arXiv subjects

Gur Lifshitz

Publications and source records attributed to Gur Lifshitz.

7 recordsLinked to original sources

arXiv2026

Tree Covers for Directed Graphs: Distance Oracles, Routing Schemes and Spanners

In a seminal work [STOC'01; JACM'05], Thorup and Zwick showed that every weighted undirected graph admits a tree cover where each vertex belongs to $O(kn^{1/k})$ trees in expectation, and every pair has a path of stretch at most $2k-1$ in some tree. In this paper, we extend this structural property to directed graphs with nonnegative edge weights. Let $\mathrm{dist}_G(u \leftrightarrows v) = \mathrm{dist}_G(u \to v) + \mathrm{dist}_G(v \to u)$ denote the roundtrip distance. For every integer $k$, we construct a collection $\mathcal{T}$ of tree-pairs, each consisting of an in-tree and an out-tree rooted at the same vertex. Each vertex belongs to $O(kn^{1/k})$ tree-pairs in expectation, and for every ordered pair $(u, v)$ some tree-pair $T_{in}\cup T_{out} \in \mathcal{T}$ satisfies $\mathrm{dist}_{T_{in}\cup T_{out}}(u \to v) \le \mathrm{dist}_G(u \to v) + (k-1)\mathrm{dist}_G(u \leftrightarrows v).$ We call this a \emph{$k$-strong (roundtrip) tree cover}. While prior techniques partition distances into geometric scales and build separate structures for each scale, our construction handles all distances simultaneously within a single, unified hierarchy. This yields the first directed tree cover for general $k$ matching the classic Thorup-Zwick stretch-size trade-off. Our cover immediately implies improved constructions of roundtrip spanners, path-reporting distance oracles, and compact routing schemes, as well as their source-wise and prioritized variants. We also provide faster algorithms for these structures: for general $k \ge 4$, all the above objects can be constructed in $\widetilde{O}(\min\{mn, n^{2.5}\})$ expected time, breaking the $mn$ barrier in dense directed graphs.

cs.DS↗

arXiv2026

Optimal (Randomized) Compact Routing Schemes

Compact routing schemes route messages along short paths while every vertex stores little information. We study the trade-off between the stretch and the storage of compact routing schemes in undirected graphs, in four models: labeled and name-independent schemes, with average or worst-case local storage. We improve the best known stretch in all four models. Our main result is a labeled routing scheme with expected stretch $2k-1$ and $\widetilde O(n^{1/k})$ average local storage, in which only the routing is randomized. Under the Erdős girth conjecture, no scheme with this storage has stretch below $2k-1$, so our scheme is optimal, up to polylogarithmic factors in the storage. With deterministic routing, we obtain stretch $\approx2.47k$, the best known deterministic stretch with average storage. Worst-case local storage is the classical setting of compact routing. The leading coefficient $4$ in the stretch $4k-5$ of Thorup and Zwick~[SPAA'01] was improved only by Chechik~[PODC'13], to $\approx3.68$ in unweighted graphs. In unweighted graphs, we obtain expected stretch $(\frac83k+O(1),O(k))$, and deterministic stretch $\approx3.05k$. Name-independent routing is harder, since the source knows only the name of the destination. The best known stretch with $\widetilde O(n^{1/k})$ local storage in unweighted graphs was $96k$, by Abraham, Gavoille, and Malkhi~[DISC'04]. We improve it by more than a factor of $8$. With average storage, we obtain expected stretch $4k-5$, which matches the labeled scheme of Thorup and Zwick. With worst-case storage in unweighted graphs, we obtain expected stretch $(8k+O(1),O(k))$. All our schemes for unweighted graphs extend to weighted graphs with a small loss in the stretch, and still improve the previous unweighted bounds.

