Kirkpatrick (1983) optimization by simulated annealing

Luiz Alberto Carvalho

Outline

Kirkpatrick (1983) optimization by simulated annealing

Abstract

In this article we briefly review the central constructs in combinatorial optimization and in statistical mechanics and then develop the similarities between the two fields. We show how the Metropolis algorithm for approximate numerical simulation of the behavior of a manybody system at a finite temperature provides a natural tool for bringing the techniques of statistical mechanics to bear on optimization.

Figures (7)

Fig. 1. Distribution of total number of pins required in two-way partition of a micro- processor at various temperatures. Arrow in- dicates best solution obtained by rapid quenching as opposed to annealing. If the a, are completely uncorrelated, it can be shown (/3) that this Hamilto- nian has a spin glass phase at low tem- peratures. This implies for the associated magnetic problem that there are many degenerate ‘‘ground states’’ of nearly equal energy and no obvious symmetry. The magnetic state of a spin glass is very stable at low temperatures (/4), so the ground states have energies well below the energies of the random high-tempera- ture states, and transforming one ground state into another will usually require considerable rearrangement. Thus this analogy has several implications for opti- mization of partition:

Fig. 2. Construction of a horizontal net-cross- ing histogram.

Fig. 3. Ninety-eight chips on a ceramic module from the IBM 3081. Chips are identified by number (1 to 100, with 20 and 100 absent) and function. The dark squares comprise an adder, the three types of squares with ruled lines are chips that control and supply data to the adder, the lightly dot- ted chips perform logical arithmetic (bitwise AND, OR, and so on), and the open squares denote general-purpose registers, which serve both arithmetic units. The numbers at the left and lower edges of the module image are the vertical and horizontal net-crossing histograms, respectively. (a) Original chip placement; (b) a configuration at T = 10,000; (c) T = 1250; (d) a zero-temperature result.

Fig. 4. Specific heat as a function of tempera- ture for the design of Fig. 3, a to d. The peak of the histogram provides a lower bound to the amount of wire that must be provided in the worst case, since each net requires at least one wiring

Fig. 5. Examples of (a) L-shaped and (b) Z- shaped wire rearrangements. To measure congestion at the same

Fig. 6 (left). Wire density in the 98-chip module with the connections randomly assigned to perimeter routes. Chips are in the original placement. Fig. 7 (right). Wire density after simulated annealing of the wire routing, using Z-shaped moves.

Fig. 9. Results at four temperatures for a clustered 400-city traveling salesman problem. The points are uniformly distributed in nine regions. (a) T = 1.2, a = 2.0567; (b) T = 0.8, a = 1.515; (c) T = 0.4, a = 1.055; (d) T = 0.0, a = 0.7839. To assess the value of annealing in wiring this model, we studied an ensem- ble of randomly situated connections, under various statistical assumptions. Here we consider routing wires for the 98 chips on a module considered earlier. First, we show in Fig. 6 the arrangement of wire that results from assigning each wire to an L-shaped path, choosing ori- entations at random. The thickness of the links is proportional to the number of wires on each link. The congested area that gave rise to the peaks in the histo- grams discussed above is seen in the wiring just below and to the right of the center of the module. The maximum numbers of wires along a single link in Fig. 6 are 173 (x direction) and 143 (y direction), so the design is also aniso- tropic. Various ways of rearranging the wiring paths were studied. Monte Carlo annealing with Z-moves gave the best solution, shown in Fig. 7. In this exam-

References (34)

References and Notes
E. L. Lawlor, Combinatorial Optimization (Holt, Rinehart & Winston, New York, 1976).
A. V. Aho, J. E. Hopcroft, J. D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, Mass., 1974).
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP- Completeness (Freeman, San Francisco, 1979).
R. Karp, Math. Oper. Res. 2, 209 (1977).
B. Dunham, Synthese 15, 254 (1963).
S. Lin, Networks 5, 33 (1975).
For a concise and elegant presentation of the basic ideas of statistical mechanics, see E. Shro- dinger, Statistical Thermodynamics (Cambridge Univ. Press, London, 1946).
N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, J. Chem. Phys. 21, 1087 (1953).
G. Toulouse, Commun. Phys. 2, 115 (1977).
For review articles, see C. Castellani, C. DiCas- tro, L. Peliti, Eds., Disordered .Systems and Localization (Springer, New York, 1981).
J. Soukup, Proc. IEEE 69, 1281 (1981).
M. A. Breuer, Ed., Design Automation of Digi- tal Systems (Prentice-Hall, Engelwood Cliffs, N.J., 1972).
D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975);
S. Kirkpatrick and D. Sherrington, Phys. Rev. B 17, 4384 (1978).
A. P. Young and S. Kirkpatrick, Phys. Rev. B 25, 440 (1982).
B. Mandelbrot, Fractals: Form, Chance, and Dimension (Freeman, San Francisco, 1979), pp. 237-239.
S. Kirkpatrick, Phys. Rev. B 16, 4630 (1977).
C. Davis, G. Maley, R. Simmons, H. Stoller, R. Warren, T. Wohr, in Proceedings of the IEEE International Conference on Circuits and Com- puters, N. B. Guy Rabbat, Ed. (IEEE, New York, 1980), pp. 669-673.
M. A. Hanan, P. K. Wolff, B. J. Agule, J. Des. Autom. Fault-Tolerant Comput. 2, 145 (1978).
M. Breuer, ibid. 1, 343 (1977).
P. W. Case, M. Correia, W. Gianopulos, W. R. Heller, H. Ofek, T. C. Raymond, R. L. Simek, C. B. Steiglitz, IBM J. Res. Dev. 25, 631 (1981).
A. J. Blodgett and D. R. Barbout, ibid. 26, 30 (1982);
A. J. Blodgett, in Proceedings of the Electronics and Computers Conference (IEEE, New York, 1980), pp. 283-285.
K. A. Chen, M. Feuer, K. H. Khokhani, N. Nan, S. Schmidt, in Proceedings of the 14th IEEE Design Automation Conference (New Or- leans, La., 1977), pp. 298-302.
K. W. Lallier, J. B. Hickson, Jr., R. K. Jackson, paper presented at the European Conference on Design Automation, September 1981.
D. Hightower, in Proceedings of the 6th IEEE Design Automation Workshop (Miami Beach, Fla., June 1969), pp. 1-24.
J. Beardwood, J. H. Halton, J. M. Hammers- ley, Proc. Cambridge Philos. Soc. 55, 299 (1959).
D. J. Resenkrantz, R. E. Steams, P. M. Lewis, SIAM (Soc. Ind. Appl. Math.) J. Comput. 6, 563 (1977).
S. Lin, Bell Syst. Tech. J. 44, 2245 (1965).
__ and B. W. Kernighan, Oper. Res. 21, 498 (1973).
V. teemy has described an approach to the traveling salesman problem similar to ours in a manuscript received after this article was sub- mitted for publication.
H. Crowder and M. W. Padberg, Manage. Sci. 26, 495 (1980).
The experience and collaborative efforts of many of our colleagues have been essential to this work. In particular, we thank J. Cooper, W. Donath, B. Dunham, T. Enger, W. Heller, J. Hickson, G. Hsi, D. Jepsen, H. Koch, R. Linsker, C. Mehanian, S. Rothman, and U. Schultz.

Kirkpatrick (1983) optimization by simulated annealing

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