# Reachability in High Treewidth Graphs

## TL;DR

This paper presents a space-efficient polynomial-time algorithm for reachability in directed graphs with bounded treewidth, using the graph's tree decomposition to achieve significantly reduced space complexity.

## Contribution

It introduces a novel algorithm that decides reachability in polynomial time with space complexity proportional to the treewidth times log n, improving over previous methods.

## Key findings

- Algorithm runs in polynomial time.
- Space complexity is O(w log n).
- Applicable to graphs with bounded treewidth.

## Abstract

Reachability is the problem of deciding whether there is a path from one vertex to the other in the graph. Standard graph traversal algorithms such as DFS and BFS take linear time to decide reachability however their space complexity is also linear. On the other hand, Savitch's algorithm takes quasipolynomial time although the space bound is $O(\log^2 n)$. Here, we study space efficient algorithms for deciding reachability that runs simultaneously in polynomial time. In this paper, we show that given an $n$ vertex directed graph of treewidth $w$ along with its tree decomposition, there exists an algorithm running in polynomial time and $O(w\log n)$ space, that solves reachability in the graph.