The Repeater Tree Construction Problem

Transcription

1 The Repeter Tree Constrution Problem C. Brtoshek 1, S. Held 1, J. Mßberg 1, D. Rutenbh 2, nd J. Vygen 1 1 Forshungsinstitut für Diskrete Mthemtik, Universität Bonn, Lennéstr. 2, D Bonn, Germny, emils: {brtosh,held,mssberg,vygen}@or.uni-bonn.de 2 Institut für Mthemtik, TU Ilmenu, Postfh , D Ilmenu, Germny emil: {dieter.rutenbh}@tu-ilmenu.de Abstrt A tree-like substruture on omputer hip whose tsk it is to rry signl from soure iruit to possibly mny sink iruits nd whih onsists only of wires nd so-lled repeter iruits is lled repeter tree. We present mthemtil formultion of the optimiztion problems relted to the onstrution of suh repeter trees. Furthermore, we prove theoretil properties of simple itertive proedure for these problems whih ws suessfully pplied in prtie. Keywords: VLSI design; repeter tree; Steiner tree; minimum spnning tree AMS subjet lssifition: 05C05, 05C85, 68W25, 68W35 1 Introdution During every omputtion yle of modern highly omplex omputer hip millions of signls hve to trvel between iruits t different lotions on the hip re. While for most of these signls the distnes re reltively smll nd n be bridged by pure metl onnetion between the iruits, there re still mny signls whih hve to trvel reltively long distne. Elementry physil onsidertions [5] imply tht the dely of n eletril signl propgting long metl onnetion pproximtely grows qudrtilly with the trversed distne. Trditionlly, the iruit dely dominted the wire dely nd this qudrti growth did not represent problem. Nowdys though, due to the ontinuous shrinking of feture sizes [4, 10], n ever growing prt of the totl dely is used by wires, nd long metl onnetions hve to be split into severl prts by inserting so-lled repeters. These repeters just evlute the boolen identity funtion nd serve no logil purpose within the omputtion of the hip. Their tsk is only to linerize the dely s funtion of the distne. It is estimted [11] tht for the upoming 45nm nd 32nm tehnologies up to 35% nd 70%, respetively, of ll iruits on hip might hve to be repeters. A tree-like substruture on hip whose tsk it is to rry signl from soure iruit to possibly mny sink iruits nd whih onsists only of wires nd repeters is lled repeter tree. In [2, 3] we proposed lgorithms for the onstrution of repeter tree topologies nd for the tul insertion of repeter iruits into these topologies. During this reserh we oneived simple yet reltively urte dely model whih llows onise mthemtil 1

2 Figure 1: Qulity of the dely model formultion of the repeter tree problem. The purpose of the present pper is to present this formultion, to explin the min optimiztion gols, nd to prove some theoretil properties of the lgorithms in [2, 3]. 2 The Repeter Tree Problem An instne of the repeter tree (topology problem onsists of soure r R 2, finite non-empty set S R 2 of sinks, required rrivl time s R for every sink s S, nd two numbers, d R >0. A fesible solution of suh n instne is rooted tree T = (V (T, E(T with vertex set {r} S I where I R 2 is set of S 1 points suh tht r is the root of T nd hs extly one hild, the elements of I re the internl verties of T nd hve extly two hildren eh, nd the elements of S re the leves of T. In [2, 3] suh fesible solution ws lled repeter tree topology, beuse the number, types, nd positions of the tul repeters re not yet determined. The optimiztion gols for repeter tree re relted to the wiring, to the number of repeter iruits, nd to the timing. We ssume tht every edge e = (u, v E(T of T is relized long pth between the two points u nd v in the plne whih is shortest with respet to some norm on R 2. Furthermore, we ssume tht repeters re inserted in reltively uniform wy into ll wires in order to linerize the dely within the repeter tree. Hene the wiring nd lso the number of repeter iruits needed for the physil reliztion of the edge e re proportionl to u v. For the entire repeter tree topology, this result in totl ost of l(t := u v. (u,v E(T 2

