arXiv subjects

Avi Kadria

Publications and source records attributed to Avi Kadria.

11 recordsLinked to original sources

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.

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arXiv2026

A Tight Space Lower Bound for Pivot-Based Distance Oracles

An approximate distance oracle (ADO) is a compact data structure that answers distance queries within a multiplicative \emph{stretch} $k$. The classical oracle of Thorup and Zwick~[STOC'01, JACM'05] has stretch $k=2\ell-1$ and uses $O(\ell n^{1+1/\ell})$ space, for every integer $\ell\ge1$. This trade-off is optimal for dense graphs under the Erdős girth conjecture. However, for sparse graphs ($m=O(n)$), their lower bound yields nothing beyond $Ω(m)$ space. The main result of this paper is a nearly tight lower bound for pivot-based $1$-hop ADOs in sparse graphs. A pivot-based $1$-hop ADO gives every vertex $\tilde{O}(1)$ pivots and returns as an estimate the length of a walk of the form $u\to p_u \to p_v \to v$, where $p_u$ is a pivot of $u$, $p_v$ is a pivot of $v$, and $p_u\to p_v$ is a stored distance. We observe that all the state-of-the-art ADOs with $\tilde{O}(1)$ query time are in fact pivot-based $1$-hop ADOs. We prove that every \emph{pivot-based} $1$-hop ADO with \emph{real} stretch $k$ requires $\widetildeΩ(m^{1+2/(k+1)})$ space, under the Erdős girth conjecture for graphs with constant degree. This matches the best known ADOs with $\tilde{O}(1)$ query time, at every known stretch, up to polylogarithmic factors. We also extend our lower bound to pivot-based ADOs with any constant number $β$ of hops, showing that they require $\widetildeΩ(m^{1+2/(β(k+1))})$ space. In addition, we obtain a tight lower bound for two-hop hopsets and improved lower bounds for general $β$-hop hopsets.

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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 $k-1$.

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arXiv2026

Deterministic Girth Approximation in Weighted Undirected Graphs

We give deterministic algorithms for approximating the girth~$g$ of an undirected graph with $n$ vertices, $m$ edges, and nonnegative real edge lengths. For every integer $k\geq2$, our algorithm returns a cycle of length at most $\frac{2k}{3}g$ in $O((m+n^{1+2/k})k\log n)$ worst-case time. For every constant~$k$, this matches the stretch and the expected running time of the randomized algorithms of Kadria, Roditty, Sidford, Vassilevska Williams, and Zwick [SODA'23, SICOMP'26] and Kadria, Roditty, and Vassilevska Williams [ESA'26]. In general, it loses only a factor of $O(\min\{k,\log n\})$ in the term that depends on~$m$. The best previous deterministic trade-off, of Kadria, Roditty, Sidford, Vassilevska Williams, and Zwick [SODA'22], has stretch~$k$ instead of $\frac{2k}{3}$ at the same exponent (for even~$k$) and requires integer edge lengths. In particular, for $k=3$ we obtain a deterministic $2$-approximation in $O((m+n^{5/3})\log n)$ time for real edge lengths, whereas the deterministic algorithm of Ducoffe [SIDMA'21] for real lengths has stretch $2+\varepsilon$. Our main technical idea is that a deterministic construction of the Thorup--Zwick hierarchy used by these algorithms may stop exploring as soon as it finds a cycle: the approximation analysis only needs clusters truncated at half the best cycle certificate found, so balls stopped in this way need not be hit. Together with a degree-weighted selection rule, this bounds the work of each level by its number of edges and stored ball entries. More generally, a ball need not be hit whenever the analysis only uses clusters up to a radius bounded by a certificate that the algorithm has already found.

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arXiv2026

A Faster Undirected Single-Source Shortest Path Algorithm

The single-source shortest paths (SSSP) problem in graphs with non-negative edge weights is one of the most classic problems in algorithms. For decades, the best known running time in the comparison-addition model was the $O(m+n\log n)$ bound of Dijkstra's algorithm with Fibonacci heaps. Recently, Duan, Mao, Shu, and Yin (FOCS'23) gave a randomized $O(m\log^{1/2} n \log\log^{1/2} n)$-time algorithm for SSSP in weighted undirected graphs. For weighted directed graphs, Duan, Mao, Mao, Shu, and Yin (STOC'25) gave an $O(m\log^{2/3} n)$-time algorithm for SSSP. Very recently, Duan, Mao, Shu, and Yin (ICALP'26) obtained an algorithm for directed graphs whose running time matches the $O(m\log^{1/2} n \log\log^{1/2} n)$ time of the undirected case. In this paper, we present a faster algorithm for SSSP in weighted undirected graphs, giving the first improvement in running time since the FOCS'23 breakthrough of Duan, Mao, Shu, and Yin. Our algorithm runs in $O(m\log^{1/2} n \log\log^{1/4} n \log\log\log^{1/4} n)$ time, improving the previous running time by a factor of $(\frac{\log\log n}{\log\log\log n})^{1/4}$. Our main contribution is a simple and efficient tool that computes, for every vertex, its distance to the nearest vertex in a random sample; this tool may be of independent interest.

