arXiv subjects

Asaf Petruschka

Publications and source records attributed to Asaf Petruschka.

12 recordsLinked to original sources

arXiv2026

Fast Metric Decompositions in High Dimension

Metric decompositions are a fundamental tool in the design of algorithms involving distances. We study fast algorithms for sampling from probabilistic metric decompositions of $n$-point sets in $\ell_\infty$ and $\ell_2$ spaces of high dimension $d$. For $\ell_\infty$, we design a padded-decomposition algorithm that runs in time $\tilde{O}(nd^2)$, which is near-linear in $n$, and achieves padding parameter $\tilde{O}(\log n)$. Our algorithm constructs a new sparse neighborhood cover that is based on geometric properties of $\ell_\infty$ [Indyk, JCSS'01], and utilizes recent reductions between covers and decompositions [Conroy and Filtser, STOC'25]. For $\ell_2$, we design a separating-decomposition algorithm that achieves near optimal separation $\tilde{O}(\sqrt{\log n})$ in almost-linear time $n^{1+o(1)}$. Our bounds improve over known algorithms with similar running time by a factor $Ω(\sqrt{\log n})$, and the techniques have additional applications to spanners and nearest-neighbor search.

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arXiv2026

Space-Optimal Sensitivity Oracles for Single-Source Mincuts

We study Single-Source Mincut Sensitivity Oracles: compact data structures that, when queried with an edge e, report those affected vertices whose mincut value to source $s$ changes upon the insertion or failure of e. Insertion queries were treated by Baswana, Gupta, and Knollmann [Algorithmica '22], who showed an extremely compact oracle with only O(n) space. In this work, we consider edge failure queries, which are of even greater interest, but far more challenging. The current-best approaches give O(n^2) space: either using n-1 fixed-pair oracles of O(n) space each, based on the Picard-Queyranne representation [MPS '80], or using the O(n^2) space all-pairs oracle by Baswana and Pandey [SODA '22]. -Our key result is an optimal O(n) space single-source mincut sensitivity oracle for edge failure queries. It reports the set of affected vertices in O(n) time, thus matching the state-of-the-art bounds for the insertion case. -Additionally, we provide oracles with near-optimal query times at the cost of increasing the space to O(n^{1.5}). They can determine if any given vertex is affected by an insertion/failure of an edge in O(log n) time, or reports all affected vertices in amortized O(\log^3 n) time per vertex. Such oracles of subquadratic space were previously unknown, even for insertion. Our main technical contribution is in establishing novel and intricate connections between two seemingly distant objects, representing two different families of mincuts. The first is the DAG representation of farthest mincuts to the source, which was the central tool introduced by Baswana, Gupta, and Knollmann. The second is the Connectivity Carcass for Steiner mincuts of Dinitz and Vainshtein [STOC '94], which generalizes well-known cactus representations of global mincuts. Our work demonstrates the relatively unexplored potential of the carcass beyond its obvious Steiner mincuts scope.

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arXiv2026

New Oracles and Labeling Schemes for Vertex Cut Queries

We study the succinct representations of vertex cuts by centralized oracles and labeling schemes. For an undirected $n$-vertex graph $G = (V,E)$ and integer parameter $f \geq 1$, the goal is supporting vertex cut queries: Given $F \subseteq V$ with $|F| \leq f$, determine if $F$ is a vertex cut in $G$. In the centralized data structure setting, it is required to preprocess $G$ into an $f$-vertex cut oracle that can answer such queries quickly, while occupying only small space. In the labeling setting, one should assign a short label to each vertex in $G$, so that a cut query $F$ can be answered by merely inspecting the labels assigned to the vertices in $F$. While the ``$st$ cut variants'' of the above problems have been extensively studied and are known to admit very efficient solutions, the basic (global) ``cut query'' setting is essentially open (particularly for $f > 3$). This work achieves the first significant progress on these problems: [$f$-Vertex Cut Labels:] Every $n$-vertex graph admits an $f$-vertex cut labeling scheme, where the labels have length of $\tilde{O}(n^{1-1/f})$ bits (when $f$ is polylogarithmic in $n$). This nearly matches the recent lower bound given by Long, Pettie and Saranurak (SODA 2025). [$f$-Vertex Cut Oracles:] For $f=O(\log n)$, every $n$-vertex graph $G$ admits $f$-vertex cut oracle with $\tilde{O}(n)$ space and $\tilde{O}(2^f)$ query time. We also show that our $f$-vertex cut oracles for every $1 \leq f \leq n$ are optimal up to $n^{o(1)}$ factors (conditioned on plausible fine-grained complexity conjectures). If $G$ is $f$-connected, i.e., when one is interested in \emph{minimum} vertex cut queries, the query time improves to $\tilde{O}(f^2)$, for any $1 \leq f \leq n$.

