Recski, András

Szeszlér, Dávid

Repository of the Academy's Library

3-dimensional RoutingAndra´s RecskiDepartment of Computer Science andInformation TheoryBudapest University of Technology andEconomicsH-1521 Budapest, Hungaryrecski@cs.bme.huDa´vid Szeszle´r∗Department of Computer Science andInformation TheoryBudapest University of Technology andEconomicsH-1521 Budapest, Hungaryszeszler@cs.bme.huAbstract: Consider a single planar grid, or two parallel planar grids of size w × n.The vertices of the grids are called terminals and pairwise disjoint sets of terminalsare called nets. We aim at routing all nets in a cubic grid (above the single grid,or between the two grids holding the terminals) in a vertex-disjoint way. However,to ensure solvability, it is allowed to extend the length and the width of the originalgrid to w′ = sw and n′ = sn by introducing s− 1 pieces of empty rows and columnsbetween every two consecutive rows and columns containing the terminals. Hencethe routing is to be realized in a cubic grid of size (s · n)× (s ·w)× h. The objectiveis to minimize the height h.The above problems are motivated by VLSI design, where technological improve-ments of the past two decades motivated the research of routing problems with a“real” 3-dimensional flavour. In this paper we survey a few recent results.Keywords: VLSI design, 3-dimensional routing1 IntroductionTraditionally, the detailed routing phase of the design of VLSI (Very Large Scale Integrated)circuits was considered as a 2-dimensional problem, gradually extended to 2, 3, . . . layers.Even within this problem, single row routing and channel routing are the better understoodsubproblems, where the terminals are placed on one side, or two opposite sides, respectively, ofa rectangular circuit board, and the routing is to be realized on a few planar layers. Since thelength n of the board is fixed by the row(s) of terminals and the number of layers k is fixed, theobjective is to minimize the width w of the routing.The specification of a (2-dimensional) single row routing or channel routing problem instanceinvolves a family of pairwise disjoint subsets of the terminals, called nets. By a routing we meanan assignment of pairwise vertex-disjoint Steiner-trees in the 3-dimensional grid (of size w×n×k)to each net, such that the assigned tree connects the terminals of the corresponding net.Thus in single row routing and channel routing the inputs are essentially one-dimensional(one or two lists of terminals) and the output is essentially two-dimensional (a fixed number of∗Research is supported by grant Nr. OTKA 67651 of the Hungarian National Science Fund1planar layers). However, as technology permits more and more layers, a ”real” 3-dimensionalapproach becomes reasonable. The research of 3-dimensional routing started in the 1980s andthere are plenty of deep results in this area, see [1, 2, 4, 5, 6, 8, 9, 10, 15, 17], for example.However, most of these results embed certain ”universal-purpose” graphs (like n-permuters,n-rearrangeable permutation networks, shuﬄe-exchange graphs) into the 3-dimensional grid,ensuring that pairs of terminals can be connected, moreover, in some papers along edge-disjointpaths.The results surveyed in this paper allow multiterminal nets as well, and ensure vertex disjointpaths (or Steiner-trees) for the interconnections of the terminals within each net. We considerthe 3-dimensional analogues of single row routing and channel routing. That is, terminals occupycertain vertices of a w×n planar grid (single active layer routing problem) or two parallel w×ngrids (3-dimensional channel routing) and the third dimension (above the single grid or betweenthe two grids) is for interconnections only. The objective is to minimize the height h of therouting.One can easily see that a routing can be impossible with an arbitrary height (even in smallinstances, like a single active layer problem of size 4 × 1 or 2 × 2). Therefore it is allowed toextend the length and the width of the original grids containing the terminals to w′ = sw · wand n′ = sn · n, respectively. sw and sn, called the spacing, are given constants. Extending thedimensions of the grids is done by introducing empty rows and columns into the (one or two)planar grids holding the terminals. Hence the final routing is realized in a cubic grid of size(sw · w)× (sn · n)× h.As mentioned above, the results of this paper are motivated by the research of VLSI design,where 3-dimensional routing has been a much