arXiv subjects

Yinzhan Xu

Publications and source records attributed to Yinzhan Xu.

At least 19 recordsLinked to original sources

arXiv2026

More Asymmetry Yields Faster Matrix Multiplication

We present a new improvement on the laser method for designing fast matrix multiplication algorithms. The new method further develops the recent advances by [Duan, Wu, Zhou FOCS 2023] and [Vassilevska Williams, Xu, Xu, Zhou SODA 2024]. Surprisingly the new improvement is achieved by incorporating more asymmetry in the analysis, circumventing a fundamental tool of prior work that requires two of the three dimensions to be treated identically. Together with recent advances in numerical optimization [Dupont et al. arXiv:2608.16884], the method yields a new bound on the square matrix multiplication exponent $ω<2.371177$, improved from the previous bound of $ω<2.371552$. We also improve the bounds of the exponents for multiplying rectangular matrices of various shapes.

cs.DS↗

arXiv2026

Almost Optimal Multiple Source Shortest Paths and Reachability

Given a graph, computing distances and reachabilities from a small set of vertices to the whole graph is an important primitive both in theory and in practice. In undirected unweighted graphs, while computing single-source shortest path (SSSP) requires $O(n^2)$ time in dense graphs, all-pairs shortest paths (APSP) can be computed in $\hat{O}(n^ω) = O(n^{2.372})$ time [Seidel '95] providing significant savings over running $n$ SSSP instances separately. However, if one needs to compute multiple-source shortest paths (MSSP) from a set of $n^σ$ vertices, the previously best known running time was $\hat{O}(\min\{n^ω, n^{2 + σ}\})$: either compute APSP or run SSSP from each source. On the other hand, MSSP is only as hard as computing Boolean matrix product (BMM) between an $n^σ\times n$ matrix and $n \times n$ matrix, leaving a significant gap. Our first main result is an almost optimal algorithm for MSSP on undirected unweighted graphs running in $\hat{O}(n^{ω(σ, 1, 1)})$ time, which gives a smooth interpolation between the SSSP and APSP algorithms. The main technical tool behind our result is a novel graph decomposition, which may be of independent interest. Next, we study the multiple-source reachability problem, where we need to determine whether a given set of $n^σ$ vertices can reach each of the vertices in a given directed graph. Multiple-source reachability can also be solved in $\hat{O}(\min\{n^ω, n^{2 + σ}\})$ time, with the same lower bound from rectangular BMM. We give an optimal algorithm that runs in $\hat{O}(n^{ω(σ, 1, 1)})$ time, again matching the running time for BMM. Our algorithm for multiple-source reachability can be generalized to MSSP on DAGs. As an application, we provide an $O(n^{2.084})$ time algorithm for computing an $\widetilde{O}(n)$-size shortcut set that reduces diameter to $O(n^{1/3})$.

cs.DS↗

arXiv2026

Furthest Pair Requires Quadratic Time in Superconstant Dimension under SETH

Several fundamental problems in computational geometry admit algorithms with running time $f(d) \cdot n^{2-Θ(1/d)}$ for $n$ points in $d$ dimensions, making them among the most prominent examples of barely subquadratic computation. Notable members of this class include Furthest Pair, Bichromatic Closest Pair, (Bichromatic) Maximum Innter Product, and Hopcroft's Problem. Chen [Theory Comput. 2020] proved that, assuming the Strong Exponential Time Hypothesis (SETH), these problems require $n^{2-o(1)}$ time when the dimension satisfies $d=2^{Θ(\log^* n)}$. We extend this lower bound to all efficiently constructible dimensions $d=ω(1)$. Thus, assuming SETH, the dependence of the best known algorithms on the dimension is essentially unavoidable. The proof utilizes techniques in OpenAI's recent disproof of the Erdos unit distance conjecture. The proof was initially discovered by ChatGPT 5.5 Pro. The authors have validated and substantially edited the proof to improve the presentation.

