Incremental Topological Ordering and Cycle Detection with Predictions
Samuel McCauley, Benjamin Moseley, Aidin Niaparast, Shikha Singh
TL;DR
This work tackles maintaining an incremental topological ordering and cycle detection in directed graphs via learning-augmented methods. It introduces a coarse-grained prediction model and two data structures: an ideal, decomposition-based Ideal Learned Ordering with provable worst-case guarantees adapted to prediction quality, and a practical Learned DFS Ordering (LDFS) that achieves $O(m\eta)$ total time, with per-edge cost $O(\eta)$. Theoretical results show smooth interpolation between ideal and worst-case performance, and experiments on real temporal DAGs demonstrate substantial speedups using modest training data and robustness to prediction errors. Overall, the paper bridges theory and practice in dynamic graphs by leveraging predictions to improve both asymptotic guarantees and empirical performance, highlighting potential for broader learning-augmented data-structure design.
Abstract
This paper leverages the framework of algorithms-with-predictions to design data structures for two fundamental dynamic graph problems: incremental topological ordering and cycle detection. In these problems, the input is a directed graph on $n$ nodes, and the $m$ edges arrive one by one. The data structure must maintain a topological ordering of the vertices at all times and detect if the newly inserted edge creates a cycle. The theoretically best worst-case algorithms for these problems have high update cost (polynomial in $n$ and $m$). In practice, greedy heuristics (that recompute the solution from scratch each time) perform well but can have high update cost in the worst case. In this paper, we bridge this gap by leveraging predictions to design a learned new data structure for the problems. Our data structure guarantees consistency, robustness, and smoothness with respect to predictions -- that is, it has the best possible running time under perfect predictions, never performs worse than the best-known worst-case methods, and its running time degrades smoothly with the prediction error. Moreover, we demonstrate empirically that predictions, learned from a very small training dataset, are sufficient to provide significant speed-ups on real datasets.