investigated topic in the past two decades. How-ever, as it is very often the case with algorithms having provably good performance in VLSIrouting, our constructions are not intended for designing specific, ready-to-use routing patternsfor given specifications. Contrarily, we rather aim at understanding the nature of the problem,exploring the capacity of the 3D grid for solving such problems and thus helping heuristics tobe designed and tested.2 Basic DefinitionsAssume that two parallel grids of size w × n (consisting of w rows and n columns) are given.The vertices of these grids are called terminals. A net N is a set of terminals. A 3-dimensionalchannel routing problem (or 3DCRP for short) is a set N = {N1, N2, . . . , Nt} of pairwise disjointnets. n and w are the length and the width of the routing problem, respectively. By a singleactive layer routing problem (or SALRP for short) we mean the special case of the 3DCRPwhere one of the two grids (say, the top grid) is empty (that is, nets do not contain terminalsof the top grid).By a spacing of sw in direction w we are going to mean that we introduce sw − 1 piecesof extra columns between every two consecutive columns (and also to the right hand side ofthe rightmost column) of the original grids. This way the width of the grids is extended tow′ = sw · w. A spacing of sn in direction n is defined analogously.A solution with a given spacing sw and sn of a routing problem N = {N1, N2, . . . , Nt} isa set T = {T1, T2, . . . , Tt} of pairwise vertex-disjoint Steiner-trees in the cubic grid of size(w · sw)× (n · sn)×h (between the two parallel planar grids containing the terminals) such that2the terminal set of Ti is Ni for every 1 ≤ i ≤ t. The Steiner-trees Ti are called wires. The heighth of the routing is to be minimized.3 Single Active Layer RoutingWe start by two straightforward observations.Lemma 1 For any given n there exists a routing problem that cannot be solved with height hsmaller than n2sw.Proof: Let, for simplicity, the width and the length be even, let w = 2a and n = 2b. Considerthe following example (the idea is very similar to that in [3, 7]). Suppose that each net consistsof two terminals in central-symmetric position as shown in Figure 1.1ttn+1 n+2tt2t2ntn−1tantn b+1ttbtbtb+1 tn−1 tnt2ntantn+2 tn+1t2 t1t2n−1t2n−1eFigure 1:The number of nets is an. Since each net is cut into two by the central vertical line e, anyrouting with width w′ = sw · w and height h must satisfy w′h ≥ an. Therefore h ≥ (w/2w′)n,hence h ≥ n2sw. Lemma 2 If sw ≥ 2 and sn ≥ 2 then every routing problem can be solved with height h ≤wn2.Proof: We assign a separate layer to each net. For every terminal we introduce a vertical (thatis, parallel with the height) wire segment to connect the terminal with the layer of its net. Theinterconnection of the terminals of each net can now be performed trivially on its layer usingthe extra rows and columns guaranteed by the spacing in both directions.Since 1-terminal nets can be disregarded, the number of nets is at most 12nw thus h ≤ wn2follows immediately. It seems that the nature of the single active layer routing problem depends fundamentally onwhether the assumptions of the above lemma are true (that is, both sw ≥ 2 and sn ≥ 2) or not.3An alternative interpretation of Lemma 2 is that if we fix w then there is a routing of heighth = O(n), provided that sw, sn ≥ 2. The truth of the same statement is not at all obvious inthe sn = 1 case. However, the following result shows that it is still essentially true.Theorem 3 (A. Recski and D. Szeszle´r, 2001)[12] If sw ≥ 8 then for any fixed value of w andfor any n a single active layer routing problem can always be solved in time t = O(n) and withheight h = O(n) such that the length n is preserved or increased by at most one.We emphasize that the above result considers w as fixed and obtains a bound for the heightas a function of n only. The proof of Theorem 3 is highly technical, it is based on decomposingthe SALRP instance into(w2)(2-dimensional) channel routing instances. However, the proofinvolves an algorithm that gives t = O(w3n) and h = O(wn); details can be found in [12].We will assume sw, sn ≥ 2 in the sequel. Thus the trivial lower and upper bounds ofLemmata 1 and 2 provide the gap to be filled. For example, in the n = w case we have the lowerbound of h = Ω(n) in the worst case and the upper bound of h = O(n2). The first major steptowards narrowing this gap was the following result, which was proved by elaborate