arXiv subjects

Christian Wulff-Nilsen

Publications and source records attributed to Christian Wulff-Nilsen.

At least 19 recordsLinked to original sources

arXiv2026

A Near-Optimal Offline Algorithm for Dynamic All-Pairs Shortest Paths in Planar Digraphs

In the planar, dynamic All-Pairs Shortest Paths (APSP) problem, a planar, weighted digraph $G$ undergoes a sequence of edge weight updates and the goal is to maintain a data structure on $G$, that can quickly answer distance queries between any two vertices $x,y \in V(G)$. The currently best algorithms for this problem require $\tilde{O}(n^{2/3})$ worst-case update and query time, while conditional lower bounds show that either update or query time $n^{0.5-δ}$ is needed for any constant $δ> 0$. In this article, we present the first algorithm with near-optimal $\tilde{O}(\sqrt{n})$ worst-case update and query time for the offline setting, where the update sequence is given initially. This result is obtained by giving the first offline dynamic algorithm for maintaining dense distance graphs (DDGs) faster than recomputing from scratch after each update. Further, we also present an \emph{online} algorithm for the incremental APSP problem with $\tilde{O}(\sqrt{n})$ worst-case update/ query time. This allows us to reduce the online dynamic APSP problem to the online decremental APSP problem, which constitutes partial progress even for the online version of this notorious problem.

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arXiv2025

Negative-Weight Single-Source Shortest Paths in Near-linear Time

We present a randomized algorithm that computes single-source shortest paths (SSSP) in $O(m\log^8(n)\log W)$ time when edge weights are integral and can be negative. This essentially resolves the classic negative-weight SSSP problem. The previous bounds are $\tilde O((m+n^{1.5})\log W)$ [BLNPSSSW FOCS'20] and $m^{4/3+o(1)}\log W$ [AMV FOCS'20]. Near-linear time algorithms were known previously only for the special case of planar directed graphs [Fakcharoenphol and Rao FOCS'01]. In contrast to all recent developments that rely on sophisticated continuous optimization methods and dynamic algorithms, our algorithm is simple: it requires only a simple graph decomposition and elementary combinatorial tools. In fact, ours is the first combinatorial algorithm for negative-weight SSSP to break through the classic $\tilde O(m\sqrt{n}\log W)$ bound from over three decades ago [Gabow and Tarjan SICOMP'89].

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arXiv2024

Fully Dynamic Exact Edge Connectivity in Sublinear Time

Given a simple $n$-vertex, $m$-edge graph $G$ undergoing edge insertions and deletions, we give two new fully dynamic algorithms for exactly maintaining the edge connectivity of $G$ in $\tilde{O}(n)$ worst-case update time and $\tilde{O}(m^{1-1/31})$ amortized update time, respectively. Prior to our work, all dynamic edge connectivity algorithms either assumed bounded edge connectivity, guaranteed approximate solutions, or were restricted to edge insertions only. Our results provide an affirmative answer to an open question posed by Thorup [Combinatorica'07].

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arXiv2023

VC Set Systems in Minor-free (Di)Graphs and Applications

A recent line of work on VC set systems in minor-free (undirected) graphs, starting from Li and Parter, who constructed a new VC set system for planar graphs, has given surprising algorithmic results. In this work, we initialize a more systematic study of VC set systems for minor-free graphs and their applications in both undirected graphs and directed graphs (a.k.a digraphs). More precisely: - We propose a new variant of Li-Parter set system for undirected graphs. - We extend our set system to $K_h$-minor-free digraphs and show that its VC dimension is $O(h^2)$. - We show that the system of directed balls in minor-free digraphs has VC dimension at most $h-1$. - On the negative side, we show that VC set system constructed from shortest path trees of planar digraphs does not have a bounded VC dimension. The highlight of our work is the results for digraphs, as we are not aware of known algorithmic work on constructing and exploiting VC set systems for digraphs.

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arXiv2023

A Simple Algorithm for Multiple-Source Shortest Paths in Planar Digraphs

Given an $n$-vertex planar embedded digraph $G$ with non-negative edge weights and a face $f$ of $G$, Klein presented a data structure with $O(n\log n)$ space and preprocessing time which can answer any query $(u,v)$ for the shortest path distance in $G$ from $u$ to $v$ or from $v$ to $u$ in $O(\log n)$ time, provided $u$ is on $f$. This data structure is a key tool in a number of state-of-the-art algorithms and data structures for planar graphs. Klein's data structure relies on dynamic trees and the persistence technique as well as a highly non-trivial interaction between primal shortest path trees and their duals. The construction of our data structure follows a completely different and in our opinion very simple divide-and-conquer approach that solely relies on Single-Source Shortest Path computations and contractions in the primal graph. Our space and preprocessing time bound is $O(n\log |f|)$ and query time is $O(\log |f|)$ which is an improvement over Klein's data structure when $f$ has small size.

