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

Daniel Platnick, Richard Anthony Valenzano

TL;DR

The performance of BrFS and constant-depth restarting random walks is analyzed to better understand when each is best suited for finding exits to a plateau/local minima and it is proved RRW will be faster than BrFS on trees if there are enough goals at that 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.

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

TL;DR

The performance of BrFS and constant-depth restarting random walks is analyzed to better understand when each is best suited for finding exits to a plateau/local minima and it is proved RRW will be faster than BrFS on trees if there are enough goals at that 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.

Paper Structure

This paper contains 7 sections, 5 theorems, 10 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Breadth-First Search (BrFS)
  4. Constant-Depth Restarting Random Walks (RRW)
  5. General Expected Runtime Analysis
  6. Comparing BrFS and RRW on Trees
  7. Discussion and Future Work

Key Result

Theorem 3.1

If $\mathcal{S}$ has $N_{\mathcal{O}}$ unique vertices with a level less than $d^*$, and $g \geq 1$ goal vertices uniformly distributed amongst the $N_{d^*} \geq g$ unique vertices at the goal level, then

Figures (1)

  • Figure 1: Comparing BrFS and RRW with different depth errors ($e$) on a directed tree with a branching factor of 4. Figure \ref{['fig:goal_behaviour']} shows the expected goal tests when the goal depth is 6 for different numbers of goals. Figure \ref{['fig:cutoff_behaviour']} shows how the crossover point changes at different goal levels. The crossover point as computed empirically is shown as a dashed line, while the bound from Theorem \ref{['thm:brfs_vs_rrw']} is a solid line. Figure \ref{['fig:density_behaviour']} shows how the goal density crossover changes with the goal level.

Theorems & Definitions (8)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 4.1
  • Corollary 4.2
  • Theorem 4.1
  • proof