Particle swarm-based optimal partitioning algorithm for combinational CMOS circuits

Abstract

This paper presents a swarm intelligence based approach to optimally partition combinational CMOS circuits for pseudoexhaustive testing. The partitioning algorithm ensures reduction in the number of test vectors required to detect faults in VLSI circuits. The algorithm is based on the circuit's maximum primary input cone size (N) and minimum fanout (F) values to decide the location and number of partitions. Particle swarm optimization (PSO) is used to determine the optimal values of N and F to minimize the number of test vectors, the number of partitions, and the increase in critical path delay due to the added partitions. The proposed algorithm has been applied to the ISCAS’85 benchmark circuits and the results are compared to other partitioning approaches, showing that the PSO partitioning algorithm produces similar results, approximately one-order of magnitude faster.

Introduction

The advancement in VLSI semiconductor technology has led to a phenomenal development in electronic systems. With the reduction in device sizes, a very large number of transistors can fit onto a single chip. However, as the chip density increases, the probability of defects occurring in a chip increases as well. The quality, reliability, and cost are directly related to the intensity of product testing. As a result, testing has become a major concern. All possible irredundant faults can be found only through exhaustive testing of a circuit. However, a purely combinational digital circuit with M inputs requires 2^M test vectors for exhaustive testing, while a sequential circuit may require even more test vectors. As M increases, the applicability of all 2^M test vectors becomes impractical and alternative approaches are needed.

The pseudoexhaustive testing approach has been proposed by McCluskey (McCluskey and Bozorgui-Nesbat, 1981). In this method, the circuit is partitioned into a number of subcircuits, which considerably reduces the required number of test vectors. Archambeau (Archambeau and McCluskey, 1984) demonstrated that when single-output partitions are used, all detectable combinational faults within each partition are detected by such a test. The PIFAN algorithm for partitioning was developed by (Shaer et al., 2000a, Shaer et al., 2000b). It is based on the primary input (PI) cone and FANout (FO) values of each node in the circuit. This algorithm was automated and improved upon later by the authors and called I-PIFAN (Shaer and Dib, 2002). This method has been found to be more effective than the previously suggested approaches such as partition design methodology and simulated annealing (Al-Arian and Bolling, 1994; Shperling and McCluskey, 1987). I-PIFAN seeks to minimize the number of test vectors, the number of partitions, and the increase in critical path delay.

I-PIFAN is based on two constant inputs, maximum node fanin size, N, and minimum partitioning FO value, F. A node is partitionable if it is not an inverter or buffer and has a PI value greater than one. If a node's FO value is greater than or equal to F and the node is partitionable, a partition is inserted. The inputs of any node whose PI value is greater than N are processed for partitioning to reduce the node's PI cone. The algorithm performs an exhaustive test of all combinations of constants, N and F, over some specified range, fully partitioning the circuit for each combination of N and F, and producing a table of possible circuit partitions, from which the best result can be selected.

A recently developed algorithm known as particle swarm optimization (PSO) that emerges and allies itself to evolutionary algorithms based on simulation of the behavior of a flock of birds or school of fish, has proven to have great potential for optimization problems (Kennedy and Eberhart, 2001; Kennedy and Eberhart, 1995; Venayagamoorthy and Gudise, 2004; Gudise and Venayagamoorthy, 2004). Swarm algorithms differ from evolutionary algorithms most importantly in both metaphorical explanation and how they work. PSO is able to find optimal solutions for both single-objective and multi-objective problems (Clerc and Kennedy, 2002; Coello et al., 2004). In this paper, PSO is used to efficiently determine the optimal values of N and F for the I-PIFAN partitioning algorithm, without necessitating an exhaustive test or limiting the values to a specified range, resulting in an order of magnitude speedup; and the authors refer to this approach as PSO–PIFAN.

The paper is organized as follows. The partitioning algorithm is presented in Section 2. The PSO algorithm is described in Section 3. Section 4 presents PSO-based partitioning and the fitness function formulation. Section 5 describes the results obtained using PSO–PIFAN and compares with those obtained from the I-PIFAN approach, for the standard ISCAS’85 benchmark circuits (Brglez and Fujiwara, 1985). Finally, the conclusions and future work are presented in Section 6. The nomenclature used throughout this paper is summarized in Table 1.