cs.CG↗

arXiv2026

A Unified Lower Bound on the Noisy Query Complexity of Boolean Functions

We study the query complexity of Boolean functions $f: \{0, 1\}^n \rightarrow \{0, 1\}$ in the noisy query model introduced by Feige, Raghavan, Peleg and Upfal [SICOMP 1994]. In this model, an algorithm can adaptively query the bits of an input vector, but each query result is independently flipped with constant probability $p \in (0, 1/2)$; repeated queries are allowed. The noisy query complexity $\mathsf{N}_p(f)$ of a function $f$ is defined as the minimum expected number of queries needed to compute $f(x)$ with error probability at most $1/3$, for the worst case input $x$. We prove a general lower bound on $\mathsf{N}_p(f)$ based on degree statistics of certain subgraphs of the Boolean hypercube. This is the first general lower bound beyond those implied by the simple observation that $\mathsf{N}_p(f)$ is lower bounded by the randomized query complexity. We show that this recovers (up to a constant factor) most previously known lower bounds on the noisy query complexity of Boolean functions, providing a unified framework for understanding these results and simplifying the proofs in several cases. Furthermore, this resolves in the affirmative an open problem of Gu, Li and Xu [COLT 2025] that $\mathsf{N}_p(f) = Ω(\mathsf{I}(f) \log \mathsf{I}(f))$, where $\mathsf{I}(f)$ denotes the total influence of $f$. We also apply our general lower bound to obtain tight bounds on the noisy query complexity for several new functions.

cs.DS↗

arXiv2026

Deterministic Monotone Min-Plus Product and Convolution

The Monotone Min-Plus Product problem is a useful primitive that has seen many algorithmic applications over the past decade. In this problem, we are given two $n\times n$ integer matrices $A$ and $B$, where each row of $B$ is a monotone non-decreasing sequence of integers from $\{1,\dots,n\}$, and the goal is to compute their Min-Plus product, defined as the $n\times n$ matrix $C$ with $C_{i,j} = \min_{k}\{A_{i,k} + B_{k,j}\}$. The fastest known algorithm for this task [Chi, Duan, Xie, and Zhang, STOC'22] runs in $n^{(ω+3)/2+o(1)} = O(n^{2.686})$ time, significantly improving over the brute-force cubic algorithm. However, its main disadvantage is that it requires randomization, which is then inherited by all downstream applications. Our main result is a deterministic algorithm for Monotone Min-Plus product with the same time complexity $n^{(ω+3)/2+o(1)} = O(n^{2.686})$ as its randomized counterpart, improving upon the previous deterministic bound $O(n^{2.875})$ [Gu, Polak, Vassilevska Williams, and Xu, ICALP'21]. Our derandomization also applies to previously studied extensions and variants (e.g., [Dürr, IPL'23]), including rectangular matrices, bounded range $[n^μ]$, and column-monotone matrices. As an immediate consequence, we derandomize state-of-the-art algorithms for multiple problems, including Language Edit Distance, RNA Folding, Optimum Stack Generation, unweighted Tree Edit Distance, Batched Range Mode, and Approximate All-Pairs Shortest Paths. Our techniques also yield a deterministic algorithm for the Monotone Min-Plus Convolution problem that runs in $n^{1.5+o(1)}$ time, nearly matching the best known randomized time complexity $\widetilde{O}(n^{1.5})$ [Chi, Duan, Xie, and Zhang, STOC'22]. This algorithm can be used to derandomize state-of-the-art algorithms for Jumbled Indexing for binary strings and several variants of Knapsack.

cs.DS↗

arXiv2026

On the Computational Hardness of Transformers

The transformer has revolutionized modern AI across language, vision, and beyond. It consists of $L$ layers, each running $H$ attention heads in parallel and feeding the combined output to the subsequent layer. In attention, the input consists of $N$ tokens, each a vector of dimension $m$. The attention mechanism involves multiplying three $N \times m$ matrices, applying softmax to an intermediate product. Several recent works have advanced our understanding of the complexity of attention. Known algorithms for transformers compute each attention head independently. This raises a fundamental question that has recurred throughout TCS under the guise of ``direct sum'' problems: can multiple instances of the same problem be solved more efficiently than solving each instance separately? Many answers to this question, both positive and negative, have arisen in fields spanning communication complexity and algorithm design. Thus, we ask whether transformers can be computed more efficiently than $LH$ independent evaluations of attention. In this paper, we resolve this question in the negative, and give the first non-trivial computational lower bounds for multi-head multi-layer transformers. In the small embedding regime ($m = N^{o(1)}$), computing $LH$ attention heads separately takes $LHN^{2 + o(1)}$ time. We establish that this is essentially optimal under SETH. In the large embedding regime ($m = N$), one can compute $LH$ attention heads separately using $LHN^{ω+ o(1)}$ arithmetic operations (plus exponents), where $ω$ is the matrix multiplication exponent. We establish that this is optimal, by showing that $LHN^{ω- o(1)}$ arithmetic operations are necessary when $ω> 2$. Our lower bound in the large embedding regime relies on a novel application of the Baur-Strassen theorem, a powerful algorithmic tool underpinning the famous backpropagation algorithm.

