Abstract. An ordered tree is a tree in which each node’s incident edges are cyclically ordered; think of the tree as being embedded in the plane. Let A and B be two ordered trees. The edit distance between A and B is the minimum cost of a sequence of operations (contract an edge, uncontract an edge, modify the label of an edge) needed to transform A into B. WegiveanO(n 3 log n) algorithm to compute the edit distance between two ordered trees. 1

Several papers describe linear time algorithms to preprocess a tree, such that one can answer subsequent nearest common ancestor queries in constant time. Here, we survey these algorithms and related results. A common idea used by all the algorithms for the problem is that a solution for complete balanced binary trees is straightforward. Furthermore, for complete balanced binary trees we can easily solve the problem in a distributed way by labeling the nodes of the tree such that from the labels of two nodes alone one can compute the label of their nearest common ancestor. Whether it is possible to distribute the data structure into short labels associated with the nodes is important for several applications such as routing. Therefore, related labeling problems have received a lot of attention recently.

We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semi-duality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log 3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n 2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n 2 log 3 n). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner. 1 Introduction The minimum cut problem has been studied for many years as a fundamental graph optimization problem with numerous applications. Initially, th...

This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.-R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate post-office problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divide-and-conquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...

A policy describes the conditions under which an action is permitted or forbidden. We show that a fragment of (multi-sorted) first-order logic can be used to represent and reason about policies. Because we use first-order logic, policies have a clear syntax and semantics. We show that further restricting the fragment results in a language that is still quite expressive yet is also tractable. More precisely, questions about entailment, such as `May Alice access the file?', can be answered in time that is a low-order polynomial (indeed, almost linear in some cases), as can questions about the consistency of policy sets. We also give a brief overview of a prototype that we have built whose reasoning engine is based on the logic and whose interface is designed for non-logicians, allowing them to enter both policies and background information, such as `Alice is a student', and to ask questions about the policies.

The problem of finding sections of code that either are identical or are related by the systematic renaming of variables or constants can be modeled in terms of parameterized strings (p-strings) and parameterized matches (p- matches) [Baker93a]. P-strings are strings over two alphabets, one of which represents parameters. Two p-strings are a parameterized match (p-match) if one pstring is obtained by renaming the parameters of the other by a one-to-one function. In this paper, we investigate parameterized pattern matching via parameterized suffix trees (p-suffix trees), defined in [Baker93a]. We give two algorithms for constructing p-suffix trees: one (eager) that runs in linear time for fixed alphabets, and another that uses auxiliary data structures and runs in O(nlog (n)) time for variable alphabets, where n is input length. We show that using a psuffix tree for a pattern p-string P, it is possible to search for all p-matches of P within a text p-string T in space linear in ï P ï...

This paper gives algorithms for network problems that work by scaling the numeric parameters. Assume all parameters are integers. Let n, m, and N denote the number of vertices, number of edges, and largest parameter of the network, respectively. A scaling algorithm for maximum weight matching on a bipartite graph runs in O(n3 % log N) time. For appropriate N this improves the traditional Hungarian method, whose most efftcient implementation is O(n(m + n log n)). The speedup results from finding augmenting paths in batches. The matching algorithm gives similar improvements for the following problems: single-source shortest paths for arbitrary edge lengths (Bellman’s algorithm); maximum weight degree-constrained subgraph; minimum cost flow on a cl network. Scaling gives a simple maximum value flow algorithm that matches the best known bound (Sleator and Tarjan’s algorithm) when log N = O(log n). Scaling also gives a good algorithm for shortest paths on a directed graph with nonnegative edge lengths (Dijkstra’s algorithm).

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A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably e#cient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree. The method for maintaining the closest pair of points can be extended to the maintenance of closest pair of other distance functions which allows us to maintain the closest pair of a set of moving objects with similar sizes and of a set of points on a smooth manifold.

known that the planarity of a graph can be tested by an algorithm that uses only O(n + e) steps, where n =1 V I and e =1 E I. The quest for algorithms of low complexity has lead to many intriguing problems in the study of graph algorithms. For example, no efficient, i.e., polynomial time algorithm is presently known for the graph layout problem discussed above. In fact, the MIN-CUT LINEAR ARRANGEMENT problem is known to be NP- complete. It is common in the theory of graph algorithms to study problems for special classes of graphs also, and to show that the special graphs are easier to deal with algorithmically. For example, the MIN-CUT LINEAR ARRANGEMENT problem is solvable by an O(n log n) algorithm for the case of n-node trees. 1.2 Computer Representations of Graphs. 3 (a) a grap 6 1 2 3 4 5 6 o 1 o o o 1 1 o 1 o o 1 o 1 o o 1 o o o o o 1 o o o 1 1 o 1 o 1 1 o 1 o (b) the adjacency matrix of Figure 3: In this survey we will give an overview of the common paradigms and r