310(a)X
1 f
9 f
1361 MX
(->)174 987 oc
1 f
1448(0\))X
1557(the)X
1702(upper)X
1939(bound)X
2196(on)X
2 f
0 1452(A)N
1 f
68(\()X
2 f
97(N)X
1 f
163(\))X
214(goes)X
397(to:)X
8 s
9 f
440 1738(a)N
480 MX
(->)174 987 oc
543 1726 MXY
(sr)214 549 oc
10 f
578(d)X
588(d)X
1 f
578 1738(2)N
2 f
625(/)X
1 f
648(2)X
11 s
500 1662(lim)N
2 f
701(a)X
8 s
1752(k)Y
9 f
783(=)X
1 f
818(0)X
16 s
9 f
759 1688(S)N
2 f
8 s
774 1579(m)N
1 f
11 s
849 1662(2)N
2 f
8 s
1627(k)Y
1 f
11 s
9 f
928 1662(a)N
8 s
1 f
983 1627(2)N
2 f
(k)S
9 f
1053(-)X
1 f
1088(2)X
2 f
(m)S
1 f
11 s
2 f
9 f
1195 1662(+)N
1 f
1265(2)X
2 f
8 s
1627(m)Y
9 f
1365(+)X
1 f
1400(1)X
2 f
11 s
9 f
745 1902(=)N
1 f
8 s
9 f
815 1978(a)N
855 MX
(->)174 987 oc
918 1966 MXY
(sr)214 549 oc
10 f
953(d)X
963(d)X
1 f
953 1978(2)N
2 f
1000(/)X
1 f
1023(2)X
11 s
875 1902(lim)N
2 f
1076(am)X
1 f
9 f
1190(a)X
2 f
8 s
9 f
1245 1867(-)N
1 f
1280(2)X
2 f
(m)S
1 f
11 s
2 f
9 f
1387 1902(+)N
1 f
1457(2)X
2 f
8 s
1867(m)Y
9 f
1557(+)X
1 f
1592(1)X
11 s
2 f
9 f
745 2129(=)N
1 f
2 f
815(aN)X
1 f
939(log)X
2 f
1066(N)X
1 f
2 f
9 f
1154(+)X
1 f
1224(2)X
2 f
(N)S
1 f
2 f
1378(.)X
200 2384(Lower)N
478(bound.)X
1 f
795(Let)X
2 f
966(N)X
9 f
1039(=)X
1 f
1087(2)X
2 f
8 s
2349(m)Y
1 f
11 s
1184 2384(,)N
1259(and)X
1439(let)X
2 f
1582(c)X
1 f
1675(be)X
1812(the)X
1974(maximal)X
0 2496(amount)N
297(of)X
402(cross-over.)X
843(We)X
997(obtain)X
1250(a)X
1321(lower)X
1553(bound)X
1805(on)X
1924(the)X
2063(layout)X
0 2608(area)N
169(for)X
295(a)X
358(complete)X
706(binary)X
955(tree)X
1111(for)X
2 f
1237(any)X
1388(layout)X
1 f
1609(,)X
1655(so)X
2 f
1757(also)X
1 f
1928(the)X
2060(H-tree)X
0 2720(layout.)N
289(For)X
435(each)X
2 f
620(i)X
1 f
645(,)X
691(1)X
2 f
9 f
(\243)S
2 f
783(i)X
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822(\243)X
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870(m)X
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933(,)X
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979(A)X
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1040(\(2)X
2 f
8 s
1113 2685(i)N
1 f
11 s
1138 2720(\))N
1191(is)X
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2 f
1404(minimum)X
1754(layout)X
1998(area)X
1 f
2182(for)X
0 2832(a)N
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455(binary)X
726(tree)X
2 f
904(T)X
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953 2849(i)N
1 f
11 s
1024 2832(with)N
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8 s
2797(i)Y
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11 s
1344 2832(leaves,)N
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561(c)X
1 f
626(cross-over)X
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1168(a)X
1 f
1237(aspect)X
1482(ratio.)X
1713(Imagine,)X
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0 3056(sake)N
190(of)X
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439(argument,)X
829(a)X
903(\(nonexistent\))X
1403(layout)X
1659(for)X
2 f
1796(T)X
8 s
1845 3073(m)N
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11 s
1898 3056(,)N
1955(such)X
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218(maximal)X
585(subtree)X
896(determined)X
1350(by)X
1494(a)X
1589(node)X
1816(of)X
2 f
1945(T)X
8 s
1994 3185(m)N
1 f
11 s
2103 3168(takes)N
0 3280(minimal)N
327(area.)X
548(Selecting)X
908(wires)X
1129(from)X
1331(the)X
1470(maximal)X
1811(lengths)X
2098(paths)X
0 3392(in)N
95(these)X
301(subtrees,)X
636(we)X
763(sum)X
935(their)X
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1327(while)X
1548(taking)X
1794(care)X
1965(that)X
2123(each)X
0 3504(such)N
183(wire)X
361(is)X
442(counted)X
743(only)X
922(once.)X
1154(Let)X
2 f
1294(T)X
8 s
1343 3521(i)N
1 f
11 s
1390 3504(be)N
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0 3616(of)N
2 f
95(i)X
9 f
134(+)X
1 f
182(1)X
248(levels)X
476(with)X
655(the)X
785(root)X
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1035(level)X
1229(0)X
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1902(level)X
2 f
2096(i)X
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2121(.)X
3 f
200 3756(Claim)N
457(1.)X
1 f
572(There)X
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892(a)X
959(path)X
1139(in)X
2 f
1236(T)X
8 s
1285 3773(i)N
1 f
11 s
1338 3756(of)N
1439(at)X
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1731(2)X
2 f
(i)S
1 f
1828(edges)X
2056(and)X
2211(of)X
0 3877(length)N
245(at)X
333(least)X
12 s
9 f
519 3858 MXY
(sr)214 549 oc
10 f
572(d)X
589(dddd)X
2 f
11 s
572 3877(A)N
1 f
640(\(2)X
2 f
8 s
713 3842(i)N
1 f
11 s
738 3877(\))N
805(in)X
898(the)X
1030(layout.)X
1319(The)X
1480(sum)X
1650(of)X
1746(the)X
1877(areas)X
2080(of)X
2176(the)X
0 3989(wires)N
212(in)X
303(such)X
486(a)X
547(path)X
721(exceeds)X
2 f
1021(aA)X
1 f
1133(\(2)X
2 f
8 s
1206 3954(i)N
1 f
11 s
1231 3989(\))N
2 f
1267(/)X
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1299(2)X
2 f
(ic)S
1 f
1407(.)X
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200 4129(Proof)N
426(of)X
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1 f
805(By)X
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2211(of)X
0 4241(the)N
132(smallest)X
446(convex)X
724(area)X
894(containing)X
2 f
1291(T)X
8 s
1340 4258(i)N
1 f
11 s
1389 4241(it)N
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1952(that)X
2108(there)X
0 4362(is)N
84(a)X
148(path)X
325(of)X
423(length)X
12 s
9 f
670 4343 MXY
(sr)214 549 oc
10 f
723(d)X
740(dddd)X
2 f
11 s
723 4362(A)N
1 f
791(\(2)X
2 f
8 s
864 4327(i)N
1 f
11 s
889 4362(\))N
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1617(1)X
2 f
9 f
(\243)S
2 f
1716(j)X
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1741 4379(i)N
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9 f
1766 4362(\243)N
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1814(2)X
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(i)S
1 f
1909(wires.)X
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0 4474(sum)N
201(of)X
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690(of)X
817(the)X
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1222(in)X
1344(such)X
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1650(path)X
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1967(therefore)X
9 f
0 4586(W)N
1 f
68(\()X
2 f
97(aA)X
1 f
209(\(2)X
2 f
8 s
282 4551(i)N
1 f
11 s
307 4586(\))N
2 f
343(/)X
375(c)X
421(j)X
8 s
446 4603(i)N
1 f
11 s
471 4586(\).)N
558(\(Assume)X
906(that)X
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1401(of)X
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1654(wires)X
1881(in)X
1987(the)X
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0 4698(can)N
150(be)X
261(distributed)X
667(over)X
2 f
851(c)X
1 f
918(levels,)X
1174(so)X
1280(as)X
1381(to)X
1477(keep)X
2 f
1670(A)X
1 f
1738(\(2)X
2 f
8 s
1811 4663(i)N
1 f
11 s
1836 4698(\))N
1892(as)X
1992(small)X
2211(as)X
0 4810(possible.\))N
10 f
362(`)X
3 f
200 4950(Claim)N
467(2.)X
1 f
592(Let)X
2 f
747(P)X
1 f
839(be)X
960(a)X
1037(path)X
1227(through)X
2 f
1539(T)X
8 s
1588 4967(i)N
1 f
11 s
1613 4950(.)N
1695(At)X
1821(each)X
2020(level)X
2 f
2237(j)X
1 f
2262(,)X
0 5062(2)N
9 f
(\243)S
2 f
99(j)X
1 f
9 f
131(\243)X
2 f
179(i)X
1 f
204(,)X
255(there)X
460(are)X
596(at)X
689(least)X
880(2)X
952(maximal)X
1290(complete)X
1642(binary)X
1895(subtrees)X
2211(of)X
2 f
0 5174(i)N
9 f
39(-)X
2 f
94(j)X
9 f
133(+)X
1 f
181(1)X
253(levels)X
487(with)X
672(roots)X
876(at)X
968(level)X
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1175(j)X
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1200(,)X
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1688(no)X
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1805(node)X
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396(P)X
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450(.)X
535(Moreover,)X
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1093(2\()X
2 f
1166(i)X
9 f
1205(-)X
1 f
1253(1\))X
1367(complete)X
1731(binary)X
1996(subtrees)X
0 5398(concerned)N
391(are)X
528(pairwise)X
856(disjoint)X
1152(as)X
1255(well.)X
1481(\(This)X
1697(is)X
1787(easy)X
1974(to)X
2074(verify)X
0 5510(from)N
193(a)X
254(simple)X
512(picture.\))X
200 5650(By)N
327(Claims)X
602(1)X
670(and)X
821(2,)X
911(we)X
1037(can)X
1183(give)X
1359(a)X
1422(lower)X
1647(bound)X
1892(on)X
2005(the)X
2138(area)X
2 f
0 5771(A)N
1 f
68(\(2)X
2 f
8 s
141 5736(i)N
1 f
11 s
166 5771(\))N
229(of)X
2 f
336(T)X
8 s
385 5788(i)N
1 f
11 s
444 5771(by)N
566(adding)X
840(the)X
982(minimal)X
1312(possible)X
1635(area)X
1815(of)X
1921(a)X
12 s
9 f
1993 5752 MXY
(sr)214 549 oc
10 f
2046(d)X
2063(dddd)X
2 f
11 s
2046 5771(A)N
1 f
2114(\(2)X
2 f
8 s
2187 5736(i)N
1 f
11 s
2212 5771(\))N
2255(-)X
