arXiv subjects

Yuri Rabinovich

Publications and source records attributed to Yuri Rabinovich.

11 recordsLinked to original sources

arXiv2026

Deterministic Online Embedding of Metric Spaces into Low Dimensional Spaces

We study online embeddings of metric spaces into Euclidean spaces of a constant dimension $d>1$, against an adaptive adversary. While the case of $d=1$ is well understood, for higher dimensions little is known. In particular, even for $d=2$ it remains unknown whether the worst-case distortion grows exponentially with the number of exposed points, as it does in the case for the line, or whether it is polynomial, as in the case for unbounded $d$. Our first result is about fixed {\em solid} graphs, i.e., $K_5$, whose edges are solid intervals, equipped with the shortest-path metric. We show that if the input points arrive from such a metric space, they can indeed be online-embedded into ${\mathbb R}^2$ with a polynomial distortion. This refutes the previously believed conjecture that the topological non-embeddability of $K_5$ into the plane could be exploited for establishing exponential lower bounds. The second results is about online embeddings of tree metrics of a certain type, including, e.g., ultrametrics and HST's. Somewhat surprisingly, we show that for metrics from this class the worst-case online embedding into ${\mathbb R}^d$ is not much worse that the offline embedding, both being $n^{Θ(1/d)}$, and this holds even when $d = Θ(\log n)$. This is in a stark contrast to the more common situation where the online-offline gap is typically huge, and even exponential. This result allows us to transfer results about probabilistic embeddings of metrics into HST's to low-dimensional Euclidean spaces, in an almost optimal possible manner.

cs.DS↗

arXiv2023

Online embedding of metrics

We study deterministic online embeddings of metrics spaces into normed spaces and into trees against an adaptive adversary. Main results include a polynomial lower bound on the (multiplicative) distortion of embedding into Euclidean spaces, a tight exponential upper bound on embedding into the line, and a $(1+ε)$-distortion embedding in $\ell_\infty$ of a suitably high dimension.

cs.CG↗

arXiv2019

Large Simple d-Cycles in Simplicial Complexes

We show that the size of the largest simple d-cycle in a simplicial d-complex $K$ is at least a square root of $K$'s density. This generalizes a well-known classical result of Erdős and Gallai \cite{EG59} for graphs. We use methods from matroid theory applied to combinatorial simplicial complexes.

math.CO↗

arXiv2019

Hamiltonian and Pseudo-Hamiltonian Cycles and Fillings In Simplicial Complexes

We introduce and study a $d$-dimensional generalization of Hamiltonian cycles in graphs - the Hamiltonian $d$-cycles in $K_n^d$ (the complete simplicial $d$-complex over a vertex set of size $n$). Those are the simple $d$-cycles of a complete rank, or, equivalently, of size $1 + {{n-1} \choose d}$. The discussion is restricted to the fields $F_2$ and $Q$. For $d=2$, we characterize the $n$'s for which Hamiltonian $2$-cycles exist. For $d=3$ it is shown that Hamiltonian $3$-cycles exist for infinitely many $n$'s. In general, it is shown that there always exist simple $d$-cycles of size ${{n-1} \choose d} - O(n^{d-3})$. All the above results are constructive. Our approach naturally extends to (and in fact, involves) $d$-fillings, generalizing the notion of $T$-joins in graphs. Given a $(d-1)$-cycle $Z^{d-1} \in K_n^d$, ~$F$ is its $d$-filling if $\partial F = Z^{d-1}$. We call a $d$-filling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size ${{n-1} \choose d}$. If a Hamiltonian $d$-cycle $Z$ over $F_2$ contains a $d$-simplex $σ$, then $Z\setminus σ$ is a a Hamiltonian $d$-filling of $\partial σ$ (a closely related fact is also true for cycles over $Q$). Thus, the two notions are closely related. Most of the above results about Hamiltonian $d$-cycles hold for Hamiltonian $d$-fillings as well.

math.CO↗

arXiv2017

Rational Polygons: Odd Compression Ratio and Odd Plane Coverings

Let P be a polygon with rational vertices in the plane. We show that for any finite odd-sized collection of translates of P, the area of the set of points lying in an odd number of these translates is bounded away from 0 by a constant depending on P alone. The key ingredient of the proof is a construction of an odd cover of the plane by translates of P. That is, we establish a family F of translates of P covering (almost) every point in the plane a uniformly bounded odd number of times.

