Expected Runtime Comparisons Between Breadth-First Search and Constant-Depth Restarting Random Walks

TL;DR

This paper compares the expected runtime of breadth-first search and restarting random walks in finding goals on trees, providing theoretical bounds and conditions for when each method is more efficient.

Contribution

It derives formal expected runtime bounds for both methods and identifies the conditions under which restarting random walks outperform breadth-first search.

Findings

RRW is faster than BrFS when enough goals are present at the goal depth.

The crossover point depends linearly on branching factor, goal depth, and error in walk depth.

The bounds highlight the exponential growth of tree size with branching factor and goal depth.

Abstract

When greedy search algorithms encounter a local minima or plateau, the search typically devolves into a breadth-first search (BrFS), or a local search technique is used in an attempt to find a way out. In this work, we formally analyze the performance of BrFS and constant-depth restarting random walks (RRW) -- two methods often used for finding exits to a plateau/local minima -- to better understand when each is best suited. In particular, we formally derive the expected runtime for BrFS in the case of a uniformly distributed set of goals at a given goal depth. We then prove RRW will be faster than BrFS on trees if there are enough goals at that goal depth. We refer to this threshold as the crossover point. Our bound shows that the crossover point grows linearly with the branching factor of the tree, the goal depth, and the error in the random walk depth, while the size of the tree grows exponentially in branching factor and goal depth. Finally, we discuss the practical implications and applicability of this bound.