# Average-case analysis of the Modified Harmonic algorithm

• Prakash Ramanan
• Kazuhiro Tsuga
Session 3 Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 241)

## Abstract

In this paper, we analyze the average-case performance of the Modified Harmonic algorithm for bin packing. We first analyze the average-case performance for arbitrary distribution of item sizes over (0, 1]. This analysis is based on the following result. Let f1 and f2 be two linear combinations of random variables {ni} n=1 k , where the Nis have a joint multinomial distribution for each $$n = \sum\limits_{i = 1}^k {N_i }$$. Let E(f1)≠0, and E(f2)↮0. Then $$\mathop {\lim }\limits_{n \to \infty }$$E(max(f1, f2))/n=$$\mathop {\lim }\limits_{n \to \infty }$$max(E(f1), E(f2))/n. We then consider the special case when the item sizes are uniformly distributed over (0, 1], and obtain optimal values for the parameters of the algorithm. For these values of the parameters, the average-case performance ratio is less than 1.19. This compares well with the performance ratio 1.2865... of the Harmonic algorithm.

## Keywords

Performance Ratio Item Size Harmonic Algorithm Joint Multinomial Distribution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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