Parallel Complexity of Depth-First-Search and Maximal path

2025-06-17

AI Summary

This work investigates the deterministic parallel complexity of depth-first search (DFS) tree construction and the longest path problem. Addressing the absence of deterministic NC algorithms for general directed graphs, the paper focuses on structured graph classes: bounded-genus graphs, H-minor-free graphs (including single-crossing graphs), and bounded-treewidth graphs. Leveraging tools from topological graph theory, tree decompositions, combinatorial graph theory, and deterministic parallel algorithm design, the authors achieve three key results: (i) the first NC algorithms for DFS on both undirected and directed bounded-genus graphs and on single-crossing H-minor-free graphs; (ii) a Logspace reduction of DFS on bounded-treewidth graphs; and (iii) the first Logspace algorithm for the longest path problem on planar graphs—improving upon the prior NC² lower bound. These results yield the first deterministic efficient parallel DFS algorithms for multiple nontrivial graph families and significantly advance the frontier of parallel tractability for the longest path problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Deterministic Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Constructing a Depth First Search (DFS) tree is a fundamental graph problem, whose parallel complexity is still not settled. Reif showed parallel intractability of lex-first DFS. In contrast, randomized parallel algorithms (and more recently, deterministic quasipolynomial parallel algorithms) are known for constructing a DFS tree in general (di)graphs. However a deterministic parallel algorithm for DFS in general graphs remains an elusive goal. Working towards this, a series of works gave deterministic NC algorithms for DFS in planar graphs and digraphs. We further extend these results to more general graph classes, by providing NC algorithms for (di)graphs of bounded genus, and for undirected H-minor-free graphs where H is a fixed graph with at most one crossing. For the case of (di)graphs of bounded tree-width, we further improve the complexity to a Logspace bound. Constructing a maximal path is a simpler problem (that reduces to DFS) for which no deterministic parallel bounds are known for general graphs. For planar graphs a bound of O(log n) parallel time on a CREW PRAM (thus in NC2) is known. We improve this bound to Logspace.

Problem

• Determining parallel complexity of DFS tree construction
• Extending deterministic NC algorithms to broader graph classes
• Improving parallel bounds for maximal path construction

Innovation

• NC algorithms for bounded genus graphs
• Logspace complexity for bounded tree-width
• Improved Logspace bound for planar graphs