Generalized Sorting with Predictions Revisited

Abstract

This paper presents a novel algorithm for the generalized sorting problem with predictions, which involves determining a total ordering of an underlying directed graph using as few probes as possible. Specifically, we consider the problem of sorting an undirected graph with predicted edge directions. Our proposed algorithm is a Monte Carlo approach that has a polynomial-time complexity, which uses $O(n\log w+w)$ probes with probability at least $1-e^{-\varTheta (n)}$, where n is the number of vertices in the graph and w is the number of mispredicted edges. Our approach involves partitioning the vertices of the graph into O(w) disjoint verified directed paths, which can reduce the number of probes required. Lu et al. [11] introduced a bound of $O(n\log n + w)$ for the number of probes, which was the only known result in this setting. Our bound reduces the factor $O(\log n)$ to $O(\log w)$.

This research was partially supported by the Hong Kong RGC grants 17201220, 17202121 and 17203122.