Christos Levcopoulos

Fast Greedy Algorithms for Constructing Sparse Geometric Spanners

Given a set V of n points in

TSP with neighborhoods of varying size

\IR^d

Quasi-greedy triangulations approximating the minimum weight triangulation

and a real constant t>1, we present the first O(nlog n)-time algorithm to compute a geometric t-spanner on V. A geometric t-spanner on V is a connected graph G = (V,E) with edge weights equal to the Euclidean distances between the endpoints, and with the property that, for all

A balanced search tree with O (1) worst case update time

u,v\in V

Approximate distance oracles for geometric graphs

, the distance between u and v in G is at most t times the Euclidean distance between u and v. The spanner output by the algorithm has O(n) edges and weight

Ther Are Planar Graphs Almost as Good as the Complete Graphs and as Short as Minimum Spanning Trees

O(1)\cdot wt(MST)

There are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees

, and its degree is bounded by a constant.

Efficient algorithms for constructing fault-tolerant geometric spanners

In TSP with neighborhoods we are given a set of objects in the plane, called neighborhoods, and we seek the shortest tour that visits all neighborhoods. Until now constant-factor approximation algorithms have been known only for cases where the objects are of approximately the same size. We present the first polynomial-time constant-factor approximation algorithm for disjoint convex fat objects of arbitrary size. We also show that the problem is APX-hard and cannot be approximated within a factor of 391/390 in polynomial time, unless P = NP.

Improved Greedy Algorithms for Constructing Sparse Geometric Spanners

This article settles the following two longstanding open problems:?What is the worst case approximation ratio between the greedy triangulation and the minimum weight triangulation??Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum?The answer to the first question is that the known lower bound is tight. The second question is answered in the affirmative by using a slight modification of anO(nlogn) algorithm for the greedy triangulation. We also derive some other interesting results. For example, we show that a constant-factor approximation of the minimum weight convex partition can be obtained within the same time bounds.

An ω(√ n ) lower bound for the nonoptimality of the greedy triangulation

SummaryIn this paper a new data structure is described for performing member and neighbor queries in O(logn) time that allows for O(1) worst-case update time once the position of the inserted or deleted element is known. In this way previous solutions that achieved only O(1) amortized time or O(log*n) worst-case time are improved. The method is based on a combinatorial result on the height of piles that are split after some fixed number of insertions. This combinatorial result is interesting in its own right and might have other applications as well.