Dynamic graph connectivity with improved worst case update time and sublinear space

TL;DR

This paper introduces a more efficient fully dynamic graph connectivity algorithm with improved worst-case update time and sublinear space, matching the space efficiency of streaming algorithms.

Contribution

It presents a new dynamic graph connectivity data structure with faster worst-case update time and sublinear space, improving upon previous polylogarithmic time solutions.

Findings

Update time reduced to O(log^4 n) per operation.

Queries answered in O(log n / log log n) time with high probability.

Uses O(n log^2 n) words of space, matching streaming algorithm efficiency.

Abstract

This paper considers fully dynamic graph algorithms with both faster worst case update time and sublinear space. The fully dynamic graph connectivity problem is the following: given a graph on a fixed set of n nodes, process an online sequence of edge insertions, edge deletions, and queries of the form "Is there a path between nodes a and b?" In 2013, the first data structure was presented with worst case time per operation which was polylogarithmic in n. In this paper, we shave off a factor of log n from that time, to O(log^4 n) per update. For sequences which are polynomial in length, our algorithm answers queries in O(log n/\log\log n) time correctly with high probability and using O(n \log^2 n) words (of size log n). This matches the amount of space used by the most space-efficient graph connectivity streaming algorithm. We also show that 2-edge connectivity can be maintained using O(n log^2 n) words with an amortized update time of O(log^6 n).