Improved Space efficient linear time algorithms for BFS, DFS and applications

TL;DR

This paper introduces space-efficient algorithms and data structures for fundamental graph algorithms like BFS and DFS, significantly reducing their space complexity while maintaining linear time performance, with practical applications demonstrated.

Contribution

The authors develop new data structures and algorithms that reduce space usage for BFS and DFS to near-linear bits, improving upon previous methods and enabling efficient graph analysis.

Findings

BFS implemented in O(m+n) time using at most 2n+o(n) bits.

Further space reduction of BFS to 1.585n+o(n) bits.

DFS implemented in O(m+n) time with O(n log(m/n)) bits.

Abstract

Recent work by Elmasry et al. (STACS 2015) and Asano et al. (ISAAC 2014), reconsidered classical fundamental graph algorithms focusing on improving the space complexity. We continue this line of work focusing on space. Our first result is a simple data structure that can maintain any subset $S$ of a universe of $n$ elements using $n+o(n)$ bits and support in constant time, apart from the standard insert, delete and membership queries, the operation {\it findany} that finds and returns any element of the set (or outputs that the set is empty). Using this we give a BFS implementation that takes $O(m+n)$ time using at most $2n+o(n)$ bits. Later, we further improve the space requirement of BFS to at most $1.585n + o(n)$ bits. We demonstrate the use of our data structure by developing another data structure using it that can represent a sequence of $n$ non-negative integers $x_1, x_2, \ldots x_n$ using at most $\sum_{i=1}^n x_i + 2n + o(\sum_{i=1}^n x_i+n)$ bits and, in constant time, determine whether the $i$-th element is $0$ or decrement it otherwise. We also discuss an algorithm for finding a minimum weight spanning tree of a weighted undirected graph using at most $n+o(n)$ bits. We also provide an implementation for DFS that takes $O(m+n)$ time and $O(n \lg(m/n))$ bits. Using this DFS algorithm and other careful implementations, we can test biconnectivity, 2-edge connectivity, and determine cut vertices, bridges etc among others, essentially within the same time and space bounds required for DFS. These improve the space required for earlier implementations from $\Omega (n\lg n)$ bits.