When Clusters Meet Partitions: Dennis J.-H. Huang and Andrew B. Kahng. UCLA Computer Science Department, Los Angeles, CA USA

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1 When Clusters Meet Partitions: New Density-Base Methos for Circuit Decomposition Dennis J.-H. Huang an Anrew B. Kahng UCLA Computer Science Department, Los Angeles, CA USA Abstract Top-own partitioning has focuse on minimum cut or ratio cut objectives, while bottom-up clustering has focuse on ensity-base objectives. In seeking a more unie perspective, we propose a new sum of ensities measure for multi-way circuit ecomposition, where the ensity of a subhypergraph is the ratio of the number of eges to the number of noes in the subhypergraph. Fining a k-way partition that maximizes the sum of k subhypergraph ensities is NP-har, but an ecient ow-base metho can n the optimal (maximumensity) subhypergraph in a given hypergraph. Base on this metho, we evelop a heuristic which in practice has less than 0% error from an optimal sum of ensities ecomposition. Other results suggest that ensity-base heuristics can capture cut-base objectives, whereas the converse woul seem icult. Introuction A funamental task in VLSI layout applications is ning a \natural" ecomposition of the netlist hypergraph into k isjoint subsets. Two types of objectives have been use. First, minimum cut objectives minimize the total number of signal nets crossing between ierent components, with possible size constraints on the components; the Fiuccia-Mattheyses (FM) heuristic [4] is a leaing example. More recent research has focuse on ratio cut bipartitioning [4], which minimizes C(P;P2) jp jjp 2j where C(P;P2) is the number of signal nets crossing between the two partitions P an P2. This metric is \natural" in that it aresses both the minimum cut an the partition size balance objectives. Secon, ensity-base ecompositions intuitively view a cluster as a ensely connecte group of circuit elements. The goal is to ientify \goo" clusters, typically in bottom-up fashion. To assess whether a cluster is goo, several clustering metrics have been propose, e.g., Cong an Smith [3] ene a measure of cluster jej C(n;2) ensity as, where jej is the number of eges an n is the number of noes in the cluster. This metric is biase to small clusters since C(n; 2) increases quaratically in n (a cluster with one ege an two noes (i.e., a 2-clique) woul have high ensity). Many clustering objectives are icult to evaluate, let alone optimize. The main theme of our work is that (cut-base) partitioning an (ensity-base) clustering have noobvi- ous common groun for \mile" values of k number of components in the circuit ecomposition. Yet, ue to their common application to hierarchical circuit layout, these approaches arguably shoul \meet in the mile" (e.g., in mapping a system esign onto amultiple-fpga prototyping architecture, or in highlevel synthesis applications). Thus, we propose a new maximum sum of ensities objective fork-way circuit ecomposition. Formally, the ensity of a hypergraph H =(V; E) is the ratio of the number of hypereges to the number of noes, jej i.e., jv j. Given a noe subset V 0 V, the ensity of the sub-hypergraph H 0 =(V 0 ;E 0 ) inuce by V 0 is the number of hypereges in E that are completely containe in E 0, ivie by thenumber of noes in V 0. The main problem that we aress is: The k-way Maximum Sum of Densities (k-msd) Problem: Given a hypergraph H(V; E), ivie V into k partitions, P;P2;:::;Pk, such that the sum of the ensities of the inuce subhypergraphs is maximum. In aressing the k-msd problem, we will also iscuss a more basic problem: The Maximum Density Subhypergraph (MDS) Problem: Given ahypergraph H(V; E), n a subhypergraph of H with largest ensity. max sum of ensities cut min ratio cut max sum of ensities cut min ratio cut Figure : (a) The minimum ratio cut gives an uneven partitioning, while the maximum sum of ensities cut gives a more balance an natural partitioning. (b) When the circuit structure is change, the maximum sum of ensities cut shifts accoringly, while the minimum ratio cut remains the same. Our hope is that the k-msd objective can lea to

