Improved Bounds for Shortest Paths in Dense Distance Graphs

We study the problem of computing shortest paths in so-called dense distance graphs, a basic building block for designing efficient planar graph algorithms. Let G be a plane graph with a distinguished set partial{G} of boundary vertices lying on a constant number of faces of G. A distance clique of G is a complete graph on partial{G} encoding all-pairs distances between these vertices. A dense distance graph is a union of possibly many unrelated distance cliques. Fakcharoenphol and Rao [Fakcharoenphol and Rao, 2006] proposed an efficient implementation of Dijkstra\u27s algorithm (later called FR-Dijkstra) computing single-source shortest paths in a dense distance graph. Their algorithm spends O(b log^2{n}) time per distance clique with b vertices, even though a clique has b^2 edges. Here, n is the total number of vertices of the dense distance graph. The invention of FR-Dijkstra was instrumental in obtaining such results for planar graphs as nearly-linear time algorithms for multiple-source-multiple-sink maximum flow and dynamic distance oracles with sublinear update and query bounds. At the heart of FR-Dijkstra lies a data structure updating distance labels and extracting minimum labeled vertices in O(log^2{n}) amortized time per vertex. We show an improved data structure with O((log^2{n})/(log^2 log n)) amortized bounds. This is the first improvement over the data structure of Fakcharoenphol and Rao in more than 15 years. It yields improved bounds for all problems on planar graphs, for which computing shortest paths in dense distance graphs is currently a bottleneck

Approximate Circular Pattern Matching

We investigate the complexity of approximate circular pattern matching (CPM, in short) under the Hamming and edit distance. Under each of these two basic metrics, we are given a length-n text T, a length-m pattern P, and a positive integer threshold k, and we are to report all starting positions (called occurrences) of fragments of T that are at distance at most k from some cyclic rotation of P. In the decision version of the problem, we are to check if there is any such occurrence. All previous results for approximate CPM were either average-case upper bounds or heuristics, with the exception of the work of Charalampopoulos et al. [CKP+, JCSS'21], who considered only the Hamming distance. For the reporting version of the approximate CPM problem, under the Hamming distance we improve upon the main algorithm of [CKP+, JCSS'21] from O(n+(n/m) k4) to O(n+(n/m) k3 log log k) time; for the edit distance, we give an O(nk2)-time algorithm. Notably, for the decision versions and wide parameter-ranges, we give algorithms whose complexities are almost identical to the state-of-the-art for standard (i.e., non-circular) approximate pattern matching: For the decision version of the approximate CPM problem under the Hamming distance, we obtain an O(n + (n/m) k2 log k/ log log k)-time algorithm, which works in O(n) time whenever k = O( p mlog log m/logm). In comparison, the fastest algorithm for the standard counterpart of the problem, by Chan et al. [CGKKP, STOC'20], runs in O(n) time only for k = O(√ m). We achieve this result via a reduction to a geometric problem by building on ideas from [CKP+, JCSS'21] and Charalampopoulos et al. [CKW, FOCS'20]. For the decision version of the approximate CPM problem under the edit distance, the O(nk log3 k) runtime of our algorithm near matches the O(nk) runtime of the Landau-Vishkin algorithm [LV, J. Algorithms'89] for approximate pattern matching under edit distance; the latter algorithm remains the fastest known for k = Ω(m2/5). As a stepping stone, we propose an O(nk log3 k)-time algorithm for solving the Longest Prefix k-Approximate Match problem, proposed by Landau et al. [LMS, SICOMP'98], for all k ∈ {1, , k}. Our algorithm is based on Tiskin's theory of seaweeds [Tiskin, Math. Comput. Sci.'08], with recent advancements (see Charalampopoulos et al. [CKW, FOCS'22]), and on exploiting the seaweeds' relation to Monge matrices. In contrast, we obtain a conditional lower bound that suggests a polynomial separation between approximate CPM under the Hamming distance over the binary alphabet and its non-circular counterpart. We also show that a strongly subquadratic-time algorithm for the decision version of approximate CPM under edit distance would refute the Strong Exponential Time Hypothesis

Near-Optimal Distance Emulator for Planar Graphs

Given a graph G and a set of terminals T, a distance emulator of G is another graph H (not necessarily a subgraph of G) containing T, such that all the pairwise distances in G between vertices of T are preserved in H. An important open question is to find the smallest possible distance emulator. We prove that, given any subset of k terminals in an n-vertex undirected unweighted planar graph, we can construct in O~(n) time a distance emulator of size O~(min(k^2,sqrt{k * n})). This is optimal up to logarithmic factors. The existence of such distance emulator provides a straightforward framework to solve distance-related problems on planar graphs: Replace the input graph with the distance emulator, and apply whatever algorithm available to the resulting emulator. In particular, our result implies that, on any unweighted undirected planar graph, one can compute all-pairs shortest path distances among k terminals in O~(n) time when k=O(n^{1/3})

