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Pat Morin

Publications and source records attributed to Pat Morin.

At least 73 records · Page 4Linked to original sources
arXiv2017

Layered Separators in Minor-Closed Graph Classes with Applications

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as $Ω(\sqrt{n})$ in graphs with $n$ vertices. This is the case for planar graphs, and more generally, for proper minor-closed classes. We study a special type of graph separator, called a "layered separator", which may have linear size in $n$, but has bounded size with respect to a different measure, called the "width". We prove, for example, that planar graphs and graphs of bounded Euler genus admit layered separators of bounded width. More generally, we characterise the minor-closed classes that admit layered separators of bounded width as those that exclude a fixed apex graph as a minor. We use layered separators to prove $\mathcal{O}(\log n)$ bounds for a number of problems where $\mathcal{O}(\sqrt{n})$ was a long-standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results with a $\mathcal{O}(\log n)$ bound on the nonrepetitive chromatic number of graphs excluding a fixed topological minor, and a $\log^{ \mathcal{O}(1)}n$ bound on the queue-number of graphs excluding a fixed minor. Only for planar graphs were $\log^{ \mathcal{O}(1)}n$ bounds previously known. Our results imply that every $n$-vertex graph excluding a fixed minor has a 3-dimensional grid drawing with $n\log^{ \mathcal{O}(1)}n$ volume, whereas the previous best bound was $\mathcal{O}(n^{3/2})$.

math.CO↗
arXiv2017

Encoding Arguments

Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is effective, but the underlying probabilistic machinery can be daunting. "Encoding arguments" provide an alternative presentation in which probabilistic reasoning is encapsulated in a "uniform encoding lemma". This lemma provides an upper bound on the probability of an event using the fact that a uniformly random choice from a set of size $n$ cannot be encoded with fewer than $\log_2 n$ bits on average. With the lemma, the argument reduces to devising an encoding where bad objects have short codewords. In this expository article, we describe the basic method and provide a simple tutorial on how to use it. After that, we survey many applications to classic problems from discrete mathematics and computer science. We also give a generalization for the case of non-uniform distributions, as well as a rigorous justification for the use of non-integer codeword lengths in encoding arguments. These latter two results allow encoding arguments to be applied more widely and to produce tighter results.

cs.IT↗
arXiv2017

Array Layouts for Comparison-Based Searching

We attempt to determine the best order and search algorithm to store $n$ comparable data items in an array, $A$, of length $n$ so that we can, for any query value, $x$, quickly find the smallest value in $A$ that is greater than or equal to $x$. In particular, we consider the important case where there are many such queries to the same array, $A$, which resides entirely in RAM. In addition to the obvious sorted order/binary search combination we consider the Eytzinger (BFS) layout normally used for heaps, an implicit B-tree layout that generalizes the Eytzinger layout, and the van Emde Boas layout commonly used in the cache-oblivious algorithms literature. After extensive testing and tuning on a wide variety of modern hardware, we arrive at the conclusion that, for small values of $n$, sorted order, combined with a good implementation of binary search is best. For larger values of $n$, we arrive at the surprising conclusion that the Eytzinger layout is usually the fastest. The latter conclusion is unexpected and goes counter to earlier experimental work by Brodal, Fagerberg, and Jacob (SODA~2003), who concluded that both the B-tree and van Emde Boas layouts were faster than the Eytzinger layout for large values of $n$. Our fastest C++ implementations, when compiled, use conditional moves to avoid branch mispredictions and prefetching to reduce cache latency.

cs.DS↗
arXiv2016

Spanning Trees in Multipartite Geometric Graphs

Let $R$ and $B$ be two disjoint sets of points in the plane where the points of $R$ are colored red and the points of $B$ are colored blue, and let $n=|R\cup B|$. A bichromatic spanning tree is a spanning tree in the complete bipartite geometric graph with bipartition $(R,B)$. The minimum (respectively maximum) bichromatic spanning tree problem is the problem of computing a bichromatic spanning tree of minimum (respectively maximum) total edge length. 1. We present a simple algorithm that solves the minimum bichromatic spanning tree problem in $O(n\log^3 n)$ time. This algorithm can easily be extended to solve the maximum bichromatic spanning tree problem within the same time bound. It also can easily be generalized to multicolored point sets. 2. We present $Θ(n\log n)$-time algorithms that solve the minimum and the maximum bichromatic spanning tree problems. 3. We extend the bichromatic spanning tree algorithms and solve the multicolored version of these problems in $O(n\log n\log k)$ time, where $k$ is the number of different colors (or the size of the multipartition in a complete multipartite geometric graph).

