Partitioning Optimization by Recursive Moves of Hierarchically Built Clusters Roman Bazylevych, Ihor Podolskyy and Lubov Bazylevych Lviv Polytechnic National University, University of Information Technology and Management, 12 Stephan Bandera Street, Lviv, 79013, Ukraine Tel: +380322582578 rbaz@polynet.lviv.ua Abstract The paper presents a general approach for partitioning optimization based on the hierarchical clustering by the Optimal Circuit Reduction (OCR) method. This method has proved to be robust, effective and efficient tool to identify the hierarchical clusters circuit structure. For initial partitioning and its optimization the Optimal Circuit Reduction Trees are used. Recursive moves (transfers and exchanges) of hierarchically built clusters and their groups of arbitrary sizes are performed for optimization. Some new efficient procedures to escape from the local optima are suggested. I. INTRODUCTION Of keen interest in Physical Design Automation is the problem of developing the partitioning algorithms for millions of elements that give good solutions in a reasonable time and can be used for other problems. The best way to solve this problem is to use the hierarchical clustering approach. For this reason recognizing the clusters in the circuit appears to be a relevant problem. The Optimal Circuit Reduction (OCR) method [1-4] reveals the hierarchical clusters circuit structure. This approach is used for initial partitioning and its optimization. II. PREVIOUS WORKS Most likely, the first proposal to use free hierarchical clustering for partitioning was in [1]. It was further developed in [2] and used for packaging and placement [3-5] with very good results. Much more later enforced hierarchical clustering was used for hypergraph partitioning also with good results [6,12,13]. Additional information about clustering approach in partitioning is published in [7-13]. III. PROBLEM FORMULATION It is necessary to obtain a partitioning P* = {P 1 * ,…, P k * } of the set of elements P = { p 1 , … , p n } so that the quality function is optimized: Q(P*) → opt Q(P i ), P i ,∈ D Here P i is arbitrary partitioning in feasible region D of given constraints, k – the number of partitions. The solution should satisfy the following conditions: (∀P i ∈ P)[P ={p i1 ,…, p ini }, p ij ∈ P; i=1, ... , k; j=1, … , n i ]; (∀ P i ∈P)(P i ≠∅), (∀(P i ,P j )∈P)[P i ∩P j =∅]; k ni 1 i 1 j ij P p = = = , where n i is the number of elements of i-th partitions. To recognize the better clusters to transfer from one partition to the other or to exchange between partitions in iterative improvement process we need to have hierarchically built clusters circuit structure. To generate such structure we use the OCR method [1,2,3], which creates the mathematical model of the circuit structure – the Optimal Reduction Tree (ORT) T R . IV. PARTITIONING OPTIMIZATION It is possible to start optimization from some initial solution received by any constructive method or from a randomly generated one. We recommend that the OCR method should be used for initial solution. Some possibilities are suggested in [1,2,5,6]. Here we study the possibility of this methodology to improve randomly generated solutions. We select two basic strategies of partitioning optimization: pairwise and group. The first performs optimization for two initial partitions. A group optimization allows to perform a few partitions simultaneously. Let us have some initial partitioning P = {P 1 ,…, P k } with quality Q( P ~ ) for some given criterion. The purpose of optimization is to get partitioning P * = {P 1 *,…, P k * } with the best value of criterion. The offered algorithms realize the iterative process of solution improvement. For the pairwise optimization algorithms partitions P i and P j with the criterion Q(P i , P j ) are given. It is necessary to get the optimized partitions P i * and P j * with the better value of criterion: Q(P i *,P j *) Q(P i , P j ). The problem lies in such redistribution of elements between partitions that it should improve the criterion. There are two ways possible to do this. P3 P2 T R 1 T R 2 P1 P1 P4 Fig. 1. Optimization procedures 1-4244-1161-0/07/$25.00 ©2007 IEEE