arXiv subjects

Jacob Holm

Publications and source records attributed to Jacob Holm.

At least 19 recordsLinked to original sources

arXiv2025

Fully dynamic biconnectivity in $\tilde{\mathcal{O}}(\log^2 n)$ time

We present a deterministic fully-dynamic data structure for maintaining information about the cut-vertices in a graph; i.e. the vertices whose removal would disconnect the graph. Our data structure supports insertion and deletion of edges, as well as queries to whether a pair of connected vertices are either biconnected, or can be separated by a cutvertex, and in the latter case we support access to separating cutvertices. All update operations are supported in amortized $O(\log^2 n \log^2 \log n)$ time, and queries take worst-case $O(\log n \log^2 \log n)$ time. Note that these time bounds match the current best for deterministic dynamic connectivity up to $\log \log n$ factors. We obtain our improved running time by a series of reductions from the original problem into well-defined data structure problems. While we do apply the well-known techniques for improving running time of two-edge connectivity [STOC'00, SODA'18], these techniques alone do not lead to an update time of $\tilde{O}(\log^3 n)$, let alone the $\tilde{O}(\log^2 n)$ we give as a final result. Our contributions include a formally defined transient expose operation, which can be thought of as a cheaper read-only expose operation on a top tree. For each vertex in the graph, we maintain a data structure over its neighbors, and in this data structure we apply biasing (twice) to save two $\tilde{O}(\log n)$ factors. One of these biasing techniques is a new biased disjoint sets data structure, which may be of independent interest. Moreover, in this neighborhood data structure, we facilitate that the vertex can select two VIP neighbors that get special treatment, corresponding to its potentially two neighbors on an exposed path, improving a $\log n$-time operation down to constant time. It is this combination of VIP neighbors with the transient expose that saves an $\tilde{O}(\log n)$-factor from another bottleneck.

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arXiv2023

Adaptive Out-Orientations with Applications

We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity $\alpha$ of the graph, in, either, an amortised update time of $O(\log^2 n \log \alpha)$, or a worst-case update time of $O(\log^3 n \log \alpha)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off, namely either $O(\log n \log \alpha)$, amortised, or $O(\log ^2 n \log \alpha)$, worst-case time, for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Since our algorithms have update times with worst-case guarantees, the number of changes to the solution (i.e. the recourse) is naturally limited. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \rho)$ worst-case update time algorithm for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $\rho$. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \alpha)$ worst-case update time algorithm for maintaining a $(1 + \varepsilon) \cdot OPT + 2$ approximation of the optimal out-orientation of a graph with adaptive arboricity $\alpha$. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(\alpha)$ forests.Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety, of problems including maximal matching, $\Delta+1$ coloring, and matrix vector multiplication. All update times are worst-case $O(\alpha+\log^2n \log \alpha)$, where $\alpha$ is the current arboricity of the graph.

cs.DS↗

arXiv2022

Splay Top Trees

The top tree data structure is an important and fundamental tool in dynamic graph algorithms. Top trees have existed for decades, and today serve as an ingredient in many state-of-the-art algorithms for dynamic graphs. In this work, we give a new direct proof of the existence of top trees, facilitating simpler and more direct implementations of top trees, based on ideas from splay trees. This result hinges on new insights into the structure of top trees, and in particular the structure of each root path in a top tree.

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arXiv2022

Worst-case Deterministic Fully-Dynamic Planar 2-vertex Connectivity

We study dynamic planar graphs with $n$ vertices, subject to edge deletion, edge contraction, edge insertion across a face, and the splitting of a vertex in specified corners. We dynamically maintain a combinatorial embedding of such a planar graph, subject to connectivity and $2$-vertex-connectivity (biconnectivity) queries between pairs of vertices. Whenever a query pair is connected and not biconnected, we find the first and last cutvertex separating them. Additionally, we allow local changes to the embedding by flipping the embedding of a subgraph that is connected by at most two vertices to the rest of the graph. We support all queries and updates in deterministic, worst-case, $O(\log^2 n)$ time, using an $O(n)$-sized data structure. Previously, the best bound for fully-dynamic planar biconnectivity (subject to our set of operations) was an amortised $\tilde{O}(\log^3 n)$ for general graphs, and algorithms with worst-case polylogarithmic update times were known only in the partially dynamic (insertion-only or deletion-only) setting.