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arXiv2026

Color Fault-Tolerant Distance Preservers: Õptimal Size in Conditionally Õptimal Time

We revisit the problem of fault-tolerant (FT) distance preservers, when failure events in the network admit a form of correlation modeled as color faults. FT distance preservers are sparse subgraphs that preserve distances between specified pairs of vertices, even after some edge or vertex failures occur. In the classical fault model, any set of at most $k$ edges or vertices might fail (where $k \geq 1$ is a given parameter). Despite extensive research, the classical model admits significant and tantalizing gaps, both in terms of sparsity bounds and of algorithmic efficiency. In this work, we study the problem in the recently introduced color fault-tolerant (CFT) model: the given graph $G=(V,E)$ has arbitrary colors on its edges/vertices where each color appears at most $k$ times, and is susceptible to color faults, where the failure of color $c$ causes all the $c$-colored elements to crash. Our main contribution is in the multi-source setting, where $G$ has a source-set $S \subseteq V$, and the CFT preserver should preserve $S \times V$ distances under any single color fault. We show the following results (where $n = |V|$, $m = |E|$): - There exists a CFT distance preserver $H$ of $G$ with $\tilde{O}(n^{2 - \frac{1}{k+1}} \cdot |S|^{\frac{1}{k+1}} )$ edges. - The above sparsity bound is worst-case optimal up to polylogarithmic terms. - There is a combinatorial randomized algorithm that produces a preserver $H$ whose size meets the above optimal sparsity bound, with running time of $\tilde{O}(m \cdot n^{1 - \frac{1}{k+1}} \cdot |S|^{\frac{1}{k+1}})$. - The above running time is conditionally optimal: a polynomial improvement would refute the combinatorial Boolean Matrix Multiplication (BMM) conjecture. Furthermore, the running time remains optimal even if we only require mild sparsification to $m^{1-ε}$ edges.

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arXiv2025

Near-Optimal Vertex Fault-Tolerant Labels for Steiner Connectivity

We present a compact labeling scheme for determining whether a designated set of terminals in a graph remains connected after any $f$ (or less) vertex failures occur. An $f$-FT Steiner connectivity labeling scheme for an $n$-vertex graph $G=(V,E)$ with terminal set $U \subseteq V$ provides labels to the vertices of $G$, such that given only the labels of any subset $F \subseteq V$ with $|F| \leq f$, one can determine if $U$ remains connected in $G-F$. The main complexity measure is the maximum label length. The special case $U=V$ of global connectivity has been recently studied by Jiang, Parter, and Petruschka, who provided labels of $n^{1-1/f} \cdot \mathrm{poly}(f,\log n)$ bits. This is near-optimal (up to $\mathrm{poly}(f,\log n)$ factors) by a lower bound of Long, Pettie and Saranurak. Our scheme achieves labels of $|U|^{1-1/f} \cdot \mathrm{poly}(f, \log n)$ for general $U \subseteq V$, which is near-optimal for any given size $|U|$ of the terminal set. To handle terminal sets, our approach differs from Jiang et al. We use a well-structured Steiner tree for $U$ produced by a decomposition theorem of Duan and Pettie, and bypass the need for Nagamochi-Ibaraki sparsification.

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arXiv2024

Fault-Equivalent Lowest Common Ancestors

Let $T$ be a rooted tree in which a set $M$ of vertices are marked. The lowest common ancestor (LCA) of $M$ is the unique vertex $\ell$ with the following property: after failing (i.e., deleting) any single vertex $x$ from $T$, the root remains connected to $\ell$ if and only if it remains connected to some marked vertex. In this note, we introduce a generalized notion called $f$-fault-equivalent LCAs ($f$-FLCA), obtained by adapting the above view to $f$ failures for arbitrary $f \geq 1$. We show that there is a unique vertex set $M^* = \operatorname{FLCA}(M,f)$ of minimal size such after the failure of any $f$ vertices (or less), the root remains connected to some $v \in M$ iff it remains connected to some $u \in M^*$. Computing $M^*$ takes linear time. A bound of $|M^*| \leq 2^{f-1}$ always holds, regardless of $|M|$, and holds with equality for some choice of $T$ and $M$.