probabilisticmethods.Theorem 4 (A. Aggarwal, J. Kleinberg and D. P. Williamson, 2000) [2] If each net consists oftwo terminals only then the nets of an n× n SALRP can be partitioned into O(n log2 n) classessuch that each class of nets can be routed on a copy of the grid (of size n× n).An easy corollary of this theorem is that if sw = sn = 2 and each net consists of two terminalsonly then every SALRP can be solved with height h = O(n log2 n). To show this, we assign alayer to each partition class and we place the routings of the partition classes into the “new” rowsand colums provided by the spacing (in both directions); now the terminals can be connected tothe appropriate layer by a “long” vertical wire segment, and (at most) two additional 1-unit-longsegments are needed for each terminal to connect this segment to the routing of its net.Figure 2 illustrates the above idea. The nets of the SALRP with w = n = 3 on the left handside are first partitioned into 2 classes and each class of nets is routed on a single layer. Thenthe two routings are combined to give a solution of the original problem with sw = sn = 2 andheight h = 2. (This illustration also shows that this construction works for multiterminal netsas well.)If we want to improve further on the upper bound of h = O(n log2 n), we could try to reducethe necessary number of partition classes in Theorem 4; it is an open problem whether this ispossible. However, the following result shows that we can still achieve a height of h = O(n) ifwe allow two layers (intstead of one) for the routing of each partition class.Theorem 5 (A. Recski and D. Szeszle´r, 2007)[13] Assume that a SALRP instance is given suchthat each net contains two terminals only. Then(1) the nets can be partitioned into ⌊32n⌋ classes such that each class of nets can be routed asa separate SALRP with sn = ⌈w2n⌉, sw = 1 and height h = 2;(2) there is a routing with sn = ⌈w2n⌉+ 1, sw = 2 and height h = 3n.4331 11221 112233hnw12 2111’1’1’2’332’3’3’Figure 2:As mentioned above, the second statement is a corollary of the first one. To prove the firststatement, we have to observe that a SALRP is trivially solvable on 2 layers if the increasedlength n · sn is at least the number of nets and each row contains either one terminal or twoterminals belonging to the same net. From this we can obtain the necessary partition by edge-coloring a suitably defined conflict graph (whose edges correspond to the nets). The coefficient of32comes from Shannon’s classic theorem [16] claiming that the chromatic index of a multigraphis at most 32∆. Details can be found in [13].Since the role of n and w is symmetric, we can also formulate the following corollary ofTheorem 5 for the sw = sn = 2 case.Corollary 6 (A. Recski and D. Szeszle´r, 2007)[13] Assume that a SALRP instance is givensuch that each net contains two terminals only.(1) if n ≤ w ≤ 2n then there is a routing with sn = sw = 2 and height h = 3n;(2) there is a routing with sn = sw = 2 and height h = 3max(n,w).In Theorem 5 and Corollary 6 we only considered SALRP instances where each net is aterminal pair. However, allowing multiterminal nets is a crucial point in VLSI design. If we5wanted to apply the proof method of Theorem 5 for the multiterminal case in a straightforwardway, we would come across the problem of edge-coloring hypergraphs, where an analogue ofShannon’s theorem does not exist. However, if we break up each t-terminal net into t − 1terminal pairs then the construction of Theorem 5 works with the only difference being thatthe maximum degree in the corresponding conflict graph, and thus the height of the obtainedrouting is doubled.Theorem 7 (A. Recski and D. Szeszle´r, 2007)[13] Any SALRP can be solved with(1) sn = ⌈w2n⌉+ 1, sw = 2 and height h = 6n;(2) sn = sw = 2 and height h = 6max(n,w).We mention that the constructions in the proof of Theorems 5 and 7 can be realized by apolynomial algorithm. More precisely, the running time is O(t · (w+n)), where t is the numberof nets (and w and n denote the width and the length, respectively). The size of the input isA = w · n, thus if we make the (sensible) assumption that w = Θ(n) holds then the runningtime is O(A32 ).4 3-dimensional Channel RoutingThe constructions in the proof of Theorems 5 and 7 have the following, very useful property:for each terminal t the vertical (that is, parallel with the height) line ofthe grid intersecting t is occupied by a single ”long” vertical wire segment(which, of course, ends in t).