We give new methods for maintaining a data structure that supports ray shooting and shortest path queries in a dynamically-changing connected planar subdivision S. Our approach is based on a new dynamic method for maintaining a balanced decomposition of a simple polygon via geodesic triangles. We maintain such triangulations by viewing their dual trees as balanced trees. We show that rotations in these trees can be implemented via a simple "diagonal swapping" operation performed on the corresponding geodesic triangles, and that edge insertion and deletion can be implemented on these trees using operations akin to the standard split and splice operations. We also maintain a dynamic point location structure on the geodesic triangulation, so that we may implement ray shooting queries by first locating the ray's endpoint and then walking along the ray from geodesic triangle to geodesic triangle until we hit the boundary of some region of S. The shortest path between two points in the same ...

A pair in a string is the occurrence of the same substring twice. A pair is maximal if the two occurrences of the substring cannot be extended to the left and right without making them different. The gap of a pair is the number of characters between the two occurrences of the substring. In this paper we present methods for finding all maximal pairs under various constraints on the gap. In a string of length n we can find all maximal pairs with gap in an upper and lower bounded interval in time O(n log n + z) where z is the number of reported pairs. If the upper bound is removed the time reduces to O(n+z). Since a tandem repeat is a pair where the gap is zero, our methods can be seen as a generalization of finding tandem repeats. The running time of our methods equals the running time of well known methods for finding tandem repeats.

A polyhedron is any set that can be obtained from the open halfspaces by a finite number of set complement and set intersection operations. We give an efficient and complete algorithm for intersecting two three--dimensional polyhedra, one of which is convex. The algorithm is efficient in the sense that its running time is bounded by the size of the inputs plus the size of the output times a logarithmic factor. The algorithm is complete in the sense that it can handle all inputs and requires no general position assumption. We also describe a novel data structure that can represent all three--dimensional polyhedra (the set of polyhedra representable by all previous data structures is not closed under the basic boolean operations).

We present and evaluate several new optimization and implementation techniques for string sorting. In particular, we study a recently published radix sorting algorithm, Forward radixsort, that has a provably good worst-case behavior. Our experimental results indicate that radix sorting is considerably faster (often more than twice as fast) than comparison-based sorting methods. This is true even for small input sequences. We also show that it is possible to implement a radix sort with good worst-case running time without sacrificing average-case performance. Our implementations are competitive with the best previously published string sorting algorithms. Code, test data, and test results are available from the World Wide Web. 1. Introduction Radix sorting is a simple and very efficient sorting method that has received too little attention. A common misconception is that a radix sorting algorithm either has to inspect all the characters of the input or use an inordinate amount of extra...

In this paper so-called discrete loops are introduced which narrow the gap between general loops (e.g. while- or repeat-loops) and for-loops. Alt- hough discrete loops can be used for applications that would otherwise require general loops, discrete loops are known to complete in any case. Furthermore it is possible to determine the number of iterations of a discrete loop, while this is trivial to do for for-loops and extremely difficult for general loops. Thus discrete loops form an ideal frame-work for determining the worst case timing behavior of a program and they are especially useful in implementing real-time systems and proving such systems correct.

B-trees with relaxed balance have been defined to facilitate fast updating on shared-memory asynchronous parallel architectures. To obtain this, rebalancing has been uncoupled from the updating such that extensive locking can be avoided in connection with updates. We analyze B-trees with relaxed balance, and prove that each update gives rise to at most blog a (N=2)c + 1 rebalancing operations, where a is the degree of the Btree, and N is the bound on its maximal size since it was last in balance. Assuming that the size of nodes are at least twice the degree, we prove that rebalancing can be performed in amortized constant time. So, in the long run, rebalancing is constant time on average, even if any particular update could give rise to logarithmic time rebalancing. We also prove that the amount of rebalancing done at any particular level decreases exponentially going from the leaves towards the root. This is important since the higher up in the tree a lock due to a rebalancing operat...

Abstract Evolutionary trees describing the relationship for a set of species are central in evolutionarybiology, and quantifying differences between evolutionary trees is therefore an important task. The quartet distance is a distance measure between trees previously proposed by Estabrook,McMorris and Meacham. The quartet distance between two unrooted evolutionary trees is the number of quartet topology differences between the two trees, where a quartet topologyis the topological subtree induced by four species. In this paper, we present an algorithm for computing the quartet distance between two unrooted evolutionary trees of n species, whereall internal nodes have degree three, in time O(n log n). The previous best algorithm for theproblem uses time O(n2).

We describe an efficient fair queuing scheme, Leap Forward Virtual Clock , that provides endto -end delay bounds almost identical to that of PGPS fair queuing, along with throughput fairness. Our scheme can be implemented with a worst-case time O(log log N ) per packet (inclusive of sorting costs), which improves upon all previously known schemes that achieve guaranteed delay and throughput fairness. As its name suggests, our scheme is based on Zhang's virtual clock. While the original virtual clock scheme does not achieve throughput fairness, we can modify it with a simple leap forward mechanism that keeps the server clock from lagging too far behind the packet tags. We prove that our scheme guarantees a fair share of the available bandwidth to each of the backlogged users, while precisely matching the delay bounds of PGPS schemes. In order to improve computational efficiency, we introduce a "coarsened" version of our scheme in which all tags assume values from a set of O(N ) integers...

What is the smallest constant c so that the planar point location queries can be answered in c log 2 n + o(log n) steps (i.e. point-line comparisons) in the worst case? In SODA 97 Goodrich, Orletsky, and Ramaiyer [6] showed that c = 2 is possible using linear space and conjectured this to be optimal. We disprove this conjecture and show that c = 1 can be achieved. Moreoever by giving upper and lower bounds we show that without space restrictions the worst case query complexity of planar point location differs from log 2 n + 2 p log 2 n at most by an additive factor of (1=2)log 2 log 2 n +O(1). For the case of linear space we show the query complexity to be bounded by log 2 n + 2 p log 2 n +O(log 1=4 n).

The purpose of this paper is to show that recursive procedures can be used for implementing real-time applications without harm, if a few conditions are met. These conditions ensure that upper bounds for space and time requirements can be derived at compile time. Moreover they are simple enough such that many important recursive algorithms can be implemented, for example Mergesort or recursive tree-traversal algorithms. In addition,