Theory and Techniques for Synthesizing a Family of Graph Algorithms

TL;DR

This paper develops a theoretical framework for efficiently synthesizing a family of graph algorithms, including BFS, by leveraging dominance relations to reduce space complexity and systematically derive solutions for key problems.

Contribution

It introduces a new theory of Efficient BFS and a recursive schema, enabling systematic derivation of algorithms for shortest path and spanning tree problems.

Findings

The theory reduces BFS space requirements.

Systematic derivation of shortest path and spanning tree algorithms.

Reveals connections between different graph problems.

Abstract

Although Breadth-First Search (BFS) has several advantages over Depth-First Search (DFS) its prohibitive space requirements have meant that algorithm designers often pass it over in favor of DFS. To address this shortcoming, we introduce a theory of Efficient BFS (EBFS) along with a simple recursive program schema for carrying out the search. The theory is based on dominance relations, a long standing technique from the field of search algorithms. We show how the theory can be used to systematically derive solutions to two graph algorithms, namely the Single Source Shortest Path problem and the Minimum Spanning Tree problem. The solutions are found by making small systematic changes to the derivation, revealing the connections between the two problems which are often obscured in textbook presentations of them.