arXiv subjects
Publications and source records attributed to Longhui Yin.
This paper presents a new deterministic algorithm for single-source shortest paths (SSSP) on real non-negative edge-weighted directed graphs, with running time $O(m\sqrt{\log n}+\sqrt{mn\log n\log \log n})$, which is $O(m\sqrt{\log n\log \log n})$ for sparse graphs. This improves the recent breakthrough result of $O(m\log^{2/3} n)$ time for directed SSSP algorithm [Duan, Mao, Mao, Shu, Yin 2025].
We give a deterministic $O(m\log^{2/3}n)$-time algorithm for single-source shortest paths (SSSP) on directed graphs with real non-negative edge weights in the comparison-addition model. This is the first result to break the $O(m+n\log n)$ time bound of Dijkstra's algorithm on sparse graphs, showing that Dijkstra's algorithm is not optimal for SSSP.
In undirected graphs with real non-negative weights, we give a new randomized algorithm for the single-source shortest path (SSSP) problem with running time $O(m\sqrt{\log n \cdot \log\log n})$ in the comparison-addition model. This is the first algorithm to break the $O(m+n\log n)$ time bound for real-weighted sparse graphs by Dijkstra's algorithm with Fibonacci heaps. Previous undirected non-negative SSSP algorithms give time bound of $O(m\alpha(m,n)+\min\{n\log n, n\log\log r\})$ in comparison-addition model, where $\alpha$ is the inverse-Ackermann function and $r$ is the ratio of the maximum-to-minimum edge weight [Pettie & Ramachandran 2005], and linear time for integer edge weights in RAM model [Thorup 1999]. Note that there is a proposed complexity lower bound of $\Omega(m+\min\{n\log n, n\log\log r\})$ for hierarchy-based algorithms for undirected real-weighted SSSP [Pettie & Ramachandran 2005], but our algorithm does not obey the properties required for that lower bound. As a non-hierarchy-based approach, our algorithm shows great advantage with much simpler structure, and is much easier to implement.
A $k$-vertex connectivity oracle for undirected $G$ is a data structure that, given $u,v\in V(G)$, reports $\min\{k,\kappa(u,v)\}$, where $\kappa(u,v)$ is the pairwise vertex connectivity between $u,v$. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov shows that a data structure of total size $\tilde{O}(kn)$ can even be encoded as a $\tilde{O}(k)$-bit labeling scheme so that vertex-connectivity queries can be answered in $\tilde{O}(k)$ time. The construction time is polynomial, but unspecified. In this paper we address the top three complexity measures: Space, Query Time, and Construction Time. We give an $\Omega(kn)$-bit lower bound on any vertex connectivity oracle. We construct an optimal-space connectivity oracle in max-flow time that answers queries in $O(\log n)$ time, independent of $k$.
In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts. As a consequence of these investigations, we exhibit a simple $O(\kappa n)$-space data structure that can quickly answer pairwise $(\kappa+1)$-connectivity queries in a $\kappa$-connected graph. We also show how to compute the "closest" $\kappa$-cut to every vertex in near linear $\tilde{O}(m+poly(\kappa)n)$ time.
Cardinality estimation is perhaps the simplest non-trivial statistical problem that can be solved via sketching. Industrially-deployed sketches like HyperLogLog, MinHash, and PCSA are mergeable, which means that large data sets can be sketched in a distributed environment, and then merged into a single sketch of the whole data set. In the last decade a variety of sketches have been developed that are non-mergeable, but attractive for other reasons. They are simpler, their cardinality estimates are strictly unbiased, and they have substantially lower variance. We evaluate sketching schemes on a reasonably level playing field, in terms of their memory-variance product (MVP). E.g., a sketch that occupies $5m$ bits and whose relative variance is $2/m$ (standard error $\sqrt{2/m}$) has an MVP of $10$. Our contributions are as follows. Cohen and Ting independently discovered what we call the Martingale transform for converting a mergeable sketch into a non-mergeable sketch. We present a simpler way to analyze the limiting MVP of Martingale-type sketches. We prove that the \Martingale{} transform is optimal in the non-mergeable world, and that \Martingale{} \fishmonger{} in particular is optimal among linearizable sketches, with an MVP of $H_0/2 \approx 1.63$. E.g., this is circumstantial evidence that to achieve 1\% standard error, we cannot do better than a 2 kilobyte sketch. \Martingale{} \fishmonger{} is neither simple nor practical. We develop a new mergeable sketch called \Curtain{} that strikes a nice balance between simplicity and efficiency, and prove that \Martingale{} \Curtain{} has limiting $\MVP\approx 2.31$. It can be updated with $O(1)$ memory accesses and it has lower empirical variance than \Martingale{} \LogLog, a practical non-mergeable version of HyperLogLog.