Dynamic Detours

2026-05-04

AI Summary

This work addresses three fundamental path-existence queries in dynamic undirected graphs: (1) whether there exists a $(u,v)$-path of length at least $k$, (2) whether there is a detour from $u$ to $v$ that is at least $k$ longer than the shortest path, and (3) whether there exists a $(u,v)$-path of prescribed parity. The paper proposes an efficient dynamic data structure supporting edge insertions and deletions, whose core innovation lies in a “lazy edge insertion” mechanism combined with localized maintenance of biconnected components. This approach circumvents known conditional lower bounds for short-path queries. Leveraging advanced dynamic graph techniques and refined amortized analysis, the structure achieves amortized update and query times of $2^{O(k^3)} \log n + O(\log^2 n \log^2 \log n)$ for the first two query types, while parity-constrained path queries are answered in $O(\log^2 n \log^2 \log n)$ time.

Abstract

Fix a parameter $k\in \mathbf{N}$. We give dynamic data structures that for a fully dynamic undirected graph $G$, updated over time by edge insertions and edge deletions, can answer the following queries: - Long $(u,v)$-path: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of length at least $k$? - Long $(u,v)$-detour: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of length at least $\text{dist}_G(u,v)+k$? - Even/odd $(u,v)$-path: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of even/odd length? The amortized time of executing an update or answering a query is $2^{O(k^3)} \log n + O(\log^2 n \log^2 \log n)$ in the first two cases, and $O(\log^2 n \log^2 \log n)$ in the last, where $n$ is the number of vertices of $G$. The first result is in sharp contrast with known conditional lower bounds for reporting paths of length at most $k$. Specifically, there is no data structure supporting queries about $(u,v)$-paths of length at most two in time $n^{o(1)}$ unless the Triangle Conjecture fails. Our main technical contribution is a mechanism of "delayed edge insertion" that works locally on the level of biconnected components.

Problem

• dynamic graph
• long path
• detour
• parity path
• connectivity query

Innovation

• dynamic data structures
• long path queries
• detour queries
• delayed edge insertion
• biconnected components