arXiv subjects

Ariel Sapir

Publications and source records attributed to Ariel Sapir.

7 recordsLinked to original sources

arXiv2026

First Near-Additive All-Pairs Approximate Shortest Paths in Decremental Weighted Graphs

We present the first four All-Pairs Approximate Shortest Paths (APASP) algorithms for decremental weighted undirected graphs. We extend the hierarchy induced edge sets of Baswana and Kavitha [FOCS'06 and SICOMP'10] to the decremental setting. At any point in time, for any $u,v$, $W_{i}$ denotes the weight of the $i$-th heaviest edge on any shortest path between them in the current graph. For any $\varepsilon>0$, our first algorithm maintains $(1+\varepsilon,2W_{2})$-APASP in $\tilde{O}(n^{\frac{3}{2}+o(1)}m^{\frac{1}{2}})$ total update time, matching a single static computation of Cohen and Zwick [SODA'97 and JALG'01], up to a subpolynomial factor. It also improves the stretch of the $(2+\varepsilon)$-APASP algorithm of Dory, Forster, Nazari and de Vos [ICALP'24]. Our second algorithm maintains $(1+\varepsilon,2W_{1}+2W_{2})$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time, improving upon Chechik's $(3+\varepsilon)$-APASP algorithm [FOCS'18] for $m=Ω(n^{\frac{11}{6}})$. We also obtain two $(1+\varepsilon,2\cdot(k-1)W_{1})$-APASP algorithms with total update times $\tilde{O}(n^{2-\frac{1}{k}+o(1)}m^{\frac{1}{k}})$ and $\tilde{O}(n^{2+\frac{2}{3k-2}+o(1)})$, adapting the static $+2\sum_{i=1}^{k-1}W_{i}$-APASP algorithms of Cohen and Zwick [SODA'97 and JALG'01] and Roditty and Sapir [FSTTCS'25]. For unweighted graphs, we present $(1+\varepsilon,2)$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time and $(1+\varepsilon,2\cdot(k-1))$-APASP in $\tilde{O}(n^{2+\frac{1}{3k-4}+o(1)})$. Combined with the results of Dory, Forster, Nazari and de Vos [ICALP'24], these match the fastest combinatorial static bounds of Dor, Halperin and Zwick [FOCS'96 and SICOMP'00] up to subpolynomial factors.

cs.DS↗

arXiv2026

Smallest Multiplicative Stretch for an $α$-APASP Below $\widetilde O \left(n^ω\right)$ Runtime

Dor, Halperin and Zwick [FOCS'96; SICOMP'00] showed that $(2-\varepsilon)$-APASP is at least as hard as BMM for every fixed $0<\varepsilon<1$. Zwick [FOCS'98; JACM'02] gave a $2$-APASP algorithm in $\widetilde O(n^{ω+o(1)}\log W_{\max})$ runtime, where $W_{\max}$ is the maximum edge weight. Bringmann, K{ü}nnemann and W{ę}grzycki [STOC'19] removed this logarithmic factor. Using OpenAI's recent bound $ω\leq\frac{9}{4}$ gives $\widetilde O\left(n^{\frac{9}{4}+o(1)}\right)$ time. We present two weighted undirected $2$-APASP algorithms. The first uses $(\min,\max)$-products. Earlier rectangular FMM bounds of Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'24] and Alman, Duan, Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'25] give $\widetilde O(n^{2.36999465})$ runtime, below the previous $ω\leq2.371177$ of Dupont, Eisenberger, Kozlovskii, Mehrabian, Ruiz, See, Zhou, Alman, Vassilevska Williams and Balog [arXiv'26]. Current rectangular FMM bounds give $\widetilde O(n^{2.34549151})$, independent of $W_{\max}$. For integer weights, our second algorithm has $\widetilde O(n^{ω-\frac{(ω-2)^2}{ω-1}+o(1)}\log^{O(1)}W_{\max})$ runtime, with exponent below $ω$ whenever $ω>2$. The rectangular bounds of Uzna{ń}ski [arXiv'26] or of Amanov, van den Berg and Zuiddam [arXiv'26] give $\widetilde O(n^{2.17157288}\log^{O(1)}W_{\max})$.

