arXiv subjects

Eva Rotenberg

Publications and source records attributed to Eva Rotenberg.

At least 55 records · Page 3Linked to original sources

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 $α$ of the graph, in, either, an amortised update time of $O(\log^2 n \log α)$, or a worst-case update time of $O(\log^3 n \log α)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off, namely either $O(\log n \log α)$, amortised, or $O(\log ^2 n \log α)$, worst-case time, for the problem of maintaining an edge-orientation with at most $O(α+ \log n)$ 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 $O(\varepsilon^{-6}\log^3 n \log ρ)$ worst-case update time algorithm for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $ρ$. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log α)$ 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 $α$. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(α)$ forests.Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety, of problems including maximal matching, $Δ+1$ coloring, and matrix vector multiplication. All update times are worst-case $O(α+\log^2n \log α)$, where $α$ is the current arboricity of the graph.

cs.DS↗

arXiv2023

Simple and Robust Dynamic Two-Dimensional Convex Hull

The convex hull of a data set $P$ is the smallest convex set that contains $P$. In this work, we present a new data structure for convex hull, that allows for efficient dynamic updates. In a dynamic convex hull implementation, the following traits are desirable: (1) algorithms for efficiently answering queries as to whether a specified point is inside or outside the hull, (2) adhering to geometric robustness, and (3) algorithmic simplicity.Furthermore, a specific but well-motivated type of two-dimensional data is rank-based data. Here, the input is a set of real-valued numbers $Y$ where for any number $y\in Y$ its rank is its index in $Y$'s sorted order. Each value in $Y$ can be mapped to a point $(rank, value)$ to obtain a two-dimensional point set. In this work, we give an efficient, geometrically robust, dynamic convex hull algorithm, that facilitates queries to whether a point is internal. Furthermore, our construction can be used to efficiently update the convex hull of rank-ordered data, when the real-valued point set is subject to insertions and deletions. Our improved solution is based on an algorithmic simplification of the classical convex hull data structure by Overmars and van Leeuwen~[STOC'80], combined with new algorithmic insights. Our theoretical guarantees on the update time match those of Overmars and van Leeuwen, namely $O(\log^2 |P|)$, while we allow a wider range of functionalities (including rank-based data). Our algorithmic simplification includes simplifying an 11-case check down to a 3-case check that can be written in 20 lines of easily readable C-code. We extend our solution to provide a trade-off between theoretical guarantees and the practical performance of our algorithm. We test and compare our solutions extensively on inputs that were generated randomly or adversarially, including benchmarking datasets from the literature.

cs.CG↗

arXiv2023

Triangulations Admit Dominating Sets of Size $2n/7$

We show that every planar triangulation on $n>10$ vertices has a dominating set of size $n/7=n/3.5$. This approaches the $n/4$ bound conjectured by Matheson and Tarjan [MT'96], and improves significantly on the previous best bound of $17n/53\approx n/3.117$ by Špacapan [Š'20]. From our proof it follows that every 3-connected $n$-vertex near-triangulation (except for 3 sporadic examples) has a dominating set of size $n/3.5$. On the other hand, for 3-connected near-triangulations, we show a lower bound of $3(n-1)/11\approx n/3.666$, demonstrating that the conjecture by Matheson and Tarjan [MT'96] cannot be strengthened to 3-connected near-triangulations. Our proof uses a penalty function that, aside from the number of vertices, penalises vertices of degree 2 and specific constellations of neighbours of degree 3 along the boundary of the outer face. To facilitate induction, we not only consider near-triangulations, but a wider class of graphs (skeletal triangulations), allowing us to delete vertices more freely. Our main technical contribution is a set of attachments, that are small graphs we inductively attach to our graph, in order both to remember whether existing vertices are already dominated, and that serve as a tool in a divide and conquer approach. Along with a well-chosen potential function, we thus both remove and add vertices during the induction proof. We complement our proof with a constructive algorithm that returns a dominating set of size $\le 2n/7$. Our algorithm has a quadratic running time.