cs.DS↗

arXiv2026

Roundtrip All-Pairs Shortest Paths in Directed Graphs

We study approximation algorithms for All-Pairs Shortest Paths (APSP) in directed graphs under the roundtrip metric. In directed graphs, even finite-factor approximation of ordinary one-way distances must distinguish reachable from unreachable vertex pairs, and therefore computes transitive closure, a problem equivalent to Boolean matrix multiplication (BMM) under standard reductions. The roundtrip distance, defined as $\text{dist}_G(u \leftrightarrows v) = \text{dist}_G(u \to v) + \text{dist}_G(v \to u)$, offers a natural and robust relaxation. It measures the cost of traveling from $u$ to $v$ and back, satisfies the triangle inequality, and is inherently symmetric. While this metric has become central to the study of directed roundtrip spanners and compact routing, the fundamental problem of all-pairs roundtrip distance approximation has remained largely unexplored. In this work, we present a comprehensive study of roundtrip APSP and distance oracles, presenting several new lower and upper bounds. On the algorithmic side, our main result is a combinatorial algorithm that computes a stretch-$3$ approximation to all-pairs roundtrip distances in $\widetilde{O}(n^2 + \min\{m\sqrt{n},\, m^{1/3}n^{5/3}\})$ time. For edge weights in $[1,W]$, we also obtain a $(3+\varepsilon)$-approximation in $\widetilde O((n^{2.037}+nm^{2/3})\log(nW)/\varepsilon)$ time and a $(4+\varepsilon)$-approximation in $\widetilde O(n^{2.032}\log(nW)/\varepsilon)$ time using fast matrix multiplication. On the hardness side, we show that any stretch below $3$ would imply a breakthrough for BMM. We also obtain additional tradeoffs between stretch and running time.

cs.DS↗

arXiv2026

Reading between the levels: new tools for the Thorup-Zwick hierarchy and their applications

In this paper, we develop three new analytical tools for the Thorup-Zwick hierarchy: the Mixed-Level Growth, Two-Level Growth, and Long-Short Growth lemmas. Using these tools, we obtain improved approximate distance oracles (ADOs), and algorithms for $n$-pairs shortest paths ($n$-PSP), multiple-source shortest paths (MSSP), and $ST$-shortest paths, as well as new hopsets. One of our main results is a $(\frac 65 k+O(1))$-stretch ADO with $\tilde O(mn^{1/k})$ preprocessing time and space (where $\tilde O$ hides polylogarithmic factors), and $\tilde O(n^{1/k})$ query time, breaking, for sparse graphs, the $(2k-O(1))$-stretch barrier for ADOs with $\tilde O(n^{1+1/k})$ space and $\tilde O(n^{1/k})$ query time. In unweighted graphs, we also obtain a $(\frac 65 k+O(1),O(k))$-approximate ADO with $\tilde O(n^{1+1/k})$ space and $\tilde O(n^{1/k})$ query time. This also yields a $(\frac 65 k+O(1))$-approximation algorithm for $m$-pairs shortest paths ($m$-PSP) in $\tilde O(mn^{1/k})$ time, answering the question of Chechik, Hoch, and Lifshitz [SODA'25] of whether the $2k-O(1)$ barrier for $n$-PSP in $\tilde O(mn^{1/k})$ time can be broken. For unweighted graphs, we give a new analysis of the existing $\tilde O(mn^{1/k}+n^{1+2/k})$-time $n$-PSP algorithm of [Chechik et al., SODA'25], and show that, without any modification, it is a $(k,k-1)$-approximation, i.e., $d(u,v)\le\hat d(u,v)\le k\cdot d(u,v)+k-1$ for every queried pair. This improves its multiplicative factor from $\frac 43 k+O(1)$ to $k$, and the leading coefficient $k$ is tight for this algorithm. Moreover, for graphs with $m=Ω(n^{1+1/k})$, this answers the question of Dalirrooyfard et al. [FOCS'22] up to the additive $k-1$.

