Simultaneous Time-Space Upper Bounds for Certain Problems in Planar Graphs

TL;DR

This paper presents polynomial time algorithms with sublinear space bounds for shortest paths, red-blue path problems, and perfect matching in planar graphs, advancing space-efficient solutions for key planar graph problems.

Contribution

It introduces new polynomial time, sublinear space algorithms for shortest paths, red-blue path problems, and perfect matchings in planar graphs, improving space efficiency.

Findings

Shortest path in planar graphs with O(n^{1/2+ε}) space

Red-blue path problem in planar DAGs with O(n^{1/2+ε}) space

Matching and Hall-obstacle problems with same time-space bounds as shortest path

Abstract

In this paper, we show that given a weighted, directed planar graph $G$, and any $\epsilon >0$, there exists a polynomial time and $O(n^{\frac{1}{2}+\epsilon})$ space algorithm that computes the shortest path between two fixed vertices in $G$. We also consider the {\RB} problem, which states that given a graph $G$ whose edges are colored either red or blue and two fixed vertices $s$ and $t$ in $G$, is there a path from $s$ to $t$ in $G$ that alternates between red and blue edges. The {\RB} problem in planar DAGs is {\NL}-complete. We exhibit a polynomial time and $O(n^{\frac{1}{2}+\epsilon})$ space algorithm (for any $\epsilon >0$) for the {\RB} problem in planar DAG. In the last part of this paper, we consider the problem of deciding and constructing the perfect matching present in a planar bipartite graph and also a similar problem which is to find a Hall-obstacle in a planar bipartite graph. We show the time-space bound of these two problems are same as the bound of shortest path problem in a directed planar graph.