We show how to maintain centers and medians for a collection of dynamic trees where edges may be inserted and deleted and node and edge weights may be changed. All updates are supported in O(log n) time, where n is the size of the tree(s) involved in the update.

We introduce top trees as a design of a new simpler interface for data structures maintaining information in a fully-dynamic forest. We demonstrate how easy and versatile they are to use on a host of different applications. For example, we show how to maintain the diameter, center, and median of each tree in the forest. The forest can be updated by insertion and deletion of edges and by changes to vertex and edge weights. Each update is supported in O(log n) time, where n is the size of the tree(s) involved in the update. Also, we show how to support nearest common ancestor queries and level ancestor queries with respect to arbitrary roots in O(log n) time. Finally, with marked and unmarked vertices, we show how to compute distances to a nearest marked vertex. The later has applications to approximate nearest marked vertex in general graphs, and thereby to static optimization problems over shortest path metrics. Technically speaking, top trees are easily implemented either with Frederickson’s topology trees [Ambivalent Data Structures for Dynamic 2-Edge-Connectivity and k Smallest Spanning Trees, SIAM J. Comput. 26 (2) pp. 484–538, 1997] or with Sleator and Tarjan’s dynamic

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It is well known that optimal server placement is NP-hard. We present an approximate model for the case when both clients and servers are dense, and propose a simple server allocation and placement algorithm based on high-rate vector quantization theory. The key idea is to regard the location of a request as a random variable with probability density that is proportional to the demand at that location, and the problem of server placement as source coding, i.e., to optimally map a source value (request location) to a codeword (server location) to minimize distortion (network cost). This view has led to a joint server allocation and placement algorithm that has a time-complexity that is linear in the number of clients. Simulations are presented to illustrate its performance.

We consider the problem of placing k identical resources in a graph where each vertex is associated with a nonnegative weight representing the frequency of requests issued by that vertex for the resource. We define the cost of a placement as the sum over all vertices of their distances to the closest resource weighted by their weights. The optimal placement is the placement with least cost among all placements. We give an algorithm for placing optimally k resources on a "growing" line. The algorithm starts with an empty line. At each step a new vertex is appended to the line and the algorithm has to recompute the optimal placement of the k resources. Our algorithm processes each new vertex in O(k) amortized time. As a corollary, we obtain an algorithm that computes the optimal placement of k resources in an n-vertex line in time O(kn), which is optimal for constant k. 1 Introduction We consider the problem of placing k identical resources in a graph. Each vertex v of the graph is ass...

In this note, we present linear time algorithms for computing the median set of plane triangulations with inner vertices of degree 6 and plane quadrangulations with inner vertices of degree 4: 1.

Contents 1 Introduction 3 1.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Organisation of this thesis . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Readers prerequisites . . . . . . . . . . . . . . . . . . . . . . . . 5 I Top-trees 6 2 Introduction to part I 7 2.1 Top-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Top-trees 10 3.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Black box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Applications 36 4.1 Maxweight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Global search . . . . . . . . . . . . . . . . . . . . . . . .

The k-median problem is a classical facility location problem. We consider the k-median problem for directed trees, motivated by the problem of locating proxies on the World Wide Web. The two main results of the paper are an O(n log n) time algorithm for k = 2 and an O(n log² n) time algorithm for k = 3. The previously known upper bounds for these two cases were O(n²).

The p-median problem on a tree T is to find a set S of p vertices on T that minimize the sum of distances from T's vertices to S. For this problem, Tamir [14] had an O(pn )-time algorithm, while Gavish and Sridhar [6] had an O(nlog n)-time algorithm for the case of p=2. In this paper, we study two generalizations of the 2-median problem, which are obtained by imposing constraints on a 2-median: one is to limit their eccentricity while the other is to limit their distance. We solve both generalizations in O(nlogn) time, improving the previous ones from O(n ). We also study cases when linear time algorithms exist for the 2-median problem and the two generalizations. For example, we solve all three in linear time when edge lengths and vertex weights are all polynomially bounded integers. Finally, we consider the relaxation of the two generalized problems by allowing 2-medians on any position of edges, instead of just on vertices, and we give O(nlogn)-time algorithms for them.

This paper deals with facility location problems with pos/neg weights in trees. We consider two different objective functions which model two different ways to handle obnoxious facilities. If we minimize the overall sum of the minimum weighted distances of the vertices from the facilities, the optimal solution has nice combinatorial properties, e.g., vertex optimality. These properties can be exploited to derive an O(n²) algorithm for trees, an O(n log n) algorithm for stars and a linear algorithm for paths, for the pos/neg 2-median problem on a network with n vertices. For the p-median problem with pos/neg weights on a path we give an O(pn²) algorithm. If we minimize the overall sum of the weighted minimum distances of the vertices from the facilities, we can show that there exist a finite set of O(n³) points in the tree which contains the locations of facilities in an optimal solution. This leads to an O(n³) algorithm for finding the optimum 2-medians in a tree. T...