cs.DS↗

arXiv2026

Weighted Emulators with Local Heaviest Edges Stretch for Undirected Graphs

We introduce a generalized family of $\left( 2\cdot \left\lfloor \frac{k}{2} \right\rfloor-1, 2\cdot \left\lceil \frac{k}{2} \right\rceil \cdot W_{1} +\max\left\{0,2\cdot\left(\left\lceil\frac{k}{2}\right\rceil-2\right)\right\}\cdot W_{2} \right)$-emulators with $\tilde O \left(n^{1+\frac{1}{k}}\right)$ edges, for any $k\in\mathbb{N}$, where $W_{i}$ is the $i$th heaviest edge on a shortest path between two vertices. Our construction generalizes the $+2W_{1}$-spanner of size $\tilde O\left(n^{\frac{3}{2}}\right)$ and the $+4W_{1}$-emulator of size $\tilde O \left(n^{\frac{4}{3}}\right)$, both by Elkin, Gitlitz and Neiman [DISC'21 and DICO'23]. When $k$ is even, these are $\left(k-1,k\cdot W_{1} + \left(k-4\right)\cdot W_{2}\right)$-emulators and when $k$ is odd, these are $\left(k-2,\left(k+1\right)\cdot W_{1} + \left(k-3\right) \cdot W_{2}\right)$-emulators. Our framework not only expands known constructions for weighted graphs but also yields an improved stretch over state of the art emulators and spanners for unweighted graphs within a specific distance regime. In particular, for all vertex pairs separated by a distance of $δ\leq O\left(3^{k^{2}}\right)$, our construction improves upon the seminal additive $+\tilde O\left(δ^{1-\frac{1}{k}}\right)$-emulator of size $\tilde O\left(n^{1+\frac{1}{2^{k+1}-1}}\right)$ by Thorup and Zwick [SODA'06].

cs.DS↗

arXiv2026

Additive, Near-Additive, and Multiplicative Approximations for APSP in Weighted Undirected Graphs: Trade-offs and Algorithms

We present a $+2\sum_{i=1}^{k+1}{W_i}$-APASP algorithm for dense weighted graphs with runtime $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$, where $W_{i}$ is the weight of an $i^{th}$ heaviest edge on a shortest path. Dor, Halperin and Zwick [FOCS'96, SICOMP'00] had two algorithms for the commensurate unweighted $+2\cdot\left( k+1\right)$-APASP: $\tilde O\left(n^{2-\frac{1}{k+2}}m^{\frac{1}{k+2}}\right)$ runtime for sparse graphs and $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$ runtime for dense graphs. Cohen and Zwick [SODA'97, JALG'01] adapted the sparse variant to weighted graphs: $+2\sum_{i=1}^{k+1}{W_i}$-APASP algorithm in the same runtime. We show an algorithm for dense weighted graphs. For nearly additive APASP, we present a $\left(1+\varepsilon,\min{\left\{2W_1,4W_{2}\right\}}\right)$-APASP algorithm with $\tilde O\left(\left(\frac{1}{\varepsilon}\right)^{O\left(1\right)}\cdot n^{2.15135313}\cdot\log W\right)$ runtime. This improves the $\left(1+\varepsilon,2W_1\right)$-APASP of Saha and Ye [SODA'24]. For multiplicative APASP, we show a framework of $\left(\frac{3\ell +4}{\ell + 2}+\varepsilon\right)$-APASP algorithms, reducing the runtime of Akav and Roditty [ESA'21] for dense graphs and generalizing the $\left(2+\varepsilon\right)$-APASP algorithm of Dory et al [SODA'24]. Our base case is a $\left(\frac{7}{3}+\varepsilon\right)$-APASP in $\tilde O\left(\left(\frac{1}{\varepsilon}\right)^{O\left(1\right)}\cdot n^{2.15135313}\cdot \log W\right)$ runtime, improving the $\frac{7}{3}$-APASP algorithm of Baswana and Kavitha [FOCS'06, SICOMP'10] for dense graphs. Finally, we "bypass" an $\tilde Ω\left(n^ω\right)$ conditional lower bound by Dor, Halperin, and Zwick for $α$-APASP with $α< 2$, by allowing an additive term (e.g. $\left(\frac{6k+3}{3k+2},\sum_{i=1}^{k+1}W_{i}\right)$-APASP in $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$ runtime).