0 5883(length)N
247(path)X
425(in)X
2 f
521(T)X
8 s
570 5900(i)N
1 f
11 s
622 5883(to)N
718(twice)X
936(the)X
1071(sum)X
1245(for)X
2 f
1381(j)X
1 f
1433(ranges)X
1689(from)X
1887(2)X
1958(through)X
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2259(i)X
1 f
0 5995(of)N
95(the)X
225(areas)X
427(of)X
522(subtrees)X
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832(T)X
8 s
881 6012(i)N
9 f
909(-)X
2 f
949(j)X
1 f
11 s
974 5995(:)N
2 f
599 6209(A)N
1 f
667(\(2)X
2 f
8 s
740 6174(i)N
1 f
11 s
765 6209(\))N
2 f
816(>)X
1 f
984 6270(2)N
2 f
(ci)S
919 6157(aA)N
1 f
1031(\(2)X
2 f
8 s
1104 6122(i)N
1 f
11 s
1129 6157(\))N
10 f
905 6183(h)N
907(hhhhhh)X
2 f
9 f
1193 6209(+)N
1 f
1255(2)X
2 f
8 s
1304 6299(j)N
9 f
1332(=)X
1 f
1367(2)X
16 s
9 f
1311 6235(S)N
2 f
8 s
1340 6126(i)N
11 s
1399 6209(A)N
1 f
1467(\(2)X
2 f
8 s
1540 6174(i)N
9 f
1568(-)X
2 f
1608(j)X
1 f
11 s
1633 6209(\))N
2 f
1662(.)X
1 f
0 6477(Unfolding)N
384(this)X
534(inequality)X
910(we)X
1034(obtain)X
2 f
475 6709(A)N
1 f
543(\(2)X
2 f
8 s
616 6674(i)N
1 f
11 s
641 6709(\))N
2 f
692(>)X
1 f
2 f
8 s
778 6799(j)N
9 f
806(=)X
1 f
841(0)X
16 s
9 f
785 6735(S)N
2 f
8 s
776 6626(i)N
9 f
804(-)X
1 f
839(1)X
11 s
950 6770(2)N
2 f
(c)S
1 f
1047(\()X
2 f
1076(i)X
9 f
1115(-)X
2 f
1170(j)X
1 f
1202(\))X
2 f
895 6639(aF)N
8 s
998 6656(j)N
11 s
1023 6639(A)N
1 f
1091(\(2)X
2 f
8 s
1164 6604(i)N
9 f
1192(-)X
2 f
1232(j)X
1 f
11 s
1257 6639(\))N
10 f
881 6683(h)N
903(hhhhhhhhh)X
2 f
9 f
1329 6709(+)N
1 f
2 f
1548 6770(c)N
1421 6639(aF)N
8 s
1519 6656(i)N
11 s
1544 6639(A)N
1 f
1612(\(1\))X
10 f
1407 6683(h)N
1419(hhhhhhh)X
1 f
1779 6709(,)N
0 6949(with)N
2 f
196(F)X
8 s
255 6966(j)N
1 f
11 s
319 6949(the)N
2 f
473(j)X
1 f
498(-th)X
635(element)X
955(of)X
1068(the)X
1216(Fibonacci-like)X
1766(sequence)X
2128(gen-)X
0 7061(erated)N
255(by)X
383(the)X
531(recurrence)X
941(relation)X
2 f
1251(F)X
8 s
1305 7078(i)N
1 f
11 s
2 f
9 f
1352 7061(=)N
1 f
2 f
1422(F)X
8 s
1476 7078(i)N
9 f
1504(-)X
1 f
1539(1)X
2 f
11 s
9 f
1578 7061(+)N
1 f
1626(2)X
2 f
(F)S
8 s
1724 7078(i)N
9 f
1752(-)X
1 f
1787(2)X
11 s
1865 7061(with)N
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2061(F)X
1 f
8 s
2125 7078(0)N
2 f
11 s
9 f
2178 7061(=)N
1 f
2240(1)X
0 7173(and)N
2 f
149(F)X
1 f
8 s
213 7190(1)N
2 f
11 s
9 f
266 7173(=)N
1 f
328(0.)X
438(Therefore,)X
2 f
3330 1022(F)N
8 s
3384 1039(i)N
1 f
11 s
2 f
9 f
3431 1022(=)N
1 f
3683 1083(3)N
3523 970(2)N
2 f
8 s
935(i)Y
11 s
9 f
3606 970(+)N
1 f
3668(2\()X
2 f
9 f
3741(-)X
1 f
3789(1\))X
2 f
8 s
3862 935(i)N
1 f
11 s
10 f
3509 996(h)N
3548(hhhhhhhh)X
9 f
3930 1005(~)N
3930 1027(~)N
1 f
4034 1084(3)N
4022 971(2)N
2 f
8 s
936(i)Y
1 f
11 s
10 f
4008 997(h)N
4016(hh)X
2 f
4156 1023(.)N
1 f
2612 1262(Substituting)N
2 f
3062(A)X
1 f
3130(\(2)X
2 f
8 s
3203 1227(i)N
1 f
11 s
3228 1262(\))N
2 f
9 f
3271(=)X
2 f
3333(g)X
1 f
3391(\()X
2 f
3420(i)X
1 f
3452(\)2)X
2 f
8 s
3525 1227(i)N
1 f
11 s
3572 1262(in)N
3663(the)X
3793(inequality)X
4169(above)X
4401(yields)X
2 f
3259 1472(g)N
1 f
3317(\()X
2 f
3346(i)X
1 f
3378(\))X
2 f
3429(>)X
1 f
3532 1533(6)N
2 f
(c)S
3552 1420(a)N
1 f
10 f
3519 1446(h)N
3541(hh)X
2 f
8 s
3664 1562(j)N
9 f
3692(=)X
1 f
3727(1)X
16 s
9 f
3671 1498(S)N
2 f
8 s
3700 1389(i)N
11 s
3849 1533(j)N
3781 1420(g)N
1 f
3839(\()X
2 f
3875(j)X
1 f
3907(\))X
10 f
3767 1446(h)N
3773(hhhh)X
2 f
9 f
3979 1472(+)N
1 f
4071 1533(3)N
2 f
(c)S
4091 1420(a)N
1 f
10 f
4058 1446(h)N
4080(hh)X
1 f
4220 1472(,)N
2612 1712(which)N
2849(is)X
2930(satis\256ed)X
3241(for)X
2 f
3365(g)X
1 f
3423(\()X
2 f
3452(i)X
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3484(\))X
2 f
9 f
3527(\316)X
1 f
9 f
3604(W)X
1 f
3672(\()X
2 f
3701(i)X
1 f
8 s
9 f
3736 1677(e)N
11 s
1 f
3771 1712(\))N
3822(with)X
9 f
4001(e)X
2 f
4040(>a)X
4150(/)X
1 f
4182(6)X
2 f
(c)S
1 f
4265(.)X
4309(Hence,)X
2 f
3308 1880(A)N
1 f
3376(\()X
2 f
3405(N)X
1 f
3471(\))X
2 f
9 f
3522(\316)X
1 f
9 f
3607(W)X
1 f
3675(\()X
2 f
3704(N)X
1 f
3784(log)X
2 f
8 s
3897 1845(a)N
3934(/)X
1 f
3957(6)X
2 f
(c)S
11 s
4038 1880(N)N
1 f
4104(\))X
2 f
4177(.)X
1 f
2612 2048(Crudely)N
2936(derived,)X
3263(considering)X
3715(only)X
3913(volume)X
4219(without)X
4530(considera-)X
2612 2160(tions)N
2807(of)X
2903(placement)X
3289(and)X
3439(routing,)X
3738(this)X
3888(lower)X
4110(bound)X
4352(is)X
4433(yet)X
4563(nonlinear)X
2612 2272(and)N
2775(re\257ects)X
3065(both)X
3258(the)X
3403(necessary)X
3781(in\257uence)X
4141(of)X
4251(the)X
4396(aspect)X
4653(ratio)X
2 f
4852(a)X
1 f
2612 2384(and)N
2791(the)X
2951(cross-over)X
3368(coef\256cient)X
2 f
3792(c)X
1 f
3831(.)X
3905(It)X
4011(seems)X
4278(likely)X
4531(that)X
4715(more)X
2612 2496(sophisticated)N
3097(arguments)X
3488(for)X
3615(\256xed)X
3816(aspect)X
4061(ratio)X
2 f
4248(a)X
4306(>)X
1 f
4365(0)X
4434(will)X
4597(raise)X
4788(the)X
2612 2608(lower)N
2834(bound)X
3076(to)X
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3405(the)X
3535(upper)X
3757(bound)X
9 f
3999(W)X
1 f
4067(\()X
2 f
4096(N)X
1 f
8 s
4165 2573(1)N
2 f
9 f
(+)S
1 f
9 f
4232(e)X
11 s
1 f
4267 2608(\).)N
3 f
2612 2832(5.2.)N
2788(Radical)X
3099(Signal)X
3357(Propagation)X
3844(Delay)X
1 f
2812 2972(For)N
2956(radical)X
3219(signal)X
3453(propagation)X
3897(delay)X
4111(proportional)X
4570(to)X
4662(the)X
2 f
4793(r)X
1 f
4827(th)X
2612 3084(root)N
2777(of)X
2873(the)X
3004(length)X
2 f
3248(L)X
1 f
3320(of)X
3416(the)X
3547(wire,)X
3748(the)X
2 f
3879(area)X
1 f
4063(taken)X
4277(by)X
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4518(wire)X
4696(needs)X
2612 3196(be)N
2 f
2717(aL)X
1 f
8 s
2820 3161(2)N
2 f
9 f
(-)S
1 f
2887(1)X
2 f
2924(/)X
1 f
2947(2)X
2 f
(r)S
1 f
11 s
3011 3196(,)N
3055(for)X
3179(some)X
3387(constant)X
2 f
3703(a)X
1 f
(,)S
3791(by)X
3901(\(15\).)X
2 f
2812 3336(Upper)N
3059(bound.)X
1 f
3347(The)X
3508(upper)X
3732(bound)X
3976(on)X
2 f
4088(A)X
1 f
4156(\()X
2 f
4185(N)X
1 f
4251(\))X
4304(is)X
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4808(by)X
2612 3448(using)N
2826(the)X
2 f
2957(H)X
1 f
3020(-tree)X
3203(layout)X
3446(\(without)X
3767(overlap\))X
4082(again.)X
4339(Let)X
2 f
4479(N)X
1 f
2 f
9 f
4567(=)X
1 f
4637(2)X
2 f
8 s
3413(m)Y
1 f
11 s
4734 3448(.)N
4778(Let)X
2612 3560(again)N
2831(the)X
2967(quotient)X
3285(between)X
3607(the)X
3744(lengths)X
4028(of)X
4130(the)X
4267(wires)X
4486(at)X
4579(two)X
4740(con-)X
2612 3672(secutive)N
2928(levels)X
2 f
3161(k)X
9 f
3214(+)X
1 f
3262(1)X
3333(and)X
2 f
3487(k)X
1 f
3553(be)X
9 f
3663(a)X
1 f
3718(,)X
3767(0)X
2 f
(<)S
1 f
9 f
3870(a)X
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3925(<)X
1 f
3984(1,)X
4077(with)X
4261(the)X
4396(root)X
4564(at)X
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4852(0)X
2612 3784(and)N
2762(the)X
2893(leaves)X
3136(at)X
3223(level)X
2 f
3418(m)X
1 f
3481(.)X
3526(Considering)X
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4111(H-tree)X
4359(layout,)X
4626(it)X
4700(is)X
4783(not)X
2612 3896(to)N
2721(dif\256cult)X
3041(to)X
3150(see)X
3302(that,)X
3496(with)X
3692(aspect)X
3951(ratio)X
2 f
4152(aL)X
8 s
9 f
4255 3861(-)N
1 f
4290(1)X
2 f
4327(/)X
1 f
4350(2)X
2 f
(r)S
1 f
11 s
4453 3896(for)N
4594(wires)X
4823(of)X
2612 4008(length)N
2 f
2855(L)X
1 f
2926(\(0)X
2 f
2999()X
1 f
3143(0,)X
3052 5276(0)N
2 f
3110(<)X
1 f
9 f
3183(a)X
2 f
3252(<)X
1 f
3378 5337(2)N
9 f
3347 5207 MXY
(sr)214 549 oc
10 f
3395(d)X
3409(d)X
1 f
3395 5224(2)N
10 f
3333 5250(h)N
3334(hhh)X
1 f
3518 5276(,)N
3052 5564(0)N
2 f
3110(<)X
3183(a)X
3248(<)X
1 f
3321(\()X
3372 5625(12)N
2 f
(r)S
9 f
3508(+)X
1 f
3556(2)X
2 f
(k)S
9 f
3653(-)X
1 f
3701(2)X
3416 5512(2)N
2 f
(r)S
9 f
3508(+)X
2 f
3556(k)X
9 f
3609(-)X
1 f
3657(1)X
10 f
3358 5538(h)N
3362(hhhhhhhhh)X
1 f
3766 5564(\))N
8 s
3892 5528(4)N
2 f
(r)S
1 f
3817 5446(2)N
2 f
(r)S
9 f
3884(+)X
2 f
3919(k)X
9 f
3957(-)X
1 f
3992(1)X
10 f
3801 5465(h)N
3816(hhhhhhh)X
11 s
1 f
4053 5564(\(1)N
2 f
9 f
4140(-)X
1 f
4224 5625(12)N
2 f
(r)S
9 f
4360(+)X
1 f
4408(2)X
2 f
(k)S
9 f
4505(-)X
1 f
4553(2)X
4246 5512(4)N
2 f
(r)S
9 f
4338(+)X
1 f
4386(2)X
2 f
(k)S
9 f
4483(-)X
1 f
4531(2)X
10 f
4210 5538(h)N
4214(hhhhhhhhh)X
1 f
4618 5564(\).)N
2612 5804(For)N
2 f
2759(r)X
1 f
9 f
2800 MX
(->)174 987 oc
2887(\245)X
1 f
2975(we)X
3102(obtain)X
3348(the)X
3481(upper)X
3706(bound)X
3951(on)X
2 f
4064(a)X
1 f
4134(we)X
4262(saw)X
4424(above)X
4660(for)X
4788(the)X
2612 5916(logarithmic)N
3050(delay)X
3271(case.)X
3496(Note)X
3697(that)X
3860(again)X
4080(for)X
9 f
4211(a)X
4280 MX
(->)174 987 oc
4381 5899 MXY
(sr)214 549 oc
10 f
4429(d)X
4443(d)X
1 f
4429 5916(2)N
2 f
4494(/)X
1 f
4526(2)X
4599(we)X
4730(have)X
2612 6028(that)N
2 f
2767(a)X
1 f
9 f
2832 MX
(->)174 987 oc
1 f
2933(0.)X
3043(For)X
3187(given)X
9 f
3405(a)X