cs.CC↗

arXiv2025

Breaking the $n^{1.5}$ Additive Error Barrier for Private and Efficient Graph Sparsification via Private Expander Decomposition

We study differentially private algorithms for graph cut sparsification, a fundamental problem in algorithms, privacy, and machine learning. While significant progress has been made, the best-known private and efficient cut sparsifiers on $n$-node graphs approximate each cut within $\widetilde{O}(n^{1.5})$ additive error and $1+γ$ multiplicative error for any $γ> 0$ [Gupta, Roth, Ullman TCC'12]. In contrast, "inefficient" algorithms, i.e., those requiring exponential time, can achieve an $\widetilde{O}(n)$ additive error and $1+γ$ multiplicative error [Eli{á}{š}, Kapralov, Kulkarni, Lee SODA'20]. In this work, we break the $n^{1.5}$ additive error barrier for private and efficient cut sparsification. We present an $(\varepsilon,δ)$-DP polynomial time algorithm that, given a non-negative weighted graph, outputs a private synthetic graph approximating all cuts with multiplicative error $1+γ$ and additive error $n^{1.25 + o(1)}$ (ignoring dependencies on $\varepsilon, δ, γ$). At the heart of our approach lies a private algorithm for expander decomposition, a popular and powerful technique in (non-private) graph algorithms.

cs.DS↗

arXiv2025

Optimal Graph Reconstruction by Counting Connected Components in Induced Subgraphs

The graph reconstruction problem has been extensively studied under various query models. In this paper, we propose a new query model regarding the number of connected components, which is one of the most basic and fundamental graph parameters. Formally, we consider the problem of reconstructing an $n$-node $m$-edge graph with oracle queries of the following form: provided with a subset of vertices, the oracle returns the number of connected components in the induced subgraph. We show $Θ(\frac{m \log n}{\log m})$ queries in expectation are both sufficient and necessary to adaptively reconstruct the graph. In contrast, we show that $Ω(n^2)$ non-adaptive queries are required, even when $m = O(n)$. We also provide an $O(m\log n + n\log^2 n)$ query algorithm using only two rounds of adaptivity.

cs.DS↗

arXiv2025

Subquadratic Algorithms and Hardness for Attention with Any Temperature

Despite the popularity of the Transformer architecture, the standard algorithm for computing Attention suffers from quadratic time complexity in context length $n$. Alman and Song [NeurIPS 2023] showed that when the head dimension $d = Θ(\log n)$, subquadratic Attention is possible if and only if the inputs have small entries bounded by $B = o(\sqrt{\log n})$ in absolute values, under the Strong Exponential Time Hypothesis ($\mathsf{SETH}$). Equivalently, subquadratic Attention is possible if and only if the softmax is applied with high temperature for $d=Θ(\log n)$. Running times of these algorithms depend exponentially on $B$ and thus they do not lead to even a polynomial-time algorithm outside the specific range of $B$. This naturally leads to the question: when can Attention be computed efficiently without strong assumptions on temperature? Are there fast attention algorithms that scale polylogarithmically with entry size $B$? In this work, we resolve this question and characterize when fast Attention for arbitrary temperatures is possible. First, for all constant $d = O(1)$, we give the first subquadratic $\tilde{O}(n^{2 - 1/d} \cdot \mathrm{polylog}(B))$ time algorithm for Attention with large $B$. Our result holds even for matrices with large head dimension if they have low rank. In this regime, we also give a similar running time for Attention gradient computation, and therefore for the full LLM training process. Furthermore, we show that any substantial improvement on our algorithm is unlikely. In particular, we show that even when $d = 2^{Θ(\log^* n)}$, Attention requires $n^{2 - o(1)}$ time under $\mathsf{SETH}$. Finally, in the regime where $d = \mathrm{poly}(n)$, we show that the standard algorithm is optimal under popular fine-grained complexity assumptions.