(∗)This property implies the following observation:Lemma 8 If sw ≥ 2 and sn ≥ 4 then every 3DCRP can be solved with height h = 6max(n,w).Proof: Let us move the terminals of the top grid by two units in a direction parallel with thelength n and then project these terminals on the bottom grid. This way we obtain a SALRPwith sw, sn ≥ 2, which can be solved with height h = 6max(n,w) by Theorem 7. This solutioncan be modified in a straightforward way to give a solution of the original 3DCRP by the aboveproperty (∗). Therefore we will only consider the 3DCRP problem with sw = sn = 2 in the sequel.Property (∗) also enables us to reduce the problem to a very special case: a 3DCRP instanceis called bipartite if each net contains two terminals only, one on the top grid and the otherone on the bottom grid. Indeed, if we solve the two SALRP instances defined by the top gridand the bottom grid separately, the remaining task is nothing but a bipartite 3DCRP instance.(This bipartite 3DCRP instance is obtained by choosing one terminal on the top grid and oneon the bottom grid from every net that has terminals on both grids arbitrarily.) Again, property(∗) enables us to combine any solution of the bipartite 3DCRP instance with the two SALRPsoultions to give a routing of the original 3DCRP instance.6Theorem 9 (A. Reiss and D. Szeszle´r, 2005)[14](1) Every bipartite 3DCRP can be solved with sw = sn = 2 and height h = 3max(n,w).(2) Every 3DCRP can be solved with sw = sn = 2 and height h = 15max(n,w).The second statement is a corollary of the first one, using Theorem 7 and the above obser-vation. The first statement is proved by a careful construction that again uses the edge-coloringof a conflict graph (which is, in this case, bipartite) and a (2-dimensional) channel routingalgorithm of Recski and Strzyzewski [11]. Details can be found in [14].Again, the construction of the proof of Theorem 9 can be realized by a polynomial algorithmwith a running time of O(t · (w + n)) (which is O(A32 ) if w = Θ(n) is assumed).References[1] Aggarwal, A., M. Klawe, D. Lichtenstein, N. Linial and A. Wigderson, A lowerbound on the area of permutation layouts, Algorithmica (1991) 6, 241-255.[2] Aggarwal, A., J. Kleinberg and D. P. Williamson, Node-disjoint paths on the meshand a new trade-off in VLSI layout, SIAM J. Computing (2000) 29, 1321-1333.[3] Boros, E., A. Recski and F. Wettl, Unconstrained multilayer switchbox routing,Ann. Oper. Res. (1995) 58, 481-491.[4] Cutler, M. and Y. Shiloach, Permutation layout, Networks (1978) 8, 253-278.[5] Enbody, R.J., G. Lynn and K. H. Tan, Routing the 3-D chip, Proc. 28th ACM/IEEEDesign Automation Conf. (1991) 132-137 .[6] Games, R. A., Optimal book embeddings of the FFT, Benes, and barrel shifter networks,Algorithmica (1986) 1, 233-250.[7] Hambrusch, S. E., Channel routing in overlap models, IEEE Trans. Computer-AidedDesign of Integrated Circ. Syst. (1985) CAD-4, 23-30.[8] Leighton, T., Complexity issues in VLSI: Optimal layouts for the shuﬄe-exchange graphand other networks, The MIT Press, Cambridge, MA. (1983)[9] Leighton, T. and A. L. Rosenberg, Three-dimensional circuit layouts, SIAM J.Computing (1986) 15,793-813 .[10] Leighton, T., S. Rao and A. Srinivasan, New algorithmic aspects of the Local Lemmawith applications to routing and partitioning, Proc. Tenth Annual ACM-SIAM Symp. onDiscrete Algorithms ACM/SIAM, New York and Philadelphia (1999) 643-652.[11] Recski A. and F. Strzyzewski, Vertex-disjoint channel routing on two layers, Integerprogramming and combinatorial optimization, 397-405, Ravi Kannan andW.R. Pulleyblank,eds., University of Waterloo Press (1990).[12] Recski, A. and D. Szeszle´r, Routing vertex disjoint Steiner-trees in a cubic grid — anapplication in VLSI, Discrete Applied Mathematics (2007) 155, 44-52.7[13] Recski, A. and D. Szeszle´r, 3-dimensional single active layer routing, Discrete andComputational Geometry, 318-329, Lecture Notes in Computer Science 2098, Springer,Berlin (2001).[14] Reiss, A. and D. Szeszle´r, 3-dimensional Channel Routing, Proc. 4th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications (2005), 409-415.[15] Rosenberg, A. L., Three-dimensional VLSI: a case study, J. ACM. (1983) 30, 397-416.[16] Shannon, C., A theorem on coloring the lines of a network, J. Math. Phys. (1949) 28,148-151.[17] Shirakawa, I., Some comments on permutation layout, Networks (1980) 10, 179-182.8

3-dimensional Routing

Abstract

Similar works

Full text

Available Versions

Repository of the Academy's Library