cs.DS↗

arXiv2026

$(α, β)$ Spanners and Hybrid Spanners with Nearly Tight Bounds

For an $n$-vertex undirected, unweighted graph $G=(V,E)$ and a positive integer $k$, we present new spanner constructions with $O_k(n^{1+1/k})$ edges that achieve nearly optimal guarantees for all distances $d\le k$. Specifically, we construct a spanner $H\subseteq G$ with $O(n^{1+1/k}+(k+d\log d)n)$ edges, ensuring that any pair at original distance at most $d$ satisfies $\mathrm{dist}_H(u,v)\le 2k+O(d\log d)$. Equivalently, the multiplicative stretch for pairs at distance $d$ is $2k/d+O(\log d)$. In particular, setting $d=k/\log k$ yields an $(O(\log k),O(k))$-spanner with $O(n^{1+1/k}+kn)$ edges. For comparison, Ben-Levy and Parter (SODA'20) obtained, for every fixed $\varepsilon>0$ and sufficiently large $k$, an $(O(k^\varepsilon),O_\varepsilon(k))$-spanner with $O_{\varepsilon,k}(n^{1+1/k})$ edges. Our result improves the multiplicative stretch from $O(k^\varepsilon)$ to $O(\log k)$ while keeping the additive term linear in $k$, bringing us closer to the goal of $(O(1),O(k))$-spanners. Furthermore, Ben-Levy and Parter obtained multiplicative stretch $O_\varepsilon(k/d)$ for distances $d\le k^{1-\varepsilon}$, for every fixed $\varepsilon>0$, and an explicit bound of $7k/d$ for $d\le\sqrt{k}/2$. We achieve $2k/d+O(\log d)$, which is $(2+o(1))k/d$ whenever $d=o(k/\log k)$. Our second result is an improved construction of $k$-hybrid spanners, which guarantee stretch $2k-1$ for adjacent pairs and $k$ for non-adjacent pairs. Parter's original construction uses $O(k^2 n^{1+1/k})$ edges; we achieve the same guarantees with $O(n^{1+1/k}+kn)$ edges, removing the $k^2$ factor from the $n^{1+1/k}$ term. For every fixed $k$, our edge bound is optimal up to a constant factor under Erdős' girth conjecture.

cs.DS↗

arXiv2026

Girth Approximations in the CONGEST Model

This paper advances the state of the art in girth approximation within the CONGEST model. Manoharan and Ramachandran [PODC '24] provided the first significant improvement in girth approximation in over a decade. We build on this momentum and make progress on all fronts: we provide a unified family of algorithms yielding girth approximation-round tradeoffs for undirected networks; we obtain improved bounds for directed networks; and we establish better lower bounds for directed and undirected weighted networks. Together, these results substantially narrow the remaining complexity gaps across all settings. Specifically, for networks with $n$ nodes and hop-diameter $D$, we show that one can compute, with high probability: $(1)$ An $f$-approximation for unweighted undirected girth in $\tilde{O}(n^{1/f}+D)$ rounds, for every constant integer $f>2$, $(2)$ A $(2k-1+o(1))$-approximation for weighted undirected girth in $\tilde{O}(n^{(k+1)/(2k+1)}+D)$ rounds, for every constant integer $k>1$, and $(3)$ A $2$-approximation for directed unweighted girth, and a $(2+\varepsilon)$-approximation for directed weighted girth, both in $\tilde{O}(n^{2/3}+D)$ rounds. We also prove new lower bounds for directed networks and for undirected weighted networks: for every integer $k > 2$ and $\varepsilon>0$, assuming the Erdős-Simonovits' even cycle conjecture (and unconditionally for $k\in\{3,4,6\}$), any $(k-\varepsilon)$-approximation for the girth requires $\tildeΩ(n^{k/(2k-1)})$ rounds, even when $D = O(\log n)$.

cs.DS↗

arXiv2025

Spectral Graph Neural Networks are Incomplete on Graphs with a Simple Spectrum

Spectral features are widely incorporated within Graph Neural Networks (GNNs) to improve their expressive power, or their ability to distinguish among non-isomorphic graphs. One popular example is the usage of graph Laplacian eigenvectors for positional encoding in MPNNs and Graph Transformers. The expressive power of such Spectrally-enhanced GNNs (SGNNs) is usually evaluated via the k-WL graph isomorphism test hierarchy and homomorphism counting. Yet, these frameworks align poorly with the graph spectra, yielding limited insight into SGNNs' expressive power. We leverage a well-studied paradigm of classifying graphs by their largest eigenvalue multiplicity to introduce an expressivity hierarchy for SGNNs. We then prove that many SGNNs are incomplete even on graphs with distinct eigenvalues. To mitigate this deficiency, we adapt rotation equivariant neural networks to the graph spectra setting to propose a method to provably improve SGNNs' expressivity on simple spectrum graphs. We empirically verify our theoretical claims via an image classification experiment on the MNIST Superpixel dataset and eigenvector canonicalization on graphs from ZINC.

cs.LG↗