cs.LG↗

arXiv2025

New Separations and Reductions for Directed Preservers and Hopsets

We study distance preservers, hopsets, and shortcut sets in $n$-node, $m$-edge directed graphs, and show improved bounds and new reductions for various settings of these problems. Our first set of results is about exact and approximate distance preservers. We give the following bounds on the size of directed distance preservers with $p$ demand pairs: 1) $\tilde{O}(n^{5/6}p^{2/3} + n)$ edges for exact distance preservers in unweighted graphs; and 2) $Ω(n^{2/3}p^{2/3})$ edges for approximate distance preservers with any given finite stretch, in graphs with arbitrary aspect ratio. Additionally, we establish a new directed-to-undirected reduction for exact distance preservers. We show that if undirected distance preservers have size $O(n^λp^μ + n)$ for constants $λ, μ> 0$, then directed distance preservers have size $O\left( n^{\frac{1}{2-λ}}p^{\frac{1+μ-λ}{2-λ}} + n^{1/2}p + n\right).$ As a consequence of the reduction, if current upper bounds for undirected preservers can be improved for some $p \leq n$, then so can current upper bounds for directed preservers. Our second set of results is about directed hopsets and shortcut sets. For hopsets in directed graphs, we prove that the hopbound is: 1) $Ω(n^{2/9})$ for $O(m)$-size shortcut sets, improving the previous $Ω(n^{1/5})$ bound [Vassilevska Williams, Xu and Xu, SODA 2024]; 2) $Ω(n^{2/7})$ for $O(m)$-size exact hopsets in unweighted graphs, improving the previous $Ω(n^{1/4})$ bound [Bodwin and Hoppenworth, FOCS 2023]; and 3) $Ω(n^{1/2})$ for $O(n)$-size approximate hopsets with any given finite stretch, in graphs with arbitrary aspect ratio. This result establishes a separation between this setting and $O(n)$-size approximate hopsets for graphs with polynomial aspect ratio, which have hopbound $\widetilde{O}(n^{1/3})$ [Bernstein and Wein, SODA 2023].

cs.DS↗

arXiv2025

Faster Weighted and Unweighted Tree Edit Distance and APSP Equivalence

The tree edit distance (TED) between two rooted ordered trees with $n$ nodes labeled from an alphabet $Σ$ is the minimum cost of transforming one tree into the other by a sequence of valid operations consisting of insertions, deletions and relabeling of nodes. The tree edit distance is a well-known generalization of string edit distance and has been studied since the 1970s. Years of steady improvements have led to an $O(n^3)$ algorithm [DMRW 2010]. Fine-grained complexity casts light onto the hardness of TED showing that a truly subcubic time algorithm for TED implies a truly subcubic time algorithm for All-Pairs Shortest Paths (APSP) [BGMW 2020]. Therefore, under the popular APSP hypothesis, a truly subcubic time algorithm for TED cannot exist. However, unlike many problems in fine-grained complexity for which conditional hardness based on APSP also comes with equivalence to APSP, whether TED can be reduced to APSP has remained unknown. In this paper, we resolve this. Not only we show that TED is fine-grained equivalent to APSP, our reduction is tight enough, so that combined with the fastest APSP algorithm to-date [Williams 2018] it gives the first ever subcubic time algorithm for TED running in $n^3/2^{Ω(\sqrt{\log{n}})}$ time. We also consider the unweighted tree edit distance problem in which the cost of each edit is one. For unweighted TED, a truly subcubic algorithm is known due to Mao [Mao 2022], later improved slightly by Dürr [Dürr 2023] to run in $O(n^{2.9148})$. Their algorithm uses bounded monotone min-plus product as a crucial subroutine, and the best running time for this product is $\tilde{O}(n^{\frac{3+ω}{2}})\leq O(n^{2.6857})$ (where $ω$ is the exponent of fast matrix multiplication). In this work, we close this gap and give an algorithm for unweighted TED that runs in $\tilde{O}(n^{\frac{3+ω}{2}})$ time.

