Proceedings of the 3rd international workshop on Discrete algorithms and methods for mobile computing and communications, 1999
We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes ar... more We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes are points in the plane and two nodes can communicate if the distance between them is less than some fixed unit. We describe the first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guaranty that a packet is delivered to its destination. These algorithms can be extended to yield algorithms for broadcasting and geocasting that do not require packet duplication. A byproduct of our results is a simple distributed protocol for extracting a planar subgraph of a unit graph. We also present simulation results on the performance of our algorithms. *This work was partly funded by the Natural Sciences and En& neering Research Council of Canada. pcmission to m&c digital or hard copies of all or part of this work for personal or classroom USC is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the tint Page. To coPY otherwise, to republish, to post on servers or to redistribute to lists. requires prior specific permission and/or a fee. DIAL M 99 Seattle WA ES.4
For every n ∈ N, there is a straight-line drawing Dn of a planar graph on n vertices such that in... more For every n ∈ N, there is a straight-line drawing Dn of a planar graph on n vertices such that in any crossing-free straight-line drawing of the graph, at most O(n .4982) vertices lie at the same position as in Dn. This improves on an earlier bound of O(√ n) by Goaoc et al. [6].
We give a new local test, called a Half-Space Proximal or HSP test, for extracting a sparse direc... more We give a new local test, called a Half-Space Proximal or HSP test, for extracting a sparse directed or undirected subgraph of a given unit disk graph. The HSP neighbors of each vertex are unique, given a fixed underlying unit disk graph. The HSP test is a fully distributed, computationally simple algorithm that is applied independently to each vertex of a unit disk graph. The directed spanner obtained by this test is shown to be strongly connected, has out-degree at most six, its dilation is at most 2π + 1, contains the minimum weight spanning tree as its subgraph and, unlike the Yao graph, it is rotation invariant. Since no coordinate assumption is needed to determine the HSP nodes, the test can be applied in any metric space.
We analyze several perfect-information combinatorial games played on planar triangulations. We in... more We analyze several perfect-information combinatorial games played on planar triangulations. We introduce three broad categories of such games: constructing, transforming, and marking triangulations. In various situations, we develop polynomial-time algorithms to determine who wins a given game under optimal play, and to find a winning strategy. Along the way, we show connections to existing combinatorial games such as Kayles.
We say that a first order formula Φ defines a graph G if Φ is true on G and false on every graph ... more We say that a first order formula Φ defines a graph G if Φ is true on G and false on every graph G ′ non-isomorphic with G. Let D(G) be the minimal quantifier rank of a such formula. We prove that, if G is a tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar graph, then D(G) = O(log n), where n denotes the order of G. This bound is optimal up to a constant factor. If h is a constant, for connected graphs with no minor K h and degree O(√ n/ log n), we prove the bound D(G) = O(√ n). This result applies to planar graphs and, more generally, to graphs of bounded genus. Our proof techniques are based on the characterization of the quantifier rank as the length of the Ehrenfeucht game on non-isomorphic graphs. We use the separator theorems to design a winning strategy for Spoiler in this game.
We consider combinatorial and algorithmic aspects of the well-known paradigm "killing two birds w... more We consider combinatorial and algorithmic aspects of the well-known paradigm "killing two birds with one stone". We define a stage graph as follows: vertices are points from a planar point set, and {u, u} is an edge if and only if the (infinite, straight) line segment joining u to u intersects a given line segment, called a stage. We show that a graph is a stage graph if and only if it is a permutation graph. The characterization results in a compact linear space representation of stage graphs. This has been exploited for designing improved algorithms for maximum matching in permutation graphs, two processor task scheduling for dependency graphs known to be permutation graphs, and dominance-related problems for planar point sets. We show that a maximum matching in permutation graphs can be computed in Q(nlog* n) time, where n is the number of vertices. We provide simple optimal sequential and parallel algorithms for several dominance related problems for planar point sets.
A geometric network is a distributed network where each processor is identified by two numbers, r... more A geometric network is a distributed network where each processor is identified by two numbers, representing the coordinates of the point in the plane where the processor is located. The edges of the network correspond to straight line segments such that no two of them intersect. In this paper we introduce the study of distributed computing in geometric networks. We study several computational geometry problems from the distributed computing point of view, such as finding convex hulls of geometric networks and identification of the external face. In particular, we obtain a O(n log 2 n) message complexity algorithm to find the convex hull of a planar geometric graph, and a O(n log n) algorithm to identify the external face of a geometric graph. We also prove that the message complexity of leader election in an asynchronous geometric ring of n processors is Ω(n log n).