Decremental Single-Source Reachability in Planar Digraphs

In this paper we show a new algorithm for the decremental single-source reachability problem in directed planar graphs. It processes any sequence of edge deletions in $O(n\log^2{n}\log\log{n})$ total time and explicitly maintains the set of vertices reachable from a fixed source vertex. Hence, if all edges are eventually deleted, the amortized time of processing each edge deletion is only $O(\log^2 n \log \log n)$, which improves upon a previously known $O(\sqrt{n})$ solution. We also show an algorithm for decremental maintenance of strongly connected components in directed planar graphs with the same total update time. These results constitute the first almost optimal (up to polylogarithmic factors) algorithms for both problems. To the best of our knowledge, these are the first dynamic algorithms with polylogarithmic update times on general directed planar graphs for non-trivial reachability-type problems, for which only polynomial bounds are known in general graphs

Approximate circular pattern matching

We investigate the complexity of approximate circular pattern matching (CPM, in short) under the Hamming and edit distance. Under each of these two basic metrics, we are given a length-n text T, a length-m pattern P, and a positive integer threshold k, and we are to report all starting positions (called occurrences) of fragments of T that are at distance at most k from some cyclic rotation of P. In the decision version of the problem, we are to check if there is any such occurrence. All previous results for approximate CPM were either average-case upper bounds or heuristics, with the exception of the work of Charalampopoulos et al. [CKP+, JCSS'21], who considered only the Hamming distance. For the reporting version of the approximate CPM problem, under the Hamming distance we improve upon the main algorithm of [CKP+, JCSS'21] from O(n+(n/m) k4) to O(n+(n/m) k3 log log k) time; for the edit distance, we give an O(nk2)-time algorithm. Notably, for the decision versions and wide parameter-ranges, we give algorithms whose complexities are almost identical to the state-of-the-art for standard (i.e., non-circular) approximate pattern matching: For the decision version of the approximate CPM problem under the Hamming distance, we obtain an O(n + (n/m) k2 log k/ log log k)-time algorithm, which works in O(n) time whenever k = O( p mlog log m/logm). In comparison, the fastest algorithm for the standard counterpart of the problem, by Chan et al. [CGKKP, STOC'20], runs in O(n) time only for k = O(√ m). We achieve this result via a reduction to a geometric problem by building on ideas from [CKP+, JCSS'21] and Charalampopoulos et al. [CKW, FOCS'20]. For the decision version of the approximate CPM problem under the edit distance, the O(nk log3 k) runtime of our algorithm near matches the O(nk) runtime of the Landau-Vishkin algorithm [LV, J. Algorithms'89] for approximate pattern matching under edit distance; the latter algorithm remains the fastest known for k = Ω(m2/5). As a stepping stone, we propose an O(nk log3 k)-time algorithm for solving the Longest Prefix k-Approximate Match problem, proposed by Landau et al. [LMS, SICOMP'98], for all k ∈ {1, , k}. Our algorithm is based on Tiskin's theory of seaweeds [Tiskin, Math. Comput. Sci.'08], with recent advancements (see Charalampopoulos et al. [CKW, FOCS'22]), and on exploiting the seaweeds' relation to Monge matrices. In contrast, we obtain a conditional lower bound that suggests a polynomial separation between approximate CPM under the Hamming distance over the binary alphabet and its non-circular counterpart. We also show that a strongly subquadratic-time algorithm for the decision version of approximate CPM under edit distance would refute the Strong Exponential Time Hypothesis

Max s,ts,t-Flow Oracles and Negative Cycle Detection in Planar Digraphs

We study the maximum $s,t$-flow oracle problem on planar directed graphs where the goal is to design a data structure answering max $s,t$-flow value (or equivalently, min $s,t$-cut value) queries for arbitrary source-target pairs $(s,t)$. For the case of polynomially bounded integer edge capacities, we describe an exact max $s,t$-flow oracle with truly subquadratic space and preprocessing, and sublinear query time. Moreover, if $(1-\epsilon)$-approximate answers are acceptable, we obtain a static oracle with near-linear preprocessing and $\tilde{O}(n^{3/4})$ query time and a dynamic oracle supporting edge capacity updates and queries in $\tilde{O}(n^{6/7})$ worst-case time. To the best of our knowledge, for directed planar graphs, no (approximate) max $s,t$-flow oracles have been described even in the unweighted case, and only trivial tradeoffs involving either no preprocessing or precomputing all the $n^2$ possible answers have been known. One key technical tool we develop on the way is a sublinear (in the number of edges) algorithm for finding a negative cycle in so-called dense distance graphs. By plugging it in earlier frameworks, we obtain improved bounds for other fundamental problems on planar digraphs. In particular, we show: (1) a deterministic $O(n\log(nC))$ time algorithm for negatively-weighted SSSP in planar digraphs with integer edge weights at least $-C$. This improves upon the previously known bounds in the important case of weights polynomial in $n$, and (2) an improved $O(n\log{n})$ bound on finding a perfect matching in a bipartite planar graph.Comment: Extended abstract to appear in SODA 202