cs.DS↗

arXiv2025

Tight Bounds for Noisy Computation of High-Influence Functions, Connectivity, and Threshold

In the noisy query model, the (binary) return value of every query (possibly repeated) is independently flipped with some fixed probability $p \in (0, 1/2)$. In this paper, we obtain tight bounds on the noisy query complexity of several fundamental problems. Our first contribution is to show that any Boolean function with total influence $Ω(n)$ has noisy query complexity $Θ(n\log n)$. Previous works often focus on specific problems, and it is of great interest to have a characterization of noisy query complexity for general functions. Our result is the first noisy query complexity lower bound of this generality, beyond what was known for random Boolean functions [Reischuk and Schmeltz, FOCS 1991]. Our second contribution is to prove that Graph Connectivity has noisy query complexity $Θ(n^2 \log n)$. In this problem, the goal is to determine whether an undirected graph is connected using noisy edge queries. While the upper bound can be achieved by a simple algorithm, no non-trivial lower bounds were known prior to this work. Last but not least, we determine the exact number of noisy queries (up to lower order terms) needed to solve the $k$-Threshold problem and the Counting problem. The $k$-Threshold problem asks to decide whether there are at least $k$ ones among $n$ bits, given noisy query access to the bits. We prove that $(1\pm o(1)) \frac{n\log (\min\{k,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}$ queries are both sufficient and necessary to achieve error probability $δ= o(1)$. Previously, such a result was only known when $\min\{k,n-k+1\}=o(n)$ [Wang, Ghaddar, Zhu and Wang, arXiv 2024]. We also show a similar $(1\pm o(1)) \frac{n\log (\min\{k+1,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}$ bound for the Counting problem, where one needs to count the number of ones among $n$ bits given noisy query access and $k$ denotes the answer.

cs.DS↗

arXiv2024

Faster Algorithms for Text-to-Pattern Hamming Distances

We study the classic Text-to-Pattern Hamming Distances problem: given a pattern $P$ of length $m$ and a text $T$ of length $n$, both over a polynomial-size alphabet, compute the Hamming distance between $P$ and $T[i\, .\, . \, i+m-1]$ for every shift $i$, under the standard Word-RAM model with $Θ(\log n)$-bit words. - We provide an $O(n\sqrt{m})$ time Las Vegas randomized algorithm for this problem, beating the decades-old $O(n \sqrt{m \log m})$ running time [Abrahamson, SICOMP 1987]. We also obtain a deterministic algorithm, with a slightly higher $O(n\sqrt{m}(\log m\log\log m)^{1/4})$ running time. Our randomized algorithm extends to the $k$-bounded setting, with running time $O\big(n+\frac{nk}{\sqrt{m}}\big)$, removing all the extra logarithmic factors from earlier algorithms [Gawrychowski and Uznański, ICALP 2018; Chan, Golan, Kociumaka, Kopelowitz and Porat, STOC 2020]. - For the $(1+ε)$-approximate version of Text-to-Pattern Hamming Distances, we give an $\tilde{O}(ε^{-0.93}n)$ time Monte Carlo randomized algorithm, beating the previous $\tilde{O}(ε^{-1}n)$ running time [Kopelowitz and Porat, FOCS 2015; Kopelowitz and Porat, SOSA 2018]. Our approximation algorithm exploits a connection with $3$SUM, and uses a combination of Fredman's trick, equality matrix product, and random sampling; in particular, we obtain new results on approximate counting versions of $3$SUM and Exact Triangle, which may be of independent interest. Our exact algorithms use a novel combination of hashing, bit-packed FFT, and recursion; in particular, we obtain a faster algorithm for computing the sumset of two integer sets, in the regime when the universe size is close to quadratic in the number of elements. We also prove a fine-grained equivalence between the exact Text-to-Pattern Hamming Distances problem and a range-restricted, counting version of $3$SUM.