A finite poset P(X,<) on a set X={x 1 ,...,x m } is an angle order (regular n-gon order) if the e... more A finite poset P(X,<) on a set X={x 1 ,...,x m } is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and having parallel sides) such that x i <x j if and only if the angular region (regular n-gon) for x i is contained in the region (regular n-gon) for x j. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using equliateral triangles with the same orientation. This result does not generalizes to higher dimensions. We will prove that there is a partial order of dimension 4 with 14 elements which is not a regular n-gon order regardless of the value of n. Finally, we prove that partial orders of dimension 3 are regular n-gon orders for n≥3.
The problem of traversal of planar subdivisions or other graph-like structures without using mark... more The problem of traversal of planar subdivisions or other graph-like structures without using mark bits is central to many real-world applications [7, 8, 11, 12, 13, 17, 18]. The first such algorithms developed were able to traverse triangulated subdivisions [10]. Later these algorithms were extended to traverse vertices of an arrangement or a convex polytope [3]. The research progress culminated to an algorithm that can traverse any planar subdivision [6, 9]. In this paper, we extend the notion of planar subdivision to quasi-planar subdivision in which we allow many edges to cross each other. We generalize the algorithm from [9] to traverse any quasi-planar subdivision that satisfies a simple geometric requirement. If we use techniques from [6] the worst case running time of our algorithm is O(|E| log |E|); matching the running time of the traversal algorithm for planar subdivisions [6].
International Journal of Computational Geometry & Applications, 1998
We provide the first tight bound for covering an orthogonal polygon with n vertices and h holes w... more We provide the first tight bound for covering an orthogonal polygon with n vertices and h holes with vertex floodlights (guards with restricted angle of vision). In particular, we provide tight bounds for the number of orthogonal floodlights, placed at vertices or on the boundary, sufficient to illuminate the interior or the exterior of an orthogonal polygon with holes. Our results lead directly to very simple linear, and thus optimal, algorithms for computing a covering of an orthogonal polygon.
Let P 1 , ..., P k be a collection of disjoint point sets in 2 in general position. We prove that... more Let P 1 , ..., P k be a collection of disjoint point sets in 2 in general position. We prove that for each 1 ≤ i ≤ k we can find a plane spanning tree T i of P i such that the edges of T 1 , ..., T k intersect at most kn(k −1)(n−k)+ (k)(k−1) 2 , where n is the number of points in P 1 ∪ ... ∪ P k. If the intersection of the convex hulls of P 1 , ..., P k is non empty, we can find k spanning cycles such that their edges intersect at most (k − 1)n times, this bound is tight. We also prove that if P and Q are disjoint point sets in general position, then the minimum weight spanning trees of P and Q intersect at most 7n times, where |P ∪ Q| = n (the weight of an edge is its length).
We show that vertex floodlights of angle 7r suffice to illuminate any polygon, and that no angle ... more We show that vertex floodlights of angle 7r suffice to illuminate any polygon, and that no angle less than r suffices for all polygons. A vertexfloodlight of angle (Y is an internal cone of light of aperture no more than a with apex at a vertex. No vertex may have more than one floodlight.
We examine the following problem. Given a square C we want a hinged dissection of C into congruen... more We examine the following problem. Given a square C we want a hinged dissection of C into congruent squares and a colouring of the edges of these smaller squares with k colours such that we can transform the original square into another with its perimeter coloured with colour i, for all i in f1;. .. ; kg. We have the restriction that the moves have to be realizable in the plane, so when swinging the pieces no overlappings are allowed. We show a solution for k colours that uses p 2 pieces, with p an even number and at least 2k þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 À k p , this by using a necklace made of the p 2 pieces and an ingenious way to wrap it into a square.
For a graph G and integer k ≥ 1, we define the token graph F k (G) to be the graph with vertex se... more For a graph G and integer k ≥ 1, we define the token graph F k (G) to be the graph with vertex set all k-subsets of V (G), where two vertices are adjacent in F k (G) whenever their symmetric difference is a pair of adjacent vertices in G. Thus vertices of F k (G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.
Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses... more Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv (B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 ,. .. , p n and S a set of m points contained in the interior of P , m ≤ n − 1. Then there is a convex decomposition {P 1 ,. .. , P n } of P such that all points from S lie on the boundaries of P 1 ,. .. , P n , and each P i contains a whole edge of P on its boundary.
