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Elad Tzalik

Publications and source records attributed to Elad Tzalik.

12 recordsLinked to original sources
arXiv2026

The Structure of Cycles in Projective Geometry over $\mathbb{F}_q$

A classical geometric result says that every nonzero cycle of the mod-$2$ incidence map from $d$-subsets to $(d-1)$-subsets of $[n]$ has support at least $d+1$, with equality attained by the boundary of a simplex on $d+1$ vertices. We prove an analogous result for the subspace lattice of $\mathbb{F}_q^n$, determining the minimum support size of a nonzero $d$-cycle over a field $K$ of characteristic $p \mid q+1$. Surprisingly, the boundary of a $(d+1)$-space is not always optimal. Shorter cycles occur for $d=1$, and for $d=2$ when $n\ge4$, and otherwise, the boundary of a $(d+1)$-space is shortest. For $d\ge4$, we prove a gap-stability result: every cycle with support less than $(2-10/q)$ times the minimum is a multiple of the boundary of a $d+1$ space. We also construct support-controlled cones, yielding a direct geometric analysis of the dimensions in which the homology groups of the subspace incidence complex vanish and explicit lower bounds on its coboundary expansion. The degree-$1$ expansion estimate is an ingredient in the stability theorem.

math.CO↗
arXiv2026

Topological expansion for posets and the homological $k$-connectivity of random $q$-complexes

We study high dimensional expansion beyond simplicial complexes (posets) and focus on $q$-complexes which are complexes whose basic building blocks are linear spaces. We show that the complete $q$-complex (consists of all subspaces of a given linear space) may have non-trivial homology groups and therefore some techniques for simplicial complexes fail. We develop new techniques to work bypass this. In particular: (i) We describe a new construction of cones and use it to determine when the homology of the complete $q$-complex is trivial. We use this construction to prove the "projective support dimension conjecture" conjectured by Mnukhin and Siemons. (ii) We define topological high dimensional expansion for posets, and show that the complete $q$-complex has linear (in the number of lines) coboundary expansion. (iii) We define the $q$-Linial-Meshulam model of random $q$-complexes and prove a sharp threshold for the connectivity of random $q$-complexes.

math.CO↗
arXiv2026

Greedy Completion for Weighted $(α,β)$-Spanners

We study $(α,β)$-spanners for weighted graphs. We propose a simple greedy completion procedure which starts from a sparse initial graph, and repeatedly fixes pairs of vertices with a bad stretch, generalizing Knudsen's additive completion [SWAT '14]. As an application, we construct $(k,k-1)$-spanners for weighted graphs of size $\tilde{O}(n^{1+1/k})$, which were previously unknown.

cs.DS↗
arXiv2026

Determining a Points Configuration from a Subset of the Pairwise Distances

We study rigidity without assuming general position. Given $n$ distinct labelled points and a set $\mathcal{P}\subseteq \binom{[n]}{2}$ of revealed pairs, we ask when the corresponding distances determine the configuration up to isometry. On the line, we prove an extremal result: if $|\mathcal{P}|=Ω(n^{3/2})$, then there is an induced globally rigid subgraph on $Ω(|\mathcal{P}|/n)$ vertices. In other words, any dense enough graph will contain a subset of labels whose locations can be determined from their distances up to isometry. To prove this, we establish a graph-theoretic result, which may be of independent interest: a dense graph in which every non-edge has few common neighbours contains a clique of size $Ω(|E|/n)$. We also study random revealed pairs. For every labelled configuration $V$ of distinct points in $\mathbb{R}$, if each pair is revealed independently with probability $p=C\ln n/n$, where $C>1$, then the revealed distances determine $V$ w.h.p. We prove a similar result for $d\ge1$ under the mild non-degeneracy assumption that every subcollection of more than $τn$ points of $V\subseteq\mathbb R^d$ affinely spans $\mathbb R^d$, for some fixed $0<τ<1$. In this case, every $C>1/(1-τ)$ suffices. The same ideas also settle the weak-threshold form of a conjecture of Girão et al. for a giant reconstructable component, and substantially improve in this direction the work of Barnes et al. establishing such a component for $p>n^{-2/(d+4)}$.