3 The dely of the signl strting t the root nd trvelling through T to the sinks hs two omponents. Let E[r, s] denote the set of edges on the pth P in T between the root r nd some sink s S. The linerized dely long the edges of P is modelled by d u v. (u,v E[r,s] Furthermore, every internl vertex on P orresponds to bifurtion whih uses n dditive dely of long P. For the entire pth P, these dditionl delys sum up to ( E[r, s] 1. In prtie there is sometimes ertin degree of freedom how to distribute the dditionl dely used by bifurtion to the two brnhes [9]. Altogether, we estimte the dely of the signl long P by the sum of these two omponents. Assuming tht the signl strts t time 0 t the root, the slk t some sink s S in T is estimted by σ(t, s := s d u v ( E[r, s] 1 nd the worst slk equls (u,v E[r,s] σ(t := min{σ(t, s s S}. The restritions on the number of hildren of the root nd the internl verties of T imply tht the number of sinks ontributes logrithmilly to the dely, whih orresponds to physil experiene. The ury of our simple dely estimtion is shown in Figure 1, whih ompres our estimtion with the rel physil dely one the repeter tree hs been relized nd optimized. The prmeters nd d re tehnology-dependent. For the 65nm tehnology their vlues re bout = 20ps nd d = 220ps/mm. In priniple, repeter tree topology is eptble with respet to timing if σ(t is nonnegtive, i.e. the signl rrives t every sink s S not lter thn s. Nevertheless, in order to ount for inurte estimtions nd mnufturing vrition, the worst slk σ(t should hve t lest some resonble positive vlue σ min or should even be mximized. We n formulte three min optimiztion senrios: Determine T suh tht (O1 σ(t is mximized, or (O2 l(t is minimized, or (O3 for suitble onstnts α, β, σ min > 0, the expression is mximized. α min{σ(t, σ min } βl(t While senrio (O1 is resonble for instnes whih re very timing ritil, senrio (O2 is resonble for very timing unritil instnes. Senrio (O3 is probbly the prtilly most relevnt one. In the next setion, we will show tht (O1 n be solved extly in polynomil time. In ontrst to tht, (O2 is hrd even for restrited hoies of the norm suh s the l 1 -norm, sine it is essentilly the Steiner tree problem [6]. 3

4 3 A Simple Proedure nd its Properties In [2, 3] we onsidered the following very simple proedure for the onstrution of repeter tree topologies. Choose sink s 1 S; V (T 1 {r, s 1 }; E(T 1 {(r, s 1 }; T 1 (V (T 1, E(T 1 ; n S ; for i = 2 to n do Choose sink s i S \ {s 1, s 2,..., s i 1 }, n edge e i = (u, v E(T i 1, nd n internl vertex x i R 2 ; V (T i V (T i 1. {x i }. {s i }; E(T i (E(T i 1 \ {(u, v} {(u, x i, (x i, v, (x i, s i }; T i (V (T i, E(T i ; end The proedure inserts the sinks one by one ording to some order s 1, s 2,..., s n strting with tree ontining only the root r nd the first sink s 1. The sinks s i for i 2 re inserted by subdividing n edge e i with new internl vertex x i nd onneting x i to s i. The behviour of the proedure lerly depends on the hoie of the order, the hoie of the edge e i, nd the hoie of the point x i R 2. In view of the lrge number of instnes whih hve to be solved in n eptble time [2, 3] the simpliity of the bove proedure is n importnt dvntge for its prtil pplition. Furthermore, implementing suitble rules for the hoie of s i, e i, nd x i llows to pursue nd blne vrious prtil optimiztion gols. We present two vrints (P1 nd (P2 of the proedure orresponding to the bove optimiztion senrios (O1 nd (O2, respetively. (P1 The sinks re inserted in n order of non-inresing ritility, where the ritility of sink s S is quntified by ( s d r s. (Note tht this is the estimted worst slk of repeter tree topology ontining only the one sink s. Sine sink s n be ritil beuse its required rrivl time s is smll nd/or beuse its distne r s to the root is lrge, this is resonble mesure for its ritility. During the i-th exeution of the for-loop, the new internl vertex x i is lwys hosen t the sme position s r formlly this turns V (T i into multiset nd the edge e i is hosen suh tht σ(t i is mximized. (P2 s 1 is hosen suh tht r s 1 = min{ r s s S} nd during the i-th exeution of the for-loop, s i, e i = (u, v, nd x i re hosen suh tht is minimized. l(t i = l(t i 1 + u x i + x i v + x i s i u v 4