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arXiv2022

Constructing Light Spanners Deterministically in Near-Linear Time

Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [CW18] improved the state-of-the-art for light spanners by constructing a $(2k-1)(1+ε)$-spanner with $O(n^{1+1/k})$ edges and $O_ε(n^{1/k})$ lightness. Soon after, Filtser and Solomon [FS19] showed that the classic greedy spanner construction achieves the same bounds The major drawback of the greedy spanner is its running time of $O(mn^{1+1/k})$ (which is faster than [CW16]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness $Ω_ε(kn^{1/k})$, even when randomization is used. The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an $O_ε(n^{2+1/k+ε'})$ time spanner construction which achieves the state-of-the-art bounds. Our second result is an $O_ε(m + n\log n)$ time construction of a spanner with $(2k-1)(1+ε)$ stretch, $O(\log k\cdot n^{1+1/k})$ edges and $O_ε(\log k\cdot n^{1/k})$ lightness. This is an exponential improvement in the dependence on $k$ compared to the previous result with such running time. Finally, for the important special case where $k=\log n$, for every constant $ε>0$, we provide an $O(m+n^{1+ε})$ time construction that produces an $O(\log n)$-spanner with $O(n)$ edges and $O(1)$ lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any $k = ω(1)$. To achieve our constructions, we show a novel deterministic incremental approximate distance oracle, which may be of independent interest.

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arXiv2021

Optimal Approximate Distance Oracle for Planar Graphs

A $(1+ε)$-approximate distance oracle of an edge-weighted graph is a data structure that returns an approximate shortest path distance between any two query vertices up to a $(1+ε)$ factor. Thorup (FOCS 2001, JACM 2004) and Klein (SODA 2002) independently constructed a $(1+ε)$-approximate distance oracle with $O(n\log n)$ space, measured in number of words, and $O(1)$ query time when $G$ is an undirected planar graph with $n$ vertices and $ε$ is a fixed constant. Many follow-up works gave $(1+ε)$-approximate distance oracles with various trade-offs between space and query time. However, improving $O(n\log n)$ space bound without sacrificing query time remains an open problem for almost two decades. In this work, we resolve this problem affirmatively by constructing a $(1+ε)$-approximate distance oracle with optimal $O(n)$ space and $O(1)$ query time for undirected planar graphs and fixed $ε$. We also make substantial progress for planar digraphs with non-negative edge weights. For fixed $ε> 0$, we give a $(1+ε)$-approximate distance oracle with space $o(n\log(Nn))$ and $O(\log\log(Nn)$ query time; here $N$ is the ratio between the largest and smallest positive edge weight. This improves Thorup's (FOCS 2001, JACM 2004) $O(n\log(Nn)\log n)$ space bound by more than a logarithmic factor while matching the query time of his structure. This is the first improvement for planar digraphs in two decades, both in the weighted and unweighted setting.

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arXiv2021

Near-Optimal Distance Oracles for Vertex-Labeled Planar Graphs

Given an undirected $n$-vertex planar graph $G=(V,E,ω)$ with non-negative edge weight function $ω:E\rightarrow \mathbb R$ and given an assigned label to each vertex, a vertex-labeled distance oracle is a data structure which for any query consisting of a vertex $u$ and a label $λ$ reports the shortest path distance from $u$ to the nearest vertex with label $λ$. We show that if there is a distance oracle for undirected $n$-vertex planar graphs with non-negative edge weights using $s(n)$ space and with query time $q(n)$, then there is a vertex-labeled distance oracle with $\tilde{O}(s(n))$ space and $\tilde{O}(q(n))$ query time. Using the state-of-the-art distance oracle of Long and Pettie, our construction produces a vertex-labeled distance oracle using $n^{1+o(1)}$ space and query time $\tilde O(1)$ at one extreme, $\tilde O(n)$ space and $n^{o(1)}$ query time at the other extreme, as well as such oracles for the full tradeoff between space and query time obtained in their paper. This is the first non-trivial exact vertex-labeled distance oracle for planar graphs and, to our knowledge, for any interesting graph class other than trees.

