Incremental Approximate Single-Source Shortest Paths with Predictions

TL;DR

We study maintaining $(1+\epsilon)$-approximate single-source shortest paths in incremental graphs with a predictions-based warm start. Our approach builds an offline recursive framework that informs a online learned algorithm, achieving a prediction-dependent runtime of $\tilde{O}\big(m \min_{\tau} \{\tau+|\text{HIGH}(\tau)|\} \log W / \epsilon\big)$ and extending to All-Pairs Shortest Paths. A foundational offline problem is solved with efficient maintenance of alive edges and time-indexed distance lists, enabling robust online performance. The results show ideal learned behavior (consistent, robust, smooth) and provably near-optimal performance under perfect predictions, with graceful degradation under prediction errors and strong worst-case guarantees when predictions are poor.

Abstract

The algorithms-with-predictions framework has been used extensively to develop online algorithms with improved beyond-worst-case competitive ratios. Recently, there is growing interest in leveraging predictions for designing data structures with improved beyond-worst-case running times. In this paper, we study the fundamental data structure problem of maintaining approximate shortest paths in incremental graphs in the algorithms-with-predictions model. Given a sequence $σ$ of edges that are inserted one at a time, the goal is to maintain approximate shortest paths from the source to each vertex in the graph at each time step. Before any edges arrive, the data structure is given a prediction of the online edge sequence $\hatσ$ which is used to ``warm start'' its state. As our main result, we design a learned algorithm that maintains $(1+ε)$-approximate single-source shortest paths, which runs in $\tilde{O}(m η\log W/ε)$ time, where $W$ is the weight of the heaviest edge and $η$ is the prediction error. We show these techniques immediately extend to the all-pairs shortest-path setting as well. Our algorithms are consistent (performing nearly as fast as the offline algorithm) when predictions are nearly perfect, have a smooth degradation in performance with respect to the prediction error and, in the worst case, match the best offline algorithm up to logarithmic factors. As a building block, we study the offline incremental approximate single-source shortest-paths problem. In this problem, the edge sequence $σ$ is known a priori and the goal is to efficiently return the length of the shortest paths in the intermediate graph $G_t$ consisting of the first $t$ edges, for all $t$. Note that the offline incremental problem is defined in the worst-case setting (without predictions) and is of independent interest.

Figures (2)

• Figure 1: The recursion tree of the algorithm. Each node $x$ in the tree is associated with an interval $[\ell,r]$ such that $x=(\ell+r)/2$. The depth of a node is the number of nodes in the path from the root $m/2$ to that node. For example, the depth of node $x=6$ is 3, and its corresponding interval is $[4,8]$.
• Figure 2: An illustration of the subproblems that get rebuilt during one edge insertion. In this example, at time $t=4$, the edge $e_4$ was predicted to arrive but edge $e_8$ has arrived. So the algorithm moves edge $e_8$ from position $t'=8$ to position $t=4$. The algorithm then rebuilds all the subproblems with $t \leq (\ell+r)/2 < t'$ (colored dark gray) and their descendants (colored light gray) from top to bottom.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof