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Breadth-First Search vs. Restarting Random Walks for Escaping Uninformed Heuristic Regions

Daniel Platnick, Dawson Tomasz, Eamon Earl, Sourena Khanzadeh, Richard Valenzano

TL;DR

This work analyzes how to escape Uninformed Heuristic Regions (UHRs) in state-space search by comparing BrFS and Restarting Random Walks (RRWs). It derives exact and bound-based expressions for the expected runtimes of BrFS and RRW^C_ℓ, establishing a crossover condition based on the exit probability $p_g$ and the structure of the search space, including goal depth and distribution. It then extends Enforced Hill-Climbing (EHC) with RRW-based escapes (EHC-RRW^C_ℓ and EHC-RRW^L), provides theoretical guarantees for EHC-complete domains, and validates the theory with experiments on PDDL planning benchmarks. The results show that RRWs can offer runtime and memory benefits in certain topologies (notably unbounded-UHRs) and that the choice between BrFS and RRWs depends on domain structure and goal-density, informing algorithm portfolios for planning under uncertainty.

Abstract

Greedy search methods like Greedy Best-First Search (GBFS) and Enforced Hill-Climbing (EHC) often struggle when faced with Uninformed Heuristic Regions (UHRs) like heuristic local minima or plateaus. In this work, we theoretically and empirically compare two popular methods for escaping UHRs in breadth-first search (BrFS) and restarting random walks (RRWs). We first derive the expected runtime of escaping a UHR using BrFS and RRWs, based on properties of the UHR and the random walk procedure, and then use these results to identify when RRWs will be faster in expectation than BrFS. We then evaluate these methods for escaping UHRs by comparing standard EHC, which uses BrFS to escape UHRs, to variants of EHC called EHC-RRW, which use RRWs for that purpose. EHC-RRW is shown to have strong expected runtime guarantees in cases where EHC has previously been shown to be effective. We also run experiments with these approaches on PDDL planning benchmarks to better understand their relative effectiveness for escaping UHRs.

Breadth-First Search vs. Restarting Random Walks for Escaping Uninformed Heuristic Regions

TL;DR

This work analyzes how to escape Uninformed Heuristic Regions (UHRs) in state-space search by comparing BrFS and Restarting Random Walks (RRWs). It derives exact and bound-based expressions for the expected runtimes of BrFS and RRW^C_ℓ, establishing a crossover condition based on the exit probability and the structure of the search space, including goal depth and distribution. It then extends Enforced Hill-Climbing (EHC) with RRW-based escapes (EHC-RRW^C_ℓ and EHC-RRW^L), provides theoretical guarantees for EHC-complete domains, and validates the theory with experiments on PDDL planning benchmarks. The results show that RRWs can offer runtime and memory benefits in certain topologies (notably unbounded-UHRs) and that the choice between BrFS and RRWs depends on domain structure and goal-density, informing algorithm portfolios for planning under uncertainty.

Abstract

Greedy search methods like Greedy Best-First Search (GBFS) and Enforced Hill-Climbing (EHC) often struggle when faced with Uninformed Heuristic Regions (UHRs) like heuristic local minima or plateaus. In this work, we theoretically and empirically compare two popular methods for escaping UHRs in breadth-first search (BrFS) and restarting random walks (RRWs). We first derive the expected runtime of escaping a UHR using BrFS and RRWs, based on properties of the UHR and the random walk procedure, and then use these results to identify when RRWs will be faster in expectation than BrFS. We then evaluate these methods for escaping UHRs by comparing standard EHC, which uses BrFS to escape UHRs, to variants of EHC called EHC-RRW, which use RRWs for that purpose. EHC-RRW is shown to have strong expected runtime guarantees in cases where EHC has previously been shown to be effective. We also run experiments with these approaches on PDDL planning benchmarks to better understand their relative effectiveness for escaping UHRs.

Paper Structure

This paper contains 26 sections, 10 theorems, 20 equations, 3 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Search Tasks and State-Space Topologies
  4. Search Algorithms and Methods
  5. Breadth-First Search (BrFS).
  6. Enforced Hill-Climbing (EHC).
  7. Restarting Random Walks (RRWs).
  8. The Luby Restart Policy.
  9. Expected Runtime Analysis
  10. Expected Runtimes for BrFS and RRW$^C_\ell$
  11. Comparative Analysis
  12. Understanding the Bounds
  13. Enforced Hill-Climbing with RRWs
  14. Properties of EHC and EHC-RRW Variants
  15. Empirical Evaluation
  16. ...and 11 more sections

Key Result

Theorem 3.1

If $\mathcal{T}$ has $g \geq 1$ goal states uniformly distributed among the $|S_{d^*}|$ states at the goal depth, then

Figures (3)

  • Figure 1: BrFS and RRW$^C_\ell$ with unbiased random walks on a directed tree with a branching factor of 4.
  • Figure 2: Per-problem runtime comparison between EHC and EHC-RRW$^\mathcal{L}$ with $m=1$.
  • Figure 3: Memory usage comparison between EHC and the EHC-RRW variants.

Theorems & Definitions (19)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary B.1
  • proof
  • ...and 9 more