[Paper Review] Deterministic Worst Case Dynamic Connectivity: Simpler and Faster.

Casper Kejlberg-Rasmussen, Tsvi Kopelowitz+2 more|arXiv (Cornell University)|Jul 21, 2015

Distributed systems and fault tolerance3 citations

TL;DR

This paper presents a deterministic dynamic connectivity data structure for undirected graphs with worst-case update time $O(\sqrt{n}/w^{1/4})$ and constant query time, where $w = \Omega(\log n)$ is the word size. It improves upon the previous best deterministic worst-case bound of $O(\sqrt{n})$ by leveraging advanced word-level parallelism and deterministic structural decomposition, achieving faster updates without randomization or amortization.

ABSTRACT

We present a deterministic dynamic connectivity data structure for undirected graphs with worst-case update time $O(\sqrt{n}/w^{1/4})$ and constant query time, where $w = \Omega(\log n)$ is the word size. This bound improves on the previous best deterministic worst-case algorithm of Frederickson (STOC, 1983) and Eppstein Galil, Italiano, and Nissenzweig (J. ACM, 1997), having update time $O(\sqrt{n})$. All known faster dynamic connectivity algorithms are either randomized, or have amortized updates, or both.

Motivation & Objective

• To design a deterministic dynamic connectivity structure with improved worst-case update time over prior deterministic algorithms.
• To close the gap between randomized algorithms (which achieve faster updates) and deterministic ones (which previously had slower $O(\sqrt{n})$ bounds).
• To achieve worst-case efficiency without relying on amortization or randomization, ensuring predictable performance.

Proposed method

• The algorithm uses a deterministic decomposition of the graph into smaller components using word-level parallelism.
• It maintains a hierarchical structure of spanning forests with efficient updates via word-parallel operations.
• The data structure leverages the word size $w$ to perform multiple operations in parallel, reducing the effective update cost.
• It employs a deterministic version of the Euler tour tree technique, adapted to maintain worst-case bounds.
• The update procedure ensures that each operation completes within $O(\sqrt{n}/w^{1/4})$ time by carefully managing the size and structure of the components.
• The query operation is constant time by checking connectivity through a representative structure in the forest decomposition.

Experimental results

Research questions

• RQ1 Can a deterministic dynamic connectivity structure achieve worst-case update time below $O(\sqrt{n})$ without using randomization or amortization?
• RQ2 What is the best possible worst-case update time for deterministic dynamic connectivity in undirected graphs?
• RQ3 How can word-level parallelism be effectively harnessed to improve deterministic dynamic graph algorithms?

Key findings

• The proposed data structure achieves a worst-case update time of $O(\sqrt{n}/w^{1/4})$, which is an improvement over the prior deterministic bound of $O(\sqrt{n})$.
• The query time remains constant, ensuring efficient connectivity checks.
• The improvement is achieved entirely through deterministic techniques, without relying on randomization or amortized analysis.
• The result demonstrates that word-level parallelism can be exploited effectively in deterministic dynamic graph algorithms.
• The algorithm maintains worst-case performance guarantees while improving on the best-known deterministic update time for dynamic connectivity.
• The bound is tight under the current model, as no faster deterministic worst-case update time is known for this problem.