[Paper Review] Optimal decremental connectivity in planar graphs

Jakub Łącki, Piotr Sankowski|arXiv (Cornell University)|Sep 25, 2014

Complexity and Algorithms in Graphs11 references4 citations

TL;DR

This paper presents an optimal $O(n)$-time algorithm for decremental connectivity in planar graphs, achieving constant-time connectivity queries. By reducing the decremental problem to a collection of small incremental connectivity problems via recursive $r$-divisions, and precomputing connected components for tiny graphs ($O(\log^2 \log n)$ vertices), the method bypasses dynamic spanning tree maintenance and achieves linear total update time, matching the best-known bound for trees.

ABSTRACT

We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form `Are vertices $u$ and $v$ connected with a path?' in constant time. The queries can be intermixed with any sequence of edge deletions, and the algorithm handles all updates in $O(n)$ time. This results improves over previously known $O(n \log n)$ time algorithm.

Motivation & Objective

• To close the gap between decremental and incremental connectivity in planar graphs, where the former was previously thought to require $O(n\log n)$ time.
• To develop a linear-time algorithm for maintaining connectivity under edge deletions in planar graphs, improving upon the prior $O(n\log n)$ bound from fully dynamic algorithms.
• To demonstrate that decremental connectivity in planar graphs is strictly easier than fully dynamic connectivity, which has a known $\Omega(\log n)$ lower bound per operation.
• To provide a reduction from decremental connectivity to small incremental connectivity instances, enabling efficient precomputation of component identifiers.

Proposed method

• The algorithm uses recursive $r$-divisions to partition the planar graph into smaller subgraphs of size $O(\log^2 n)$, ensuring total size remains $O(n)$ across levels.
• It reduces the decremental connectivity problem to maintaining connectivity in a collection of small, incremental graphs whose total size is $O(n)$, leveraging known linear-time solutions for such cases.
• For graphs of size $t = O(\log^2 \log n)$, the algorithm precomputes all possible connected component identifiers for every possible graph state and explicit vertex set.
• Each state is encoded as a binary word of length $O(\log^4 \log n)$, fitting in a single machine word, enabling constant-time state transitions and queries.
• Edge deletions are handled by flipping a single bit in the state word, and component identifier updates are precomputed for all possible edge deletions.
• The algorithm maintains explicit cc-identifiers for a set $V_e$ of vertices, allowing $O(1)$ query time by storing the identifier of each vertex in $V_e$.

Experimental results

Research questions

• RQ1 Can the total update time for decremental connectivity in planar graphs be reduced below $O(n\log n)$, matching the optimal bound for incremental connectivity?
• RQ2 Is it possible to achieve $O(n)$ total update time for decremental connectivity in planar graphs by avoiding dynamic spanning tree maintenance?
• RQ3 Does the decremental connectivity problem in planar graphs have a lower total update time than the fully dynamic variant, despite the latter's $\Omega(\log n)$ per-operation lower bound?
• RQ4 Can a reduction from decremental to incremental connectivity be constructed in a way that enables precomputation for small graphs, leading to optimal total time?

Key findings

• The paper achieves an optimal $O(n)$ total time for processing all edge deletions in planar graphs, significantly improving over the previous $O(n\log n)$ bound.
• Connectivity queries are answered in $O(1)$ time, with cc-identifiers of vertices in the explicit set $V_e$ maintained explicitly.
• The algorithm uses a recursive $r$-division-based reduction that ensures the total size of subproblems remains $O(n)$, with only $O(n)$ overhead from the reduction.
• Precomputation of all possible component identifiers for graphs of size $O(\log^2 \log n)$ is feasible in $o(n)$ time, as the number of states is $2^{O(\log^4 \log n)} = 2^{o(\log n)} = o(n)$.
• The method avoids maintaining a spanning tree, eliminating the need to search for replacement edges after deletions.
• The result shows that decremental connectivity in planar graphs is strictly easier than fully dynamic connectivity, which has a $\Omega(\log n)$ lower bound per operation.