Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs

We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in O~(mn^(4/3) log{W}/epsilon) total time (where the edge weights are from [1,W]) and explicitly maintains a (1+epsilon)-approximate distance matrix. For a fixed epsilon>0, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is o(n^2) regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC\u2702, Bernstein STOC\u2713] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the O~(*) notation). Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of O~(n/d) vertices which hit a shortest path between each pair of vertices, provided it has hop-length Omega(d). We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions

Cross-Document Pattern Matching

We study a new variant of the string matching problem called cross-document string matching, which is the problem of indexing a collection of documents to support an efficient search for a pattern in a selected document, where the pattern itself is a substring of another document. Several variants of this problem are considered, and efficient linear-space solutions are proposed with query time bounds that either do not depend at all on the pattern size or depend on it in a very limited way (doubly logarithmic). As a side result, we propose an improved solution to the weighted level ancestor problem

Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time

We give a nearly linear time randomized approximation scheme for the Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an undirected edge-weighted graph $G$ on $m$ edges and $\epsilon > 0$, the algorithm outputs in $O(m \log^4n /\epsilon^2)$ time, with high probability, a $(1+\epsilon)$-approximation to the Held-Karp bound on the metric TSP instance induced by the shortest path metric on $G$. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the $O(m^2 \log^2(m)/\epsilon^2)$ running time achieved previously by Garg and Khandekar. The LP solution can be used to obtain a fast randomized $\big(\frac{3}{2} + \epsilon\big)$-approximation for metric TSP which improves upon the running time of previous implementations of Christofides' algorithm

Incremental and Decremental Maintenance of Planar Width

We present an algorithm for maintaining the width of a planar point set dynamically, as points are inserted or deleted. Our algorithm takes time O(kn^epsilon) per update, where k is the amount of change the update causes in the convex hull, n is the number of points in the set, and epsilon is any arbitrarily small constant. For incremental or decremental update sequences, the amortized time per update is O(n^epsilon).Comment: 7 pages; 2 figures. A preliminary version of this paper was presented at the 10th ACM/SIAM Symp. Discrete Algorithms (SODA '99); this is the journal version, and will appear in J. Algorithm

Dynamic Dominators and Low-High Orders in DAGs

We consider practical algorithms for maintaining the dominator tree and a low-high order in directed acyclic graphs (DAGs) subject to dynamic operations. Let G be a directed graph with a distinguished start vertex s. The dominator tree D of G is a tree rooted at s, such that a vertex v is an ancestor of a vertex w if and only if all paths from s to w in G include v. The dominator tree is a central tool in program optimization and code generation, and has many applications in other diverse areas including constraint programming, circuit testing, biology, and in algorithms for graph connectivity problems. A low-high order of G is a preorder of D that certifies the correctness of D, and has further applications in connectivity and path-determination problems. We first provide a practical and carefully engineered version of a recent algorithm [ICALP 2017] for maintaining the dominator tree of a DAG through a sequence of edge deletions. The algorithm runs in O(mn) total time and O(m) space, where n is the number of vertices and m is the number of edges before any deletion. In addition, we present a new algorithm that maintains a low-high order of a DAG under edge deletions within the same bounds. Both results extend to the case of reducible graphs (a class that includes DAGs). Furthermore, we present a fully dynamic algorithm for maintaining the dominator tree of a DAG under an intermixed sequence of edge insertions and deletions. Although it does not maintain the O(mn) worst-case bound of the decremental algorithm, our experiments highlight that the fully dynamic algorithm performs very well in practice. Finally, we study the practical efficiency of all our algorithms by conducting an extensive experimental study on real-world and synthetic graphs