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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 guarantee that a… (More)

We consider online routing algorithms for routing between the vertices of embedded planar straight line graphs. Our results include (1) two deterministic memoryless routing algorithms, one that works for all Delaunay triangulations and the other that works for all regular triangula-tions, (2) a randomized memoryless algorithm that works for all… (More)

A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queue-number. A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in Z 3 and the edges… (More)

We consider online routing algorithms for finding paths between the vertices of plane graphs. We show (1) there exists a routing algorithm for arbitrary triangulations that has no memory and uses no randomization, (2) no equivalent result is possible for convex subdivisions, (3) there is no competitive online routing algorithm under the Euclidean distance… (More)

An abstract NP-hard covering problem is presented and xed-parameter tractable algorithms for this problem are described. The running times of the algorithms are expressed in terms of three parameters: n, the number of elements to be covered, k, the number of sets allowed in the covering, and d, the combinatorial dimension of the problem. The rst algorithm… (More)

An exact formula is given for the maximum number of edges in a graph that admits a three-dimensional grid-drawing contained in a given bounding box. A three-dimensional (straight-line) grid-drawing of a graph represents the vertices by distinct points in Z 3 , and represents each edge by a line-segment between its endpoints that does not intersect any other… (More)

The detour and spanning ratio of a graph embedded in measure how well approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe " ! $ # & % ') (time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we obtain " ! $ # & % 1 0 2) (time algorithms… (More)

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n ǫ vertices fixed. We answer this question in the affirmative with ǫ = 1/4. The previous best known… (More)

The maximum detour and spanning ratio of an embedded graph G are values that measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(n log n) time algorithms for computing the maximum detour and spanning ratio of a planar polygonal path. These algorithms solve open problems posed in at… (More)