Jorge Urrutia

Routing with guaranteed delivery in ad hoc wireless networks

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 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.

Art Gallery and Illumination Problems

Abstract In 1973, Victor Klee posed the following question: How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naive question of combinatorial geometry has, since its formulation, stimulated a plethora of papers, surveys and a book, most of them written in the last fifteen years. The first result in this area, due to V. Chvatal, asserts that n 3 guards are occasionally necessary and always sufficient to guard an art gallery represented by a simple polygon with n vertices. Since ChvataFs result, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this paper, we survey most of these results.

On minimizing the sum ofensor movements for barrier coverage of a line segment

A set of sensors establishes barrier coverage of a given line segment if every point of the segment is within the sensing range of a sensor. Given a line segment I, n mobile sensors in arbitrary initial positions on the line (not necessarily inside I) and the sensing ranges of the sensors, we are interested in finding final positions of sensors which establish a barrier coverage of I so that the sum of the distances traveled by all sensors from initial to final positions is minimized. It is shown that the problem is NP complete even to approximate up to constant factor when the sensors may have different sensing ranges. When the sensors have an identical sensing range we give several efficient algorithms to calculate the final destinations so that the sensors either establish a barrier coverage or maximize the coverage of the segment if complete coverage is not feasible while at the same time the sum of the distances traveled by all sensors is minimized. Some open problems are also mentioned.

Integer Sets with Distinct Sums and Differences and Carrier Frequency Assignments for Nonlinear Repeaters

The problem of assigning n carrier frequencies so as to avoid certain types (third and fifth order) of intermodulation interference is discussed. For the third-order case, close upper and lower bounds on the optimal solution are established; and close to optimal solutions are given for n (previously, suboptimal solutions were known only for n \leq 23 ). For the fifth-order case, it is shown that some existing results can be applied to this problem, and suboptimal solutions obtained by this construction are given for n \leq 17 (no solutions were known previously).

Immobilizing a polytope

We say that a polygon P is immobilized by a set of points I on its boundary if any rigid motion of P in the plane causes at least one point of I to penetrate the interior of P. Three immobilization points are always sufficient for a polygon with vertices in general positions, but four points are necessary for some polygons with parallel edges. An O(n log n) algorithm that finds a set of 3 points that immobilize a given polygon with vertices in general positions is suggested. The algorithm becomes linear for convex polygons. Some results are generalized for d-dimensional polytopes, where 2d points are always sufficient and sometimes necessary to immobilize. When the polytope has vertices in general position d+1 points are sufficient to immobilize.

Augmenting the connectivity of geometric graphs

Let G be a connected plane geometric graph with n vertices. In this paper, we study bounds on the number of edges required to be added to G to obtain 2-vertex or 2-edge connected plane geometric graphs. In particular, we show that for G to become 2-edge connected, 2n3 additional edges are required in some cases and that 6n7 additional edges are always sufficient. For the special case of plane geometric trees, these bounds decrease to n2 and 2n3, respectively.

Simple alternating path problem

Abstract Let A be a set of 2 n points in general position on a plane, and suppose n of the points are coloured red while the remaining are coloured blue. An alternating path P of A is a sequence p 1 , p 2 ,…, p 2 n of points of A such that p 2 i is blue and p 2 i +1 is red. P is simple if it does not intersect itself. We determine the condition under which there exists a simple alternating path P of A for the case when the 2 n points are the vertices of a convex polygon. As a consequence an O( n 2 ) algorithm to find such an alternating path (if it exists) is obtained.

The Floodlight Problem

Given three angles summing to 2π, given n points in the plane and a tripartition k1 + k2 + k3 = n, we can tripartition the plane into three wedges of the given angles so that the i-th wedge contains ki of the points. This new result on dissecting point sets is used to prove that lights of specified angles not exceeding π can be placed at n fixed points in the plane to illuminate the entire plane if and only if the angles sum to at least 2π. We give O(nlog n) algorithms for both these problems.

Finding a minimum independent dominating set in a permutation graph

Abstract We give a linear time reduction of the problem of finding a minimum independent dominating set in a permutation graph, into that of finding a shortest maximal increasing subsequence. We then give an O( n log 2 n )-time algorithm for solving the second (and hence the first) problem. This improves on the O ( n 3 )-time algorithm given in [4] for solving the problem of finding a minimum independent dominating set in a permutation graph.

Local algorithms for dominating and connected dominating sets of unit disk graphs with location aware nodes

Many protocols in ad-hoc networks use dominating and connected dominating sets, for example for broadcasting and routing. For large ad hoc networks the construction of such sets should be local in the sense that each node of the network should make decisions based only on the information obtained from nodes located a constant number of hops from it. In this paper we use the location awareness of the network, i.e. the knowledge of position of nodes in the plane to provide local, constant approximation, deterministic algorithms for the construction of dominating and connected dominating sets of a Unit Disk Graph (UDG). The size of the constructed set, in the case of the dominating set, is shown to be 5 times the optimal, while for the connected dominating set 7.453 + Ɛ the optimal, for any arbitrarily small Ɛ > 0. These are to our knowledge the first local algorithms whose time complexities and approximation bounds are independent of the size of the network.