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arXiv2026

Faster Algorithms for $(2k-1)$-Stretch Distance Oracles

Let $G=(V, E)$ be an undirected $n$-vertices $m$-edges graph with non-negative edge weights. In this paper, we present three new algorithms for constructing a $(2k-1)$-stretch distance oracle with $O(n^{1+\frac{1}{k}})$ space. The first algorithm runs in $\Ot(\max(n^{1+2/k}, m^{1-\frac{1}{k-1}}n^{\frac{2}{k-1}}))$ time, and improves upon the $\Ot(\min(mn^{\frac{1}{k}},n^2))$ time of Thorup and Zwick [STOC 2001, JACM 2005] and Baswana and Kavitha [FOCS 2006, SICOMP 2010], for every $k > 2$ and $m=Ω(n^{1+\frac{1}{k}+\eps})$. This yields the first truly subquadratic time construction for every $2 < k < 6$, and nearly resolves the open problem posed by Wulff-Nilsen [SODA 2012] on the existence of such constructions. The two other algorithms have a running time of the form $\Ot(m+n^{1+f(k)})$, which is near linear in $m$ if $m=Ω(n^{1+f(k)})$, and therefore optimal in such graphs. One algorithm runs in $\Ot(m+n^{\frac32+\frac{3}{4k-6}})$-time, which improves upon the $\Ot(n^2)$-time algorithm of Baswana and Kavitha [FOCS 2006, SICOMP 2010], for $3 < k < 6$, and upon the $\Ot(m+n^{\frac{3}{2}+\frac{2}{k}+O(k^{-2})})$-time algorithm of Wulff-Nilsen [SODA 2012], for every $k\geq 6$. This is the first linear time algorithm for constructing a $7$-stretch distance oracle and a $9$-stretch distance oracle, for graphs with truly subquadratic density.\footnote{with $m=n^{2-\eps}$ for some $\eps > 0$.} The other algorithm runs in $\Ot(\sqrt{k}m+kn^{1+\frac{2\sqrt{2}}{\sqrt{k}}})$ time, (and hence relevant only for $k\ge 16$), and improves upon the $\Ot(\sqrt{k}m+kn^{1+\frac{2\sqrt{6}}{\sqrt{k}}+O(k^{-1})})$ time algorithm of Wulff-Nilsen [SODA 2012] (which is relevant only for $k\ge 96$). ...

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arXiv2026

Tighter bounds for weighted and unweighted shortest cycle approximation

We study the problem of approximating the length of a shortest cycle in a given graph, known as the girth of the graph. The state-of-the-art approximation algorithms for unweighted graphs by Kadria et al. [SODA'22] and Roditty and Trabelsi [arXiv'25] achieve the following trade-off: for every integer $k\geq 2$, there is an $\tilde{O}(n^{1+2/k})$ time algorithm that achieves a $(2k/3)$-approximation for the girth in unweighted $n$-node graphs. The first result of this paper is to achieve the same trade-off for $m$-edge, $n$-node graphs with non-negative real edge weights: a $2k/3$-approximation algorithm running in $\tilde{O}(m+n^{1+2/k})$ time. The dependence on $m$ is unavoidable in weighted graphs. Our result improves on the work of Kadria et al.~[SODA'23] and Ducoffe [ICALP'19 and SIDMA'21], who were only able to achieve such a trade-off for some values of $k$. We also prove new fine-grained lower bounds for girth approximation and related problems in unweighted graphs.

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arXiv2026

Improved Approximation Algorithms for n-Pairs Shortest Paths

Let $G = (V, E)$ be a graph with $n = |V|$ nodes and $m = |E|$ edges. The $t$-Pairs Shortest Paths problem, introduced by Cohen [FOCS'93; SICOMP'99], asks to approximate the distances between $t$ prespecified pairs of vertices. Recently, this problem has received renewed attention, particularly in the case where $t = Θ(n)$: the $n$-Pairs Shortest Paths problem. In this setting, new algorithms and conditional lower bounds have been developed by Dalirrooyfard, Jin, Vassilevska Williams, and Wein [FOCS'22], and Chechik, Hoch, and Lifshitz [SODA'25]. In this paper, we present the first algorithm for the $n$-Pairs Shortest Paths problem in \textit{weighted} undirected graphs that achieves a $(2 - α)k$-approximation, for constant $α> 0$, that runs in $\tilde{O}(mn^{1/k} + n^{1 + 2/k})$ time. Specifically, we present a $1.622k$-approximation, improving upon the $(2k - 3)$-approximation of Chechik, Hoch, and Lifshitz [SODA'25] for graphs that are not super sparse, which answers in the affirmative the open question posed by them. We also develop improved approximation algorithms with better tradeoffs for unweighted graphs and dense weighted graphs that improve upon the results of Dalirrooyfard \etal~and Chechik, Hoch, and Lifshitz. Our main technical contribution is the new \textit{heavy-edge} technique. Using this technique, we transform an algorithm with an approximation guarantee that depends on $W_{uv}$, the weight of the heaviest edge on the shortest path between $u$ and $v$, into an algorithm with purely multiplicative approximation that does not depend on $W_{uv}$.