cs.DS↗

arXiv2024

All-Hops Shortest Paths

Let $G=(V,E,w)$ be a weighted directed graph without negative cycles. For two vertices $s,t\in V$, we let $d_{\le h}(s,t)$ be the minimum, according to the weight function $w$, of a path from $s$ to $t$ that uses at most $h$ edges, or hops. We consider algorithms for computing $d_{\le h}(s,t)$ for every $1\le h\le n$, where $n=|V|$, in various settings. We consider the single-pair, single-source and all-pairs versions of the problem. We also consider a distance oracle version of the problem in which we are not required to explicitly compute all distances $d_{\le h}(s,t)$, but rather return each one of these distances upon request. We consider both the case in which the edge weights are arbitrary, and in which they are small integers in the range $\{-M,\ldots,M\}$. For some of our results we obtain matching conditional lower bounds.

cs.DS↗

arXiv2024

New Applications of 3SUM-Counting in Fine-Grained Complexity and Pattern Matching

The 3SUM problem is one of the cornerstones of fine-grained complexity. Its study has led to countless lower bounds, but as has been sporadically observed before -- and as we will demonstrate again -- insights on 3SUM can also lead to algorithmic applications. The starting point of our work is that we spend a lot of technical effort to develop new algorithms for 3SUM-type problems such as approximate 3SUM-counting, small-doubling 3SUM-counting, and a deterministic subquadratic-time algorithm for the celebrated Balog-Szemerédi-Gowers theorem from additive combinatorics. As consequences of these tools, we derive diverse new results in fine-grained complexity and pattern matching algorithms, answering open questions from many unrelated research areas. Specifically: - A recent line of research on the "short cycle removal" technique culminated in tight 3SUM-based lower bounds for various graph problems via randomized fine-grained reductions [Abboud, Bringmann, Fischer; STOC '23] [Jin, Xu; STOC '23]. In this paper we derandomize the reduction to the important 4-Cycle Listing problem. - We establish that \#3SUM and 3SUM are fine-grained equivalent under deterministic reductions. - We give a deterministic algorithm for the $(1+ε)$-approximate Text-to-Pattern Hamming Distances problem in time $n^{1+o(1)} \cdot ε^{-1}$. - In the $k$-Mismatch Constellation problem the input consists of two integer sets $A, B \subseteq [N]$, and the goal is to test whether there is a shift $c$ such that $|(c + B) \setminus A| \leq k$ (i.e., whether $B$ shifted by $c$ matches $A$ up to $k$ mismatches). For moderately small $k$ the previously best running time was $\tilde O(|A| \cdot k)$ [Cardoze, Schulman; FOCS '98] [Fischer; SODA '24]. We give a faster $|A| \cdot k^{2/3} \cdot N^{o(1)}$-time algorithm in the regime where $|B| = Θ(|A|)$.

cs.DS↗

arXiv2024

Differentially Private Gomory-Hu Trees

Given an undirected, weighted $n$-vertex graph $G = (V, E, w)$, a Gomory-Hu tree $T$ is a weighted tree on $V$ such that for any pair of distinct vertices $s, t \in V$, the Min-$s$-$t$-Cut on $T$ is also a Min-$s$-$t$-Cut on $G$. Computing a Gomory-Hu tree is a well-studied problem in graph algorithms and has received considerable attention. In particular, a long line of work recently culminated in constructing a Gomory-Hu tree in almost linear time [Abboud, Li, Panigrahi and Saranurak, FOCS 2023]. We design a differentially private (DP) algorithm that computes an approximate Gomory-Hu tree. Our algorithm is $\varepsilon$-DP, runs in polynomial time, and can be used to compute $s$-$t$ cuts that are $\tilde{O}(n/\varepsilon)$-additive approximations of the Min-$s$-$t$-Cuts in $G$ for all distinct $s, t \in V$ with high probability. Our error bound is essentially optimal, as [Dalirrooyfard, Mitrović and Nevmyvaka, NeurIPS 2023] showed that privately outputting a single Min-$s$-$t$-Cut requires $Ω(n)$ additive error even with $(1, 0.1)$-DP and allowing for a multiplicative error term. Prior to our work, the best additive error bounds for approximate all-pairs Min-$s$-$t$-Cuts were $O(n^{3/2}/\varepsilon)$ for $\varepsilon$-DP [Gupta, Roth and Ullman, TCC 2012] and $O(\sqrt{mn} \cdot \text{polylog}(n/δ) / \varepsilon)$ for $(\varepsilon, δ)$-DP [Liu, Upadhyay and Zou, SODA 2024], both of which are implied by differential private algorithms that preserve all cuts in the graph. An important technical ingredient of our main result is an $\varepsilon$-DP algorithm for computing minimum Isolating Cuts with $\tilde{O}(n / \varepsilon)$ additive error, which may be of independent interest.