2 \natural" circuit ecompositions, i.e., our work is similar in spirit to that of Wei an Cheng [4], who in 989 propose ratio cut as a \more natural" ecomposition objective (ve years later, a large fraction of partitioning research now aresses some form of ratio cut). There are easy examples for which ratio cut oes not give a \natural" circuit bipartition, while maximizing the sum of the two partition ensities gives a more natural bipartition. Figure shows that the ratio cut objective can overemphasize the reuction in net cut, while ignoring the internal structure of the partitions. The maximum sum of ensities cut will shift as the internal structure of the graph changes, while the minimum ratio cut remains the same. Figure 2: (a) Cut objectives pertain to the cutsize at the cluster bounary, while (b) ensity objectives pertain to the implie area or wirelength insie the cluster. A seconary motivation is suggeste in Figure 2: the cut objective has an external view of the circuit components, while the ensity objective has an internal view. Cutsize at the bounary of a component has been correlate with area an wirelength via the Rent parameter analysis, but the number of terminals aects area an wirelength only in a probabilistic sense, since shorter than expecte routes may be possible. Density measures may yiel less trivial lower bouns on layout area an wirelength. A nal motivation, note in Section 4 below, is that maximizing the sum of ensities can capture both cut-base an ensity-base objectives, unlike any cut-base formulation. 2 An Optimal Maximum Density Subhypergraph Solution Many researchers have solve the maximum ensity subhypergraph problem when the input is a graph [, 6]; we use MDSG to enote this special case. The usual transformation is to a series of minimum cut computations, which are accomplishe using maximum ow techniques. Picar an Queyranne [] formulate MDSG as a special case of 0- fractional programming, an use O(n) ow computations to solve the prob- There are also supercial similarities between the k-msd an certain cut-base objectives. If the given graph is a complete graph, chain, or cycle, maximizing the sum of ensities will generate the same bipartition as minimizing the ratio cut. If we restrict the cluster sizes to be all ientical, then a maximum sum of ensities solution will also be a minimum cut solution. lem. Golberg [6] evelope a ierent solution which requires only O(log n) ow computations. In the work which irectly le to our approach, Kortsarz an Peleg [8] solve the MDSG problem for a graph using the \Provisioning Problem" formulation of Lawler [9]. The Provisioning Problem: Suppose there are n items x;x2;:::;xn to select from, with selection of item xi incurring cost ci > 0. Suppose there are m sets of items S;S2;:::;Sm that are known to confer special benets. Agiven item may becontaine in several ierent sets. If all of the items in set Sj are selecte, then a benet bj > 0 is obtaine. The objective isto maximize f total benet g,ftotal cost g for the set of selecte items. Rhys [2] an Picar [0]showe that the Provisioning formulation can be transforme into a maximum ow problem in a network. The network is a bipartite graph with noes nsi on one sie representing the sets Si, an noes nxj on the other sie representing the items xj. A irecte ege with innite capacity is rawn from nsi to nxj if Si contains xj. The source is connecte to each noensi by an ege having capacity bi, an the sink has a connection from each noe nxj via an ege having capacity cj. Eges in the minimum cut set which are incient from noes in fnxj g will correspon to the items selecte in the optimum Provisioning solution. We may exten the algorithm in [8] to solve the MDS problem for a hypergraph in polynomial time. The solution of the MDS problem via a series of Provisioning problem instances is as follows:. Let each noe be an item, an let each hyperege in the netlist be a set containing all items corresponing to its noes. 2. Suppose we want tocheck whether there exists a subhypergraph with ensity or higher, i.e., we seek a noe subset U with je(u)j juj (here, je(u)j enotes the number of hypereges completely containe in noe set U). This may be restate as je(u)j,ju j0. 3. This is exactly the Provisioning formulation with the cost of each item being an the benet of each set being. 4. Since the ierence between two istinct ensities is at least n(n,) [6], we can use binary search to guess the maximum ensity. Only O(log n) ow computations are neee to etermine the optimal solution. This sequence of steps immeiately yiels our Hyper- MDS algorithm for the MDS formulation. Figure 3 shows the transformation from a hypergraph to a ow network instance. Note that the signal nets N;:::;N4 correspon to the \sets".