cs.CG↗
arXiv2016

The Price of Order

We present tight bounds on the spanning ratio of a large family of ordered $θ$-graphs. A $θ$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ= 2 π/m$. An ordered $θ$-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer $k \geq 1$, ordered $θ$-graphs with $4k + 4$ cones have a tight spanning ratio of $1 + 2 \sin(θ/2) / (\cos(θ/2) - \sin(θ/2))$. We also show that for any integer $k \geq 2$, ordered $θ$-graphs with $4k + 2$ cones have a tight spanning ratio of $1 / (1 - 2 \sin(θ/2))$. We provide lower bounds for ordered $θ$-graphs with $4k + 3$ and $4k + 5$ cones. For ordered $θ$-graphs with $4k + 2$ and $4k + 5$ cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered $θ$-graphs have worse spanning ratios than unordered $θ$-graphs. Finally, we show that, unlike their unordered counterparts, the ordered $θ$-graphs with 4, 5, and 6 cones are not spanners.

cs.CG↗
arXiv2014

Compatible Connectivity-Augmentation of Planar Disconnected Graphs

Motivated by applications to graph morphing, we consider the following \emph{compatible connectivity-augmentation problem}: We are given a labelled $n$-vertex planar graph, $\mathcal{G}$, that has $r\ge 2$ connected components, and $k\ge 2$ isomorphic planar straight-line drawings, $G_1,\ldots,G_k$, of $\mathcal{G}$. We wish to augment $\mathcal G$ by adding vertices and edges to make it connected in such a way that these vertices and edges can be added to $G_1,\ldots,G_k$ as points and straight-line segments, respectively, to obtain $k$ planar straight-line drawings isomorphic to the augmentation of $\mathcal G$. We show that adding $Θ(nr^{1-1/k})$ edges and vertices to $\mathcal{G}$ is always sufficient and sometimes necessary to achieve this goal. The upper bound holds for all $r\in\{2,\ldots,n\}$ and $k\ge 2$ and is achievable by an algorithm whose running time is $O(nr^{1-1/k})$ for $k=O(1)$ and whose running time is $O(kn^2)$ for general values of $k$. The lower bound holds for all $r\in\{2,\ldots,n/4\}$ and $k\ge 2$.

cs.CG↗
arXiv2014

The $θ_5$-graph is a spanner

Given a set of points in the plane, we show that the $θ$-graph with 5 cones is a geometric spanner with spanning ratio at most $\sqrt{50 + 22 \sqrt{5}} \approx 9.960$. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most $\sqrt{50 + 22 \sqrt{5}}$ times the Euclidean distance between the vertices. We also give a lower bound on the spanning ratio of $\frac{1}{2}(11\sqrt{5} -17) \approx 3.798$.

cs.CG↗
arXiv2014

Top-Down Skiplists

We describe todolists (top-down skiplists), a variant of skiplists (Pugh 1990) that can execute searches using at most $\log_{2-\varepsilon} n + O(1)$ binary comparisons per search and that have amortized update time $O(\varepsilon^{-1}\log n)$. A variant of todolists, called working-todolists, can execute a search for any element $x$ using $\log_{2-\varepsilon} w(x) + o(\log w(x))$ binary comparisons and have amortized search time $O(\varepsilon^{-1}\log w(w))$. Here, $w(x)$ is the "working-set number" of $x$. No previous data structure is known to achieve a bound better than $4\log_2 w(x)$ comparisons. We show through experiments that, if implemented carefully, todolists are comparable to other common dictionary implementations in terms of insertion times and outperform them in terms of search times.

cs.DS↗
arXiv2014

Towards Tight Bounds on Theta-Graphs

We present improved upper and lower bounds on the spanning ratio of $θ$-graphs with at least six cones. Given a set of points in the plane, a $θ$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $θ=2π/m$, and adds an edge to the `closest' vertex in each cone. We show that for any integer $k \geq 1$, $θ$-graphs with $4k+2$ cones have a spanning ratio of $1+2\sin(θ/2)$ and we provide a matching lower bound, showing that this spanning ratio tight. Next, we show that for any integer $k \geq 1$, $θ$-graphs with $4k+4$ cones have spanning ratio at most $1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2))$. We also show that $θ$-graphs with $4k+3$ and $4k+5$ cones have spanning ratio at most $\cos(θ/4)/(\cos(θ/2)-\sin(3θ/4))$. This is a significant improvement on all families of $θ$-graphs for which exact bounds are not known. For example, the spanning ratio of the $θ$-graph with 7 cones is decreased from at most 7.5625 to at most 3.5132. These spanning proofs also imply improved upper bounds on the competitiveness of the $θ$-routing algorithm. In particular, we show that the $θ$-routing algorithm is $(1+2\sin(θ/2)/(\cos(θ/2)-\sin(θ/2)))$-competitive on $θ$-graphs with $4k+4$ cones and that this ratio is tight. Finally, we present improved lower bounds on the spanning ratio of these graphs. Using these bounds, we provide a partial order on these families of $θ$-graphs. In particular, we show that $θ$-graphs with $4k+4$ cones have spanning ratio at least $1+2\tan(θ/2)+2\tan^2(θ/2)$. This is somewhat surprising since, for equal values of $k$, the spanning ratio of $θ$-graphs with $4k+4$ cones is greater than that of $θ$-graphs with $4k+2$ cones, showing that increasing the number of cones can make the spanning ratio worse.

cs.CG↗
arXiv2013

Average Stretch Factor: How Low Does It Go?