math.MG↗
arXiv2026

New Greedy Spanners and Applications

We present a simple greedy procedure to compute an $(α,β)$-spanner for a graph $G$. We then show that this procedure is useful for building fault-tolerant spanners, as well as spanners for weighted graphs. Our first main result is an algorithm that, given a multigraph $G$, outputs an $f$ edge fault-tolerant $(k,k-1)$-spanner $H$ of size $O(fn^{1+\frac1k})$ which is tight. To our knowledge, this is the first tight result concerning the price of fault tolerance in spanners which are not multiplicative, in any model of faults. Our second main result is a new construction of a spanner for weighted graphs. We show that any weighted graph $G$ has a subgraph $H$ with $O(n^{1+\frac{1}{k}})$ edges such that any path $P$ of hop-length $\ell$ in $G$ has a replacement path $P'$ in $H$ of weighted length $\leq w(P)+(2k-2)w^{(1/2)}(P)$ where $w(P)$ is the total edge weight of $P$, and $w^{(1/2)}$ denotes the sum of the largest $\lceil \frac{\ell}{2} \rceil$ edge weights along $P$. Moreover, we show such approximation is optimal for shortest paths of hop-length $2$. To our knowledge, this is the first construction of a spanner for weighted graphs that strictly improves upon the stretch of multiplicative $(2k-1)$-spanners for all non-adjacent vertex pairs, while maintaining the same size bound. Our technique is based on using clustering and ball-growing, which are methods commonly used in designing spanner algorithms, to analyze simple greedy algorithms. This allows us to combine the flexibility of clustering approaches with the unique properties of the greedy algorithm to get improved bounds. In particular, our methods give a very short proof that the parallel greedy spanner adds $O(kn^{1+\frac{1}{k}})$ edges, improving upon known bounds.

cs.DS↗
arXiv2025

Hypercube minor-universality

A graph $G$ is $m$-minor-universal if every graph with at most $m$ edges (and no isolated vertices) is a minor of $G$. We prove that the $d$-dimensional hypercube, $Q_d$, is $Ω\left(\frac{2^d}{d}\right)$-minor-universal, and that there exists an absolute constant $C >0$ such that $Q_d$ is not $\frac{C2^d}{\sqrt{d}}$-minor-universal. Similar results are obtained in a more generalized setting, where we bound the size of minors in a product of finite connected graphs. A key component of our proof is the following claim regarding the decomposition of a permutation of a box into simpler, one-dimensional permutations: Let $n_1, \dots, n_d$ be positive integers, and define $X := [n_1] \times \dots \times [n_d]$. We prove that every permutation $σ: X \to X$ can be expressed as $σ= σ_1 \circ \dots \circ σ_{2d-1}$, where each $σ_i$ is a one-dimensional permutation, meaning it fixes all coordinates except possibly one. We discuss future directions and pose open problems.

math.CO↗
arXiv2024

Connectivity Certificate against Bounded-Degree Faults: Simpler, Better and Supporting Vertex Faults

An $f$-edge (or vertex) connectivity certificate is a sparse subgraph that maintains connectivity under the failure of at most $f$ edges (or vertices). It is well known that any $n$-vertex graph admits an $f$-edge (or vertex) connectivity certificate with $Θ(f n)$ edges (Nagamochi and Ibaraki, Algorithmica 1992). A recent work by (Bodwin, Haeupler and Parter, SODA 2024) introduced a new and considerably stronger variant of connectivity certificates that can preserve connectivity under any failing set of edges with bounded degree. For every $n$-vertex graph $G=(V,E)$ and a degree threshold $f$, an $f$-Edge-Faulty-Degree (EFD) certificate is a subgraph $H \subseteq G$ with the following guarantee: For any subset $F \subseteq E$ with $deg(F)\leq f$ and every pair $u,v \in V$, $u$ and $v$ are connected in $H - F$ iff they are connected in $G - F$. For example, a $1$-EFD certificate preserves connectivity under the failing of any matching edge set $F$ (hence, possibly $|F|=Θ(n)$). In their work, [BHP'24] presented an expander-based approach (e.g., using the tools of expander decomposition and expander routing) for computing $f$-EFD certificates with $O(f n \cdot poly(\log n))$ edges. They also provided a lower bound of $Ω(f n\cdot \log_f n)$, hence $Ω(n\log n)$ for $f=O(1)$. In this work, we settle the optimal existential size bounds for $f$-EFD certificates (up to constant factors), and also extend it to support vertex failures with bounded degrees (where each vertex is incident to at most $f$ faulty vertices). Specifically, we show that for every $n>f/2$, any $n$-vertex graph admits an $f$-EFD (and $f$-VFD) certificates with $O(f n \cdot \log(n/f))$ edges and that this bound is tight. Our upper bound arguments are considerably simpler compared to prior work, do not use expanders, and only exploit the basic structure of bounded degree edge and vertex cuts.

cs.DS↗
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.

cs.DS↗
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.

cs.DS↗
arXiv2023

On the Number of Shortest Paths in Graphs

It is proved that the number of shortest paths between two vertices of distance $t$ in a graph with degrees bounded by $Δ$ is at most $2 \cdot (\fracΔ{2})^t$. This improves upon the naïve $Δ(Δ-1) ^{t-1}$ bound.

math.CO↗
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].

cs.DS↗