cs.DS↗

arXiv2024

Fine-Grained Optimality of Partially Dynamic Shortest Paths and More

Single Source Shortest Paths ($\textrm{SSSP}$) is among the most well-studied problems in computer science. In the incremental (resp. decremental) setting, the goal is to maintain distances from a fixed source in a graph undergoing edge insertions (resp. deletions). A long line of research culminated in a near-optimal deterministic $(1 + \varepsilon)$-approximate data structure with $m^{1 + o(1)}$ total update time over all $m$ updates by Bernstein, Probst Gutenberg and Saranurak [FOCS 2021]. However, there has been remarkably little progress on the exact $\textrm{SSSP}$ problem beyond Even and Shiloach's algorithm [J. ACM 1981] for unweighted graphs. For weighted graphs, there are no exact algorithms beyond recomputing $\textrm{SSSP}$ from scratch in $\widetilde{O}(m^2)$ total update time, even for the simpler Single-Source Single-Target Shortest Path problem ($\textrm{stSP}$). Despite this lack of progress, known (conditional) lower bounds only rule out algorithms with amortized update time better than $m^{1/2 - o(1)}$ in dense graphs. In this paper, we give a tight (conditional) lower bound: any partially dynamic exact $\textrm{stSP}$ algorithm requires $m^{2 - o(1)}$ total update time for any sparsity $m$. We thus resolve the complexity of partially dynamic shortest paths, and separate the hardness of exact and approximate shortest paths, giving evidence as to why no non-trivial exact algorithms have been obtained while fast approximation algorithms are known. Moreover, we give tight bounds on the complexity of combinatorial algorithms for several path problems that have been studied in the static setting since early sixties: Node-weighted shortest paths (studied alongside edge-weighted shortest paths), bottleneck paths (early work dates back to 1960), and earliest arrivals (early work dates back to 1958).

cs.DS↗

arXiv2024

Shaving Logs via Large Sieve Inequality: Faster Algorithms for Sparse Convolution and More

In sparse convolution-type problems, a common technique is to hash the input integers modulo a random prime $p\in [Q/2,Q]$ for some parameter $Q$, which reduces the range of the input integers while preserving their additive structure. However, this hash family suffers from two drawbacks, which led to bottlenecks in many state-of-the-art algorithms: (1) The collision probability of two elements from $[N]$ is $O(\frac{\log N}{Q})$ rather than $O(\frac{1}{Q})$; (2) It is difficult to derandomize the choice of $p$; known derandomization techniques lead to super-logarithmic overhead [Chan, Lewenstein STOC'15]. In this paper, we partially overcome these drawbacks in certain scenarios, via novel applications of the large sieve inequality from analytic number theory. Consequently, we obtain the following improved algorithms for various problems (in the standard word RAM model): Sparse Nonnegative Convolution: We obtain an $O(t\log t)$-time Las Vegas algorithm that computes the convolution $A\star B$ of two nonnegative integer vectors $A,B$, where $t$ is the output sparsity $\|A\star B\|_0$. Moreover, our algorithm terminates in $O(t\log t)$ time with $1-1/\mathrm{poly}(t)$ probability. Text-to-Pattern Hamming Distances: Given a length-$m$ pattern $P$ and a length-$n$ text $T$, we obtain a deterministic $O(n\sqrt{m\log \log m})$-time algorithm that exactly computes the Hamming distance between $P$ and every length-$m$ substring of $T$. Sparse General Convolution: We also give a Monte Carlo $O(t\log t)$ time algorithm for sparse convolution with possibly negative input in the restricted case where the length $N$ of the input vectors satisfies $N\le t^{1.99}$.

cs.DS↗