5 Theorem 1 The lrgest hievble worst slk σ opt equls { σ (S := mx σ R } 2 1 (s d r s σ 1, nd (P1 genertes repeter tree topology T (P 1 with σ ( T (P 1 = σopt. Proof: Let s = s d r s for s S. Let T be n rbitrry repeter tree topology. By the definition of σ(t nd the tringle-inequlity for, we obtin E[r, s] 1 1 s d u v σ(t 1 ( s σ(t (u,v E[r,s] for every s S. Sine the unique hild of the root r is itself the root of binry subtree of T in whih eh sink s S hs depth extly E[r, s] 1, Krft s inequlity [8] implies 2 1 (s σ(t 2 E[r,s] By the definition of σ (S, this implies σ(t σ (S. Sine T ws rbitrry, we obtin σ opt σ (S. It remins to prove tht σ ( T (P 1 = σopt = σ (S, whih we will do by indution on n = S. For n = 1, the sttement is trivil. Now let n 2. Let s n be the lst sink inserted by (P1, i.e. s n = mx{ s s S}. Let S = S \ {s n }. Clim ( σ (S fr { ( } fr s s S (1 where fr(x := x x denotes the frtionl prt of x R. Proof of the lim: Note tht the definition of σ (S implies tht 1 ( s σ (S is n integer for t lest one s S. If the lim is flse, then 1 ( sn σ (S Z nd 1 ( s σ (S Z for every s S. Sine s n σ (S s σ (S for every s S, this implies nd hene 1 ( sn σ (S { 1 ( > mx s σ (S } s S 2 1 ( s σ (S ( sn σ (S. Now, for some suffiiently smll ɛ > 0, we obtin 2 1 ( s (σ (S+ɛ = 2 1 ( sn σ (S ( s σ (S 1 whih ontrdits the definition of σ (S nd ompletes the proof of the lim. 5

6 Let T (P 1 denote the tree produed by (P1 just before the insertion of the lst sink s n. ( By indution, σ T (P 1 = σ (S. First, we ssume tht there is some sink s S suh tht within T (P 1 1 E[r, s ( ] 1 < s σ (S. Choosing e n s the edge of T (P 1 leding to s, results in tree T suh tht whih implies σ ( T (P 1 = σopt = σ (S. Next, we ssume tht within T (P 1 σ (S σ opt σ ( T (P 1 σ(t = σ (S σ (S, E[r, s] 1 = 1 ( s σ (S for every s S. This implies 2 1 ( s σ (S > 2 1 ( s σ (S = 1 nd hene σ (S < σ (S. By (1, we obtin { ( σ { ( }} σ (S mx σ σ < σ (S, fr fr s s S { ( σ σ = mx σ σ < σ (S (S { (, fr fr s σ (S }} s S { = mx x x < σ (S (, fr x σ (S { ( fr s σ (S }} s S ( σ (S { ( = 1 + mx fr s σ (S } s S for = σ (S (1 δ If s S is suh tht { ( δ = mx fr s σ (S ( δ = fr s σ (S, s S }. then hoosing e n s the edge of T (P 1 leding to s, results in tree T suh tht σ (S σ opt σ ( T (P 1 σ(t = σ (S (1 δ σ (S, whih implies σ ( T (P 1 = σopt = σ (S nd ompletes the proof. Theorem 2 (P2 genertes repeter tree topology T for whih l(t is t most the totl length of minimum spnning tree on {r} S with respet to. 6

7 Proof: Let n = S nd for i = 0, 1,..., n, let T i denote the forest whih is the union of the tree produed by (P2 fter the insertion of the first i sinks nd the remining n i sinks s isolted verties. Note tht T 0 hs vertex set {r} S nd no edge, while for 1 i n, T i hs vertex set {r} S {x j 2 j i} nd 2i 1 edges. Let F 0 = (V (F 0, E(F 0 be spnning tree on V (F 0 = {r} S suh tht l(f 0 = uv E(F 0 u v is minimum. For i = 1, 2,..., n, let F i = (V (F i, E(F i rise from ( V ( T i, E(Fi 1 E ( T i by deleting n edge e E(F i 1 E(F 0 whih hs extly one endvertex in V (T i 1 suh tht F i is tree. (Note tht this uniquely determines F i. Sine (P2 hs the freedom to use the edges of F 0, the speifition of the insertion order nd the lotions of the internl verties in (P2 imply tht Sine F n = T n the proof is omplete. l(f 0 l(f 1 l(f 2... l(f n. For the l 1 -norm, the well-known result of Hwng [7] together with Theorem 2 imply tht (P2 is n pproximtion lgorithm for the l 1 -minimum Steiner tree on the set {r} S with pproximtion gurntee 3/2. We hve seen in Theorems 1 nd 2 tht different insertion orders re fvourble for different optimiztion senrios suh s (O1 nd (O2. Alon nd Azr [1] gve n exmple showing tht for the online retiliner Steiner tree problem the best pproximtion rtio we n hieve is Θ(log n/ log log n, where n is the number of terminls. Hene inserting the sinks in n order disregrding the lotions, like in (P1, n led to long Steiner trees, no mtter how we deide where to insert the sinks. The next exmple shows tht inserting the sinks in n order different from the one onsidered in (P1 but still hoosing the edge e i s in (P1 results in repeter tree topology whose worst slk n be muh smller thn the lrgest hievble worst slk. Exmple 3 Let = 1, d = 0 nd N. We onsider the following sequenes of s nd 0 s A(1 = (, 0, A(2 = (A(1,, 0, A(3 = (A(2,, 0,......, 0, }{{} 1+(2 1 1(+2 A(4 = (A(3,, 0, , 0,..., }{{} 1+(2 2 1(+2 i.e. for l 2, the sequene A(l is the ontention of A(l 1, one, nd sequene of 0 s of length 1 + ( 2 l 2 1 (