Proceedings of the 3rd international workshop on Discrete algorithms and methods for mobile computing and communications, 1999
We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes ar... more We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes are points in the plane and two nodes can communicate if the distance between them is less than some fixed unit. We describe the first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guaranty that a packet is delivered to its destination. These algorithms can be extended to yield algorithms for broadcasting and geocasting that do not require packet duplication. A byproduct of our results is a simple distributed protocol for extracting a planar subgraph of a unit graph. We also present simulation results on the performance of our algorithms. *This work was partly funded by the Natural Sciences and En& neering Research Council of Canada. pcmission to m&c digital or hard copies of all or part of this work for personal or classroom USC is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the tint Page. To coPY otherwise, to republish, to post on servers or to redistribute to lists. requires prior specific permission and/or a fee. DIAL M 99 Seattle WA ES.4
For every n ∈ N, there is a straight-line drawing Dn of a planar graph on n vertices such that in... more For every n ∈ N, there is a straight-line drawing Dn of a planar graph on n vertices such that in any crossing-free straight-line drawing of the graph, at most O(n .4982) vertices lie at the same position as in Dn. This improves on an earlier bound of O(√ n) by Goaoc et al. [6].
We give a new local test, called a Half-Space Proximal or HSP test, for extracting a sparse direc... more We give a new local test, called a Half-Space Proximal or HSP test, for extracting a sparse directed or undirected subgraph of a given unit disk graph. The HSP neighbors of each vertex are unique, given a fixed underlying unit disk graph. The HSP test is a fully distributed, computationally simple algorithm that is applied independently to each vertex of a unit disk graph. The directed spanner obtained by this test is shown to be strongly connected, has out-degree at most six, its dilation is at most 2π + 1, contains the minimum weight spanning tree as its subgraph and, unlike the Yao graph, it is rotation invariant. Since no coordinate assumption is needed to determine the HSP nodes, the test can be applied in any metric space.
We analyze several perfect-information combinatorial games played on planar triangulations. We in... more We analyze several perfect-information combinatorial games played on planar triangulations. We introduce three broad categories of such games: constructing, transforming, and marking triangulations. In various situations, we develop polynomial-time algorithms to determine who wins a given game under optimal play, and to find a winning strategy. Along the way, we show connections to existing combinatorial games such as Kayles.
We say that a first order formula Φ defines a graph G if Φ is true on G and false on every graph ... more We say that a first order formula Φ defines a graph G if Φ is true on G and false on every graph G ′ non-isomorphic with G. Let D(G) be the minimal quantifier rank of a such formula. We prove that, if G is a tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar graph, then D(G) = O(log n), where n denotes the order of G. This bound is optimal up to a constant factor. If h is a constant, for connected graphs with no minor K h and degree O(√ n/ log n), we prove the bound D(G) = O(√ n). This result applies to planar graphs and, more generally, to graphs of bounded genus. Our proof techniques are based on the characterization of the quantifier rank as the length of the Ehrenfeucht game on non-isomorphic graphs. We use the separator theorems to design a winning strategy for Spoiler in this game.
We consider combinatorial and algorithmic aspects of the well-known paradigm "killing two birds w... more We consider combinatorial and algorithmic aspects of the well-known paradigm "killing two birds with one stone". We define a stage graph as follows: vertices are points from a planar point set, and {u, u} is an edge if and only if the (infinite, straight) line segment joining u to u intersects a given line segment, called a stage. We show that a graph is a stage graph if and only if it is a permutation graph. The characterization results in a compact linear space representation of stage graphs. This has been exploited for designing improved algorithms for maximum matching in permutation graphs, two processor task scheduling for dependency graphs known to be permutation graphs, and dominance-related problems for planar point sets. We show that a maximum matching in permutation graphs can be computed in Q(nlog* n) time, where n is the number of vertices. We provide simple optimal sequential and parallel algorithms for several dominance related problems for planar point sets.
A geometric network is a distributed network where each processor is identified by two numbers, r... more A geometric network is a distributed network where each processor is identified by two numbers, representing the coordinates of the point in the plane where the processor is located. The edges of the network correspond to straight line segments such that no two of them intersect. In this paper we introduce the study of distributed computing in geometric networks. We study several computational geometry problems from the distributed computing point of view, such as finding convex hulls of geometric networks and identification of the external face. In particular, we obtain a O(n log 2 n) message complexity algorithm to find the convex hull of a planar geometric graph, and a O(n log n) algorithm to identify the external face of a geometric graph. We also prove that the message complexity of leader election in an asynchronous geometric ring of n processors is Ω(n log n).
