arXiv subjects

Oren Weimann

Publications and source records attributed to Oren Weimann.

At least 19 recordsLinked to original sources

arXiv2026

$\tilde{\text{O}}$ptimal Distributed Maximum Flow Approximation in Undirected Planar Graphs

Persistent efforts in recent years have been devoted to devising distributed algorithms for fundamental optimization problems in planar graphs. In particular, for Single-Source Shortest-Paths, there is an $\tilde O(D^2)$-rounds exact algorithm [Li, Parter STOC'19] for directed planar graphs, and an $\tilde {O}(D)$-rounds $(1+o(1))$-approximation algorithm [Rozhon, Grunau, Haeupler, Zuzic, Li STOC'22] for undirected planar graphs (where $D$ is the graph's hop-diameter). Recently [Abd-Elhaleem, Dory, Parter, Weimann PODC'25], a matching bound for the exact case was obtained for the Maximum $st$-Flow problem. Namely, an $\tilde O(D^2)$-rounds exact algorithm for directed planar graphs. However, for the approximate case, they give a $D\cdot n^{o(1)}$-rounds $(1-o(1))$-approximation algorithm for undirected planar graphs that works only for the special case where both $s$ and $t$ lie on the same face. In this paper, we remove the restriction that both $s$ and $t$ must lie on the same face (we also eliminate the $n^{o(1)}$ factor). Namely, we present the first distributed near-optimal $\tilde{O}(D)$-rounds $(1-o(1))$-approximation algorithm for Maximum $st$-Flow in general undirected planar graphs. Our main technical contribution is a distributed implementation of the classical Reif's [SICOMP'83] centralized algorithm. This is achieved by a careful recursive incision procedure on the planar dual $G^*$ of the graph $G$. It is challenging, because we need to simulate dynamic changes (incisions) over the dual graph $G^*$, while we can only communicate over the input graph $G$.

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arXiv2026

A Tight Bound for Facial Distance Patterns in Planar Graphs

Let $G$ be an undirected unweighted planar graph and let $S=(s_0,\dots,s_{k-1})$ be the vertices of a designated face, listed in cyclic order. Consider a vector that stores the distances from an arbitrary vertex $v$ to all vertices of $S$. The pattern of $v$ is obtained by taking the difference between every pair of consecutive values in this vector. Li and Parter [STOC'19] proved an upper bound of $O(k^3)$ on the number of unique patterns over all vertices of $G$. We improve this to $O(k^2)$, matching a known lower bound and settling a conjecture in [ISAAC'22]. The simple proof was found by OpenAI's GPT 5.6-Sol model. Plugging this new bound into known results has the following three immediate implications for undirected unweighted planar graphs: (1) it gives an improved compression of the Okamura-Seymour metric (2) it improves the space required by constant-time exact distance oracles, and (3) it improves the fastest distributed algorithm for computing the diameter. We further present a previously unknown and nontrivial implication: a (centralized) $\tilde{O}(n^{8/5})$-time algorithm for computing the diameter, improving over the $\tilde{O}(n^{5/3})$ algorithm of [SODA'18] which works for weighted directed planar graphs. Thus, there is currently a gap between the time for computing the diameter between weighted and unweighted planar graphs.

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arXiv2026

A Simple Distributed Deterministic Planar Separator

A balanced separator of a graph $G$ is a set of vertices whose removal disconnects the graph into connected components that are a constant factor smaller than $G$. Lipton and Tarjan [FOCS'77] famously proved that every planar graph admits a balanced separator of size $O(\sqrt{n})$, as well as a balanced separator of size $O(D)$ that is a simple path (where $D$ is $G$'s diameter). In the centralized setting, both separators can be found in linear time. In the distributed setting, $D$ is a universal lower bound for the round complexity of solving many optimization problems, so, separators of size $O(D)$ are preferable. It was not until [DISC'17] that a distributed algorithm was devised by Ghaffari and Parter to compute such an $O(D)$-size separator in $\tilde O(D)$ rounds, by adapting the Lipton-Tarjan algorithm to the distributed model. Since then, this algorithm was used in several distributed algorithms for planar graphs, e.g., [GP, DISC'17], [LP, STOC'19], [AEDPW, PODC'25]. However, the algorithm is randomized, deeming the algorithms that use it to be randomized as well. Obtaining a deterministic algorithm remained an interesting open question until [PODC'25], when a (complex) deterministic separator algorithm was given by Jauregui, Montealegre and Rapaport. We present a much simpler deterministic separator algorithm with the same (near-optimal) $\tilde O(D)$-round complexity. While previous works devised either complicated or randomized ways of transferring weights from vertices to faces of $G$, we show that a straightforward way also works: Each vertex simply transfers its weight to one arbitrary face it lies on. That's it! We note that a deterministic separator algorithm directly derandomizes the state-of-the-art distributed algorithms for classical problems on planar graphs such as single-source shortest-paths, maximum-flow, directed global min-cut, and reachability.

