The (maximum receiver-centric) interference of a geometric graph (von (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 @math has interference Theta(( n)^ 1 2 ).
log suggests that point sets with very high interference are unlikely to occur in practice. This intuition is born out by the results of Kranakis al @cite_7 , who show that high interference is unlikely to occur in random point sets in one dimension: Note that, in this one-dimensional case, the minimum spanning tree, @math , is simply a path that connects the points of @math in order, from left to right. Taken together, Part 1 of Theorems and generalize sqrtlogn to arbitrary constant dimensions @math .
@cite_7
2951358293 1001549597 2031671531 2036686829
The interference at a wireless node s can be modelled by the number of wireless nodes whose transmission ranges cover s. Given a set of positions for wireless nodes, the interference minimization problem is to assign a transmission radius (equivalently, a power level) to each node such that the resulting communication graph is connected, while minimizing the maximum interference. We consider the model introduced by von (2005), in which each transmission range is represented by a ball and edges in the communication graph are symmetric. The problem is NP-complete in two dimensions (Buchin 2008) and no polynomial-time approximation algorithm is known. Furthermore, even in one dimension (the highway model), the problem's complexity is unknown and the maximum interference of a set of n wireless nodes can be as high as Theta(sqrt(n)) (von 2005). In this paper we show how to solve the problem efficiently in settings typical for wireless ad hoc networks. In particular, we show that if node positions are represented by a set P of n points selected uniformly and independently at random over a d-dimensional rectangular region, for any fixed d, then the topology given by the closure of the Euclidean minimum spanning tree of P has maximum interference O(log n) with high probability. We extend this bound to a general class of communication graphs over a broad set of probability distributions. Next we present a local algorithm that constructs a graph from this class; this is the first local algorithm to provide an upper bound on the expected maximum interference. Finally, we discuss an empirical evaluation of our algorithm with a suite of simulation results. Let (X ) be a set of multicolored points in the plane such that no three points are collinear and each color appears on at most ( |X| 2 ) points. We show the existence of a non-crossing properly colored geometric perfect matching on (X ) (if (|X| ) is even), and the existence of a non-crossing properly colored geometric spanning tree with maximum degree at most (3 ) on (X ). Moreover, we show the existence of a non-crossing properly colored geometric perfect matching in the plane lattice. In order to prove these our results, we propose an useful lemma that gives a good partition of a sequence of multicolored points. The problem of finding a minimum spanning tree connecting n points in a k-dimensional space is discussed under three common distance metrics -- Euclidean, rectilinear, and @math . By employing a subroutine that solves the post office problem, we show that, for fixed k @math 3, such a minimum spanning tree can be found in time O( @math ), where a(k) = @math . The bound can be improved to O( @math ) for points in the 3-dimensional Euclidean space. We also obtain o( @math ) algorithms for finding a farthest pair in a set of n points and for other related problems. We study the complexity of geometric minimum spanning trees under a stochastic model of input: Suppose we are given a master set of points s1,s_2,...,sn in d-dimensional Euclidean space, where each point si is active with some independent and arbitrary but known probability pi. We want to compute the expected length of the minimum spanning tree (MST) of the active points. This particular form of stochastic problems is motivated by the uncertainty inherent in many sources of geometric data but has not been investigated before in computational geometry to the best of our knowledge. Our main results include the following. We show that the stochastic MST problem is SPHARD for any dimension d ≥ 2. We present a simple fully polynomial randomized approximation scheme (FPRAS) for a metric space, and thus also for any Euclidean space. For d=2, we present two deterministic approximation algorithms: an O(n4)-time constant-factor algorithm, and a PTAS based on a combination of shifted quadtrees and dynamic programming. We show that in a general metric space the tail bounds of the distribution of the MST length cannot be approximated to any multiplicative factor in polynomial time under the assumption that P ≠ NP. In addition to this existential model of stochastic input, we also briefly consider a locational model where each point is present with certainty but its location is probabilistic.