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arXiv2021

Best Laid Plans of Lions and Men

We study the following question dating back to J.E. Littlewood (1885-1977): Can two lions catch a man in a bounded area with rectifiable lakes? The lions and the man are all assumed to be points moving with at most unit speed. That the lakes are rectifiable means that their boundaries are finitely long. This requirement is necessary to avoid pathological examples where the man survives forever because any path to the lions is infinitely long. We show that three lions have a winning strategy against a man in a bounded region with finitely many rectifiable lakes. This is "tight" in the sense that there exists a region $R$ in the plane where the man has a strategy to survive forever. We give a rigorous description of such a region $R$; a polygonal region with holes whose exterior and interior boundaries are pairwise disjoint, simple polygons. Finally, we consider the following game played on the entire plane instead of a compact region: There is any finite number of unit speed lions and one fast man who can run with speed $1+\varepsilon$ for some value $\varepsilon>0$. Can the man always survive? We answer the question in the affirmative for any $\varepsilon>0$. By letting the number of lions tend to infinity, we furthermore show that the man can survive against any countably infinite set of lions.

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arXiv2021

Escaping an Infinitude of Lions

We consider the following game played in the Euclidean plane: There is any countable set of unit speed lions and one fast man who can run with speed $1+\varepsilon$ for some value $\varepsilon>0$. Can the man survive? We answer the question in the affirmative for any $\varepsilon>0$.

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arXiv2020

Decremental APSP in Directed Graphs Versus an Adaptive Adversary

Given a directed graph $G = (V,E)$, undergoing an online sequence of edge deletions with $m$ edges in the initial version of $G$ and $n = |V|$, we consider the problem of maintaining all-pairs shortest paths (APSP) in $G$. Whilst this problem has been studied in a long line of research [ACM'81, FOCS'99, FOCS'01, STOC'02, STOC'03, SWAT'04, STOC'13] and the problem of $(1+ε)$-approximate, weighted APSP was solved to near-optimal update time $\tilde{O}(mn)$ by Bernstein [STOC'13], the problem has mainly been studied in the context of oblivious adversaries, which assumes that the adversary fixes the update sequence before the algorithm is started. In this paper, we make significant progress on the problem in the setting where the adversary is adaptive, i.e. can base the update sequence on the output of the data structure queries. We present three new data structures that fit different settings: We first present a deterministic data structure that maintains exact distances with total update time $\tilde{O}(n^3)$. We also present a deterministic data structure that maintains $(1+ε)$-approximate distance estimates with total update time $\tilde O(\sqrt{m} n^2/ε)$ which for sparse graphs is $\tilde O(n^{2+1/2}/ε)$. Finally, we present a randomized $(1+ε)$-approximate data structure which works against an adaptive adversary; its total update time is $\tilde O(m^{2/3}n^{5/3} + n^{8/3}/(m^{1/3}ε^2))$ which for sparse graphs is $\tilde O(n^{2+1/3})$. Our exact data structure matches the total update time of the best randomized data structure by Baswana et al. [STOC'02] and maintains the distance matrix in near-optimal time. Our approximate data structures improve upon the best data structures against an adaptive adversary which have $\tilde{O}(mn^2)$ total update time [JACM'81, STOC'03].

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arXiv2020

Truly Subquadratic Exact Distance Oracles with Constant Query Time for Planar Graphs

Given an undirected, unweighted planar graph $G$ with $n$ vertices, we present a truly subquadratic size distance oracle for reporting exact shortest-path distances between any pair of vertices of $G$ in constant time. For any $\varepsilon > 0$, our distance oracle takes up $O(n^{5/3+\varepsilon})$ space and is capable of answering shortest-path distance queries exactly for any pair of vertices of $G$ in worst-case time $O(\log (1/\varepsilon))$. Previously no truly sub-quadratic size distance oracles with constant query time for answering exact all-pairs shortest paths distance queries existed.

