Fault-tolerant additive weighted geometric spanners

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance d_w(p, q) between two points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance between p and q. A graph G(S, E) is called a t-spanner for the additive weighted set S of points if for any two points p and q in S the distance between p and q in graph G is at most t.d_w(p, q) for a real number t > 1. Here, d_w(p,q) is the additive weighted distance between p and q. For some integer k \geq 1, a t-spanner G for the set S is a (k, t)-vertex fault-tolerant additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set S' \subset S with cardinality at most k, the graph G \ S' is a t-spanner for the points in S \ S'. For any given real number \epsilon > 0, we obtain the following results: - When the points in S belong to Euclidean space R^d, an algorithm to compute a (k,(2 + \epsilon))-VFTAWS with O(kn) edges for the metric space (S, d_w). Here, for any two points p, q \in S, d(p, q) is the Euclidean distance between p and q in R^d. - When the points in S belong to a simple polygon P, for the metric space (S, d_w), one algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n}) edges and another algorithm to compute a geodesic (k, (\sqrt{10} + \epsilon))-VFTAWS with O(kn(\lg{n})^2) edges. Here, for any two points p, q \in S, d(p, q) is the geodesic Euclidean distance along the shortest path between p and q in P. - When the points in $S$ lie on a terrain T, an algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n}) edges.Comment: a few update

Piecewise-linear approximations of uncertain functions

We study the problem of approximating a function F: R¿¿¿R by a piecewise-linear function [`(F)]F when the values of F at {x 1,…,x n } are given by a discrete probability distribution. Thus, for each x i we are given a discrete set yi,1,...,yi,miyi1yimi of possible function values with associated probabilities p i,j such that Pr[F(x i )¿=¿y i,j ]¿=¿p i,j . We define the error of [`(F)]F as error(F,[`(F)]) = maxi=1n E[|F(xi)-[`(F)](xi)|]\sl error(FF)=maxni=1E[F(xi)-F(xi)]. Let m=åi=1n mim=ni=1mi be the total number of potential values over all F(x i ). We obtain the following two results: (i) an O(m) algorithm that, given F and a maximum error e, computes a function [`(F)]F with the minimum number of links such that error(F,[`(F)]) £ e\sl error(FF) ; (ii) an O(n 4/3¿+¿d ¿+¿mlogn) algorithm that, given F, an integer value 1¿=¿k¿=¿n and any d¿>¿0, computes a function [`(F)]F of at most k links that minimizes error(F,[`(F)])\sl error(FF)

Geodesic Spanners for Points on a Polyhedral Terrain

Let S be a set of n points on a polyhedral terrain \scrT in \BbbR 3 , and let \varepsilon > 0 be a fixed constant. We prove that S admits a (2 + \varepsilon )-spanner with O(n log n) edges with respect to the geodesic distance. This is the first spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in \BbbR d admits an additively weighted (2 + \varepsilon )-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + \varepsilon ) and almost matches the lower bound

Geodesic spanners for points on a polyhedral terrain

Let S be a set S of n points on a polyhedral terrain T in R3, and let > 0 be a xed constant. We prove that S admits a (2 + )-spanner with O(n log n) edges with respect to the geodesic distance. This is the rst spanner with constant spanning ratio and a near-linear number of edges for points on a terrain. On our way to this result, we prove that any set of n weighted points in Rd admits an additively weighted (2 + )-spanner with O(n) edges; this improves the previously best known bound on the spanning ratio (which was 5 + ), and almost matches the lower bound

Out-of-order event processing in kinetic data structures

We study the problem of designing kinetic data structures (KDS’s for short) when event times cannot be computed exactly and events may be processed in a wrong order. In traditional KDS’s this can lead to major inconsistencies from which the KDS cannot recover. We present more robust KDS’s for the maintenance of two fundamental structures, kinetic sorting and tournament trees, which overcome the difficulty by employing a refined event scheduling and processing technique. We prove that the new event scheduling mechanism leads to a KDS that is correct except for finitely many short time intervals. We analyze the maximum delay of events and the maximum error in the structure, and we experimentally compare our approach to the standard event scheduling mechanism

Approximation algorithms for computing partitions with minimum stabbing number of rectilinear and simple polygons

Let P be a rectilinear simple polygon. The stabbing number of a partition of P into rectangles is the maximum number of rectangles stabbed by any axis-parallel line segment inside P. We present a 3-approximation algorithm for the problem of finding a partition with minimum stabbing number. It is based on an algorithm that finds an optimal partition for histograms. We also study Steiner triangulations of a simple (non-rectilinear) polygon P. Here the stabbing number is defined as the maximum number of triangles that can be stabbed by any line segment inside P. We give an O(1)-approximation algorithm for the problem of computing a Steiner triangulation with minimum stabbing number

A 3-approximation algorithm for computing partitions with minimum stabbing number of rectilinear simple polygons

Let P be a rectilinear simple polygon. The stabbing number of a partition of P into rectangles is the maximum number of rectangles stabbed by any axis-parallel segment inside P. We present a 3-approximation algorithm for fi??nding a partition with minimum stabbing number. It is based on an algorithm that ??finds an optimal partition for histograms