This paper presents a new deterministic algorithm for the single-source shortest paths (SSSP) problem on real non-negative edge-weighted directed graphs, with running time O(m√{log n} + √{mnlog nlog log n}), which is O(m√{log nlog 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].

Abstract. A [Formula: see text]- vertex connectivity oracle for an undirected graph [Formula: see text] is a data structure that, given [Formula: see text], reports [Formula: see text], where [Formula: see text] is the pairwise vertex connectivity between [Formula: see text]. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov [ Inform. Process. Lett., 112 (2012), pp. 39–43] produced a data structure of [Formula: see text] words, which can even be encoded as a [Formula: see text]-bit labeling scheme, that can answer vertex connectivity queries in [Formula: see text] time. The construction time is polynomial but unspecified. In this paper, we address the top three complexity measures. (1) Space: We prove that any [Formula: see text]-vertex connectivity oracle requires [Formula: see text] bits of space for any [Formula: see text]. This proves that the Iszak–Nutov data structure is optimal up to polylogarithmic factors for every [Formula: see text] and that the sparsifiers of Nagamochi and Ibaraki [ Algorithmica, 7 (1992), pp. 583–596] are optimal compression schemes for [Formula: see text]-vertex connectivity up to a logarithmic factor. In particular, whereas all edge connectivities can be efficiently compressed (as a weighted [Formula: see text]-edge Gomory–Hu tree), vertex connectivity admits no asymptotic compression: [Formula: see text] bits are necessary. We design a variation on Izsak and Nutov’s data structure that uses [Formula: see text] words of space. (2) Query time: We answer queries in [Formula: see text] time, improving on the [Formula: see text] time bound of Izsak and Nutov [ Inform. Process. Lett., 112 (2012), pp. 39–43]. The main idea is to build instances of [Formula: see text] data structures, with additional structure based on affine planes. This structure allows for query time that is linear in the output size, which evades some conditional lower bounds that are polynomial in the query set sizes [ 42 , 51 ]. (3) Construction time: Our data structure can be constructed in the time of [Formula: see text] max-flow computations, namely, [Formula: see text] time, using the recent near-linear time flow algorithm of [ 13 ]. The main technical contribution here is a fast algorithm to compute a [Formula: see text]-approximate Gomory–Hu tree for element connectivity in the time of [Formula: see text] max-flow computations. Element connectivity is a notion that generalizes edge and vertex connectivity.

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 nonnegative 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-hierarchybased 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∈ V(G), reports min{k,κ(u,v)}, where κ(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 [Inf. Process. Lett. 2012] shows that a data structure of total size O(knlogn), which can even be encoded as a O(klog3 n)-bit labeling scheme, can answer vertex-connectivity queries in O(klogn) time. The construction time is polynomial, but unspecified.

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.

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 $O(κn)$-space data structure that can quickly answer pairwise $(κ+1)$-connectivity queries in a $κ$-connected graph. We also show how to compute the "closest" $κ$-cut to every vertex in near linear $\tilde{O}(m+poly(κ)n)$ time.