Incremental Approximate Single-Source Shortest Paths with Predictions
This work is significant for researchers and practitioners working with dynamic graphs and shortest path computations, offering an incremental improvement to existing methods by incorporating predictions.
This paper tackles the problem of maintaining approximate shortest paths in incremental graphs using predictions. They developed a learned algorithm that maintains (1+ε)-approximate single-source shortest paths, achieving a runtime of O(m η log W/ε), where η is the prediction error and W is the heaviest edge weight.
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.