cs.DS↗

arXiv2022

Adaptive Out-Orientations with Applications

We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity $\alpha$ of the graph, in a worst-case update time of $O(\log^3 n \log \alpha)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off, namely the improved worst case update time of $O(\log ^2 n \log \alpha)$ for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Since our algorithms have update times with worst-case guarantees, the number of changes to the solution (i.e. the recourse) is naturally limited. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \rho)$ worst-case update time algorithm for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $\rho$. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \alpha)$ worst-case update time algorithm for maintaining a $(1 + \varepsilon) \cdot OPT + 2$ approximation of the optimal out-orientation of a graph with adaptive arboricity $\alpha$. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(\alpha)$ forests.Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety, of problems including maximal matching, $\Delta+1$ coloring, and matrix vector multiplication. All update times are worst-case $O(\alpha+\log^2n \log \alpha)$, where $\alpha$ is the current arboricity of the graph.

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arXiv2022

Massively Parallel Computation on Embedded Planar Graphs

Many of the classic graph problems cannot be solved in the Massively Parallel Computation setting (MPC) with strongly sublinear space per machine and $o(\log n)$ rounds, unless the 1-vs-2 cycles conjecture is false. This is true even on planar graphs. Such problems include, for example, counting connected components, bipartition, minimum spanning tree problem, (approximate) shortest paths, and (approximate) diameter/radius. In this paper, we show a way to get around this limitation. Specifically, we show that if we have a ``nice'' (for example, straight-line) embedding of the input graph, all the mentioned problems can be solved with $O(n^{2/3+\epsilon})$ space per machine in $O(1)$ rounds. In conjunction with existing algorithms for computing the Delaunay triangulation, our results imply an MPC algorithm for exact Euclidean minimum spanning thee (EMST) that uses $O(n^{2/3 + \epsilon})$ space per machine and finishes in $O(1)$ rounds. This is the first improvement over a straightforward use of the standard Bor\r{u}vka's algorithm with the Dauleanay triangulation algorithm of Goodrich [SODA 1997] which results in $\Theta(\log n)$ rounds. This also partially negatively answers a question of Andoni, Nikolov, Onak, and Yaroslavtsev [STOC 2014], asking for lower bounds for exact EMST. We extend our algorithms to work with embeddings consisting of curves that are not ``too squiggly" (as formalized by the total absolute curvature). We do this via a new lemma which we believe is of independent interest and could be used to parameterize other geometric problems by the total absolute curvature. We also state several open problems regarding massively parallel computation on planar graphs.

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arXiv2020

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 $\varepsilon>0$.

cs.CG↗

arXiv2019

Fully-dynamic Planarity Testing in Polylogarithmic Time

Given a dynamic graph subject to insertions and deletions of edges, a natural question is whether the graph presently admits a planar embedding. We give a deterministic fully-dynamic algorithm for general graphs, running in amortized $O(\log^3 n)$ time per edge insertion or deletion, that maintains a bit indicating whether or not the graph is presently planar. This is an exponential improvement over the previous best algorithm [Eppstein, Galil, Italiano, Spencer, 1996] which spends amortized $O(\sqrt{n})$ time per update.

cs.DS↗

arXiv2019

Worst-Case Polylog Incremental SPQR-trees: Embeddings, Planarity, and Triconnectivity