A finite poset P(X,<) on a set X={x 1 ,...,x m } is an angle order (regular n-gon order) if the e... more A finite poset P(X,<) on a set X={x 1 ,...,x m } is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and having parallel sides) such that x i <x j if and only if the angular region (regular n-gon) for x i is contained in the region (regular n-gon) for x j. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using equliateral triangles with the same orientation. This result does not generalizes to higher dimensions. We will prove that there is a partial order of dimension 4 with 14 elements which is not a regular n-gon order regardless of the value of n. Finally, we prove that partial orders of dimension 3 are regular n-gon orders for n≥3.
The problem of traversal of planar subdivisions or other graph-like structures without using mark... more The problem of traversal of planar subdivisions or other graph-like structures without using mark bits is central to many real-world applications [7, 8, 11, 12, 13, 17, 18]. The first such algorithms developed were able to traverse triangulated subdivisions [10]. Later these algorithms were extended to traverse vertices of an arrangement or a convex polytope [3]. The research progress culminated to an algorithm that can traverse any planar subdivision [6, 9]. In this paper, we extend the notion of planar subdivision to quasi-planar subdivision in which we allow many edges to cross each other. We generalize the algorithm from [9] to traverse any quasi-planar subdivision that satisfies a simple geometric requirement. If we use techniques from [6] the worst case running time of our algorithm is O(|E| log |E|); matching the running time of the traversal algorithm for planar subdivisions [6].
International Journal of Computational Geometry & Applications, 1998
We provide the first tight bound for covering an orthogonal polygon with n vertices and h holes w... more We provide the first tight bound for covering an orthogonal polygon with n vertices and h holes with vertex floodlights (guards with restricted angle of vision). In particular, we provide tight bounds for the number of orthogonal floodlights, placed at vertices or on the boundary, sufficient to illuminate the interior or the exterior of an orthogonal polygon with holes. Our results lead directly to very simple linear, and thus optimal, algorithms for computing a covering of an orthogonal polygon.
Let P 1 , ..., P k be a collection of disjoint point sets in 2 in general position. We prove that... more Let P 1 , ..., P k be a collection of disjoint point sets in 2 in general position. We prove that for each 1 ≤ i ≤ k we can find a plane spanning tree T i of P i such that the edges of T 1 , ..., T k intersect at most kn(k −1)(n−k)+ (k)(k−1) 2 , where n is the number of points in P 1 ∪ ... ∪ P k. If the intersection of the convex hulls of P 1 , ..., P k is non empty, we can find k spanning cycles such that their edges intersect at most (k − 1)n times, this bound is tight. We also prove that if P and Q are disjoint point sets in general position, then the minimum weight spanning trees of P and Q intersect at most 7n times, where |P ∪ Q| = n (the weight of an edge is its length).
We show that vertex floodlights of angle 7r suffice to illuminate any polygon, and that no angle ... more We show that vertex floodlights of angle 7r suffice to illuminate any polygon, and that no angle less than r suffices for all polygons. A vertexfloodlight of angle (Y is an internal cone of light of aperture no more than a with apex at a vertex. No vertex may have more than one floodlight.
We examine the following problem. Given a square C we want a hinged dissection of C into congruen... more We examine the following problem. Given a square C we want a hinged dissection of C into congruent squares and a colouring of the edges of these smaller squares with k colours such that we can transform the original square into another with its perimeter coloured with colour i, for all i in f1;. .. ; kg. We have the restriction that the moves have to be realizable in the plane, so when swinging the pieces no overlappings are allowed. We show a solution for k colours that uses p 2 pieces, with p an even number and at least 2k þ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 À k p , this by using a necklace made of the p 2 pieces and an ingenious way to wrap it into a square.
For a graph G and integer k ≥ 1, we define the token graph F k (G) to be the graph with vertex se... more For a graph G and integer k ≥ 1, we define the token graph F k (G) to be the graph with vertex set all k-subsets of V (G), where two vertices are adjacent in F k (G) whenever their symmetric difference is a pair of adjacent vertices in G. Thus vertices of F k (G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.
Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses... more Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv (B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 ,. .. , p n and S a set of m points contained in the interior of P , m ≤ n − 1. Then there is a convex decomposition {P 1 ,. .. , P n } of P such that all points from S lie on the boundaries of P 1 ,. .. , P n , and each P i contains a whole edge of P on its boundary.
Uploads
Papers by Jorge Urrutia