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arXiv2025

Distributed Maximum Flow in Planar Graphs

The dual of a planar graph $G$ is a planar graph $G^*$ that has a vertex for each face of $G$ and an edge for each pair of adjacent faces of $G$. The profound relationship between a planar graph and its dual has been the algorithmic basis for solving numerous (centralized) classical problems on planar graphs. In the distributed setting however, the only use of planar duality is for finding a recursive decomposition of $G$ [DISC 2017, STOC 2019]. We extend the distributed algorithmic toolkit to work on the dual graph $G^*$. These tools can then facilitate various algorithms on $G$ by solving a suitable dual problem on $G^*$. Given a directed planar graph $G$ with positive and negative edge-lengths and hop-diameter $D$, our key result is an $\tilde{O}(D^2)$-round algorithm for Single Source Shortest Paths on $G^*$, which then implies $\tilde{O}(D^2)$-round algorithms for Maximum $st$-Flow and Directed Global Min-Cut on $G$. Prior to our work, no $\tilde{O}(\text{poly}(D))$-round algorithm was known for those problems. We further obtain a $D\cdot n^{o(1)}$-rounds $(1-ε)$-approximation algorithm for Maximum $st$-Flow on $G$ when $G$ is undirected and $st$-planar. Finally, we give a near optimal $\tilde O(D)$-round algorithm for computing the weighted girth of $G$. The main challenges in our work are that $G^*$ is not the communication graph (e.g., a vertex of $G$ is mapped to multiple vertices of $G^*$), and that the diameter of $G^*$ can be much larger than $D$ (i.e., possibly by a linear factor). We overcome these challenges by carefully defining and maintaining subgraphs of the dual graph $G^*$ while applying the recursive decomposition on the primal graph $G$. The main technical difficulty, is that along the recursive decomposition, a face of $G$ gets shattered into (disconnected) components yet we still need to treat it as a dual node.

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arXiv2025

Faster Construction of a Planar Distance Oracle with Õ(1) Query Time

We show how to preprocess a weighted undirected $n$-vertex planar graph in $\tilde O(n^{4/3})$ time, such that the distance between any pair of vertices can then be reported in $\tilde O(1)$ time. This improves the previous $\tilde O(n^{3/2})$ preprocessing time [JACM'23]. Our main technical contribution is a near optimal construction of \emph{additively weighted Voronoi diagrams} in undirected planar graphs. Namely, given a planar graph $G$ and a face $f$, we show that one can preprocess $G$ in $\tilde O(n)$ time such that given any weight assignment to the vertices of $f$ one can construct the additively weighted Voronoi diagram of $f$ in near optimal $\tilde O(|f|)$ time. This improves the $\tilde O(\sqrt{n |f|})$ construction time of [JACM'23].

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arXiv2025

Õptimal Fault-Tolerant Labeling for Reachability and Approximate Distances in Directed Planar Graphs

We present a labeling scheme that assigns labels of size $\tilde O(1)$ to the vertices of a directed weighted planar graph $G$, such that for any fixed $\varepsilon>0$ from the labels of any three vertices $s$, $t$ and $f$ one can determine in $\tilde O(1)$ time a $(1+\varepsilon)$-approximation of the $s$-to-$t$ distance in the graph $G\setminus\{f\}$. For approximate distance queries, prior to our work, no efficient solution existed, not even in the centralized oracle setting. Even for the easier case of reachability, $\tilde O(1)$ queries were known only with a centralized oracle of size $\tilde O(n)$ [SODA 21].