1 f
3482(and)X
2 f
3631(a)X
1 f
3697(in)X
3788(appropriate)X
4211(ranges,)X
4484(that)X
4639(is,)X
3521 6272(2)N
9 f
(a)S
8 s
1 f
3620 6192(2)N
2 f
9 f
(-)S
1 f
3709 6236(2)N
2 f
(r)S
1 f
3722 6154(1)N
10 f
3694 6173(h)N
3719(hh)X
11 s
2 f
3818 6272(<)N
1 f
3899(1)X
2612 6440(and)N
2 f
2769(a)X
1 f
2843(accordingly,)X
3311(the)X
2 f
3449(upper)X
3684(bound)X
1 f
3934(on)X
2 f
4052(A)X
1 f
4120(\()X
2 f
4149(N)X
1 f
4215(\),)X
4296(consisting)X
4684(of)X
4788(the)X
2612 6552(total)N
2796(wire)X
2977(area)X
3148(plus)X
3320(node)X
3516(area)X
3687(for)X
3814(the)X
3947(H-tree)X
4196(layout,)X
4464(is)X
4548(computed)X
2612 6664(as)N
2707(follows:)X
2 f
3052 6908(A)N
1 f
3120(\()X
2 f
3149(N)X
1 f
3215(\))X
9 f
3266(\243)X
2 f
3336(a)X
8 s
6998(k)Y
9 f
3418(=)X
1 f
3453(0)X
16 s
9 f
3394 6934(S)N
8 s
3409 6825(\245)N
11 s
1 f
3484 6908(2)N
2 f
8 s
6873(k)Y
1 f
11 s
9 f
3563 6908(a)N
8 s
1 f
3618 6828(\()N
2 f
3639(k)X
9 f
3677(-)X
2 f
3712(m)X
1 f
3763(\)\(2)X
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990 4660 MXY
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670 5713 MXY
(sr)214 549 oc
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(\243)S
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1716(j)X
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(r)S
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11 s
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11 s
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9 f
2612 966 MXY
(sr)214 549 oc
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2665(d)X
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2733(\(2)X
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11 s
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(+)S
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(-)S
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4066 1341(+)N
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4205(=)X
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4240(2)X
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4213 1258(i)N
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8 s
4413 1306(i)N
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4441(-)X
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4481(j)X
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4506 1341(\))N
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4535(.)X
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2759 1853(A)N
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2827(\(2)X
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8 s
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2925 1853(\))N
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2976(>)X
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3081 1914(c)N
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10 f
3065 1827(h)N
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8 s
3163 1943(j)N
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3191(=)X
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3226(0)X
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3161 1770(i)N
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3189(-)X
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3224(1)X
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3317 1870(j)N
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11 s
3342 1853(\(2)N
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3415(i)X
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3454(-)X
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3502(2)X
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3553(j)X
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3585(\))X
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8 s
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3614 1773(-)N
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3649(1)X
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9 f
(+)S
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3738 1817(2)N
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(r)S
1 f
3751 1735(1)N
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3723 1754(h)N
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11 s
3825 1853(A)N
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8 s
3966 1818(i)N
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3994(-)X
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4034(j)X
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11 s
4059 1853(\))N
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4088 1773(1)N
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(-)S
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(r)S
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4190 1735(1)N
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4162 1754(h)N
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11 s
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4278 1853(+)N
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4489 1914(c)N
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4460 1800(i)N
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4553(\(1\))X
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4348 1827(h)N
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4720 1853(,)N
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2808(F)X
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2931 2093(the)N
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3085(j)X
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3110(-th)X
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3863(F)X
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3964 2205(=)N
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4034(F)X
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4088 2222(i)N
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4151(1)X
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11 s
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4190 2205(+)N
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4238(2)X
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(F)S
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4364(-)X
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4399(2)X
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8 s
4737 2222(0)N
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4790 2205(=)N
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4852(1)X
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8 s
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2878 2317(=)N
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2940(0.)X
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3330 2531(F)N
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3384 2548(i)N
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11 s
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3431 2531(=)N
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3683 2592(3)N
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8 s
2444(i)Y
11 s
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3606 2479(+)N
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3668(2\()X
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9 f
3741(-)X
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3789(1\))X
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8 s
3862 2444(i)N
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3509 2505(h)N
3548(hhhhhhhh)X
9 f
3930 2514(~)N
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4034 2593(3)N
4022 2480(2)N
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8 s
2445(i)Y
1 f
11 s
10 f
4008 2506(h)N
4016(hh)X
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4156 2532(.)N
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2612 2771(Substituting)N
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3469 2939(A)N
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3537(\(2)X
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8 s
3610 2904(i)N
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11 s
3635 2939(\))N
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9 f
3678(=)X
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3740(D)X
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3817(\()X
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3846(r)X
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3880(,)X
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3909(i)X
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3941(\)2)X
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8 s
4014 2904(i)N
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2612 3107(and)N
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2827 3516(D)N