3 N2 N x will be a maximum cut in the original graph G. (See Figure 4.) x x2 x3 x4 x5 x6 N3 N4 S N N2 N3 N4 x 2 x 3 x4 x 5 x t G M G 2-MinSumD Cut Figure 3: An example of a hypergraph (a) an its corresponing ow network instance (b). The hypergraph contains 6 noes an 4 signal nets N = fx;x2;x4g, N2 = fx;x5g, N3 = fx2;x4;x5g, an N4 = fx3;x5;x6g. We assign a \guess" ensity to all eges incient from the noes in the right sie. When = 4 3, the eges,!,!,!,!,! f sn4; xt; x2t; x4t; x5tg will be in the minimum cut set, implying that the maximum ensity subhypergraph is over noes fx;x2;x4;x5g. 3 The k-way Maximum Sum of Densities (k-msd) Problem In this section, we rst show that the k-msd problem is NP-har. We then present a greey heuristic for this problem which has performance ratio of 2 for the 2-MSD problem, an k for the k-msd problem, but which in practice returns solutions that are remarkably close to optimal. 3. K-MSD is NP-Har Lemma The 2-MSD problem is NP-har. Proof : The 2-MSD problem has equivalent complexity to minimizing the sum of 2 cluster ensities (2- MinSumD). This is because the 2-MSD solution for a given graph G will be the same as the 2-MinSumD solution in the complement graph G. To conrm that the 2-MSD problem is NP-har, we show that Simple Max Cut 2 [5] reuces to 2-MinSumD. Given graph G(V; E), construct G (V ;E ) with 2n noes that contains G an a separate isomorphic copy G 0 (V 0 ;E 0 )ofg. Every ege in E [E 0 has unit capacity. A one ege in G with capacity M between each corresponing pair (vi 2 V, vi 2 0 V 0 ). Note that for suciently large M, a 2-MinSumD partitioning (X; X) in G must have jxj = jxj with either vi 2 X an vi 2 0 X,orvi 2 X an 0 vi 2 X, for all i n: only such a partitioning can have 2-MinSumD value as small as 2jEj n. A partitioning (X; X) that solves 2-MinSumD in G will have sum of ensities = 2jEj,2k, where k is n the value of a maximum cut of G, an(x \ V; X \ V ) 2 Simple Max Cut is state as follows. Instance: Graph G = (V;E), positive integer K. Question: Is there a partition of V into isjoint sets V an V 2 such that the number of eges in E having one enpointinv an one enpointinv 2 is at least K? Figure 4: The construction of G Theorem The k-msd problem is NP-har for all xe k 3. Proof : For any k 3 an any given graph G, we construct a graph G containing G an k, 2 clusters which eachhave very high ensity D. For suciently large D, the k-msd solution for G must contain the k, 2 ense clusters an two clusters from G. Wethus obtain a reuction from the 2-MSD problem. 3.2 AGook-MSD Heuristic A simple heuristic algorithm for the k-msd problem is as follows. First, we obtain a ecomposition into up to k clusters via a scheme calle MDS Peeling(Figure 5), which iteratively applies the Hyper-MDS algorithm. The iea is to peel o a maximum-ensity cluster k, times, or until all noes in the graph have been exhauste. Secon, we compute a linear orering of the vertices of G, base on the k clusters obtaine from MDS Peeling. Essentially, we place these k clusters in the orer that they were foun, an apply the WINDOW orering of[2]within each cluster to place each noe into the overall orering. Thir, we apply ynamic programming to eciently split the linear orering into a k-way partitioning that is optimal, subject to the constraint that each cluster is contiguous in the orering []. We thus obtain our MDS Peeling+DPRP algorithm. As we shall see, this metho implicitly gives an upper boun for the k-msd solution, an has goo performance in practice. Observation Given a hypergraph H(V; E), let D be the ensity of the maximum ensity subgraph of H foun by the MDS Peeling phase. Then, an upper boun on the sum of ensities S in the optimal k-msd solution is S kd, an this boun is tight.

