EÆcient Evaluation Relaxation Under Integrated Fitness Functions

We present in this paper an application of the Constructive Genetic Algorithm (CGA) to the Linear Gate Assignment Problem (LGAP). The LGAP happen in very large scaling integration (VLSI) design, and can be described as a problem of assigning a set of circuit nodes (gates) in an optimal sequence, such that the layout area is minimized, as a consequence of optimizing the number of tracks necessary to cover the gates interconnection. The CGA evolves a dynamic population composed of schemata and structures and uses heuristics in fitness function definitions. 1 CGA APPLICATION TO LGAP The Constructive Genetic Algorithm (CGA) was proposed recently as an alternative to a traditional GA approach (Lorena, 2001), particularly, for evaluating schemata directly. The population, initially formed only by schemata, evolves controlled by recombination to a population of well adapted structures (schemata instantiation) and schemata. Linear gate assignment problems (LGAP) are related to gate matrix layout and programmable logic arrays folding. An example of a gate matrix and the representation used for structures and schemata follows: 1 0 1 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 ? 0 ? 0 1 ? 0 ? 0 ? 1 0 ? 0 ? 1 ? 0 1 ? 1 ? 1 ? 0 1 ? 0 ? 0 ? 1 0 ? 1 ? 1 ? 0 0 ? 0 ? 0 ? 1 0 ? 1 2 3 4 5 6 7 8 9 sj= ( 7 2 5 9 3 4 6 1 8) sk=(7 # 5 # # # 6 1 #) Gate matrix Permutation (structure) Permutation (schema) Two fitness functions are defined on the space of all schemata and structures that can be obtained using this representation. The evolution process considers the two objectives on an adaptive rejection threshold, which gives ranks to individuals and yields a dynamic population. The first function reflects the total cost of a given permutation of gates, and the other drives the evolutionary process to a population trained by a heuristic. The chosen heuristic is the 2-Opt neighborhood. The initial population is composed exclusively of schemata. Two structures and/or schemata are selected for recombination. The first is called the base (sbase) and is randomly selected out of the best ranked individuals. The second structure or schema is called the guide (sguide ) and is randomly selected out of the total population. The current labels in corresponding positions are merged. A new filling operator is proposed to comp lement a schema, substituting the # labels for gate numbers. A local search mutation is always applied to structures, no matter how they are created (after recombination or after the filling process). The search at 2-Opt neighborhood of the structure was used. The CGA for LGAP was run on Intel Pentium II (266Mhz). All best previous results comes of Microcanonical Optimization MCO approach (Linhares,1999). The CGA reached all the best results (number of tracks) for instances taken from the literature, but it appears to be more robust than other approaches. MCO CGA Problem Time (s) Tracks Time (s) Generations wire length wli 10 4 5 5.00 35 wsn 10 8 15 7.00 115 v4050 10 5 5 5.00 51 v4000 10 5 5 5.00 66 v4470 700 9 665 33.00 269 v4090 100 10 20 13.50 132 x0 700 11 755 92.57 343 w1 10 4 10 5.00 57 w2 400 14 185 19.50 283 w3 3900 18 3062 186.00 761 W4 61700 27 52246 225.0