[Paper Review] A Topological Approach to Meta-heuristics: Analytical Results on the BFS vs. DFS Algorithm Selection Problem

Tom Everitt, Marcus Hütter|arXiv (Cornell University)|Sep 9, 2015

Constraint Satisfaction and Optimization38 references3 citations

TL;DR

This paper presents a topological and probabilistic framework to analytically estimate the expected runtime of BFS and DFS in tree and graph search problems. By modeling goal distribution across levels and incorporating branching factors and path redundancy, it derives closed-form approximations for average search time, enabling informed algorithm and representation selection with strong empirical validation.

ABSTRACT

Search is a central problem in artificial intelligence, and breadth-first search (BFS) and depth-first search (DFS) are the two most fundamental ways to search. In this paper we derive estimates for average BFS and DFS runtime. The average runtime estimates can be used to allocate resources or judge the hardness of a problem. They can also be used for selecting the best graph representation, and for selecting the faster algorithm out of BFS and DFS. They may also form the basis for an analysis of more advanced search methods. The paper treats both tree search and graph search. For tree search, we employ a probabilistic model of goal distribution; for graph search, the analysis depends on an additional statistic of path redundancy and average branching factor. As an application, we use the results to predict BFS and DFS runtime on two concrete grammar problems and on the N-puzzle. Experimental verification shows that our analytical approximations come close to empirical reality.

Motivation & Objective

• To develop analytical estimates for the expected runtime of BFS and DFS in search problems.
• To enable data-driven algorithm and graph representation selection using runtime predictions.
• To extend worst-case analysis to average-case performance via probabilistic modeling of goal distribution.
• To provide theoretical foundations for meta-heuristics in search algorithm selection.
• To validate analytical approximations against empirical results on grammar problems and the N-puzzle.

Proposed method

• Model goal distribution using a probability vector over tree levels, enabling expected runtime estimation for BFS and DFS.
• Introduce a length-to-depth counter L to capture path redundancy and structural properties in graph search.
• Derive closed-form approximations for average BFS and DFS runtime in tree search using probabilistic goal levels.
• Generalize results to graph search by incorporating branching factors and the L counter for path redundancy.
• Use empirical estimation techniques for graph parameters such as branching factors and goal probabilities.
• Extend the framework to heuristic search by considering g-level branching factors and generalized backtracking.

Experimental results

Research questions

• RQ1 How can the expected runtime of BFS and DFS be analytically estimated in tree search with probabilistic goal distribution?
• RQ2 What structural graph properties—beyond depth and branching factor—impact DFS and BFS average runtime?
• RQ3 How do path redundancy and goal distribution affect the relative performance of BFS and DFS?
• RQ4 Can analytical runtime estimates predict empirical performance on real-world problems like the N-puzzle and grammar problems?
• RQ5 How can the framework be adapted to heuristic search methods such as heuristic DFS and A*?

Key findings

• The paper derives closed-form analytical approximations for expected BFS and DFS runtime in tree search based on goal probability vectors and branching factors.
• For graph search, the framework introduces a length-to-depth counter L to model path redundancy, enabling runtime estimation with bounds.
• Empirical validation on grammar problems and the N-puzzle shows strong alignment between analytical estimates and actual runtimes.
• The expected runtime estimates are significantly more informative than worst-case bounds, which are always 2^{D+1} for both algorithms.
• The model enables practical algorithm selection and graph representation choice by predicting performance differences between BFS and DFS.
• The framework provides a theoretical basis for extending analysis to heuristic search, including generalized backtracking in heuristic DFS.