IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. CAD-6, NO. 2. MARCH 1987 211

Thermodynamic Optimization of Block Placement PATRICK SIARRY, LUDOVIC BERGONZI, AND GERARD DREYFUS, MEMBER, IEEE

Abstract-This paper presents the results of a systematic investiga­

tion of the thermodynamic ("simulated annealing") method applied to

the placement of rectangular blocks on a chip. A new presentation of

the fundamental ideas underlying this technique is proposed. It is shown

that the analogies with physics, which have been at the origin of the

method, may be partially forgotten, but that they are still useful to

understand some results. Several simple examples are investigated, and

the influence of various parameters is studied. Typical complex indus­

trial applications are subsequently presented. Finally, an interactive

implementation of the thermodynamic optimization algorithm, based

on the results of the present investigation, is proposed.

I. INTRODUCTION

THE IDEA OF using the tools and concepts of statis­tical mechanics to obtain approximate solutions to

complex engineering problems was introduced by C. Shannon [ 1]. Recent advances made in the investigation of spin glasses have triggered a renewed interest in such approaches, and similar ideas have been developed simul­taneously and independently in several laboratories [2]­[7]. Although the motivations of these investigations were not identical, they were clearly inspired by similar formal analogies between optimization problems and physical phenomena.

Since these first attempts, the interest in such methods has been growing very quickly. Not surprisingly, this has led to a large number of approaches from the fields of physics, mathematics, and operations research, with a va­riety of interesting and sometimes conflicting results.

The purpose of the present paper is twofold. First, we propose a new presentation of the thermodynamic opti­mization ("simulated annealing") method in terms of an engineering problem, and we show how it is linked both to more physical approaches and to more mathematical ones. In the second part, we address the particular case of the optimization of block placement, a problem which arises both in the field of integrated circuits and in the design of printed circuit boards or of modules in hybrid technology [8], [9]. Contrary to the situation prevailing in more classical optimization problems, such as the pop­ular "traveling salesman" problem, no general analytic results are available yet. Therefore, careful experimental investigations are very important in order to determine the most suitable procedures and the critical parameters. We present the results of the thermodynamic optimization in

Manuscript received March 8. 1985: revised October 3. 1986. The authors are with Ecole Superieure de Physique et de Chimie Indus­

trielles de la Ville de Paris, Laboratoire d'Electronique, 10, rue Vauquelin, 75005 Paris, France.

IEEE Log Number 8612431.

some simple cases and show the influence of several pa­rameters. Finally, representative results obtained from in­dustrial applications are given, and typical performances are indicated. In view of the experience gained and the results shown, an "annealing simulator" is proposed in order to control the optimizing process and to decrease the consumption of CPU time.

II. THE THERMODYNAMIC OPTIMIZATION METHOD

In this section, we first present the method in a way which, in our opinion, should be readily accessible to en­gineers familiar with the design of electronic circuits, without unnecessary reference to any physical problem. We stress the hypotheses on which this method is based. In a second part, we recall the analogy with physics which, although not indispensable, gives some insight into the problems that may be encountered in practice. Fi­nally, it is shown that more mathematical approaches sub­stantiate the first presentation by putting it in a general and rigorous framework.

A. A Futuristic Computer-Aided Design System To introduce the basic ideas of the method, we make

an imaginary experiment with a hypothetical system which implies a technology which is not available now. We show that the simulated annealing method is, in fact, a way of approximating, with our present technology, a very fu­turistic CAD system.

Let us first formulate the optimization problem. We want to design an integrated circuit and manufacture it in very large quantities; we want to optimize it in some re­spect (for instance, the total connection length). There is an extremely large number of ways of arranging this cir­cuit so that it is absolutely impossible, even with a very advanced technology, to test exhaustively all the possible configurations of the cirCUIt.

Now, let us imagine that the following system is avail­able: N parallel processors, working under the supervi­sion of a master processor. Each of the processors has the ability to choose a configuration of the circuit, to compute the required wire length, and to draw the wiring diagram. We have a collection of boxes, labeled in the following way. Box number zero is intended to receive the wiring diagrams of the configurations using the smallest possible wire length to. Box number one is intended to receive the wiring diagrams of the second best configurations, with connection length II> etc. Once a processor has computed the wire length required by a configuration, it dispatches the corresponding wiring diagram to the appropriate box.

0278-0070/87/0200-0211$01.00 © 1987 IEEE