8 If the entries of A(l re onsidered s the requires rrivl times of n instne of the repeter tree topology problem, then Theorem 1 together with the hoie of nd d imply tht the lrgest hievble worst slk for this instne equls ( ( l log 2 l2 ( (2 i 2 1( i=2 For l = + 1 this is t lest 2 log 2 ( + 2. If we insert the sinks in the order s speified by the sequenes A(l, nd lwys hoose the edge into whih we insert the next internl vertex suh tht the worst slk is mximized, then the following sequene of topologies n rise: T (1 is the topology with two extly sinks t depth 2. The worst slk of T (1 is ( + 2. For l 2, T (l rises from T (l 1 by ( subdividing the edge of T (l 1 inident with the root with new vertex x, (b ppending n edge (x, y to x, ( tthing to y omplete binry tree B of depth l 2, (d tthing to one lef of B two new leves orresponding to sinks with required rrivl times nd 0, nd (e tthing to eh of the remining 2 l 2 1 mny leves of B binry tree whih hs + 2 leves, ll orresponding to sinks of rrivl times 0, whose depths in re 1, 2, 3,..., 1,, + 1, + 1. Note tht this uniquely determines T (l. Clerly, the worst slk in T (l equls (l + 1. Hene for l = + 1, the worst slk equls 2 2, whih differs pproximtely by ftor of 2 from the lrgest hievble worst slk s lulted bove. This exmple, however, does not show tht there is no online lgorithm for pproximtely mximizing the worst slk, sy up to n dditive onstnt of. It is n open question to find biriteri pproximtion lgorithm, or n lgorithm for (O3. Referenes [1] N. Alon nd Y. Azr, On-line Steiner trees in the Euliden plne, Disrete nd Computtionl Geometry 10 (1993, [2] C. Brtoshek, S. Held, D. Rutenbh, nd J. Vygen, Effiient genertion of short nd fst repeter tree topologies, in: Proeedings of the Interntionl Symposium on Physil Design (2006, [3] C. Brtoshek, S. Held, D. Rutenbh, nd J. Vygen, Fst buffering for optimizing worst slk nd resoure onsumption in repeter trees, in: Proeedings of the Interntionl Symposium on Physil Design (2009, [4] J. Cong, An interonnet-entri design flow for nnometer tehnologies, in: Proeedings of the IEEE 89 (2001, [5] W.C. Elmore, The trnsient response of dmped liner networks with prtiulr regrd to widebnd mplifiers, Journl of Applied Physis 19 (1948, [6] M.R. Grey, nd D.S. Johnson, The retiliner Steiner tree problem is NP-omplete, SIAM Journl on Applied Mthemtis 32 (1977,

9 [7] F.K. Hwng, On steiner miniml trees with retiliner distne, SIAM Journl of Applied Mthemtis 30 (1976, [8] L.G. Krft, A devie for quntizing grouping nd oding mplitude modulted pulses, Mster thesis, EE Dept., MIT, Cmbridge [9] J. Mßberg nd D. Rutenbh, Binry trees with hoosble edge lengths, to pper in Informtion Proessing Letters. [10] G.E. Moore, Crmming more omponents onto integrted iruits, Eletronis 38 (1965, [11] P. Sxen, N. Menezes, P. Cohini, nd D. Kirkptrik, The sling hllenge: n orret-by-onstrution design help?, in: Proeedings of the Interntionl Symposium on Physil Design (2003,