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arXiv2024

The Fine-Grained Complexity of Episode Matching

Given two strings $S$ and $P$, the Episode Matching problem is to find the shortest substring of $S$ that contains $P$ as a subsequence. The best known upper bound for this problem is $\tilde O(nm)$ by Das et al. (1997) , where $n,m$ are the lengths of $S$ and $P$, respectively. Although the problem is well studied and has many applications in data mining, this bound has never been improved. In this paper we show why this is the case by proving that no $O((nm)^{1-ε})$ algorithm (even for binary strings) exists, unless the Strong Exponential Time Hypothesis (SETH) is false. We then consider the indexing version of the problem, where $S$ is preprocessed into a data structure for answering episode matching queries $P$. We show that for any $τ$, there is a data structure using $O(n+\left(\frac{n}τ\right)^k)$ space that answers episode matching queries for any $P$ of length $k$ in $O(k\cdot τ\cdot \log \log n )$ time. We complement this upper bound with an almost matching lower bound, showing that any data structure that answers episode matching queries for patterns of length $k$ in time $O(n^δ)$, must use $Ω(n^{k-kδ-o(1)})$ space, unless the Strong $k$-Set Disjointness Conjecture is false. Finally, for the special case of $k=2$, we present a faster construction of the data structure using fast min-plus multiplication of bounded integer matrices.

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arXiv2023

Õptimal Fault-Tolerant Reachability Labeling in Planar Graphs

We show how to assign labels of size $\tilde O(1)$ to the vertices of a directed planar graph $G$, such that from the labels of any three vertices $s,t,f$ we can deduce in $\tilde O(1)$ time whether $t$ is reachable from $s$ in the graph $G\setminus \{f\}$. Previously it was only known how to achieve $\tilde O(1)$ queries using a centralized $\tilde O(n)$ size oracle [SODA'21].

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arXiv2023

What Else Can Voronoi Diagrams Do For Diameter In Planar Graphs?

The Voronoi diagrams technique was introduced by Cabello to compute the diameter of planar graphs in subquadratic time. We present novel applications of this technique in static, fault-tolerant, and partially-dynamic undirected unweighted planar graphs, as well as some new limitations. 1. In the static case, we give $n^{3+o(1)}/D^2$ and $\tilde{O}(n\cdot D^2)$ time algorithms for computing the diameter of a planar graph $G$ with diameter $D$. These are faster than the state of the art $\tilde{O}(n^{5/3})$ when $D n^{2/3}$. 2. In the fault-tolerant setting, we give an $n^{7/3+o(1)}$ time algorithm for computing the diameter of $G\setminus \{e\}$ for every edge $e$ in $G$ the replacement diameter problem. Compared to the naive $\tilde{O}(n^{8/3})$ time algorithm that runs the static algorithm for every edge. 3. In the incremental setting, where we wish to maintain the diameter while while adding edges, we present an algorithm with total running time $n^{7/3+o(1)}$. Compared to the naive $\tilde{O}(n^{8/3})$ time algorithm that runs the static algorithm after every update. 4. We give a lower bound (conditioned on the SETH) ruling out an amortized $O(n^{1-\varepsilon})$ update time for maintaining the diameter in *weighted* planar graph. The lower bound holds even for incremental or decremental updates. Our upper bounds are obtained by novel uses and manipulations of Voronoi diagrams. These include maintaining the Voronoi diagram when edges of the graph are deleted, allowing the sites of the Voronoi diagram to lie on a BFS tree level (rather than on boundaries of $r$-division), and a new reduction from incremental diameter to incremental distance oracles that could be of interest beyond planar graphs. Our lower bound is the first lower bound for a dynamic planar graph problem that is conditioned on the SETH.

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arXiv2023

Near-Optimal Dynamic Time Warping on Run-Length Encoded Strings

We give an $\tilde O(n^2)$ time algorithm for computing the exact Dynamic Time Warping distance between two strings whose run-length encoding is of size at most $n$. This matches (up to log factors) the known (conditional) lower bound, and should be compared with the previous fastest $O(n^3)$ time exact algorithm and the $\tilde O(n^2)$ time approximation algorithm.

