New Tradeoffs for Decremental Approximate All-Pairs Shortest Paths

TL;DR

This work advances the theory of decremental APSP by introducing four new randomized data-structures that achieve (2+ε)-type approximations and related variants with subquadratic total update times across both weighted and unweighted undirected graphs. The core strategy blends Thorup–Zwick bunches with hopsets and monotone ES-trees, augmented by heavy-light (bunch overlap) techniques and a reduction framework from mixed to multiplicative approximations to handle unweighted graphs. The paper delivers concrete update-time bounds such as - ilde{O}(m^{1/2} n^{3/2}) for weighted (2+ε)-APSP; - ilde{O}(n m^{3/4}) for (2+ε,W_{u,v})-APSP; - ilde{O}(m^{7/4}) for unweighted (2+ε)-APSP; and - ilde{O}(n^{2−1/k} m^{1/k}) for (1+ε,2(k−1))-APSP, all with constant query time. These results push subquadratic performance into sparser/densities regimes and expand the unexplored space between multiplicative (1+ε) and additive (2) approximations in the decremental setting, also inspiring static follow-ups and potential practical distance oracles. The methods blend dynamic clustering, randomized sampling, and carefully controlled lazy updates to achieve near-optimal tradeoffs under an oblivious adversary. Overall, the paper significantly broadens the toolkit for fast, approximate dynamic APSP and clarifies the landscape of achievable update-time vs. approximation tradeoffs in the decremental regime.

Abstract

We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with $m$ edges and $n$ nodes undergoing edge deletions, we provide four new approximate decremental APSP algorithms, two for weighted and two for unweighted graphs. Our first result is $(2+ ε)$-APSP with total update time $\tilde{O}(m^{1/2}n^{3/2})$ (when $m= n^{1+c}$ for any constant $0<c<1$). Prior to our work the fastest algorithm for weighted graphs with approximation at most $3$ had total $\tilde O(mn)$ update time for $(1+ε)$-APSP [Bernstein, SICOMP 2016]. Our second result is $(2+ε, W_{u,v})$-APSP with total update time $\tilde{O}(nm^{3/4})$, where the second term is an additive stretch with respect to $W_{u,v}$, the maximum weight on the shortest path from $u$ to $v$. Our third result is $(2+ ε)$-APSP for unweighted graphs in $\tilde O(m^{7/4})$ update time, which for sparse graphs ($m=o(n^{8/7})$) is the first subquadratic $(2+ε)$-approximation. Our last result for unweighted graphs is $(1+ε, 2(k-1))$-APSP, for $k \geq 2 $, with $\tilde{O}(n^{2-1/k}m^{1/k})$ total update time (when $m=n^{1+c}$ for any constant $c >0$). For comparison, in the special case of $(1+ε, 2)$-approximation, this improves over the state-of-the-art algorithm by [Henzinger, Krinninger, Nanongkai, SICOMP 2016] with total update time of $\tilde{O}(n^{2.5})$. All of our results are randomized, work against an oblivious adversary, and have constant query time.

Figures (5)

• Figure 1: Possible scenarios for the overlap between the bunches $B(u)$, $B(v)$ and the shortest path $\pi$ from $u$ to $v$.
• Figure 2: Possible scenarios for the overlap between the bunches $B(u)$ and $B(v)$ and the shortest path $\pi$ from $u$ to $v$.
• Figure 3: Possible scenarios for the overlap between the bunches $B(u)$ and $B(v)$ and the shortest path $\pi$ from $u$ to $v$.
• Figure 4: Possible scenarios for the overlap between the bunches $B(u)$ and $B(v)$ and the shortest path $\pi$ from $u$ to $v$.
• Figure 5: Possible scenario for the overlap between the bunches $B(u)$ and $B(v)$ and the shortest path $\pi$ from $u$ to $v$ where the estimate can be obtained using heavy pivots $q_u$ and $q_v$.

Theorems & Definitions (79)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Lemma 3.0
• Lemma 3.1: LN2020
• Definition 1
• Lemma 4.1
• proof