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Liam Roditty

Publications and source records attributed to Liam Roditty.

At least 19 recordsLinked to original sources
arXiv2026

First Near-Additive All-Pairs Approximate Shortest Paths in Decremental Weighted Graphs

We present the first four All-Pairs Approximate Shortest Paths (APASP) algorithms for decremental weighted undirected graphs. We extend the hierarchy induced edge sets of Baswana and Kavitha [FOCS'06 and SICOMP'10] to the decremental setting. At any point in time, for any $u,v$, $W_{i}$ denotes the weight of the $i$-th heaviest edge on any shortest path between them in the current graph. For any $\varepsilon>0$, our first algorithm maintains $(1+\varepsilon,2W_{2})$-APASP in $\tilde{O}(n^{\frac{3}{2}+o(1)}m^{\frac{1}{2}})$ total update time, matching a single static computation of Cohen and Zwick [SODA'97 and JALG'01], up to a subpolynomial factor. It also improves the stretch of the $(2+\varepsilon)$-APASP algorithm of Dory, Forster, Nazari and de Vos [ICALP'24]. Our second algorithm maintains $(1+\varepsilon,2W_{1}+2W_{2})$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time, improving upon Chechik's $(3+\varepsilon)$-APASP algorithm [FOCS'18] for $m=Ω(n^{\frac{11}{6}})$. We also obtain two $(1+\varepsilon,2\cdot(k-1)W_{1})$-APASP algorithms with total update times $\tilde{O}(n^{2-\frac{1}{k}+o(1)}m^{\frac{1}{k}})$ and $\tilde{O}(n^{2+\frac{2}{3k-2}+o(1)})$, adapting the static $+2\sum_{i=1}^{k-1}W_{i}$-APASP algorithms of Cohen and Zwick [SODA'97 and JALG'01] and Roditty and Sapir [FSTTCS'25]. For unweighted graphs, we present $(1+\varepsilon,2)$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time and $(1+\varepsilon,2\cdot(k-1))$-APASP in $\tilde{O}(n^{2+\frac{1}{3k-4}+o(1)})$. Combined with the results of Dory, Forster, Nazari and de Vos [ICALP'24], these match the fastest combinatorial static bounds of Dor, Halperin and Zwick [FOCS'96 and SICOMP'00] up to subpolynomial factors.

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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

Hardness Results for Approximate Distance Labeling in Sparse Graphs

Distance labeling schemes assign short labels to the vertices of a graph so that distances between pairs of vertices can be recovered, exactly or approximately, from the two corresponding labels alone. For sparse graphs, Kosowski, Uznański, and Viennot [PODC'2019] showed that, assuming the hardness of the Sum-Index communication problem, exact distance labeling for sparse unweighted graphs requires near-linear label size. Their construction also gives lower bounds for stretch $1+o(1)$, but does not rule out substantially smaller labels for stretch $1+\varepsilon$, where $\varepsilon>0$ is fixed. We prove lower bounds for this $(1+\varepsilon)$-stretch regime through a density/stretch tradeoff. For every constant integer $k\ge 1$, we show that any approximate distance labeling scheme with stretch strictly smaller than $1+1/k$ for unweighted graphs with $m=O(n^{1+1/k})$ edges requires maximum label size at least $\mathrm{SumIndex}(n)/2^{O_k(\sqrt{\log n})}$ for infinitely many values of $n$. Thus, under the conjecture that $\mathrm{SumIndex}(n)=n^{1-o(1)}$, the maximum label size must be $n^{1-o(1)}$. The main new ingredient is an unweighted graph construction based on $3$-AP-free sets, which replaces the weighted uniqueness mechanism of the previous Sum-Index reduction by an additive uniqueness mechanism and keeps the relevant critical paths short enough to obtain a constant multiplicative distance gap. The same construction also yields near-tight lower bounds for related $2$-hop structures, including hub labeling schemes, shortcut sets, and approximate distance hopsets.

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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

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

Weighted Emulators with Local Heaviest Edges Stretch for Undirected Graphs

We introduce a generalized family of $\left( 2\cdot \left\lfloor \frac{k}{2} \right\rfloor-1, 2\cdot \left\lceil \frac{k}{2} \right\rceil \cdot W_{1} +\max\left\{0,2\cdot\left(\left\lceil\frac{k}{2}\right\rceil-2\right)\right\}\cdot W_{2} \right)$-emulators with $\tilde O \left(n^{1+\frac{1}{k}}\right)$ edges, for any $k\in\mathbb{N}$, where $W_{i}$ is the $i$th heaviest edge on a shortest path between two vertices. Our construction generalizes the $+2W_{1}$-spanner of size $\tilde O\left(n^{\frac{3}{2}}\right)$ and the $+4W_{1}$-emulator of size $\tilde O \left(n^{\frac{4}{3}}\right)$, both by Elkin, Gitlitz and Neiman [DISC'21 and DICO'23]. When $k$ is even, these are $\left(k-1,k\cdot W_{1} + \left(k-4\right)\cdot W_{2}\right)$-emulators and when $k$ is odd, these are $\left(k-2,\left(k+1\right)\cdot W_{1} + \left(k-3\right) \cdot W_{2}\right)$-emulators. Our framework not only expands known constructions for weighted graphs but also yields an improved stretch over state of the art emulators and spanners for unweighted graphs within a specific distance regime. In particular, for all vertex pairs separated by a distance of $δ\leq O\left(3^{k^{2}}\right)$, our construction improves upon the seminal additive $+\tilde O\left(δ^{1-\frac{1}{k}}\right)$-emulator of size $\tilde O\left(n^{1+\frac{1}{2^{k+1}-1}}\right)$ by Thorup and Zwick [SODA'06].

