Path-search in the pyramid and in other graphs

TL;DR

This paper investigates the problem of efficiently identifying paths or sinks in acyclic directed graphs using minimal questions, providing complete solutions for complete t-ary trees and partial results for pyramid graphs.

Contribution

It offers a complete solution for path-search in complete t-ary trees and advances understanding of the problem in pyramid graphs, addressing a question posed by Soren Riis.

Findings

Optimal question strategies for complete t-ary trees

Partial results for pyramid graphs

Reduction of rounds needed to identify paths

Abstract

We are given an acyclic directed graph with one source, and a subset of its edges which contains exactly one outgoing edge for every non-sink 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 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, i.e. 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 non-trivial partial results.