Entropy as Computational Complexity

Abstract

If the given problem instance is partially solved, we want to minimize our effort to solve the problem using that information. In this paper we introduce the measure of entropy, H (S), for uncertainty in partially solved input data S (X) = (X₁, . . . , X_k), where X is the entire data set, and each X_i is already solved. We propose a generic algorithm that merges X_i's repeatedly, and finishes when k becomes 1. We use the entropy measure to analyze three example problems, sorting, shortest paths and minimum spanning trees. For sorting X_i is an ascending run, and for minimum spanning trees, X_i is interpreted as a partially obtained minimum spanning tree for a subgraph. For shortest paths, X_i is an acyclic part in the given graph. When k is small, the graph can be regarded as nearly acyclic. The entropy measure, H (S), is defined by regarding p_i = ¦X_i¦/¦X¦ as a probability measure, that is, H (S) = -n (p₁ log p₁ + . . . + p_k log p_k), where n = ¦X₁¦ + . . . + ¦X_k¦. We show that we can sort the input data S (X) in O (H (S)) time, and that we can complete the minimum cost spanning tree in O (m + H (S)) time, where m in the number of edges. Then we solve the shortest path problem in O (m + H (S)) time. Finally we define dual entropy on the partitioning process, whereby we give the time bounds on a generic quicksort and the shortest path problem for another kind of nearly acyclic graphs.