math.CO↗

arXiv2023

Multilevel Skeletonization Using Local Separators

In this paper we give a new, efficient algorithm for computing curve skeletons, based on local separators. Our efficiency stems from a multilevel approach, where we solve small problems across levels of detail and combine these in order to quickly obtain a skeleton. We do this in a highly modular fashion, ensuring complete flexibility in adapting the algorithm for specific types of input or for otherwise targeting specific applications. Separator based skeletonization was first proposed by Bærentzen and Rotenberg in [ACM Tran. Graphics'21], showing high quality output at the cost of running times which become prohibitive for large inputs. Our new approach retains the high quality output, and applicability to any spatially embedded graph, while being orders of magnitude faster for all practical purposes. We test our skeletonization algorithm for efficiency and quality in practice, comparing it to local separator skeletonization on the University of Groningen Skeletonization Benchmark [Telea'16].

cs.CG↗

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 $α$ of the graph, in a worst-case update time of $O(\log^3 n \log α)$. 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 $O(\log ^2 n \log α)$ for the problem of maintaining an edge-orientation with at most $O(α+ \log n)$ 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 $O(\varepsilon^{-6}\log^3 n \log ρ)$ worst-case update time algorithm for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $ρ$. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log α)$ 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 $α$. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(α)$ forests.Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety, of problems including maximal matching, $Δ+1$ coloring, and matrix vector multiplication. All update times are worst-case $O(α+\log^2n \log α)$, where $α$ is the current arboricity of the graph.

cs.DS↗

arXiv2023

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.

cs.DS↗

arXiv2022

Improved Dynamic Colouring of Sparse Graphs

Given a dynamic graph subject to edge insertions and deletions, we show how to update an implicit representation of a proper vertex colouring, such that colours of vertices are computable upon query time. We give a deterministic algorithm that uses $O(α^2)$ colours for a dynamic graph of arboricity $α$, and a randomised algorithm that uses $O(\min\{α\log α, α\log \log \log n\})$ colours in the oblivious adversary model. Our deterministic algorithm has update- and query times polynomial in $α$ and $\log n$, and our randomised algorithm has amortised update- and query time that with high probability is polynomial in $\log n$ with no dependency on the arboricity. Thus, we improve the number of colours exponentially compared to the state-of-the art for implicit colouring, namely from $O(2^α)$ colours, and we approach the theoretical lower bound of $Ω(α)$ for this arboricity-parameterised approach. Simultaneously, our randomised algorithm improves the update- and query time to run in time solely polynomial in $\log n$ with no dependency on $α$. Our algorithms are fully adaptive to the current value of the dynamic arboricity at query or update time.

cs.DS↗

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

Dynamic Embeddings of Dynamic Single-Source Upward Planar Graphs

A directed graph $G$ is upward planar if it admits a planar embedding such that each edge is $y$-monotone. Unlike planarity testing, upward planarity testing is NP-hard except in restricted cases, such as when the graph has the single-source property (i.e. each connected component only has one source). In this paper, we present a dynamic algorithm for maintaining a combinatorial embedding $\mathcal{E}(G)$ of a single-source upward planar graph subject to edge deletions, edge contractions, edge insertions upwards across a face, and single-source-preserving vertex splits through specified corners. We furthermore support changes to the embedding $\mathcal{E}(G)$ on the form of subgraph flips that mirror or slide the placement of a subgraph that is connected to the rest of the graph via at most two vertices. All update operations are supported as long as the graph remains upward planar, and all queries are supported as long as the graph remains single-source. Updates that violate upward planarity are identified as such and rejected by our update algorithm. We dynamically maintain a linear-size data structure on $G$ which supports incidence queries between a vertex and a face, and upward-linkability of vertex pairs. If a pair of vertices are not upwards-linkable, we facilitate one-flip-linkable queries that point to a subgraph flip that makes them linkable, if any such flip exists. We support all updates and queries in $O(\log^2 n)$ time.

