Efficient Catalytic Graph Algorithms

2025-09-07

AI Summary

This paper presents novel catalytic logspace algorithms for two fundamental problems on directed graphs: (1) $s$-$t$ connectivity and (2) $varepsilon$-approximate estimation of the endpoint probability after $T$ steps of a random walk. It introduces, for the first time in this model, randomized computation, together with a vertex-register-based edge-pushing mechanism and a visit-count-driven deterministic edge-traversal strategy. These techniques overcome prior limitations of lacking explicit time bounds. For $s$-$t$ connectivity, the randomized algorithm runs in $ ilde{O}(nm)$ time and the deterministic variant in $ ilde{O}(n^3 m)$. For simulating $T$-step random walks, the algorithm achieves $ ilde{O}(m T^2 / varepsilon)$ time with additive error at most $varepsilon$. The results bridge theoretical depth and hardware feasibility, significantly expanding the computational capabilities of catalytic space.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• 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

Abstract

We give fast, simple, and implementable catalytic logspace algorithms for two fundamental graph problems. First, a randomized catalytic algorithm for $s o t$ connectivity running in $widetilde{O}(nm)$ time, and a deterministic catalytic algorithm for the same running in $widetilde{O}(n^3 m)$ time. The former algorithm is the first algorithmic use of randomization in $mathsf{CL}$. The algorithm uses one register per vertex and repeatedly ``pushes'' values along the edges in the graph. Second, a deterministic catalytic algorithm for simulating random walks which in $widetilde{O}( m T^2 / varepsilon )$ time estimates the probability a $T$-step random walk ends at a given vertex within $varepsilon$ additive error. The algorithm uses one register for each vertex and increments it at each visit to ensure repeated visits follow different outgoing edges. Prior catalytic algorithms for both problems did not have explicit runtime bounds beyond being polynomial in $n$.

Problem

• Efficient catalytic logspace algorithms for graph connectivity
• Deterministic catalytic simulation of random walks with error bounds
• Explicit runtime improvements over prior polynomial-time catalytic methods

Innovation

• Randomized catalytic algorithm for connectivity
• Deterministic catalytic algorithm for random walks
• Uses one register per vertex for efficiency