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arXiv2026

Additive, Near-Additive, and Multiplicative Approximations for APSP in Weighted Undirected Graphs: Trade-offs and Algorithms

We present a $+2\sum_{i=1}^{k+1}{W_i}$-APASP algorithm for dense weighted graphs with runtime $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$, where $W_{i}$ is the weight of an $i^{th}$ heaviest edge on a shortest path. Dor, Halperin and Zwick [FOCS'96, SICOMP'00] had two algorithms for the commensurate unweighted $+2\cdot\left( k+1\right)$-APASP: $\tilde O\left(n^{2-\frac{1}{k+2}}m^{\frac{1}{k+2}}\right)$ runtime for sparse graphs and $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$ runtime for dense graphs. Cohen and Zwick [SODA'97, JALG'01] adapted the sparse variant to weighted graphs: $+2\sum_{i=1}^{k+1}{W_i}$-APASP algorithm in the same runtime. We show an algorithm for dense weighted graphs. For nearly additive APASP, we present a $\left(1+\varepsilon,\min{\left\{2W_1,4W_{2}\right\}}\right)$-APASP algorithm with $\tilde O\left(\left(\frac{1}{\varepsilon}\right)^{O\left(1\right)}\cdot n^{2.15135313}\cdot\log W\right)$ runtime. This improves the $\left(1+\varepsilon,2W_1\right)$-APASP of Saha and Ye [SODA'24]. For multiplicative APASP, we show a framework of $\left(\frac{3\ell +4}{\ell + 2}+\varepsilon\right)$-APASP algorithms, reducing the runtime of Akav and Roditty [ESA'21] for dense graphs and generalizing the $\left(2+\varepsilon\right)$-APASP algorithm of Dory et al [SODA'24]. Our base case is a $\left(\frac{7}{3}+\varepsilon\right)$-APASP in $\tilde O\left(\left(\frac{1}{\varepsilon}\right)^{O\left(1\right)}\cdot n^{2.15135313}\cdot \log W\right)$ runtime, improving the $\frac{7}{3}$-APASP algorithm of Baswana and Kavitha [FOCS'06, SICOMP'10] for dense graphs. Finally, we "bypass" an $\tilde Ω\left(n^ω\right)$ conditional lower bound by Dor, Halperin, and Zwick for $α$-APASP with $α< 2$, by allowing an additive term (e.g. $\left(\frac{6k+3}{3k+2},\sum_{i=1}^{k+1}W_{i}\right)$-APASP in $\tilde O\left(n^{2+\frac{1}{3k+2}}\right)$ runtime).

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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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arXiv2026

New algorithms for girth and cycle detection

Let $G=(V,E)$ be an unweighted undirected graph with $n$ vertices and $m$ edges. Let $g$ be the girth of $G$, that is, the length of a shortest cycle in $G$. We present a randomized algorithm with a running time of $\tilde{O}\big(\ell \cdot n^{1 + \frac{1}{\ell - \varepsilon}}\big)$ that returns a cycle of length at most $ 2\ell \left\lceil \frac{g}{2} \right\rceil - 2 \left\lfloor \varepsilon \left\lceil \frac{g}{2} \right\rceil \right\rfloor, $ where $\ell \geq 2$ is an integer and $\varepsilon \in [0,1]$, for every graph with $g = polylog(n)$. Our algorithm generalizes an algorithm of Kadria \etal{} [SODA'22] that computes a cycle of length at most $4\left\lceil \frac{g}{2} \right\rceil - 2\left\lfloor \varepsilon \left\lceil \frac{g}{2} \right\rceil \right\rfloor $ in $\tilde{O}\big(n^{1 + \frac{1}{2 - \varepsilon}}\big)$ time. Kadria \etal{} presented also an algorithm that finds a cycle of length at most $ 2\ell \left\lceil \frac{g}{2} \right\rceil $ in $\tilde{O}\big(n^{1 + \frac{1}{\ell}}\big)$ time, where $\ell$ must be an integer. Our algorithm generalizes this algorithm, as well, by replacing the integer parameter $\ell$ in the running time exponent with a real-valued parameter $\ell - \varepsilon$, thereby offering greater flexibility in parameter selection and enabling a broader spectrum of combinations between running times and cycle lengths. We also show that for sparse graphs a better tradeoff is possible, by presenting an $\tilde{O}(\ell\cdot m^{1+1/(\ell-\varepsilon)})$ time randomized algorithm that returns a cycle of length at most $2\ell(\lfloor \frac{g-1}{2}\rfloor) - 2(\lfloor \varepsilon \lfloor \frac{g-1}{2}\rfloor \rfloor+1)$, where $\ell\geq 3$ is an integer and $\varepsilon\in [0,1)$, for every graph with $g=polylog(n)$. To obtain our algorithms we develop several techniques and introduce a formal definition of hybrid cycle detection algorithms. [...]