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arXiv2024

Parks and Recreation: Color Fault-Tolerant Spanners Made Local

We provide new algorithms for constructing spanners of arbitrarily edge- or vertex-colored graphs, that can endure up to $f$ failures of entire color classes. The failure of even a single color may cause a linear number of individual edge/vertex faults. In a recent work, Petruschka, Sapir and Tzalik [ITCS `24] gave tight bounds for the (worst-case) size $s$ of such spanners, where $s=Θ(f n^{1+1/k})$ or $s=Θ(f^{1-1/k} n^{1+1/k})$ for spanners with stretch $(2k-1)$ that are resilient to at most $f$ edge- or vertex-color faults, respectively. Additionally, they showed an algorithm for computing spanners of size $\tilde{O}(s)$, running in $\tilde{O}(msf)$ sequential time, based on the (FT) greedy spanner algorithm. The problem of providing faster and/or distributed algorithms was left open therein. We address this problem and provide a novel variant of the classical Baswana-Sen algorithm [RSA `07] in the spirit of Parter's algorithm for vertex fault-tolerant spanners [STOC `22]. In a nutshell, our algorithms produce color fault-tolerant spanners of size $\tilde{O}_k (s)$ (hence near-optimal for any fixed $k$), have optimal locality $O(k)$ (i.e., take $O(k)$ rounds in the LOCAL model), can be implemented in $O_k (f^{k-1})$ rounds in CONGEST, and take $\tilde{O}_k (m + sf^{k-1})$ sequential time. To handle the considerably more difficult setting of color faults, our approach differs from [BS07, Par22] by taking a novel edge-centric perspective, instead of (FT)-clustering of vertices; in fact, we demonstrate that this point of view simplifies their algorithms. Another key technical contribution is in constructing and using collections of short paths that are "colorful at all scales", which we call "parks". These are intimately connected with the notion of spread set-systems that found use in recent breakthroughs regarding the famous Sunflower Conjecture.

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arXiv2024

Connectivity Labeling in Faulty Colored Graphs

Fault-tolerant connectivity labelings are schemes that, given an $n$-vertex graph $G=(V,E)$ and $f\geq 1$, produce succinct yet informative labels for the elements of the graph. Given only the labels of two vertices $u,v$ and of the elements in a faulty-set $F$ with $|F|\leq f$, one can determine if $u,v$ are connected in $G-F$, the surviving graph after removing $F$. For the edge or vertex faults models, i.e., $F\subseteq E$ or $F\subseteq V$, a sequence of recent work established schemes with $poly(f,\log n)$-bit labels. This paper considers the color faults model, recently introduced in the context of spanners [Petruschka, Sapir and Tzalik, ITCS'24], which accounts for known correlations between failures. Here, the edges (or vertices) of the input $G$ are arbitrarily colored, and the faulty elements in $F$ are colors; a failing color causes all edges (vertices) of that color to crash. Our main contribution is settling the label length complexity for connectivity under one color fault ($f=1$). The existing implicit solution, by applying the state-of-the-art scheme for edge faults of [Dory and Parter, PODC'21], might yield labels of $Ω(n)$ bits. We provide a deterministic scheme with labels of $\tilde{O}(\sqrt{n})$ bits in the worst case, and a matching lower bound. Moreover, our scheme is universally optimal: even schemes tailored to handle only colorings of one specific graph topology cannot produce asymptotically smaller labels. We extend our labeling approach to yield a routing scheme avoiding a single forbidden color. We also consider the centralized setting, and show an $\tilde{O}(n)$-space oracle, answering connectivity queries under one color fault in $\tilde{O}(1)$ time. Turning to $f\geq 2$ color faults, we give a randomized labeling scheme with $\tilde{O}(n^{1-1/2^f})$-bit labels, along with a lower bound of $Ω(n^{1-1/(f+1)})$ bits.

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arXiv2023

Connectivity Labeling and Routing with Multiple Vertex Failures

We present succinct labeling schemes for answering connectivity queries in graphs subject to a specified number of vertex failures. An $f$-vertex/edge fault tolerant ($f$-V/EFT) connectivity labeling is a scheme that produces succinct labels for the vertices (and possibly to the edges) of an $n$-vertex graph $G$, such that given only the labels of two vertices $s,t$ and of at most $f$ faulty vertices/edges $F$, one can infer if $s$ and $t$ are connected in $G-F$. The primary complexity measure is the maximum label length (in bits). The $f$-EFT setting is relatively well understood: [Dory and Parter, PODC 2021] gave a randomized scheme with succinct labels of $O(\log^3 n)$ bits, which was subsequently derandomized by [Izumi et al., PODC 2023] with $\tilde{O}(f^2)$-bit labels. As both noted, handling vertex faults is more challenging. The known bounds for the $f$-VFT setting are far away: [Parter and Petruschka, DISC 2022] gave $\tilde{O}(n^{1-1/2^{Θ(f)}})$-bit labels, which is linear in $n$ already for $f =Ω(\log\log n)$. In this work we present an efficient $f$-VFT connectivity labeling scheme using $poly(f, \log n)$ bits. Specifically, we present a randomized scheme with $O(f^3 \log^5 n)$-bit labels, and a derandomized version with $O(f^7 \log^{13} n)$-bit labels, compared to an $Ω(f)$-bit lower bound on the required label length. Our schemes are based on a new low-degree graph decomposition that improves on [Duan and Pettie, SODA 2017], and facilitates its distributed representation into labels. Finally, we show that our labels naturally yield routing schemes avoiding a given set of at most $f$ vertex failures with table and header sizes of only $poly(f,\log n)$ bits. This improves significantly over the linear size bounds implied by the EFT routing scheme of Dory and Parter.

