# Models and Problems of Dynamic Memory Allocation

## Abstract

We construct and analyze some stochastic models for optimal allocation of memory in fragmented arenas. The arenas, as well as the item types they carry, are of various shapes: linear, cubic, pentagonal pie, S^{2} (surface of sphere). Formulated as a Markov dynamic programming problem, the choice of optimal memory allocation closely resembles the choice of an optimal route for a telephone call. Much can be guessed about optimal operation from scrutiny of the partial ordering of the possible states reduced under symmetries; especially, in many specific examples, a natural relation B (read “better than ”) of preference can be defined among alternative allocations, and a topological fixed-point argument based on B given, to solve the allocation problem without resort to numerical solution of the Bellman equation.

## Keywords

Equivalence Class Allocation Problem Optimal Allocation Monotone Property Bellman Equation## Preview

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## References

- [1]E. G. Coffman, Jr., Kimming So, M. Hofri, and A. C. Yao, “A Stochastic Model of Binpacking,” Inf. and Control, 44 (1980), pp. 105–115, last paragraph.CrossRefGoogle Scholar
- [2]Ibid.Google Scholar
- [3]D. E. Knuth, The Art of Computer Programming: Fundamental Algorithms (Vol. 1, 2nd Ed.), Addison-Wesley, Reading, Mass., 1973, p. 435 ff.Google Scholar
- [4]V. E. Benes, “Reduction of Network States Under Symmetries,” Bell System Tech. J., 57, (1978), pp. 111–149.Google Scholar
- [5]V. E. Benes, “Programming and Control Problems Arising from Optimal Routing in Telephone Networks,” Bell Systems Tech. J. 49, (1966), pp. 1373–1438.Google Scholar
- [6]N. Dunford and J. T. Schwartz, Linear Operators: General Theory (Part I), Interscience, New York, 1958, p. 456.Google Scholar
- [7]R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, 1960, p. 294, exercise 8.Google Scholar