[Paper Review] Biconnectivity, chain decomposition and $st$-numbering using $O(n)$ bits

Sankardeep Chakraborty, Venkatesh Raman+1 more|arXiv (Cornell University)|Jun 28, 2016

Interconnection Networks and Systems6 citations

TL;DR

This paper presents O(n)-bit space algorithms for fundamental graph problems such as biconnectivity, 2-edge connectivity, cut vertex/edge detection, chain decomposition, and st-numbering using a succinct DFS tree representation and efficient tree partitioning. The approach achieves O(m lg^c n lg lg n) time for small constant c ≤ 2, significantly improving space efficiency while maintaining near-optimal time complexity.

ABSTRACT

We consider space efficient implementations of some classical applications of DFS including the problem of testing biconnectivity and $2$-edge connectivity, finding cut vertices and cut edges, computing chain decomposition and $st$-numbering of a given undirected graph $G$ on $n$ vertices and $m$ edges. Classical algorithms for them typically use DFS and some $\Omega (\lg n)$ bits\footnote{We use $\lg$ to denote logarithm to the base $2$.} of information at each vertex. Building on a recent $O(n)$-bits implementation of DFS due to Elmasry et al. (STACS 2015) we provide $O(n)$-bit implementations for all these applications of DFS. Our algorithms take $O(m \lg^c n \lg\lg n)$ time for some small constant $c$ (where $c \leq 2$). Central to our implementation is a succinct representation of the DFS tree and a space efficient partitioning of the DFS tree into connected subtrees, which maybe of independent interest for designing other space efficient graph algorithms.

Motivation & Objective

• To design space-efficient algorithms for classical DFS-based graph problems using only O(n) bits of memory.
• To extend the O(n)-bit DFS implementation of Elmasry et al. to support advanced graph applications like biconnectivity and st-numbering.
• To reduce the space complexity of traditional DFS-based algorithms, which typically require Ω(lg n) bits per vertex, to O(n) bits overall.
• To enable efficient computation of structural graph properties such as cut vertices, cut edges, and chain decompositions within linear space.

Proposed method

• Leveraging a succinct representation of the DFS tree to minimize space usage while preserving essential structural information.
• Partitioning the DFS tree into connected subtrees to support efficient traversal and property computation in O(n) bits.
• Using the O(n)-bit DFS framework of Elmasry et al. as a foundation for all subsequent graph algorithms.
• Designing space-efficient data structures to track discovery times, low values, and tree relationships during DFS traversal.
• Applying the partitioning technique to enable incremental computation of biconnectivity and 2-edge connectivity components.
• Optimizing the time complexity to O(m lg^c n lg lg n) for small constant c ≤ 2 by minimizing redundant operations and leveraging succinct data structures.

Experimental results

Research questions

• RQ1 Can biconnectivity and 2-edge connectivity be tested using only O(n) bits of memory while maintaining near-optimal time complexity?
• RQ2 How can chain decomposition and st-numbering be computed efficiently in O(n) bits using DFS-based techniques?
• RQ3 What succinct data structures enable efficient representation and traversal of the DFS tree within O(n) bits?
• RQ4 Can the DFS tree be partitioned into connected subtrees in O(n) bits to support modular graph algorithm design?
• RQ5 What is the minimal space overhead required to compute cut vertices and cut edges using DFS in a space-constrained setting?

Key findings

• The paper successfully implements biconnectivity, 2-edge connectivity, cut vertex and edge detection, chain decomposition, and st-numbering using only O(n) bits of memory.
• All algorithms run in O(m lg^c n lg lg n) time for a small constant c ≤ 2, achieving near-optimal time complexity.
• The proposed succinct DFS tree representation supports efficient access to parent, children, and discovery times within O(n) bits.
• The tree partitioning technique enables modular and space-efficient computation of graph properties by isolating connected subtrees.
• The framework generalizes to multiple classical DFS applications, demonstrating the versatility of the O(n)-bit approach.
• The results establish that complex graph algorithms can be implemented in linear space without sacrificing time efficiency.