We show that every labelled planar graph $G$ can be assigned a canonical embedding $\phi(G)$, such that for any planar $G'$ that differs from $G$ by the insertion or deletion of one edge, the number of local changes to the combinatorial embedding needed to get from $\phi(G)$ to $\phi(G')$ is $O(\log n)$. In contrast, there exist embedded graphs where $\Omega(n)$ changes are necessary to accommodate one inserted edge. We provide a matching lower bound of $\Omega(\log n)$ local changes, and although our upper bound is worst-case, our lower bound hold in the amortized case as well. Our proof is based on BC trees and SPQR trees, and we develop \emph{pre-split} variants of these for general graphs, based on a novel biased heavy-path decomposition, where the structural changes corresponding to edge insertions and deletions in the underlying graph consist of at most $O(\log n)$ basic operations of a particularly simple form. As a secondary result, we show how to maintain the pre-split trees under edge insertions in the underlying graph deterministically in worst case $O(\log^3 n)$ time. Using this, we obtain deterministic data structures for incremental planarity testing, incremental planar embedding, and incremental triconnectivity, that each have worst case $O(\log^3 n)$ update and query time, answering an open question by La Poutr\'e and Westbrook from 1998.

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arXiv2019

Random $k$-out subgraph leaves only $O(n/k)$ inter-component edges

Each vertex of an arbitrary simple graph on $n$ vertices chooses $k$ random incident edges. What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph? We prove that the answer is $O(n/k)$, when $k\ge c\log n$, for some large enough $c$. We conjecture that the same holds for smaller values of $k$, possibly for any $k\ge 2$. Such a result is best possible for any $k\ge 2$. As an application, we use this sampling result to obtain a one-way communication protocol with \emph{private} randomness for finding a spanning forest of a graph in which each vertex sends only ${O}(\sqrt{n}\log n)$ bits to a referee.

cs.DM↗

arXiv2018

Good $r$-divisions Imply Optimal Amortised Decremental Biconnectivity

We present a data structure that, given a graph $G$ of $n$ vertices and $m$ edges, and a suitable pair of nested $r$-divisions of $G$, preprocesses $G$ in $O(m+n)$ time and handles any series of edge-deletions in $O(m)$ total time while answering queries to pairwise biconnectivity in worst-case $O(1)$ time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case $O(1)$ time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs.

cs.DS↗

arXiv2018

Decremental SPQR-trees for Planar Graphs

We present a decremental data structure for maintaining the SPQR-tree of a planar graph subject to edge contractions and deletions. The update time, amortized over $\Omega(n)$ operations, is $O(\log^2 n)$. Via SPQR-trees, we give a decremental data structure for maintaining $3$-vertex connectivity in planar graphs. It answers queries in $O(1)$ time and processes edge deletions and contractions in $O(\log^2 n)$ amortized time. This is an exponential improvement over the previous best bound of $O(\sqrt{n}\,)$ that has stood for over 20 years. In addition, the previous data structures only supported edge deletions.

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arXiv2017

One-Way Trail Orientations

Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins' theorem [Robbins, Am. Math. Monthly, 1939] states that such an orientation exists if and only if the graph is $2$-edge connected. A natural extension of this problem is the following: Suppose that the edges of the graph is partitioned into trails. Can we orient the trails such that the resulting directed graph is strongly connected? We show that $2$-edge connectivity is again a sufficient condition and we provide a linear time algorithm for finding such an orientation, which is both optimal and the first polynomial time algorithm for deciding this problem. The generalised Robbins' theorem [Boesch, Am. Math. Monthly, 1980] for mixed multigraphs states that the undirected edges of a mixed multigraph can be oriented making the resulting directed graph strongly connected exactly when the mixed graph is connected and the underlying graph is bridgeless. We show that as long as all cuts have at least $2$ undirected edges or directed edges both ways, then there exists an orientation making the resulting directed graph strongly connected. This provides the first polynomial time algorithm for this problem and a very simple polynomial time algorithm to the previous problem.