math.CO↗

arXiv2015

Boundaries of Hypertrees, and Hamiltonian Cycles in Simplicial Complexes

A $d$-hypertree on $[n]$ is a maximal acyclic $d$-dimensional simplicial complex with full $(d-1)$-skeleton on the vertex set $[n]$. Alternatively, in the language of algebraic topology, it is a minimal $d$-dimensional simplicial complex $T$ (assuming full $(d-1)$-skeleton) such that $\tilde{H}_{d-1}(T;\mathbb{F})=0$. The $d$-hypertrees are a basic object in combinatorial theory of simplicial complexes. They have been studied; and yet, many of their structural aspects remain poorly understood. In this paper we study the boundaries $\partial_d T$ of $d$-hypertrees, and the fundamental $d$-cycles defined by them. Our findings include: 1. A full characterization of $\partial_d T$ over $\mathbb{F}_2$ for $d \leq 2$, and some partial results for $d \geq 3$. 2. Lower bounds on the maximum size of a largest simple $d$-cycle on $[n]$. In particular, for $d=2$, we construct a {\em Hamiltonian $d$-cycle} $H$ on $[n]$, i.e., a simple $d$-cycle of size ${{n-1} \choose d} + 1$. For $d\geq 3$, we construct a simple $d$-cycle of size ${{n-1} \choose d} - O(n^{d-2})$. 3. Observing that the maximum of the expected distance between two vertices chosen uniformly at random in a tree ($1$-hypertree) on $[n]$ is at most $\thicksim n/3$, attained on Hamiltonian paths, we ask a similar question about $d$-hypertrees. "How large can be the {\em average} size of a fundamental cycle of a $d$-hypertree $T$ (i.e., the expected size of the dependency created by adding a $d$-simplex on $[n]$, chosen uniformly at random, to $T$)?" For every $d \in \mathbb{N}$, we construct an infinite family of $d$-hypertrees $\{T\}$ with the average size of a fundamental cycle at least $c_d\, |T| \,=\, c_d\,{n-1 \choose d}$, where $c_d$ is a constant depending on the dimension $d$ alone.

math.CO↗

arXiv2015

Extremal problems on shadows and hypercuts in simplicial complexes

Let $F$ be an $n$-vertex forest. We say that an edge $e\notin F$ is in the shadow of $F$ if $F\cup\{e\}$ contains a cycle. It is easy to see that if $F$ is "almost a tree", that is, it has $n-2$ edges, then at least $\lfloor\frac{n^2}{4}\rfloor$ edges are in its shadow and this is tight. Equivalently, the largest number of edges an $n$-vertex cut can have is $\lfloor\frac{n^2}{4}\rfloor$. These notions have natural analogs in higher $d$-dimensional simplicial complexes, graphs being the case $d=1$. The results in dimension $d>1$ turn out to be remarkably different from the case in graphs. In particular the corresponding bounds depend on the underlying field of coefficients. We find the (tight) analogous theorems for $d=2$. We construct $2$-dimensional "$\mathbb Q$-almost-hypertrees" (defined below) with an empty shadow. We also show that the shadow of an "$\mathbb F_2$-almost-hypertree" cannot be empty, and its least possible density is $Θ(\frac{1}{n})$. In addition we construct very large hyperforests with a shadow that is empty over every field. For $d\ge 4$ even, we construct $d$-dimensional $\mathbb{F} _2$-almost-hypertree whose shadow has density $o_n(1)$. Finally, we mention several intriguing open questions.

math.CO↗

arXiv2015

On Connectivity of the Facet Graphs of Simplicial Complexes

The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of $d$-cycles, $d$-hypertrees and $d$-hypercuts are, respectively, $(d+1)$, $d$, and $(n-d-1)$-vertex-connected. It is also shown that the facet graph of a $d$-cycle cannot be split into more than $s$ connected components by removing at most $s$ vertices. In addition, the paper discusses various related issues, as well as an extension to cell-complexes.

math.CO↗

arXiv2014

Binary Jumbled Pattern Matching via All-Pairs Shortest Paths

In binary jumbled pattern matching we wish to preprocess a binary string $S$ in order to answer queries $(i,j)$ which ask for a substring of $S$ that is of size $i$ and has exactly $j$ 1-bits. The problem naturally generalizes to node-labeled trees and graphs by replacing "substring" with "connected subgraph". In this paper, we give an ${n^2}/{2^{Ω(\log n/\log \log n)^{1/2}}}$ time solution for both strings and trees. This odd-looking time complexity improves the state of the art $O(n^2/\log^2 n)$ solutions by more than any poly-logarithmic factor. It originates from the recent seminal algorithm of Williams for min-plus matrix multiplication. We obtain the result by giving a black box reduction from trees to strings. This is then combined with a reduction from strings to min-plus matrix multiplications.

cs.DS↗

arXiv2010

Finite Volume Spaces and Sparsification

We introduce and study finite $d$-volumes - the high dimensional generalization of finite metric spaces. Having developed a suitable combinatorial machinery, we define $\ell_1$-volumes and show that they contain Euclidean volumes and hypertree volumes. We show that they can approximate any $d$-volume with $O(n^d)$ multiplicative distortion. On the other hand, contrary to Bourgain's theorem for $d=1$, there exists a $2$-volume that on $n$ vertices that cannot be approximated by any $\ell_1$-volume with distortion smaller than $\tildeΩ(n^{1/5})$. We further address the problem of $\ell_1$-dimension reduction in the context of $\ell_1$ volumes, and show that this phenomenon does occur, although not to the same striking degree as it does for Euclidean metrics and volumes. In particular, we show that any $\ell_1$ metric on $n$ points can be $(1+ ε)$-approximated by a sum of $O(n/ε^2)$ cut metrics, improving over the best previously known bound of $O(n \log n)$ due to Schechtman. In order to deal with dimension reduction, we extend the techniques and ideas introduced by Karger and Bencz{ú}r, and Spielman et al.~in the context of graph Sparsification, and develop general methods with a wide range of applications.

cs.DS↗

arXiv2010

Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs

In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of Bǎdoiu, Indyk, and Sidiropoulos (2007) and Bǎdoiu, Demaine, Hajiaghayi, Sidiropoulos, and Zadimoghaddam (2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into an outerplanar metric. For this, we introduce a general notion of metric relaxed minor and show that if G contains an alpha-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is at meast alpha. Then, for H=K_{2,3}, we present an algorithm which either finds an alpha-relaxed minor, or produces an O(alpha)-embedding into an outerplanar metric.

math.MG↗