4 MDS Peeling Algorithm Input : Hypergraph H(V; E), an k Output: k clusters cluster ; cluster 2 ; : : : ; clusterk i=0; while (E is not empty) an (i <k) o Fin the maximum ensity subhypergraph U(V 0 ;E 0 )ofh Let E 00 be the hyperege cut set of (V 0 ;V, V 0 ) E = E, E 0, E 00 ; clusteri = V 0 V = V, V 0 i = i +; if E is not empty then clusterk = V Figure 5: MDS Peeling algorithm to generate clusters. Proof : The maximum-ensity cluster of H has ensity D, so the sum of ensities of k clusters can be at most kd. If H consists of k isjoint clusters each with ensity D, this upper boun is tight. Observation 2 The MDS Peeling+DPRP has performance ratio 2 for the 2-MSD problem. Proof : Let D be the ensity of the maximum-ensity cluster of H, an let D 0 be the ensity of the remaining cluster. The sum of ensities returne by MDS Peeling+DPRP is D+D0 = 2D 2 + D0 2D 2. In general, MDS Peeling+DPRP has a tight performance ratio of k for the k-msd problem. However, its performance seems much closer to optimal in practice. 3.3 Results for MDS Peeling + DPRP We have foun heuristic k-msd solutions using both MDS Peeling+DPRP as well as the obvious variant of the Fiuccia-Mattheyses (FM) metho. Our test cases consist of stanar partitioning benchmarks maintaine by ACM SIGDA. Our FM variant, which we call Density-FM, is stanar k-way FM with gains upate base on the sum of ensities metric; our implementation aapts coe from [3]. Table compares MDS Peeling+DPRP against the best result of 0 Density-FM runs. MDS Peeling+DPRP averages 5.29% improvementover Density-FM. We also note that MDS Peeling+DPRP can perform quite well when juge against the theoretical upper boun on the sum of ensities. Table 2 gives a more careful stuy of MDS Peeling+DPRP results for k-way clustering of the Primary an Primary2 benchmarks with a range of k values; we compare the sum of ensities with the theoretical upper boun for the k-msd solution. (Recall that the upper boun is k times the largest subgraph ensity.) Although MDS Peeling has aworst-case performance ratio of k for k-msd, the results show that it can actually be very close to optimum in practice. (For three test cases with very small, very ense subhypergraphs { Test02, Test04, Test05 { this analysis fails since kd is a very loose upper boun.) 4 Density-Base Ratio Cuts We close by noting a variation of Hyper-MDS which yiels goo 2-way ratio cut solutions. Such a result is Upper MDS Peeling %of Ckts #Clusters Boun + DPRP optimum % % Prim % % % % % % Prim % % % % Table 2: Comparison of sum of cluster ensities in the MDS Peeling+DPRP solution versus the theoretical k-msd upper boun, for large values of k. The 20- way clustering for Primary (833 noes) requires 0 secons an the 40-way clustering for Primary2 (304 noes) requires 968 secons on a Sun SPARC-0. Note that we only nee to run MDS Peeling+DPRP once to obtain results for all values of k. interesting because it shows that \ensity can capture cut", while the converse woul seem icult. Recall that the Hyper-MDS algorithm will return the ensest subhypergraph of the netlist hypergraph, where each signal net has \creit" =, an each noe has \cost" =. This has the following weaknesses with respect to minimizing the ratio cut: the metho concentrates on collecting many smaller nets into the cluster, an ignores resulting growth of the cluster bounary (i.e., nets cut). the metho oes not istinguish between creit for nets with high egree an creit for nets with low egree; however, a high-egree net will potentially cause more cuts. the metho oes not istinguish between cost of noes with high egree an cost of noes with low egree; however, a high-egree noe will potentially cause more cuts. To create a bias against nets an noes with large egree, we can ajust the net creit for each net an the noe cost for each noe. Details of the resulting heuristic, an experimental results showing that the ensity-base approach is very competitive with the best known ratio cut methos, are given in [7]. 5 Conclusions We have introuce a new sum of ensities objective that can lea to more natural circuit ecompositions than previous objectives. Our objective can furthermore be use to unify top-own cut-base partitioning with bottom-up ensity-base clustering. We have given an optimal Provisioning-base ow solution, calle Hyper-MDS, for the MDS problem; we have also propose the MDS Peeling + DPRP heuristic for the k-msd problem. Relate ensity-base formulations which may be of future interest inclue the following.