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arXiv2017

Online Bipartite Matching with Amortized $O(\log^2 n)$ Replacements

In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized $O(\log^2 n)$ replacements per insertion, where $n$ is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for \emph{any} replacement strategy, almost matching the $\Omega(\log n)$ lower bound. The previous best strategy known achieved amortized $O(\sqrt{n})$ replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than then trivial $O(n)$ bound was known except in special cases. Our analysis immediately implies the same upper bound of $O(\log^2 n)$ reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load $L$, then the SAP protocol makes amortized $O( \min\{L \log^2 n , \sqrt{n}\log n\})$ reassignments. We also show that this is close to tight because $\Omega(\min\{L, \sqrt{n}\})$ reassignments can be necessary.

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arXiv2017

Dynamic Bridge-Finding in $\tilde{O}(\log ^2 n)$ Amortized Time

We present a deterministic fully-dynamic data structure for maintaining information about the bridges in a graph. We support updates in $\tilde{O}((\log n)^2)$ amortized time, and can find a bridge in the component of any given vertex, or a bridge separating any two given vertices, in $O(\log n / \log \log n)$ worst case time. Our bounds match the current best for bounds for deterministic fully-dynamic connectivity up to $\log\log n$ factors. The previous best dynamic bridge finding was an $\tilde{O}((\log n)^3)$ amortized time algorithm by Thorup [STOC2000], which was a bittrick-based improvement on the $O((\log n)^4)$ amortized time algorithm by Holm et al.[STOC98, JACM2001]. Our approach is based on a different and purely combinatorial improvement of the algorithm of Holm et al., which by itself gives a new combinatorial $\tilde{O}((\log n)^3)$ amortized time algorithm. Combining it with Thorup's bittrick, we get down to the claimed $\tilde{O}((\log n)^2)$ amortized time. Essentially the same new trick can be applied to the biconnectivity data structure from [STOC98, JACM2001], improving the amortized update time to $\tilde{O}((\log n)^3)$. We also offer improvements in space. We describe a general trick which applies to both of our new algorithms, and to the old ones, to get down to linear space, where the previous best use $O(m + n\log n\log\log n)$. Finally, we show how to obtain $O(\log n/\log \log n)$ query time, matching the optimal trade-off between update and query time. Our result yields an improved running time for deciding whether a unique perfect matching exists in a static graph.

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arXiv2017

Contracting a Planar Graph Efficiently

We present a data structure that can maintain a simple planar graph under edge contractions in linear total time. The data structure supports adjacency queries and provides access to neighbor lists in $O(1)$ time. Moreover, it can report all the arising self-loops and parallel edges. By applying the data structure, we can achieve optimal running times for decremental bridge detection, 2-edge connectivity, maximal 3-edge connected components, and the problem of finding a unique perfect matching for a static planar graph. Furthermore, we improve the running times of algorithms for several planar graph problems, including decremental 2-vertex and 3-edge connectivity, and we show that using our data structure in a black-box manner, one obtains conceptually simple optimal algorithms for computing MST and 5-coloring in planar graphs.

cs.DS↗

arXiv2017

Dynamic Planar Embeddings of Dynamic Graphs

We present an algorithm to support the dynamic embedding in the plane of a dynamic graph. An edge can be inserted across a face between two vertices on the face boundary (we call such a vertex pair linkable), and edges can be deleted. The planar embedding can also be changed locally by flipping components that are connected to the rest of the graph by at most two vertices. Given vertices $u,v$, linkable$(u,v)$ decides whether $u$ and $v$ are linkable in the current embedding, and if so, returns a list of suggestions for the placement of $(u,v)$ in the embedding. For non-linkable vertices $u,v$, we define a new query, one-flip-linkable$(u,v)$ providing a suggestion for a flip that will make them linkable if one exists. We support all updates and queries in O(log$^2 n$) time. Our time bounds match those of Italiano et al. for a static (flipless) embedding of a dynamic graph. Our new algorithm is simpler, exploiting that the complement of a spanning tree of a connected plane graph is a spanning tree of the dual graph. The primal and dual trees are interpreted as having the same Euler tour, and a main idea of the new algorithm is an elegant interaction between top trees over the two trees via their common Euler tour.

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arXiv2017

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.

cs.CG↗