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arXiv2023

Color Fault-Tolerant Spanners

We initiate the study of spanners in arbitrarily vertex- or edge-colored graphs (with no "legality" restrictions), that are resilient to failures of entire color classes. When a color fails, all vertices/edges of that color crash. An $f$-color fault-tolerant ($f$-CFT) $t$-spanner of an $n$-vertex colored graph $G$ is a subgraph $H$ that preserves distances up to factor $t$, even in the presence of at most $f$ color faults. This notion generalizes the well-studied $f$-vertex/edge fault-tolerant ($f$-V/EFT) spanners. The size of an $f$-V/EFT spanner crucially depends on the number $f$ of vertex/edge faults to be tolerated. In the colored variants, even a single color fault can correspond to an unbounded number of vertex/edge faults. The key conceptual contribution of this work is in showing that the size (number of edges) required by an $f$-CFT spanner is in fact comparable to its uncolored counterpart, with no dependency on the size of color classes. We provide optimal bounds on the size required by $f$-CFT spanners, revealing an interesting phenomenon: while (individual) edge faults are "easier" than vertex faults in terms of spanner size, edge-color faults are "harder" than vertex-color faults. Our upper bounds are based on a generalization of the blocking set technique of [Bodwin and Patel, PODC 2019] for analyzing the (exponential-time) greedy algorithm for FT spanners. We complement them by providing efficient constructions of CFT spanners with similar size guarantees, based on the algorithm of [Dinitz and Robelle, PODC 2020].

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arXiv2023

Near-Optimal Distributed Computation of Small Vertex Cuts

We present near-optimal algorithms for detecting small vertex cuts in the CONGEST model of distributed computing. Despite extensive research in this area, our understanding of the vertex connectivity of a graph is still incomplete, especially in the distributed setting. To this date, all distributed algorithms for detecting cut vertices suffer from an inherent dependency in the maximum degree of the graph, $Δ$. Hence, in particular, there is no truly sub-linear time algorithm for this problem, not even for detecting a single cut vertex. We take a new algorithmic approach for vertex connectivity which allows us to bypass the existing $Δ$ barrier. As a warm-up to our approach, we show a simple $\widetilde{O}(D)$-round randomized algorithm for computing all cut vertices in a $D$-diameter $n$-vertex graph. This improves upon the $O(D+Δ/\log n)$-round algorithm of [Pritchard and Thurimella, ICALP 2008]. Our key technical contribution is an $\widetilde{O}(D)$-round randomized algorithm for computing all cut pairs in the graph, improving upon the state-of-the-art $O(Δ\cdot D)^4$-round algorithm by [Parter, DISC '19]. Note that even for the considerably simpler setting of edge cuts, currently $\widetilde{O}(D)$-round algorithms are known only for detecting pairs of cut edges. Our approach is based on employing the well-known linear graph sketching technique [Ahn, Guha and McGregor, SODA 2012] along with the heavy-light tree decomposition of [Sleator and Tarjan, STOC 1981]. Combining this with a careful characterization of the survivable subgraphs, allows us to determine the connectivity of $G \setminus \{x,y\}$ for every pair $x,y \in V$, using $\widetilde{O}(D)$-rounds. We believe that the tools provided in this paper are useful for omitting the $Δ$-dependency even for larger cut values.

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arXiv2022

Õptimal Dual Vertex Failure Connectivity Labels

In this paper we present succinct labeling schemes for supporting connectivity queries under vertex faults. For a given $n$-vertex graph $G$, an $f$-VFT (resp., EFT) connectivity labeling scheme is a distributed data structure that assigns each of the graph edges and vertices a short label, such that given the labels of a vertex pair $u$ and $v$, and the labels of at most $f$ failing vertices (resp., edges) $F$, one can determine if $u$ and $v$ are connected in $G \setminus F$. The primary complexity measure is the length of the individual labels. Since their introduction by [Courcelle, Twigg, STACS '07], FT labeling schemes have been devised only for a limited collection of graph families. A recent work [Dory and Parter, PODC 2021] provided EFT labeling schemes for general graphs under edge failures, leaving the vertex failure case fairly open. We provide the first sublinear $f$-VFT labeling schemes for $f \geq 2$ for any $n$-vertex graph. Our key result is $2$-VFT connectivity labels with $O(\log^3 n)$ bits. Our constructions are based on analyzing the structure of dual failure replacement paths on top of the well-known heavy-light tree decomposition technique of [Sleator and Tarjan, STOC 1981]. We also provide $f$-VFT labels with sub-linear length (in $|V|$) for any $f=o(\log\log n)$, that are based on a reduction to the existing EFT labels.

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