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arXiv2022

Improved Compression of the Okamura-Seymour Metric

Let $G=(V,E)$ be an undirected unweighted planar graph. Consider a vector storing the distances from an arbitrary vertex $v$ to all vertices $S = \{ s_1 , s_2 , \ldots , s_k \}$ of a single face in their cyclic order. The pattern of $v$ is obtained by taking the difference between every pair of consecutive values of this vector. In STOC'19, Li and Parter used a VC-dimension argument to show that in planar graphs, the number of distinct patterns, denoted $x$, is only $O(k^3)$. This resulted in a simple compression scheme requiring $\tilde O(\min \{ k^4+|T|, k\cdot |T|\})$ space to encode the distances between $S$ and a subset of terminal vertices $T \subseteq V$. This is known as the Okamura-Seymour metric compression problem. We give an alternative proof of the $x=O(k^3)$ bound that exploits planarity beyond the VC-dimension argument. Namely, our proof relies on cut-cycle duality, as well as on the fact that distances among vertices of $S$ are bounded by $k$. Our method implies the following: (1) An $\tilde{O}(x+k+|T|)$ space compression of the Okamura-Seymour metric, thus improving the compression of Li and Parter to $\tilde O(\min \{k^3+|T|,k \cdot |T| \})$. (2) An optimal $\tilde{O}(k+|T|)$ space compression of the Okamura-Seymour metric, in the case where the vertices of $T$ induce a connected component in $G$. (3) A tight bound of $x = Θ(k^2)$ for the family of Halin graphs, whereas the VC-dimension argument is limited to showing $x=O(k^3)$.

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arXiv2021

On the Fine-Grained Complexity of Parity Problems

We consider the parity variants of basic problems studied in fine-grained complexity. We show that finding the exact solution is just as hard as finding its parity (i.e. if the solution is even or odd) for a large number of classical problems, including All-Pairs Shortest Paths (APSP), Diameter, Radius, Median, Second Shortest Path, Maximum Consecutive Subsums, Min-Plus Convolution, and $0/1$-Knapsack. A direct reduction from a problem to its parity version is often difficult to design. Instead, we revisit the existing hardness reductions and tailor them in a problem-specific way to the parity version. Nearly all reductions from APSP in the literature proceed via the (subcubic-equivalent but simpler) Negative Weight Triangle (NWT) problem. Our new modified reductions also start from NWT or a non-standard parity variant of it. We are not able to establish a subcubic-equivalence with the more natural parity counting variant of NWT, where we ask if the number of negative triangles is even or odd. Perhaps surprisingly, we justify this by designing a reduction from the seemingly-harder Zero Weight Triangle problem, showing that parity is (conditionally) strictly harder than decision for NWT.

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arXiv2021

An Almost Optimal Edit Distance Oracle

We consider the problem of preprocessing two strings $S$ and $T$, of lengths $m$ and $n$, respectively, in order to be able to efficiently answer the following queries: Given positions $i,j$ in $S$ and positions $a,b$ in $T$, return the optimal alignment of $S[i \mathinner{.\,.} j]$ and $T[a \mathinner{.\,.} b]$. Let $N=mn$. We present an oracle with preprocessing time $N^{1+o(1)}$ and space $N^{1+o(1)}$ that answers queries in $\log^{2+o(1)}N$ time. In other words, we show that we can query the alignment of every two substrings in almost the same time it takes to compute just the alignment of $S$ and $T$. Our oracle uses ideas from our distance oracle for planar graphs [STOC 2019] and exploits the special structure of the alignment graph. Conditioned on popular hardness conjectures, this result is optimal up to subpolynomial factors. Our results apply to both edit distance and longest common subsequence (LCS). The best previously known oracle with construction time and size $\mathcal{O}(N)$ has slow $Ω(\sqrt{N})$ query time [Sakai, TCS 2019], and the one with size $N^{1+o(1)}$ and query time $\log^{2+o(1)}N$ (using a planar graph distance oracle) has slow $Ω(N^{3/2})$ construction time [Long & Pettie, SODA 2021]. We improve both approaches by roughly a $\sqrt N$ factor.

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arXiv2021

Fault-Tolerant Distance Labeling for Planar Graphs

In fault-tolerant distance labeling we wish to assign short labels to the vertices of a graph $G$ such that from the labels of any three vertices $u,v,f$ we can infer the $u$-to-$v$ distance in the graph $G\setminus \{f\}$. We show that any directed weighted planar graph (and in fact any graph in a graph family with $O(\sqrt{n})$-size separators, such as minor-free graphs) admits fault-tolerant distance labels of size $O(n^{2/3})$. We extend these labels in a way that allows us to also count the number of shortest paths, and provide additional upper and lower bounds for labels and oracles for counting shortest paths.

