Path Search in the Pyramid and in Other Graphs

Abstract

We are given an acyclic directed graph with one source, and a subset of its edges that contains exactly one outgoing edge for every nonsink vertex. These edges determine a unique path from the source to a sink. We can think of it as a switch in every vertex, which determines which way the water arriving to that vertex flows further. We are interested in determining either the sink as the flow arrives, or the whole path, with as few questions as possible. The questions we can ask correspond to the vertices of the graph, and the answer describes the switch, that is, tells which outgoing edge is in our given subset. Originally the problem was proposed by Soren Riis (who posed the question for pyramid graphs) in the following more general form: We are given a natural number k, and k questions can be asked in a round. The goal is to minimize the number of rounds. We completely solve this problem for complete t-ary trees. Also, for pyramid graphs we present some nontrivial partial results.