In a geometric graph, $G$, the \emph{stretch factor} between two vertices, $u$ and $w$, is the ratio between the Euclidean length of the shortest path from $u$ to $w$ in $G$ and the Euclidean distance between $u$ and $w$. The \emph{average stretch factor} of $G$ is the average stretch factor taken over all pairs of vertices in $G$. We show that, for any constant dimension, $d$, and any set, $V$, of $n$ points in $\mathbb{R}^d$, there exists a geometric graph with vertex set $V$, that has $O(n)$ edges, and that has average stretch factor $1+ o_n(1)$. More precisely, the average stretch factor of this graph is $1+O((\log n/n)^{1/(2d+1)})$. We complement this upper-bound with a lower bound: There exist $n$-point sets in $\mathbb{R}^2$ for which any graph with $O(n)$ edges has average stretch factor $1+Ω(1/\sqrt{n})$. Bounds of this type are not possible for the more commonly studied worst-case stretch factor. In particular, there exists point sets, $V$, such that any graph with worst-case stretch factor $1+o_n(1)$ has a superlinear number of edges.

cs.CG↗
arXiv2013

On Obstacle Numbers

The obstacle number is a new graph parameter introduced by Alpert, Koch, and Laison (2010). Mukkamala etal (2012) show that there exist graphs with n vertices having obstacle number in Omega(n/\log n). In this note, we up this lower bound to Omega(n/(\log\log n)^2. Our proof makes use of an upper bound of Mukkamala etal on the number of graphs having obstacle number at most h in such a way that any subsequent improvements to their upper bound will improve our lower bound.

math.CO↗
arXiv2013

Robust Geometric Spanners

Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in addition, geometric spanners. We define a property of spanners called robustness. Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices. We show that robust spanners must have a superlinear number of edges, even in one dimension. On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.

cs.CG↗
arXiv2013

Approximating Majority Depth

We consider the problem of approximating the majority depth (Liu and Singh, 1993) of a point q with respect to an n-point set, S, by random sampling. At the heart of this problem is a data structures question: How can we preprocess a set of n lines so that we can quickly test whether a randomly selected vertex in the arrangement of these lines is above or below the median level. We describe a Monte-Carlo data structure for this problem that can be constructed in O(nlog n) time, can answer queries O((log n)^{4/3}) expected time, and answers correctly with high probability.

cs.CG↗
arXiv2013

On the Average Number of Edges in Theta Graphs

Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds---with vanishing error terms---hold for theta graphs of finite sets of points that are uniformly distributed in a square. Finally, we show that the number of edges in a theta graph of points uniformly distributed in a square is concentrated around its expected value.

cs.CG↗
arXiv2013

The Fresh-Finger Property

The unified property roughly states that searching for an element is fast when the current access is close to a recent access. Here, "close" refers to rank distance measured among all elements stored by the dictionary. We show that distance need not be measured this way: in fact, it is only necessary to consider a small working-set of elements to measure this rank distance. This results in a data structure with access time that is an improvement upon those offered by the unified property for many query sequences.

cs.DS↗
arXiv2013

Crossings in Grid Drawings

We prove crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of non-crossing geometric graphs that can be drawn on such grids. In particular, we show that any geometric graph with m >= 8N edges and with vertices on a 3D integer grid of volume N, has Ω((m^2/n)\log(m/n)) crossings. In d-dimensions, with d >= 4, this bound becomes Ω(m^2/n). We provide matching upper bounds for all d. Finally, for d >= 4 the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some d-dimensional grid of volume N is n^Θ(n). In 3 dimensions it remains open to improve the trivial bounds, namely, the 2^Ω(n) lower bound and the n^O(n) upper bound.

math.CO↗
arXiv2012

A Note on Interference in Random Point Sets

The (maximum receiver-centric) interference of a geometric graph (von Rickenbach etal (2005)) is studied. It is shown that, with high probability, the following results hold for a set, V, of n points independently and uniformly distributed in the unit d-cube, for constant dimension d: (1) there exists a connected graph with vertex set V that has interference O((log n)^{1/3}); (2) no connected graph with vertex set V has interference o((log n)^{1/4}); and (3) the minimum spanning tree of $V$ has interference Theta((\log n)^{1/2}).

cs.CG↗