The Cost of Changing Edges for Diameter Computation and More

2026-08-12

AI Summary

This work studies how to efficiently update key graph metrics—such as diameter, radius, and eccentricities—under single-edge insertions or deletions, focusing on a sensitivity model where extensive preprocessing is allowed but near-real-time responses are required. For the decremental setting, it nearly resolves the sensitivity problem across all approximation regimes, introducing a data structure with preprocessing time matching that of the best static algorithms and supporting O(1) queries. In the incremental setting, it establishes theoretical lower bounds of 5/3 for undirected graphs and 2 for directed graphs on the approximation factor, and presents novel single-source eccentricity approximation algorithms that achieve these limits. Core technical contributions include sensitivity-preserving data structures, tight reductions, adaptations of static algorithms, and two new design paradigms for incremental algorithms.

Abstract

The sensitivity setting is a restricted setting for dynamic algorithms, particularly practical for scenarios where extensive preprocessing is feasible but responses to real-time modifications must be near-instantaneous before the data structure is eventually rebuilt. For graph problems, a sensitivity data structure is constructed with a preprocessing time P so that the following queries can be answered quickly, preferably in $O(1)$ time: given an edge $e$, return the answer to the problem on either $G \setminus e$ (decremental) or $G \cup e$ (incremental). In this paper, we almost entirely settle the decremental setting for the diameter and eccentricities problems in a variety of approximation regimes by matching P to the static runtime while supporting $O(1)$-time queries, thereby improving upon all previous results for a single failure [Bilò, Cohen, Friedrich, Schirneck, MFCS 2021; Bilò, Choudhary, Cohen, Friedrich, Krogmann, Schirneck, ICALP 2021]. More precisely: (1) We provide a tight reduction demonstrating that any exact distance sensitivity oracle can be used to efficiently solve decremental exact diameter and all-node eccentricities; (2) For the approximate setting, we match the runtime of all known static diameter algorithms across all sparsity settings, up to an additional $1+o(1)$ factor in approximation. Conversely, for the previously unexplored incremental setting of these problems: (3) We develop new lower bounds, demonstrating that no incremental algorithm can efficiently approximate diameter, radius, or eccentricity beyond a $5/3$ factor in undirected graphs or a $2$ factor in directed graphs; (4) We introduce two new instructive techniques and demonstrate how to utilize them to construct several new algorithms. Most notably, we develop incremental single-node eccentricity approximations for both directed and undirected graphs that match our new lower bounds.

Problem

• diameter
• eccentricity
• sensitivity
• dynamic graphs
• approximation

Innovation

• sensitivity oracle
• dynamic graph algorithms
• graph diameter
• eccentricity approximation
• incremental lower bounds