cs.DS↗

arXiv2026

Dependency Preservation May Prevent Stopping at $2NF$

Traditionally, it was accepted that a relational database can be normalized step-by-step, from a set of un-normalized tables to tables in $1NF$, then to $2NF$, then to $3NF$, then (possibly) to $BCNF$. In particular, the rule applied to a table in $1NF$ in order to transform it to a set of tables in $2NF$ seems to be too straightforward to pose any difficulty. While it is a common belief to consider, theoretically, a database to be 'better' the higher it is normalized, practical usage advocates that this may not always be true -- in some cases a database performance may increase if left in a lower normal form, without sacrificing any of the advantages of a higher normal form. It was taken for granted that a normalization process can be stopped after reaching any normal form, without proceeding to the next higher one. However, we show that, depending on the set of functional dependencies, it may be impossible to be `precisely' (in a manner to be explained in the sequel) in $2NF$. One must, in these cases, either perform the normalization from $1NF$ to $3NF$ as an indecomposable move, or settle for a normalization between $2NF$ and $3NF$. For a clear presentation and a concise characterization of the phenomena, we model the functional dependencies as a (particular type of) directed hyper-graph -- an FD-graph. The minimal setup to exhibit the phenomena requires a single composite key, and two partially overlapping chains of transitive dependencies. This can be sketched as a specific sub-hyper-graph. Thus, an FD-graph containing that sub-hyper-graph indicates that its corresponding database cannot be precisely in $2NF$.

cs.DB↗

arXiv2021

A tight negative example for MMS fair allocations

We consider the problem of allocating indivisible goods to agents with additive valuation functions. Kurokawa, Procaccia and Wang {[JACM, 2018]} present instances for which every allocation gives some agent less than her maximin share. We present such examples with larger gaps. For three agents and nine items, we design an instance in which at least one agent does not get more than a $\frac{39}{40}$ fraction of her maximin share. {Moreover, we show that there is no negative example in which the difference between the number of items and the number of agents is smaller than six, and that the gap (of $\frac{1}{40}$) of our example is worst possible among all instances with nine items.} For $n \ge 4$ agents, we show examples in which at least one agent does not get more than a $1 - \frac{1}{n^4}$ fraction of her maximin share. {In the instances designed by Kurokawa, Procaccia and Wang, the gap is exponentially small in $n$.} Our proof techniques extend to allocation of chores (items of negative value), though the quantitative bounds for chores are different from those for goods. For three agents and nine chores, we design an instance in which the MMS gap is $\frac{1}{43}$.

cs.GT↗

arXiv2019

Nonstationary iterative processes

In this paper we present iterative methods of high efficiency by the criteria of J. F. Traub and A. M. Ostrowski. We define {\it s-nonstationary iterative processes} and prove that, for any one-point iterative process without memory, such as, for example, Newton's, Halley's, Chebyshev's methods, there exists an s-nonstationary process of the same order, but of higher efficiency. We supply constructions of these methods, obtain their properties and, for some of them, also their geometric interpretation. The algorithms we present can be transformed into computer programs in straight-forward manner. The methods are demonstrated by numerical examples.

math.NA↗