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2904(\()X
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2933(r)X
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2967(,)X
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2996(i)X
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3028(\))X
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3079(>)X
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3297 3577(3)N
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(c)S
3182 3464(a)N
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3240(2)X
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8 s
9 f
3384(-)Y
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3319(1)X
2 f
9 f
(+)S
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3408 3428(2)N
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(r)S
1 f
3421 3346(1)N
10 f
3393 3365(h)N
3418(hh)X
11 s
3168 3490(h)N
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2 f
8 s
3521 3606(j)N
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3549(=)X
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3584(1)X
16 s
9 f
3528 3542(S)N
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8 s
3557 3433(i)N
11 s
3623 3516(j)N
8 s
9 f
3658 3436(-)N
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3693(1)X
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9 f
(+)S
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3782 3480(2)N
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(r)S
1 f
3795 3398(1)N
10 f
3767 3417(h)N
3792(hh)X
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11 s
3869 3516(D)N
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3946(\()X
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3975(r)X
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4009(,)X
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4045(j)X
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4077(\))X
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4106 3436(1)N
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(-)S
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4195 3480(4)N
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(r)S
1 f
4208 3398(1)N
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4180 3417(h)N
4205(hh)X
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1 f
4296 3516(2)N
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8 s
9 f
3436(-)Y
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4397 3480(4)N
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(r)S
4419 3398(j)N
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10 f
4381 3417(h)N
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11 s
2 f
9 f
4505 3516(+)N
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4575 3577(3)N
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(c)S
4595 3464(a)N
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10 f
4562 3490(h)N
4584(hh)X
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4794 3516(\(*\))N
2612 3756(Since)N
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2907(\()X
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2970(,)X
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3006(j)X
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3038(\))X
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3067(\263)X
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3221(i)X
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3253(\))X
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3522 4114(3)N
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(c)S
3407 4001(a)N
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3465(2)X
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8 s
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3921(-)Y
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3544(1)X
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(+)S
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3633 3965(2)N
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(r)S
1 f
3646 3883(1)N
10 f
3618 3902(h)N
3643(hh)X
11 s
3393 4027(h)N
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8 s
3768 4143(j)N
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3796(=)X
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3831(1)X
16 s
9 f
3775 4079(S)N
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8 s
3804 3970(i)N
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3870 4053(j)N
8 s
9 f
3905 3973(-)N
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3940(1)X
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9 f
(+)S
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4029 4017(2)N
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(r)S
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4042 3935(1)N
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4014 3954(h)N
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11 s
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4130 4053(2)N
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9 f
3973(-)Y
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4231 4017(4)N
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(r)S
4253 3935(j)N
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10 f
4215 3954(h)N
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11 s
2 f
9 f
4339 4053(+)N
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4431 4114(3)N
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(c)S
4451 4001(a)N
1 f
10 f
4418 4027(h)N
4440(hh)X
9 f
3304 4422(\263)N
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3594 4483(3)N
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(c)S
3396 4370(a)N
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3454(2)X
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8 s
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4290(-)Y
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3533(1)X
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9 f
(+)S
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3622 4334(2)N
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(r)S
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3635 4252(1)N
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11 s
3709 4370(R)N
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3777(\()X
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3806(r)X
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3847(\))X
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3382 4396(h)N
3405(hhhhhhhhhhh)X
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9 f
3941 4422(+)N
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4033 4483(3)N
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(c)S
4053 4370(a)N
1 f
10 f
4020 4396(h)N
4042(hh)X
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4182 4422(,)N
2612 4662(where)N
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4528(R)X
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4596(\()X
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4625(r)X
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4666(\).)X
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3652(r)X
9 f
3700(=)X
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3748 MX
(12)130 767 oc
1 f
3815(,)X
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2 f
2612 4914(D)N
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2689(\()X
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2718 MX
(12)130 767 oc
1 f
2785(,)X
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2814(i)X
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2846(\))X
2 f
9 f
2875(\316O)X
1 f
3001(\(1\))X
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4220(r)X
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4280(the)X
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4794(\(*\))X
2612 5138(yields:)N
2 f
8 s
3335 5370(r)N
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9 f
3365 MX
(->)174 987 oc
3428(\245)X
11 s
1 f
3345 5306(lim)N
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3474(D)X
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3551(\()X
2 f
3580(r)X
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3675(\))X
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3726(\316)X
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3811(W)X
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3879(\()X
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3908(i)X
8 s
3943 5271(a)N
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4003(6)X
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(c)S
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11 s
4070 5306(\))N
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3312(and)X
