An Almost-Greedy Search on Random Binary Vectors and Random Graphs

Abstract

A random vector is sampled from a general binary probability space and a search goes through the coordinates until a 1-coordinate is found. A search algorithm called “almost greedy” is shown to posses some novel characteristics. Its expectation is shown to be sharply bounded by four times the expectation of the optimal search. In addition an algorithm of complexity O(n4logn) finds with high probability an almost-greedy search procedure. This paper also studies connectivity testing of communication networks represented by random graphs. Suppose that some edges fell from a connected graph with accordance to a known distribution and we want to find if the resulting subgraph is connected. A single test is a check for the existence of a path between two vertices of our choice. The proportion between the expectations of an almost greedy and an optimal search procedures is proven to be sharply bounded by 4. Finally an algorithm of complexity O(n5logn) finds with high probability an almost greedy search where n is the number of vertices.