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arXiv2025

Compact routing schemes in undirected and directed graphs

In this paper, we study the problem of compact routing schemes in weighted undirected and directed graphs. \textit{For weighted undirected graphs}, more than a decade ago, Chechik [PODC'13] presented a $\approx3.68k$-stretch compact routing scheme that uses $\tilde{O}(n^{1/k}\log{D})$ local storage, where $D$ is the normalized diameter, for every $k>1$. We present a $\approx 2.64k$-stretch compact routing scheme that uses $\tilde{O}(n^{1/k})$ local storage \textit{on average} in each vertex. This is the first compact routing scheme that uses total local storage of $\tilde{O}(n^{1+1/k})$ while achieving a $c \cdot k$ stretch, for a constant $c < 3$. In real-world network protocols, messages are usually transformed as part of a communication session between two parties. Therefore, more than two decades ago, Thorup and Zwick [SPAA'01] considered compact routing schemes that establish a communication session using a handshake. In their handshake-based compact routing scheme, the handshake is routed along a $(4k-5)$-stretch path, and the rest of the communication session is routed along an optimal $(2k-1)$-stretch path. It is straightforward to improve the $(4k-5)$-stretch of the handshake to $\approx3.68k$-stretch using the compact routing scheme of Chechik [PODC'13]. We improve the handshake stretch to the optimal $(2k-1)$, by borrowing the concept of roundtrip routing from directed graphs to \textit{undirected} graphs. \textit{For weighted directed graphs}, more than two decades ago, Roditty, Thorup, and Zwick [SODA'02 and TALG'08] presented a $(4k+\eps)$-stretch compact roundtrip routing scheme that uses $\tilde{O}(n^{1/k})$ local storage for every $k\ge 3$. For $k=3$, this gives a $(12+\eps)$-roundtrip stretch using $\tilde{O}(n^{1/3})$ local storage. We improve the stretch by developing a $7$-roundtrip stretch routing scheme with $\tilde{O}(n^{1/3})$ local storage.

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arXiv2025

New approximate distance oracles and their applications

Let $G = (V, E)$ be an undirected graph with $n$ vertices and $m$ edges, and let $μ= m/n$. A \emph{distance oracle} is a data structure designed to answer approximate distance queries, with the goal of achieving low stretch, efficient space usage, and fast query time. While much of the prior work focused on distance oracles with constant query time, this paper presents a comprehensive study of distance oracles with non-constant query time. We explore the tradeoffs between space, stretch, and query time of distance oracles in various regimes. Specifically, we consider both weighted and unweighted graphs in the regimes of stretch $< 2$ and stretch $\ge 2$. Among our results, we present a new three-way trade-off between stretch, space, and query time, offering a natural extension of the classical Thorup-Zwick distance oracle [STOC'01 and JACM'05] to regimes with larger query time. Specifically, for any $0 < r < 1/2$ and integer $k \ge 1$, we construct a $(2k(1 - 2r) - 1)$-stretch distance oracle with $\tilde{O}(m + n^{1 + 1/k})$ space and $\tilde{O}(μn^r)$ query time. In addition, we demonstrate several applications of our new distance oracles to the $n$-Pairs Shortest Paths ($n$-PSP) problem and the All Nodes Shortest Cycles ($ANSC$) problem.

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arXiv2025

Improved girth approximation in weighted undirected graphs

Let $G = (V,E,\ell)$ be a $n$-node $m$-edge weighted undirected graph, where $\ell: E \rightarrow (0,\infty)$ is a real \emph{length} function defined on its edges, and let $g$ denote the girth of $G$, i.e., the length of its shortest cycle. We present an algorithm that, for any input, integer $k \geq 1$, in $O(kn^{1+1/k}\log{n} + m(k+\log{n}))$ expected time finds a cycle of length at most $\frac{4k}{3}g$. This algorithm nearly matches a $O(n^{1+1/k}\log{n})$-time algorithm of \cite{KadriaRSWZ22} which applied to unweighted graphs of girth $3$. For weighted graphs, this result also improves upon the previous state-of-the-art algorithm that in $O((n^{1+1/k}\log n+m)\log (nM))$ time, where $\ell: E \rightarrow [1, M]$ is an integral length function, finds a cycle of length at most $2kg$~\cite{KadriaRSWZ22}. For $k=1$ this result improves upon the result of Roditty and Tov~\cite{RodittyT13}.

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