Given a node x at depth d in a rooted tree LevelAncestor(x; i) returns the ancestor to x in depth d i. We show how to maintain a tree under addition of new leaves so that updates and level ancestor queries are being performed in worst case constant time. Given a forest of trees with n nodes where edges can be added, m queries and updates take O(m(m;n)) time. This solves two open problems (P.F.

Let F be a function on pairs of vertices. An F- labeling scheme is composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute F (u, v) of any two vertices u and v directly from their labels. As applications for labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. This paper investigates labeling schemes for dynamic trees. We consider two dynamic tree models, namely, the leaf-dynamic tree model in which at each step a leaf can be added to or removed from the tree and the leaf-increasing tree model in which the only topological event that may occur is that a leaf joins the tree. A general method for constructing labeling schemes for dynamic trees (under the above mentioned dynamic tree models) was previously developed in [29]. This method is based on extending an existing static tree labeling scheme to the dynamic setting. This approach fits many natural functions on trees, such as distance, separation level, ancestry relation, routing (in both the adversary and the designer port models), nearest common ancestor etc.. Their

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

We consider the problem of testing the uniqueness of maximum matchings, both in the unweighted and in the weighted case. For the unweighted case, we have two results. First, given a graph with n vertices and m edges, we can test whether the graph has a unique perfect matching, and find it if it exists, in O(m log^4 n) time. This algorithm uses a recent dynamic connectivity algorithm and an old result of Kotzig characterizing unique perfect matchings in terms of bridges. For the special case of...

We show that we can maintain up to polylogarithmic edge connectivity for a fully-dynamic graph in ~ O ( p n) time per edge insertion or deletion. Within logarithmic factors, this matches the best time bound for 1-edge connectivity. Previously, no o(n) bound was known for edge connectivity above 3, and even for 3-edge connectivity, the best update time was O(n 2=3), dating back to FOCS’92. Our algorithm maintains a concrete min-cut in terms of a pointer to a tree spanning one side of the cut plus ability to list the cut edges in O(log n) time per edge. By dealing with polylogarithmic edge connectivity, we immediately get a sampling based expected factor (1 + o(1)) approximation to general edge connectivity in ~O ( p n) time per edge insertion or deletion. This algorithm also maintains a pointer to one side of a min-cut, but if we want to list the cut edges in O(log n) time per edge, the update time increases to ~O ( p m). 1.

We present tight upper and lower bounds for the problem of constructing evolutionary trees in the experiment model. We describe an algorithm which constructs an evolutionary tree of n species in time O(nd log d n) using at most n⌈d/2⌉(log 2⌈d/2⌉−1 n+O(1)) experiments for d> 2, and at most n(log n+O(1)) experiments for d = 2, where d is the degree of the tree. This improves the previous best upper bound by a factor Θ(log d). For d = 2 the previously best algorithm with running time O(n log n) had a bound of 4n log n on the number of experiments. By an explicit adversary argument, we show an Ω(nd log d n) lower bound, matching our upper bounds and improving the previous best lower bound by a factor Θ(log d n). Central to our algorithm is the construction and maintenance of separator trees of small height, which may be of independent interest.

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.

The dynamic trees problem is that of maintaining a forest that changes over time through edge insertions and deletions. We can associate data with vertices or edges and manip-ulate this data, individually or in bulk, with operations that deal with whole paths or trees. Efficient solutions to this problem have numerous applications, particularly in algo-rithms for network flows and dynamic graphs in general. Several data structures capable of logarithmic-time dynamic tree operations have been proposed. The first was Sleator and Tarjan’s ST-tree, which represents a partition of the tree into paths. Although reasonably fast in practice, adapting ST-trees to different applications is nontrivial. Frederickson’s topology trees, Alstrup et al.’s top trees, and Acar et al.’s RC-trees are based on tree contractions: they progressively combine vertices or edges to obtain a hierarchical represen-tation of the tree. This approach is more flexible in theory, but all known implementations assume the trees have bounded degree; arbitrary trees are supported only after ternar-ization. This thesis shows how these two approaches can be combined (with very little overhead) to produce a data structure that is at least as generic as any other, very easy to

The work presented in this dissertation was done while I was enrolled as a PhD student at the IT University of Copenhagen in the 4-year PhD program. My work was funded by the EU-project ”Deep Structure, Singularities, and Computer Vision ” (IST Programme of the European Union (IST-2001-35443)). During the summer of 2003 my advisors Stephen Alstrup and Theis Rauhe left to start their own company and my advisors then became Lars Birkedal and Anna Östlin Pagh. In the period from March 2003 to September 2003 I was on paternity leave. I received my Masters Degree in January 2005. In Spring 2005 I visited Martin Farach-Colton at Rutgers University twice for a total period of two months. In the period from October 2006 to April 2007 I was on another 6 months of paternity leave. Finally, in the remaining period I came back to finish the present dissertation. I want to thank all of the inspiring people that I have worked with during my PhD. In particular, I want to thank Stephen Alstrup and Theis Rauhe for introducing me to their unique perspective on algorithms. I also want to thank Lars Birkedal, Anna Östlin Pagh, and Rasmus Pagh for tons of guidance. I am grateful to Martin Farach-Colton for the very pleasant stay at Rutgers University. Thanks to all of my co-authors:

In this paper, we consider two facility location problems on tree networks. One is the 2-radius problem, whose goal is to partition the vertex set of the given network into two non-empty subsets such that the sum of the radii of these two induced subgraphs is minimum. The other is the 2-radiian problem, whose goal is to partition the network into two non-empty subsets such that the sum of the centdian values of these two induced subgraphs is minimum. We propose an O(n)-time algorithm for the 2-radius problem on trees and an O(n log n)-time algorithm for the 2-radiian problem on trees, where n is the number of vertices in the given tree.