5 Test Algorithm Sum of Densities Average Cases k =2 k =3 k =4 k =8 k =0 k =5 k =20 Improvement Prim MDS Peeling+DPRP % Density-FM Prim2 MDS Peeling+DPRP % Density-FM Test02 MDS Peeling+DPRP % Density-FM Test03 MDS Peeling+DPRP % Density-FM Test04 MDS Peeling+DPRP % Density-FM Test05 MDS Peeling+DPRP % Density-FM Test06 MDS Peeling+DPRP % Density-FM avq.small MDS Peeling+DPRP Density-FM N.A. N.A. N.A. N.A. N.A. N.A. N.A. Table : Comparison of sum of cluster ensities for MDS Peeling+DPRP an for the best of 0 Density-FM runs. Results for avq.small benchmark using Density-FM woul require many ays of SPARC-0 time. Boune Size Maximum Density Subhypergraph (BMDS) problem : Given a hypergraph H(V; E) an an integer B, n the subhypergraph of H with maximum ensity an size B. While MDS was polynomial time solvable, BMDS is shown NP-complete by reuction from Maximum Clique. Max-Density Subhypergraph with Prescribe Noe problem : Given a hypergraph H(V; E) an a prescribe noe p, n the maximum ensity subhypergraph which contains noe p. This can be solve using the 0, fractional programming technique in [] by assigning the variable corresponing to p to, an transforming the resulting fractional expression to a series of ow computations. Max-Density Subhypergraph with Exclue Noe problem : Given a hypergraph H(V; E) an a prescribe noe p, n the maximum ensity subhypergraph which oesnotcon- tain noe p. This problem can be solve in the same time complexity as the MDS problem. We can remove noe p an all eges incient from p, an solve the MDS problem in the remaining hypergraph. 6 Acknowlegments We are grateful to Dr. Lars Hagen for his valuable comments, an to Professor Laura Sanchis for proviing k-way FMcoe[3]. Also, thanks to Dr. L. T. Liu for valuable iscussion. References [] C. J. Alpert an A. B. Kahng, \Multi-Way Partitioning Via Spacelling Curves an Dynamic Programming", 3th ACM/IEEE Design Automation Conference, 994, pp [2] C. J. Alpert an A. B. Kahng, \A General Framework for Vertex Orerings, With Applications to Netlist Clustering", Proc. IEEE Intl. Conf. on Computer- Aie Design, 994, to appear. [3] J. Cong an M. Smith, \A Parallel Bottom-up Clustering Algorithm with Applications to Circuit Partitioning in VLSI Design", 30th ACM/IEEE Design Automation Conference, 993, pp [4] C. M. Fiuccia an R. M. Mattheyses, \A Linear Time Heuristic for Improving Network Partitioning", 9th ACM/IEEE Design Automation Conference, 982, pp [5] M. R. Garey an D. S. Johnson, Computers an Intractability : A Guie to the Theory of NP- Completeness, W.H. Freeman an Company, New York, 979. [6] A. V. Golberg, \Fining a Maximum Density Subgraph", Technical Report No. UCB/CSD 84/7, May 984. [7] Dennis J.-H. Huang an Anrew B. Kahng, \When Clusters Meet Partitions: New Density-Base Methos for Circuit Decomposition", UCLA Computer Science Department, TR-94009, 994. [8] G. Kortsarz an D. Peleg, \Generating Sparse 2- spanners", Thir Scaninavian Workshop Proceeings, 992, pp [9] E. L. Lawler, Combinatorial Optimization: Networks an Matrois Holt, Rinehart an Wiston, New York, 976. [0] J.-C. Picar, \Maximal Closure of a Graph an Application to Combinatorial Problems", Management Science, Vol. 22, 976, pp [] J.-C. Picar an M. Queyranne, \A Network Flow Solution to Some Nonlinear 0- Programming Programs, with Applications to Graph Theory", Networks, Vol. 2, 982, pp [2] J. M. W. Rhys, \A Selection Problem of Share Fixe Costs an Network Flows", Management Science, Vol. 7, 970, pp [3] L. A. Sanchis, \Multiple-Way Network Partitioning", IEEE Trans. Computers, vol. 38, no., 989, pp [4] Y. C. Wei an C. K. Cheng, \Ratio Cut Partitioning for Hierarchical Designs", IEEE Trans. Computer- Aie Design, vol. 0, July 992, pp