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4372(c)X
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4411(.)X
3 f
2612 6080(5.3.)N
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3569(Period)X
1 f
2812 6220(Let)N
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3167(A)X
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3228(\()X
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3257(N)X
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3323(\))X
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6297(m)Y
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3885 6332(leaves)N
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4356(period)X
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4621(is)X
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3270 6724(e)N
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3323(\()X
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3418(\))X
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3606(\()X
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3635 6707 MXY
(sr)214 549 oc
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3780(N)X
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337 1071 MXY
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385(d)X
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68(\()X
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97 1183 MXY
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308(\))X
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1498(c)X
1684(c)Y
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1753 1708(radical)N
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(r)S
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9 s
2016(\))X
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51 1825(i)N
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(c)S
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974(\))X
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995(\307O)X
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1090(\()X
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1212(log)X
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1359(2)X
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(a)S
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1411(\))X
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1432 2122(\))N
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1743(W)X
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1792(\()X
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1813(D)X
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1869(\()X
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1959(\))X
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2028(\))X
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1130(\))X
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1884(\))X
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(r)S
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1933 2271(1)N
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1063 2722(2)N
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1092 2747(N)N
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1140(\))X
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1872(\))X
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(\))S
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(r)S
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1 f
3962(is)X
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4426(characteristic)X
2612 1648(of)N
2731(the)X
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3130(Equation)X
3495(\(17\))X
3687(is)X
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4102(using)X
4338(``Rent's)X
4671(Rule'')X
2612 1760(which)N
2857(states)X
3083(that)X
3246(the)X
3384(average)X
3687(number)X
3986(of)X
4089(terminals)X
4449(per)X
4592(complex)X
2612 1872(of)N
2 f
2730(C)X
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2834(elements)X
3193(\(in)X
3336(units,)X
3575(modules,)X
3941(cards,)X
4193(gates)X
4418(etc.\))X
4638(is)X
2 f
4741(tC)X
8 s
4835 1837(p)N
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11 s
4874 1872(,)N
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2849(t)X
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2897(is)X
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2 f
2612 2096(p)N
1 f
2689(is)X
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4759(The)X
2612 2208(analysis)N
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2612 2432(individual)N
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2612 2544(recursion)N
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2612 2656(determined)N
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3544(rule.)X
3742(This)X
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9 f
4368(l)X
4430 2639(~)N
4430 2661(~)N
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4492 2657(3)N
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9 f
(-)S
1 f
4584(2)X
2 f
(p)S
1 f
(.)S
4755(In)X
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2612 2880(placed)N
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2612 2992(high-speed)N
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3423(p)X
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3493(was)X
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2612 3104(hand)N
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3563(p)X
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3621(=)X
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3669(0.59.)X
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2 f
9 f
(\243)S
1 f
9 f
4121(l)X
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4169(<)X
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4228(3,)X
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2612 3216(of)N
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3768(distribution;)X
4232(similar)X
4509(laws)X
4701(occur)X
2612 3328(in)N
2716(contexts)X
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3429(frequencies,)X
3892(noise)X
4114(in)X
4219(transmission)X
4701(chan-)X
2612 3440(nels)N
2786(etc.)X
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3878(see)X
4021([)X
4081(Donat)X
4327(Let)X
4476(furthermore)X
2612 3552(the)N
2747(network)X
3062(be)X
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3578(so)X
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2 f
2612 3664(C)N
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2716(available)X
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4788(the)X
2612 3776(number)N
2912(of)X
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2612 3888(the)N
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2 f
4384(c)X
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4423(.)X
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9 f
4611(l\271)X
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4707(3:)X
2 f
3385 4091(c)N
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9 f
3453 4074(~)N
3453 4096(~)N
2 f
3545 4164(a)N
1 f
3603(\()X
2 f
3632(A)X
1 f
8 s
3696 4129(\(3)N
2 f
9 f
3749(-)X
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9 f
3784(l)X
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3819(\))X
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3845(/)X
1 f
3868(2)X
2 f
11 s
9 f
3907 4164(-)N
1 f
3955(1\))X
3568 4040(\()N
2 f
3597(A)X
9 f
3665(-)X
2 f
3713(C)X
1 f
3779(\)\(3)X
2 f
9 f
3881(-)X
1 f
9 f
3929(l)X
1 f
3977(\))X
10 f
3532 4066(h)N
3558(hhhhhhhhhhh)X
1 f
4094 4092(,)N
4711(\(18a\))X
2612 4314(and)N
2761(for)X
9 f
2885(l)X
2 f
9 f
2933(=)X
1 f
2981(3,)X
2 f
3488 4517(c)N
1 f
9 f
3556 4500(~)N
3556 4522(~)N
2 f
3667 4579(a)N
1 f
3732(log)X
2 f
3859(A)X
1 f
3648 4466(2\()N
2 f
3721(A)X
9 f
3789(-)X
2 f
3837(C)X
1 f
3903(\))X
10 f
3634 4492(h)N
3637(hhhhhhh)X
2 f
3997 4518(.)N
1 f
4706(\(18b\))X
2612 4757(Consequently,)N
3141(for)X
9 f
3265(l\271)X
1 f
3361(1)X
3427(&)X
9 f
3518(l\271)X
1 f
3614(3)X
3680(by)X
3790(\(16b\):)X
2 f
3014 4980(C)N
1 f
9 f
3102 4963(~)N
3102 4985(~)N
2 f
8 s
3172 5071(i)N
9 f
3200(=)X
1 f
3235(1)X
16 s
9 f
3181 5007(S)N
8 s
3177 4886 MXY
(sr)214 549 oc
10 f
3212(d)X
3229(d)X
2 f
3212 4898(A)N
11 s
3273 4981(f)N
1 f
3326(\()X
2 f
3355(i)X
1 f
3387(\))X
9 f
3438 4964(~)N
3438 4986(~)N
2 f
3617 5054(a)N
1 f
3682(\(1)X
2 f
9 f
3755(-)X
1 f
9 f
3803(l)X
1 f
3851(\))X
3894(\()X
2 f
3923(A)X
1 f
8 s
3987 5019(\(3)N
2 f
9 f
4040(-)X
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9 f
4075(l)X
1 f
4110(\))X
2 f
4136(/)X
1 f
4159(2)X
2 f
11 s
9 f
4198 5054(-)N
1 f
4246(1\))X
3530 4930(\()N
2 f
3559(A)X
9 f
3627(-)X
2 f
3675(C)X
1 f
3741(\)\(3)X
2 f
9 f
3843(-)X