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arXiv2020

Near-Optimal Decremental SSSP in Dense Weighted Digraphs

In the decremental Single-Source Shortest Path problem (SSSP), we are given a weighted directed graph $G=(V,E,w)$ undergoing edge deletions and a source vertex $r \in V$; let $n = |V|, m = |E|$ and $W$ be the aspect ratio of the graph. The goal is to obtain a data structure that maintains shortest paths from $r$ to all vertices in $V$ and can answer distance queries in $O(1)$ time, as well as return the corresponding path $P$ in $O(|P|)$ time. This problem was first considered by Even and Shiloach [JACM'81], who provided an algorithm with total update time $O(mn)$ for unweighted undirected graphs; this was later extended to directed weighted graphs [FOCS'95, STOC'99]. There are conditional lower bounds showing that $O(mn)$ is in fact near-optimal [ESA'04, FOCS'14, STOC'15, STOC'20]. In a breakthrough result, Forster et al. showed that it is possible to achieve total update time $mn^{0.9+o(1)}\log W$ if the algorithm is allowed to return $(1+ε)$-approximate paths, instead of exact ones [STOC'14, ICALP'15]. No further progress was made until Probst Gutenberg and Wulff-Nilsen [SODA'20] provided a new approach for the problem, which yields total time $\tilde{O}(\min{m^{2/3}n^{4/3}\log W, (mn)^{7/8} \log W})$. Our result builds on this recent approach, but overcomes its limitations by introducing a significantly more powerful abstraction, as well as a different core subroutine. Our new framework yields a decremental $(1+ε)$-approximate SSSP data structure with total update time $\tilde{O}(n^2 \log^4 W)$. Our algorithm is thus near-optimal for dense graphs with polynomial edge-weights. Our framework can also be applied to sparse graphs to obtain total update time $\tilde{O}(mn^{2/3} \log^3 W)$. Our main technique allows us to convert SSSP algorithms for DAGs to ones for general graphs, which we believe has significant potential to influence future work.

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arXiv2020

Fully-Dynamic All-Pairs Shortest Paths: Improved Worst-Case Time and Space Bounds

Given a directed weighted graph $G=(V,E)$ undergoing vertex insertions \emph{and} deletions, the All-Pairs Shortest Paths (APSP) problem asks to maintain a data structure that processes updates efficiently and returns after each update the distance matrix to the current version of $G$. In two breakthrough results, Italiano and Demetrescu [STOC '03] presented an algorithm that requires $\tilde{O}(n^2)$ \emph{amortized} update time, and Thorup showed in [STOC '05] that \emph{worst-case} update time $\tilde{O}(n^{2+3/4})$ can be achieved. In this article, we make substantial progress on the problem. We present the following new results: (1) We present the first deterministic data structure that breaks the $\tilde{O}(n^{2+3/4})$ worst-case update time bound by Thorup which has been standing for almost 15 years. We improve the worst-case update time to $\tilde{O}(n^{2+5/7}) = \tilde{O}(n^{2.71..})$ and to $\tilde{O}(n^{2+3/5}) = \tilde{O}(n^{2.6})$ for unweighted graphs. (2) We present a simple deterministic algorithm with $\tilde{O}(n^{2+3/4})$ worst-case update time ($\tilde{O}(n^{2+2/3})$ for unweighted graphs), and a simple Las-Vegas algorithm with worst-case update time $\tilde{O}(n^{2+2/3})$ ($\tilde{O}(n^{2 + 1/2})$ for unweighted graphs) that works against a non-oblivious adversary. Both data structures require space $\tilde{O}(n^2)$. These are the first exact dynamic algorithms with truly-subcubic update time \emph{and} space usage. This makes significant progress on an open question posed in multiple articles [COCOON'01, STOC '03, ICALP'04, Encyclopedia of Algorithms '08] and is critical to algorithms in practice [TALG '06] where large space usage is prohibitive. Moreover, they match the worst-case update time of the best previous algorithms and the second algorithm improves upon a Monte-Carlo algorithm in a weaker adversary model with the same running time [SODA '17].

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arXiv2020

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

In the decremental $(1+ε)$-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$, undergoing edge deletions, and a distinguished source $s \in V$, and we are asked to process edge deletions efficiently and answer queries for distance estimates $\widetilde{\mathbf{dist}}_G(s,v)$ for each $v \in V$, at any stage, such that $\mathbf{dist}_G(s,v) \leq \widetilde{\mathbf{dist}}_G(s,v) \leq (1+ ε)\mathbf{dist}_G(s,v)$. In the decremental $(1+ε)$-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance estimates $\widetilde{\mathbf{dist}}_G(u,v)$ for every $u,v \in V$. In this article, we consider the problems for undirected, unweighted graphs. We present a new \emph{deterministic} algorithm for the decremental $(1+ε)$-approximate SSSP problem that takes total update time $O(mn^{0.5 + o(1)})$. Our algorithm improves on the currently best algorithm for dense graphs by Chechik and Bernstein [STOC 2016] with total update time $\tilde{O}(n^2)$ and the best existing algorithm for sparse graphs with running time $\tilde{O}(n^{1.25}\sqrt{m})$ [SODA 2017] whenever $m = O(n^{1.5 - o(1)})$. In order to obtain this new algorithm, we develop several new techniques including improved decremental cover data structures for graphs, a more efficient notion of the heavy/light decomposition framework introduced by Chechik and Bernstein and the first clustering technique to maintain a dynamic \emph{sparse} emulator in the deterministic setting. As a by-product, we also obtain a new simple deterministic algorithm for the decremental $(1+ε)$-approximate APSP problem with near-optimal total running time $\tilde{O}(mn /ε)$ matching the time complexity of the sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai [FOCS 2013].