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arXiv2020

A Note on a Recent Algorithm for Minimum Cut

Given an undirected edge-weighted graph $G=(V,E)$ with $m$ edges and $n$ vertices, the minimum cut problem asks to find a subset of vertices $S$ such that the total weight of all edges between $S$ and $V \setminus S$ is minimized. Karger's longstanding $O(m \log^3 n)$ time randomized algorithm for this problem was very recently improved in two independent works to $O(m \log^2 n)$ [ICALP'20] and to $O(m \log^2 n + n\log^5 n)$ [STOC'20]. These two algorithms use different approaches and techniques. In particular, while the former is faster, the latter has the advantage that it can be used to obtain efficient algorithms in the cut-query and in the streaming models of computation. In this paper, we show how to simplify and improve the algorithm of [STOC'20] to $O(m \log^2 n + n\log^3 n)$. We obtain this by replacing a randomized algorithm that, given a spanning tree $T$ of $G$, finds in $O(m \log n+n\log^4 n)$ time a minimum cut of $G$ that 2-respects (cuts two edges of) $T$ with a simple $O(m \log n+n\log^2 n)$ time deterministic algorithm for the same problem.

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arXiv2020

Minimum Cut in $O(m\log^2 n)$ Time

We give a randomized algorithm that finds a minimum cut in an undirected weighted $m$-edge $n$-vertex graph $G$ with high probability in $O(m \log^2 n)$ time. This is the first improvement to Karger's celebrated $O(m \log^3 n)$ time algorithm from 1996. Our main technical contribution is a deterministic $O(m \log n)$ time algorithm that, given a spanning tree $T$ of $G$, finds a minimum cut of $G$ that 2-respects (cuts two edges of) $T$.

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arXiv2020

Voronoi diagrams on planar graphs, and computing the diameter in deterministic $\tilde{O}(n^{5/3})$ time

We present an explicit and efficient construction of additively weighted Voronoi diagrams on planar graphs. Let $G$ be a planar graph with $n$ vertices and $b$ sites that lie on a constant number of faces. We show how to preprocess $G$ in $\tilde O(nb^2)$ time (footnote: The $\tilde O$ notation hides polylogarithmic factors.) so that one can compute any additively weighted Voronoi diagram for these sites in $\tilde O(b)$ time. We use this construction to compute the diameter of a directed planar graph with real arc lengths in $\tilde{O}(n^{5/3})$ time. This improves the recent breakthrough result of Cabello (SODA'17), both by improving the running time (from $\tilde{O}(n^{11/6})$), and by providing a deterministic algorithm. It is in fact the first truly subquadratic {\em deterministic} algorithm for this problem. Our use of Voronoi diagrams to compute the diameter follows that of Cabello, but he used abstract Voronoi diagrams, which makes his diameter algorithm more involved, more expensive, and randomized. As in Cabello's work, our algorithm can compute, for every vertex $v$, both the farthest vertex from $v$ (i.e., the eccentricity of $v$), and the sum of distances from $v$ to all other vertices. Hence, our algorithm can also compute the radius, median, and Wiener index (sum of all pairwise distances) of a planar graph within the same time bounds. Our construction of Voronoi diagrams for planar graphs is of independent interest.

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arXiv2019

Top Tree Compression of Tries

We present a compressed representation of tries based on top tree compression [ICALP 2013] that works on a standard, comparison-based, pointer machine model of computation and supports efficient prefix search queries. Namely, we show how to preprocess a set of strings of total length $n$ over an alphabet of size $σ$ into a compressed data structure of worst-case optimal size $O(n/\log_σn)$ that given a pattern string $P$ of length $m$ determines if $P$ is a prefix of one of the strings in time $O(\min(m\log σ,m + \log n))$. We show that this query time is in fact optimal regardless of the size of the data structure. Existing solutions either use $Ω(n)$ space or rely on word RAM techniques, such as tabulation, hashing, address arithmetic, or word-level parallelism, and hence do not work on a pointer machine. Our result is the first solution on a pointer machine that achieves worst-case $o(n)$ space. Along the way, we develop several interesting data structures that work on a pointer machine and are of independent interest. These include an optimal data structures for random access to a grammar-compressed string and an optimal data structure for a variant of the level ancestor problem.

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