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9 f
3891(l)X
1 f
3939(\))X
3982(\()X
2 f
4011(A)X
1 f
8 s
4075 4895(\(1)N
2 f
9 f
4128(-)X
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9 f
4163(l)X
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4198(\))X
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4224(/)X
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4247(2)X
2 f
11 s
9 f
4286 4930(-)N
1 f
4334(1\))X
10 f
3516 4956(h)N
3540(hhhhhhhhhhhhhhhhhhhh)X
2 f
4472 4982(.)N
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4711(\(19a\))X
2612 5248(and)N
2761(for)X
9 f
2885(l)X
2 f
9 f
2933(=)X
1 f
2981(3,)X
2 f
3391 5451(C)N
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9 f
3479 5434(~)N
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2 f
3650 5513(aA)N
1 f
3769(log)X
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3896(A)X
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3571 5400(\()N
2 f
3600(A)X
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3668(-)X
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3716(C)X
1 f
3782(\)\()X
2 f
3840(A)X
9 f
3908(-)X
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3956(1\))X
10 f
3557 5426(h)N
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2 f
4094 5452(.)N
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4706(\(19b\))X
2612 5663(For)N
9 f
2756(l)X
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9 f
2804(=)X
1 f
2852(1,)X
2 f
3209 5886(C)N
1 f
9 f
3297 5869(~)N
3297 5891(~)N
2 f
8 s
3367 5977(i)N
9 f
3395(=)X
1 f
3430(1)X
16 s
9 f
3376 5913(S)N
8 s
3372 5792 MXY
(sr)214 549 oc
10 f
3407(d)X
3424(d)X
2 f
3407 5804(A)N
11 s
3468 5887(f)N
1 f
3521(\()X
2 f
3550(i)X
1 f
3582(\))X
9 f
3633 5870(~)N
3633 5892(~)N
2 f
3725 5949(a)N
1 f
3783(\()X
2 f
3812(A)X
9 f
3880(-)X
1 f
3928(1\))X
2 f
3776 5836(A)N
9 f
3844(-)X
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3892(C)X
1 f
10 f
3712 5862(h)N
3751(hhhhhh)X
1 f
4045 5888(log)N
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4172(A)X
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2 f
4277(.)X
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4711(\(19c\))X
2612 6154(\(Note:)N
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2983(l)X
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3031(<)X
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3090(1)X
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3524(c)X
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3636(1,)X
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4156(f)X
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4209(\()X
2 f
4238(i)X
1 f
4270(\))X
9 f
4299 6137(~)N
4299 6159(~)N
1 f
4347 6155(0)N
4414(also)X
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4704(small)X
2 f
2612 6266(i)N
1 f
2637(,)X
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2830(C)X
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2911(a)X
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3 f
2812 6406(Lemma)N
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3250(Let)X
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3454(\()X
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3483(i)X
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3515(\))X
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3575(be)X
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3819(\()X
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3848(i)X
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3880(\))X
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9 f
3923(=)X
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3985(Q)X
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(c)S
4053(/)X
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4124(P)X
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4179(for)X
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3476(O)X
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3539(\(log)X
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3695(A)X
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3756(\))X
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3785(.)X
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2812 7106(Since)N
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ep
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11 s 0 xH 0 xS 1 f
10 s
0 32(--)N
4323(--)X
11 s
2353 568(-)N
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2514(-)X
0 976(that)N
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0 1088(particular)N
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412(area)X
2 f
586(A)X
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668(is)X
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2 f
1837(N)X
1 f
1924(of)X
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0 1312(lem,)N
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1235(propagation)X
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0 1424(lemma,)N
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0 1536(putation)N
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1 f
666(\()X
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695(N)X
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761(\))X
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1601(N)X
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1826(N)X
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381(N)X
1 f
463(bits)X
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0 1760(wires)N
221(is)X
9 f
311(W)X
1 f
379(\(log)X
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535(N)X
1 f
601(\).)X
705(In)X
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0 1872(number)N
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193(f)X
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243(the)X
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1691(and)X
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0 2096(number)N
307(of)X
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668(and)X
833(similarly)X
1187(the)X
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0 2208(ponents,)N
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2 f
983(A)X
1 f
1067(\(the)X
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200 2572(The)N
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1 f
1585(For)X
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0 2796(width)N
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1498(-)X
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0 2908(but)N
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461(\(16c\))X
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2 f
8 s
827 3221(i)N
9 f
855(=)X
1 f
890(1)X
16 s
9 f
836 3157(S)N
8 s
832 3036 MXY
(sr)214 549 oc
10 f
867(d)X
884(d)X
2 f
867 3048(A)N
11 s
928 3131(f)N
1 f
981(\()X
2 f
1010(i)X
1 f
1042(\))X
2 f
1085(i)X
1 f
9 f
1139 3114(~)N
1139 3136(~)N
2 f
1209 3132(A)N
9 f
1277(-)X
2 f
1325(C)X
1 f
2 f
1435(.)X
1 f
2138(\(20\))X
0 3399(Thus,)N
230(for)X
2 f
371(f)X
1 f
424(\()X
2 f
453(i)X
1 f
485(\))X
2 f
9 f
528(=)X
10 f
590(Q)X
2 f
(ci)S
8 s
9 f
686 3364(-)N
1 f
9 f
721(l)X
2 f
11 s
10 f
763 3399(P)N
1 f
817(\(1)X
2 f
9 f
890(\243)X
2 f
938(i)X
9 f
977(\243)X
1 f
9 f
1025 3382 MXY
(sr)214 549 oc
10 f
1073(d)X
1097(d)X
2 f
1073 3399(A)N
1 f
1141(\))X
1202(and)X
2 f
1368(f)X
1 f
1421(\()X
2 f
1450(i)X
1 f
1482(\))X
9 f
1525 3382(~)N
1525 3404(~)N
1 f
1587 3400(0)N
1663(\()X
2 f
1692(i)X
1731(>)X
1 f
9 f
1790 3383 MXY
(sr)214 549 oc
10 f
1838(d)X
1862(d)X
2 f
1838 3400(A)N
1 f
1906(\))X
1967(and)X
2127(with)X
2 f
0 3511(A)N
1 f
54(,)X
2 f
98(C)X
1 f
179(and)X
2 f
328(c)X
1 f
389(as)X
484(above)X
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9 f
2092(l)X
2 f
9 f
2140(=)X
1 f
2188(1:)X
2 f
440 3714(c)N
1 f
9 f
508 3697(~)N
508 3719(~)N
600 3777 MXY
(sr)214 549 oc
10 f
648(d)X
672(d)X
2 f
648 3794(A)N
9 f
716(-)X
1 f
764(1)X
2 f
617 3663(A)N
9 f
685(-)X
2 f
733(C)X
1 f
10 f
587 3689(h)N
602(hhhhh)X
1 f
2138 3715(\(21\))N
2 f
420 3978(C)N
1 f
9 f
508 3961(~)N
508 3983(~)N
1 f
655 4058(2\()N
9 f
728 4041 MXY
(sr)214 549 oc
10 f
776(d)X
800(d)X
2 f
776 4058(A)N
9 f
844(-)X
1 f
892(1\))X
600 3927(\()N
2 f
629(A)X
9 f
697(-)X
2 f
745(C)X
1 f
811(\)log)X
2 f
967(A)X
1 f
10 f
586 3953(h)N
594(hhhhhhhhhh)X
2 f
1086 3979(.)N
1 f
0 4235(For)N
9 f
144(l\271)X
1 f
240(1)X
306(&)X
9 f
397(l\271)X
1 f
493(2:)X
2 f
440 4438(c)N
1 f
9 f
508 4421(~)N
508 4443(~)N
2 f
635 4511(A)N
1 f
8 s
699 4476(\(2)N
2 f
9 f
752(-)X
1 f
9 f
787(l)X
1 f
822(\))X
2 f
848(/)X
1 f
871(2)X
2 f
11 s
9 f
910 4511(-)N
1 f
958(1)X
600 4387(\(2)N
2 f
9 f
673(-)X
1 f
9 f
721(l)X
1 f
769(\)\()X
2 f
827(A)X
9 f
895(-)X
2 f
943(C)X
1 f
1009(\))X