cs.DS↗

arXiv2022

Fully-dynamic $α+ 2$ Arboricity Decomposition and Implicit Colouring

In the implicit dynamic colouring problem, the task is to maintain a representation of a proper colouring as a dynamic graph is subject to insertions and deletions of edges, while facilitating interspersed queries to the colours of vertices. The goal is to use few colours, while still efficiently handling edge-updates and responding to colour-queries. For an n-vertex dynamic graph of arboricity $α$, we present an algorithm that maintains an implicit vertex colouring with $4\cdot2^α$ colours, in amortised poly-$(\log n)$ update time, and with $O(α log n)$ worst-case query time. The previous best implicit dynamic colouring algorithm uses $2^{40α}$) colours, and has a more efficient update time of $O(\log^3 n)$ and the same query time of $O(α log n)$ [Henzinger et al'20]. For graphs undergoing arboricity $α$ preserving updates, we give a fully-dynamic $α+2$ arboricity decomposition in poly$(\log n,α)$ time, which matches the number of forests in the best near-linear static algorithm by Blumenstock and Fischer [2020] who obtain $α+2$ forests in near-linear time. Our construction goes via dynamic bounded out-degree orientations, where we present a fully-dynamic explicit, deterministic, worst-case algorithm for $\lfloor (1+\varepsilon)α\rfloor + 2$ bounded out-degree orientation with update time $O(\varepsilon^{-6}α^2 \log^3 n)$. The state-of-the-art explicit, deterministic, worst-case algorithm for bounded out-degree orientations maintains a $β\cdot α+ \log_β n$ out-orientation in $O(β^2α^2+βα\log_β n)$ time [Kopelowitz et al'13].

cs.DS↗

arXiv2022

On the Discrete Fréchet Distance in a Graph

The Fréchet distance is a well-studied similarity measure between curves that is widely used throughout computer science. Motivated by applications where curves stem from paths and walks on an underlying graph (such as a road network), we define and study the Fréchet distance for paths and walks on graphs. When provided with a distance oracle of $G$ with $O(1)$ query time, the classical quadratic-time dynamic program can compute the Fréchet distance between two walks $P$ and $Q$ in a graph $G$ in $O(|P| \cdot |Q|)$ time. We show that there are situations where the graph structure helps with computing Fréchet distance: when the graph $G$ is planar, we apply existing (approximate) distance oracles to compute a $(1+\varepsilon)$-approximation of the Fréchet distance between any shortest path $P$ and any walk $Q$ in $O(|G| \log |G| / \sqrt{\varepsilon} + |P| + \frac{|Q|}{\varepsilon } )$ time. We generalise this result to near-shortest paths, i.e. $κ$-straight paths, as we show how to compute a $(1+\varepsilon)$-approximation between a $κ$-straight path $P$ and any walk $Q$ in $O(|G| \log |G| / \sqrt{\varepsilon} + |P| + \frac{κ|Q|}{\varepsilon } )$ time. Our algorithmic results hold for both the strong and the weak discrete Fréchet distance over the shortest path metric in $G$. Finally, we show that additional assumptions on the input, such as our assumption on path straightness, are indeed necessary to obtain truly subquadratic running time. We provide a conditional lower bound showing that the Fréchet distance, or even its $1.01$-approximation, between arbitrary \emph{paths} in a weighted planar graph cannot be computed in $O((|P|\cdot|Q|)^{1-δ})$ time for any $δ> 0$ unless the Orthogonal Vector Hypothesis fails. For walks, this lower bound holds even when $G$ is planar, unit-weight and has $O(1)$ vertices.

cs.CG↗

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.

cs.CG↗

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 $\varepsilon>0$.

cs.CG↗

arXiv2020

Skeletonization via Local Separators

We propose a new algorithm for curve skeleton computation which differs from previous algorithms by being based on the notion of local separators. The main benefits of this approach are that it is able to capture relatively fine details and that it works robustly on a range of shape representations. Specifically, our method works on shape representations that can be construed as a spatially embedded graphs. Such representations include meshes, volumetric shapes, and graphs computed from point clouds. We describe a simple pipeline where geometric data is initially converted to a graph, optionally simplified, local separators are computed and selected, and finally a skeleton is constructed. We test our pipeline on polygonal meshes, volumetric shapes, and point clouds. Finally, we compare our results to other methods for skeletonization according to performance and quality.

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

Space Efficient Construction of Lyndon Arrays in Linear Time

We present the first linear time algorithm to construct the $2n$-bit version of the Lyndon array for a string of length $n$ using only $o(n)$ bits of working space. A simpler variant of this algorithm computes the plain ($n\lg n$-bit) version of the Lyndon array using only $\mathcal{O}(1)$ words of additional working space. All previous algorithms are either not linear, or use at least $n\lg n$ bits of additional working space. Also in practice, our new algorithms outperform the previous best ones by an order of magnitude, both in terms of time and space.

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 $ϕ(G)$, 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 $ϕ(G)$ to $ϕ(G')$ is $O(\log n)$. In contrast, there exist embedded graphs where $Ω(n)$ changes are necessary to accommodate one inserted edge. We provide a matching lower bound of $Ω(\log n)$ 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 Poutré and Westbrook from 1998.

cs.DS↗

arXiv2019

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↗