# Partial Solution and Entropy

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5734)

## 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) = (X 1, ..., X k ), where X is the entire data set, and each X i is already solved. 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 shortest paths, X i is an acyclic part in the given graph. For minimum spanning trees, X i is interpreted as a partially obtained minimum spanning tree for a subgraph. The entropy measure, H(S), is defined by regarding p i  = |X i |/|X| as a probability measure, that is, $$H(S)=-n\Sigma_{i=1}^{k}p_i\log p_i$$, where $$n=\Sigma_{i=1}^k|X_i|$$. Then we show that we can sort the input data S(X) in O(H(S)) time, and solve the shortest path problem in O(m + H(S)) time where m is the number of edges of the graph. Finally we show that the minimum spanning tree is computed in O(m + H(S)) time.

## Keywords

entropy complexity adaptive sort minimal mergesort ascending runs shortest paths nearly acyclic graphs minimum spanning trees

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