10 f
586 4413(h)N
611(hhhhhhhhhh)X
1 f
2138 4439(\(22\))N
2 f
420 4707(C)N
1 f
9 f
508 4690(~)N
508 4712(~)N
1 f
720 4780(\(1)N
2 f
9 f
793(-)X
1 f
9 f
841(l)X
1 f
889(\)\()X
2 f
947(A)X
1 f
8 s
1011 4745(\(2)N
2 f
9 f
1064(-)X
1 f
9 f
1099(l)X
1 f
1134(\))X
2 f
1160(/)X
1 f
1183(2)X
2 f
11 s
9 f
1222 4780(-)N
1 f
1270(1\))X
600 4656(\(2)N
2 f
9 f
673(-)X
1 f
9 f
721(l)X
1 f
769(\)\()X
2 f
827(A)X
9 f
895(-)X
2 f
943(C)X
1 f
1009(\)\()X
2 f
1067(A)X
1 f
8 s
1131 4621(\(1)N
2 f
9 f
1184(-)X
1 f
9 f
1219(l)X
1 f
1254(\))X
2 f
1280(/)X
1 f
1303(2)X
2 f
11 s
9 f
1342 4656(-)N
1 f
1390(1\))X
10 f
586 4682(h)N
596(hhhhhhhhhhhhhhhhhhhh)X
2 f
1528 4708(.)N
1 f
0 4958(For)N
9 f
144(l)X
2 f
9 f
192(=)X
1 f
240(2:)X
2 f
440 5161(c)N
1 f
9 f
508 5144(~)N
508 5166(~)N
1 f
651 5223(log)N
2 f
778(A)X
1 f
600 5110(2\()N
2 f
673(A)X
9 f
741(-)X
2 f
789(C)X
1 f
855(\))X
10 f
586 5136(h)N
589(hhhhhhh)X
1 f
2138 5162(\(23\))N
2 f
420 5425(C)N
1 f
9 f
508 5408(~)N
508 5430(~)N
719 5488 MXY
(sr)214 549 oc
10 f
767(d)X
791(d)X
2 f
767 5505(A)N
1 f
849(log)X
2 f
976(A)X
1 f
600 5374(2\()N
2 f
673(A)X
9 f
741(-)X
2 f
789(C)X
1 f
855(\)\()X
9 f
913 5357 MXY
(sr)214 549 oc
10 f
961(d)X
985(d)X
2 f
961 5374(A)N
9 f
1029(-)X
1 f
1077(1\))X
10 f
586 5400(h)N
591(hhhhhhhhhhhhh)X
2 f
1215 5426(.)N
1 f
0 5682(\(Note:)N
254(for)X
9 f
385(l)X
2 f
433(<)X
1 f
492(0)X
565(we)X
696(obtain)X
2 f
946(c)X
999(<)X
1 f
1058(1.\))X
1204(For)X
9 f
1355(l)X
2 f
1403(>)X
1 f
1462(0)X
1535(we)X
1667(have)X
2 f
1863(C)X
9 f
1943(\316)X
1 f
9 f
2020(W)X
1 f
2088(\()X
9 f
2117 5665 MXY
(sr)214 549 oc
10 f
2165(d)X
2189(d)X
2 f
2165 5682(A)N
1 f
2233(\).)X
0 5794(Thus:)N
3 f
200 5934(Lemma)N
519(2.)X
2 f
638(Let)X
789(f)X
1 f
842(\()X
2 f
871(i)X
1 f
903(\))X
2 f
963(be)X
1077(the)X
1216(wire)X
1405(length)X
1658(distribution)X
2103(func-)X
0 6046(tion)N
166(of)X
263(a)X
334(chip)X
513(layout)X
761(with)X
941(area)X
1129(A)X
1210(under)X
1442(the)X
1577(constant)X
1903(wire)X
2087(width)X
0 6158(assumption.)N
471(If)X
558(f)X
1 f
611(\()X
2 f
640(i)X
1 f
672(\))X
2 f
9 f
715(=)X
10 f
777(Q)X
2 f
(c)S
845(/)X
877(i)X
8 s
9 f
912 6123(l)N
11 s
10 f
954 6158(P)N
1 f
1003(\(1)X
2 f
9 f
1076(\243)X
2 f
1124(i)X
9 f
1163(\243)X
1211 6141 MXY
(sr)214 549 oc
10 f
1259(d)X
1283(d)X
2 f
1259 6158(A)N
1 f
1327(\))X
2 f
1383(and)X
1549(f)X
1 f
1602(\()X
2 f
1631(i)X
1 f
1663(\))X
2 f
9 f
1692 6141(~)N
1692 6163(~)N
1 f
1740 6159(0)N
1811(\()X
2 f
1840(i)X
1879(>)X
9 f
1938 6142 MXY
(sr)214 549 oc
10 f
1986(d)X
2010(d)X
2 f
1986 6159(A)N
1 f
2054(\))X
2 f
2110(for)X
2240(a)X
0 6270(constant)N
324(c)X
388(then)X
564(the)X
696(maximal)X
1023(feasible)X
1322(number)X
1614(of)X
1707(wires)X
1922(in)X
2015(the)X
2147(lay-)X
0 6382(out,)N
163(and)X
324(similarly)X
667(the)X
804(maximal)X
1136(number)X
1433(of)X
1531(gates)X
1746(and)X
1907(transistors)X
0 6494(in)N
91(the)X
221(layout,)X
486(is)X
9 f
567(W)X
1 f
635(\()X
2 f
9 f
664 6477 MXY
(sr)214 549 oc
10 f
712(d)X
736(d)X
2 f
712 6494(A)N
1 f
780(\))X
2 f
831(for)X
956(all)X
9 f
1072(l\263)X
1 f
1168(0)X
2 f
(.)S
1 f
200 6634(Recall)N
448(that)X
603(the)X
733(quotient)X
1045(of)X
1140(the)X
1270(length)X
1514(and)X
1664(width)X
1888(of)X
1984(a)X
2046(wire)X
2225(is)X
0 6746(its)N
2 f
117(aspect)X
375(ratio.)X
1 f
624(By)X
760(the)X
900(previous)X
1235(analysis,)X
1573(considering)X
2016(just)X
2176(the)X
0 6858(wire)N
201(length)X
467(distribution)X
920(while)X
1161(leaving)X
1466(free)X
1647(the)X
1801(actual)X
2058(circuit)X
0 6970(topology,)N
361(placement)X
749(and)X
901(routing)X
1180(in)X
1273(the)X
1405(layouts,)X
1706(attaining)X
2040(a)X
2103(loga-)X
0 7082(rithmic)N
305(signal)X
565(propagation)X
1035(delay)X
1275(by)X
1412(changing)X
1784(constant)X
2128(wire)X
0 7194(width)N
224(to)X
315(constant)X
631(aspect)X
873(ratio)X
1057(for)X
1181(all)X
1292(wires)X
1504(in)X
1595(a)X
1656(layout)X
1899(can)X
2043(carry)X
2245(a)X
0 7306(surprisingly)N
443(severe)X
689(penalty.)X
3 f
2812 976(Theorem.)N
2 f
3221(Let)X
3357(the)X
3488(original)X
3796(layout)X
4040(area)X
4224(be)X
4330(A)X
4407(and)X
4563(the)X
4695(origi-)X
2612 1088(nal)N
2751(amount)X
3041(of)X
3136(wires)X
3353(in)X
3448(the)X
3582(layout)X
3829(be)X
3938(C.)X
4045(For)X
4203(the)X
4336(wire)X
4518(length)X
4764(dis-)X
2612 1200(tribution)N
2967(f)X
1 f
3020(\()X
2 f
3049(i)X
1 f
3081(\))X
2 f
9 f
3124(=)X
10 f
3186(Q)X
2 f
(ci)S
8 s
9 f
3282 1165(-)N
1 f
3317(1)X
2 f
11 s
10 f
3356 1200(P)N
2 f
3416(for)X
1 f
3557(1)X
2 f
9 f
(\243)S
2 f
3649(i)X
9 f
3688(\243)X
3736 1183 MXY
(sr)214 549 oc
10 f
3784(d)X
3808(d)X
2 f
3784 1200(A)N
3891(and)X
4069(f)X
1 f
4122(\()X
2 f
4151(i)X
1 f
4183(\))X
2 f
9 f
4226 1183(~)N
4226 1205(~)N
1 f
4288 1201(0)N
2 f
4371(for)X
4513(i)X
4552(>)X
9 f
4611 1184 MXY
(sr)214 549 oc
10 f
4659(d)X
4683(d)X
2 f
4659 1201(A)N
4727(,)X
4788(the)X
2612 1312(change)N
2889(from)X
3078(constant)X
3400(wire)X
3580(width)X
3800(to)X
3892(wires)X
4106(with)X
4282(a)X
4349(constant)X
4671(aspect)X
2612 1424(ratio)N
2806(has)X
2950(the)X
3080(following)X
3437(effect.)X
1 f
2612 1564(\(i\))N
2 f
2812(Retaining)X
3182(the)X
3316(original)X
3627(amount)X
3917(of)X
4013(wires)X
4231(C)X
4317(and)X
4476(the)X
4611(original)X
2812 1676(wire)N
2994(length)X
3240(distribution)X
3678(relative)X
3973(to)X
4067(the)X
4200(layout)X
4446(area,)X
4653(that)X
4815(is,)X
2819 1788(f)N
1 f
2872(\()X
2 f
2901(i)X
1 f
2933(\))X
2 f
9 f
2976(=)X
10 f
3038(Q)X
2 f
(c)S
1 f
9 f
3106(\242)X
2 f
3128(i)X
8 s
9 f
3163 1753(-)N
1 f
3198(1)X
2 f
11 s
10 f
3237 1788(P)N
2 f
3285(for)X
1 f
3414(1)X
2 f
9 f
(\243)S
2 f
3506(i)X
9 f
3545(\243)X
3593 1771 MXY
(sr)214 549 oc
10 f
3641(d)X
3650(dd)X
2 f
3641 1788(A)N
1 f
9 f
3702(\242)X
2 f
3764(and)X
3929(f)X
1 f
3982(\()X
2 f
4011(i)X
1 f
4043(\))X
2 f
9 f
4086 1771(~)N
4086 1793(~)N
1 f
4148 1789(0)N
2 f
4218(for)X
4347(i)X
4379(>)X
9 f
4438 1772 MXY
(sr)214 549 oc
10 f
4486(d)X
4495(dd)X
2 f
4486 1789(A)N
1 f
9 f
4547(\242)X
2 f
4609(with)X
4788(the)X
2812 1900(normalization)N
3372(constant)X
3736(c)X
1 f
9 f
3782(\242)X
2 f
3869(set)X
4032(to)X
4166(its)X
4315(maximal)X
4683(value,)X
2812 2012(exponentially)N
3316(increases)X
3676(the)X
3812(required)X
4143(area)X
4332(A)X
1 f
9 f
4393(\242)X
2 f
4443(over)X
4627(the)X
4764(ori-)X
2812 2124(ginal)N
3016(area)X
3199(A.)X
1 f
2612 2264(\(ii\))N
2 f
2812(Retaining)X
3181(the)X
3314(original)X
3624(area)X
3810(A)X
3889(and)X
4047(the)X
4181(original)X
4492(wire)X
4675(length)X
2812 2376(distribution,)N
3275(that)X
3440(is,)X
3555(f)X
1 f
3608(\()X
2 f
3637(i)X
1 f
3669(\))X
2 f
9 f
3712(=)X
10 f
3774(Q)X
2 f
(c)S
1 f
9 f
3842(\242)X
2 f
3864(i)X
8 s
9 f
3899 2341(-)N
1 f
3934(1)X
2 f
11 s
10 f
3973 2376(P)N
2 f
4022(for)X
1 f
4152(1)X
2 f
9 f
(\243)S
2 f
4244(i)X
9 f
4276(\243)X
4324 2359 MXY
(sr)214 549 oc
10 f
4372(d)X
4396(d)X
2 f
4372 2376(A)N
4467(and)X
4633(f)X
1 f
4686(\()X
2 f
4715(i)X
1 f
4747(\))X
2 f
9 f
4790 2359(~)N
4790 2381(~)N
1 f
4852 2377(0)N
2 f
2812 2488(for)N
2958(i)X
2997(>)X
9 f
3056 2471 MXY
(sr)214 549 oc
10 f
3104(d)X
3128(d)X
2 f
3104 2488(A)N
3215(with)X
3411(the)X
3563(normalization)X
4102(constant)X
4445(c)X
1 f
9 f
4491(\242)X
2 f
4557(set)X
4699(to)X
4812(its)X
2812 2600(maximum)N
3185(value,)X
3430(reduces)X
3735(the)X
3875(maximal)X
4210(amount)X
4505(of)X
4605(wires)X
4827(in)X
2812 2712(the)N
2945(layout,)X
3213(and)X
3370(therefore)X
3718(the)X
3851(proportionate)X
4366(amount)X
4655(of)X
4750(use-)X
2812 2824(ful)N
3016(components)X
3546(like)X
3783(gates)X
4078(and)X
4319(transistors,)X
4827(to)X
2812 2936(C)N
1 f
9 f
2878(\242)X
2 f
9 f
2914(\316)X
2991(O)X
1 f
3054(\(log)X
2 f
3210(C)X
1 f
3276(\))X
2 f
3305(.)X
1 f
2612 3076(\(iii\))N
2 f
2812(Retaining)X
3189(the)X
3331(original)X
3650(amount)X
3948(of)X
4051(wires)X
4276(C)X
4369(\(or)X
4510(logic)X
4721(com-)X
2812 3188(ponents\))N
3163(and)X
3343(the)X
3499(original)X
3832(area)X
4041(A)X
4142(requires)X
4482(a)X
4573(thorough)X
2812 3300(change)N
3115(of)X
3233(wire)X
3439(length)X
3710(distribution)X
4173(to)X
4299(f)X
1 f
9 f
4331(\242)X
1 f
4353(\()X
2 f
4382(i)X
1 f
4414(\))X
2 f
9 f
4457(=)X
10 f
4519(Q)X
2 f
(c)S
1 f
9 f
4587(\242)X
2 f
4609(i)X
8 s
9 f
4644 3265(-l)N
11 s
10 f
4721 3300(P)N
2 f
4793(for)X
1 f
2812 3412(1)N
2 f
9 f
(\243)S
2 f
2904(i)X
9 f
2936(\243)X
2984 3395 MXY
(sr)214 549 oc
10 f
3032(d)X
3056(d)X
2 f
3032 3412(A)N
3124(and)X
3287(f)X
1 f
3340(\()X
2 f
3369(i)X
1 f
3401(\))X
2 f
9 f
3444 3395(~)N
3444 3417(~)N
1 f
3506 3413(0)N
2 f
3573(for)X
3699(i)X
3738(>)X
9 f
3797 3396 MXY
(sr)214 549 oc
10 f
3845(d)X
3869(d)X
2 f
3845 3413(A)N
3936(and)X
4091(the)X
4222(normalization)X
4740(con-)X
2812 3524(stant)N
3013(c)X
1 f
9 f
3059(\242)X
2 f
3110(set)X