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arXiv2026

New Diameter Approximations via Distance Oracle Techniques

Computing the diameter of a graph is a problem of great interest both in general algorithms research and specifically within fine-grained complexity, where it is a cornerstone hard problem. Recent work has achieved a full conditional lower bound tradeoff curve for both directed and undirected graphs. However, the best known upper bounds do not match the lower bounds. In particular, the best known approximation scheme for undirected graph diameter has not been improved. Moreover, this scheme is randomized and no similar deterministic scheme is known. Another fundamental field of research in shortest paths computation is the construction of approximate distance oracles. Thorup and Zwick [JACM'05] provided the first such distance oracle with constant query time and (conditionally) optimal space, and in the years since many advances have led to a vast toolbox of techniques and data structures. These two areas of research seem natural to combine since they both concern approximating shortest paths. However, the known diameter approximation algorithms only use a small subset of the techniques used in distance oracles research. In this work we show that in fact approximate diameter and distance oracles are intricately connected. We first demonstrate a strong connection between the current best known diameter approximation scheme of Cairo, Grossi and Rizzi ("CGR") and the $(2k-1)$-approximate distance oracle of Thorup and Zwick. This allows us to derandomize the CGR algorithm and obtain the first deterministic diameter approximation tradeoff. We further derandomize other central techniques in the field of distance oracles and use them to achieve new deterministic diameter approximation algorithms. Finally, we show how these new techniques can be used to derandomize many current best known results in various fields of shortest paths approximations.

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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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arXiv2024

On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch

For an undirected unweighted graph G = (V, E) with n vertices and m edges, let d(u, v) denote the distance from u in V to v in V in G. An (alpha, beta)-stretch approximate distance oracle (ADO) for G is a data structure that, given u, v in V, returns in constant time a value d-hat (u, v) such that d(u, v) <= d-hat (u, v) <= alpha * d(u, v) + beta, for some reals alpha > 1, beta. If beta = 0, we say that the ADO has stretch alpha. Thorup and Zwick (2005) showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Patrascu and Roditty (2010) showed that one can obtain stretch 2 using O(m^(1/3)n^(4/3)) space, and so if m is subquadratic in n, then the space usage is also subquadratic. Moreover, Patrascu and Roditty (2010) showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = O-tilde(n), based on the set-intersection hypothesis. In this paper, we investigate the minimum possible stretch achievable by an ADO as a function of the graph's maximum degree, a study motivated by the question of identifying the conditions under which an ADO can be stored with subquadratic space while still ensuring a sub-2 stretch. In particular, we show that if the maximum degree in G is Delta_G <= O(n^(1/k - epsilon)) for some 0 < epsilon <= 1/k, then there exists a (2, 1 - k)-stretch ADO for G that uses O-tilde(n^(2 - (k * epsilon) / 3)) space. For k = 2, this result implies a subquadratic sub-2 stretch ADO for graphs with Delta_G <= O(n^(1/2 - epsilon)). We provide tight lower bounds for the upper bound under the same set intersection hypothesis, showing that if Delta_G = Theta(n^(1/k)), a (2, 1 - k)-stretch ADO requires Omega-tilde(n^2) space. Moreover, we show that for constants epsilon, c > 0, a (2 - epsilon, c)-stretch ADO requires Omega-tilde(n^2) space even for graphs with Delta_G = Theta-tilde(1).

cs.DS↗
arXiv2024

The Complexity of Manipulation of k-Coalitional Games on Graphs

In many settings, there is an organizer who would like to divide a set of agents into $k$ coalitions, and cares about the friendships within each coalition. Specifically, the organizer might want to maximize utilitarian social welfare, maximize egalitarian social welfare, or simply guarantee that every agent will have at least one friend within his coalition. However, in many situations, the organizer is not familiar with the friendship connections, and he needs to obtain them from the agents. In this setting, a manipulative agent may falsely report friendship connections in order to increase his utility. In this paper, we analyze the complexity of finding manipulation in such $k$-coalitional games on graphs. We also introduce a new type of manipulation, socially-aware manipulation, in which the manipulator would like to increase his utility without decreasing the social welfare. We then study the complexity of finding socially-aware manipulation in our setting. Finally, we examine the frequency of socially-aware manipulation and the running time of our algorithms via simulation results.

cs.GT↗