Partial Solution and Entropy

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, ..., Xk), where X is the entire data set, and each Xi is already solved. We use the entropy measure to analyze three example problems, sorting, shortest paths and minimum spanning trees. For sorting Xi is an ascending run, and for shortest paths, Xi is an acyclic part in the given graph. For minimum spanning trees, Xi is interpreted as a partially obtained minimum spanning tree for a subgraph. The entropy measure, H(S), is defined by regarding pi = |Xi|/|X| as a probability measure, that is, H(S) = −nΣ k i=1pi log pi, where n = Σ i=1|Xi|. 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.