3238(to)X
3337(its)X
3451(maximum)X
3822(value.)X
4087(Each)X
9 f
4298(l)X
4360(\263)X
1 f
4422(2)X
2 f
9 f
(+e)S
2 f
4583(for)X
4716(some)X
2812 3636(small)N
9 f
3026(e)X
2 f
3065(>)X
1 f
3124(0)X
2 f
3191(depending)X
3580(only)X
3754(on)X
3864(A)X
3940(and)X
4094(a)X
4160(suf\256ces.)X
4485(For)X
4639(a)X
4705(given)X
2812 3748(network)N
3128(topology)X
3469(this)X
3629(entails)X
3897(a)X
3974(placement)X
4369(and)X
4534(routing)X
4827(of)X
2812 3860(the)N
2942(layout)X
3185(which)X
3418(may)X
3586(well)X
3756(be)X
3861(impossible)X
4260(in)X
4351(many)X
4563(cases.)X
3 f
2812 4000(Proof.)N
1 f
3102(Since)X
3334(we)X
3472(assume)X
3767(the)X
3911(circuit)X
4174(to)X
4280(be)X
4400(connected)X
4794(we)X
2612 4112(have)N
2 f
2809(A)X
2884(>)X
2957(A)X
9 f
3025(-)X
2 f
3073(C)X
3153(>)X
3226(A)X
3287(/)X
1 f
3319(2)X
3394(in)X
3494(the)X
3633(various)X
3923(equations.)X
4313(We)X
4465(also)X
4637(assume)X
2 f
2612 4224(A)N
1 f
2687(>)X
2711(>)X
2775(1.)X
2612 4364(\(i\))N
2812(Equate)X
3085(expression)X
3489(\(21\))X
3663(for)X
2 f
3793(C)X
1 f
3880(with)X
4065(expression)X
4470(\(19c\))X
4684(for)X
2 f
4815(C)X
1 f
4874(,)X
2812 4476(with)N
2 f
3031(A)X
1 f
9 f
3092(\242)X
1 f
3176(substituted)X
3621(for)X
2 f
3785(A)X
1 f
3901(in)X
4032(the)X
4202(latter.)X
4467(This)X
4685(yields)X
2812 4588(log)N
2 f
2939(A)X
1 f
9 f
3000(\242)X
2 f
9 f
3036(\316)X
1 f
9 f
3113(W)X
1 f
3181(\()X
9 f
3210 4571 MXY
(sr)214 549 oc
10 f
3258(d)X
3282(d)X
2 f
3258 4588(A)N
1 f
3326(\).)X
2612 4728(\(ii\))N
2812(Substitute)X
2 f
3207(C)X
1 f
9 f
3273(\242)X
1 f
3336(for)X
2 f
3479(C)X
1 f
3579(in)X
3689(equation)X
4035(\(21\))X
4223(and)X
4392(express)X
2 f
4697(C)X
1 f
9 f
4763(\242)X
1 f
4827(in)X
2812 4840(terms)N
3044(of)X
2 f
3153(C)X
1 f
3248(by)X
3371(eliminati)X
3687(ng)X
2 f
3810(A)X
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3899(from)X
4105(the)X
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2812 4952(and)N
2961(\(19c\).)X
2612 5092(\(iii\))N
2812(Equate)X
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3493(\(21\))X
3670(for)X
2 f
3803(C)X
1 f
3893(with)X
4081(expression)X
4488(\(19a\))X
4704(for)X
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1 f
2812 5204(\(expressions)N
3289(\(19b\))X
3516(and)X
3680(\(19c\))X
3902(contradict)X
4292(\(21\)\).)X
4526(The)X
4700(terms)X
2812 5316(\()N
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9 f
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2 f
2957(C)X
1 f
3023(\))X
3085(on)X
3206(both)X
3396(sides)X
3605(cancel)X
3863(each)X
4057(other.)X
4294(Solving)X
9 f
4603(l)X
1 f
4685(yields)X
9 f
2812 5428(l)N
2 f
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1 f
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2 f
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1 f
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1 f
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4290 MX
(->)174 987 oc
4377(\245)X
1 f
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2 f
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1 f
4740(con-)X
2812 5540(stant.)N
3023(Every)X
3255(distribution)X
3685(with)X
3864(exponent)X
4209(equal)X
4422(or)X
4517(larger)X
4744(than)X
2812 5652(this)N
9 f
2962(l)X
1 f
3032(suf\256ces.)X
10 f
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2812 5792(We)N
2956(observe)X
3251(that)X
3406(in)X
3497(case)X
3670(\(i\))X
3775(of)X
3871(the)X
4002(Theorem)X
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4474(wires)X
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2806(that)X
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3926(as)X
4031(in)X
4132(the)X
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2612 6352(Finally,)N
2912(in)X
3007(case)X
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3576(Theorem)X
3920(the)X
4054(subject)X
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4582(topology)X
2612 6464(may)N
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4488(distribution)X
2612 6716(the)N
2750(analog)X
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3256(Theorem)X
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3825(with)X
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2612 6940(decreases)N
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4159(that)X
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2612 7052(A)N
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4352(However,)X
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2612 7276(implies)N
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3280(Rent)X
3470(constant)X
2 f
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9 f
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4120 7259(~)N
4120 7281(~)N
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4182 7277(3)N
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0 32(--)N
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11 s
2353 568(-)N
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0 976(therefore)N
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1105(satisfy)X
1358(Rent's)X
1611(Rule.)X
1823(The)X
1983(reader)X
2225(is)X
0 1088(invited)N
285(to)X
393(analyse)X
697(the)X
845(relations)X
1189(for)X
1331(different)X
1674(values)X
1939(of)X
9 f
2052(l)X
1 f
2100(.)X
2162(We)X
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291(one)X
452(more)X
667(case,)X
874(the)X
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1903(with)X
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2218(2,)X
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352(interesting)X
764(because)X
1081(it)X
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1671(with)X
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318(For)X
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1477(the)X
1617(Theorem)X
1967(yield)X
2176(the)X
0 1536(following:)N
0 1676(\(i\))N
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9 f
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297(\316)X
1 f
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374(W)X
1 f
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8 s
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1 f
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877(.)X
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1 f
9 f
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2 f
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302(\316)X
379(O)X
1 f
442(\()X
9 f
471 1799 MXY
(sr)214 549 oc
10 f
519(d)X
535(dddddd)X
2 f
519 1816(C)N
1 f
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726(C)X
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850(.)X
0 1956(\(iii\))N
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740(a)X
829(change)X
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1336(new)X
1532(distribution)X
1990(function)X
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207 2068(f)N
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(sr)214 549 oc
10 f
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1380 2051(~)N
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1428 2069(0)N
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1535(i)X
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1626 2052 MXY
(sr)214 549 oc
10 f
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1674 2069(A)N
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1805(with)X
1996(the)X
2138(new)X
200 2180(normalization)N
720(constant)X
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1089(\242)X
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1140(set)X
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1866(value.)X
2108(Each)X
9 f
200 2292(l\263)N
1 f
296(3)X
362(suf\256ces.)X
200 2432(Network)N
540(topologies,)X
964(which)X
1213(can)X
1369(be)X
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1799(with)X
1990(constant)X
0 2544(width)N
223(wires,)X
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316(in)X
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860(\(or)X
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2245(a)X
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1935(with)X
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160(large)X
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363(l)X
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411(,)X
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834(propagation)X
1277(delay.)X
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200 3916(Exercise.)N
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569(Do)X
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1925(.)X
10 s
0 8088(--)N
4323(--)X
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