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arXiv2020

Decremental SSSP in Weighted Digraphs: Faster and Against an Adaptive Adversary

Given a dynamic digraph $G = (V,E)$ undergoing edge deletions and given $s\in V$ and $ε>0$, we consider the problem of maintaining $(1+ε)$-approximate shortest path distances from $s$ to all vertices in $G$ over the sequence of deletions. Even and Shiloach (J.~ACM'$81$) give a deterministic data structure for the exact version of the problem with total update time $O(mn)$. Henzinger et al. (STOC'$14$, ICALP'$15$) give a Monte Carlo data structure for the approximate version with improved total update time $ O(mn^{0.9 + o(1)}\log W)$ where $W$ is the ratio between the largest and smallest edge weight. A drawback of their data structure is that they only work against an oblivious adversary, meaning that the sequence of deletions needs to be fixed in advance. This limits its application as a black box inside algorithms. We present the following $(1+ε)$-approximate data structures: (1) the first data structure is Las Vegas and works against an adaptive adversary; it has total expected update time $\tilde O(m^{2/3}n^{4/3})$ for unweighted graphs and $\tilde O(m^{3/4}n^{5/4}\log W)$ for weighted graphs, (2) the second data structure is Las Vegas and assumes an oblivious adversary; it has total expected update time $\tilde O(\sqrt m n^{3/2})$ for unweighted graphs and $\tilde O(m^{2/3}n^{4/3}\log W)$ for weighted graphs, (3) the third data structure is Monte Carlo and is correct w.h.p.~against an oblivious adversary; it has total expected update time $\tilde O((mn)^{7/8}\log W) = \tilde O(mn^{3/4}\log W)$. Each of our data structures can be queried at any stage of $G$ in constant worst-case time; if the adversary is oblivious, a query can be extended to also report such a path in time proportional to its length. Our update times are faster than those of Henzinger et al.~for all graph densities.

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arXiv2019

Decremental Strongly-Connected Components and Single-Source Reachability in Near-Linear Time

Computing the Strongly-Connected Components (SCCs) in a graph $G=(V,E)$ is known to take only $O(m + n)$ time using an algorithm by Tarjan from 1972[SICOMP 72] where $m = |E|$, $n=|V|$. For fully-dynamic graphs, conditional lower bounds provide evidence that the update time cannot be improved by polynomial factors over recomputing the SCCs from scratch after every update. Nevertheless, substantial progress has been made to find algorithms with fast update time for \emph{decremental} graphs, i.e. graphs that undergo edge deletions. In this paper, we present the first algorithm for general decremental graphs that maintains the SCCs in total update time $\tilde{O}(m)$, thus only a polylogarithmic factor from the optimal running time. Previously such a result was only known for the special case of planar graphs [Italiano et al, STOC 2017]. Our result should be compared to the formerly best algorithm for general graphs achieving $\tilde{O}(m\sqrt{n})$ total update time by Chechik et.al. [FOCS 16] which improved upon a breakthrough result leading to $O(mn^{0.9 + o(1)})$ total update time by Henzinger, Krinninger and Nanongkai [STOC 14, ICALP 15]; these results in turn improved upon the longstanding bound of $O(mn)$ by Roditty and Zwick [STOC 04]. All of the above results also apply to the decremental Single-Source Reachability (SSR) problem, which can be reduced to decrementally maintaining SCCs. A bound of $O(mn)$ total update time for decremental SSR was established already in 1981 by Even and Shiloach [JACM 1981]. Using a well known reduction, we can maintain the reachability of pairs $S \times V$, $S \subseteq V$ in fully-dynamic graphs with update time $\tilde{O}(\frac{|S|m}{t})$ and query time $O(t)$ for all $t \in [1,|S|]$; this generalizes an earlier All-Pairs Reachability where $S = V$ [Łącki, TALG 2013].

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arXiv2017

Greedy spanners are optimal in doubling metrics

We show that the greedy spanner algorithm constructs a $(1+ε)$-spanner of weight $ε^{-O(d)}w(\mathrm{MST})$ for a point set in metrics of doubling dimension $d$, resolving an open problem posed by Gottlieb. Our result generalizes the result by Narasimhan and Smid who showed that a point set in $d$-dimension Euclidean space has a $(1+ε)$-spanner of weight at most $ε^{-O(d)}w(\mathrm{MST})$. Our proof only uses the packing property of doubling metrics and thus implies a much simpler proof for the same result in Euclidean space.

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