This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.

. Ambivalent data structures are presented for several problems on undirected graphs. These data structures are used in finding the k smallest spanning trees of a weighted undirected graph in O(m log #(m, n) + min{k 3/2 ,km 1/2 }) time, where m is the number of edges and n the number of vertices in the graph. The techniques are extended to find the k smallest spanning trees in an embedded planar graph in O(n + k(log n) 3 ) time. Ambivalent data structures are also used to dynamically maintain 2-edge-connectivity information. Edges and vertices can be inserted or deleted in O(m 1/2 ) time, and a query as to whether two vertices are in the same 2-edge-connected component can be answered in O(log n) time, where m and n are understood to be the current number of edges and vertices, respectively. Key words. analysis of algorithms, data structures, embedded planar graph, fully persistent data structures, k smallest spanning trees, minimum spanning tree, on-line updating, topology tr...

In previous work we introduced sparsification, a technique that transforms fully dynamic algorithms for sparse graphs into ones that work on any graph, with a logarithmic increase in complexity. In this work we describe an improvement on this technique that avoids the logarithmic overhead. Using our improved sparsification technique, we keep track of the following properties: minimum spanning forest, best swap, connectivity, 2-edge-connectivity, and bipartiteness, in time O(n 1/2 ) per edge insertion or deletion; 2-vertex-connectivity and 3-vertex-connectivity, in time O(n) per update; and 4-vertex-connectivity, in time O(n#(n)) per update.

We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log 2 n) work per modification. No previous dynamic geometric MST algorithm was known.

We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a singlesource shortest path tree to changes in edge costs, and to analyze the sensitivity of a minimum cost network flow. The algorithm is simple and practical. It uses the properties of a planar embedding, combined with a heap-ordered queue data structure. Let G = (V; E) be a planar graph, either directed or undirected, with n vertices and m = O(n) edges. Each edge e 2 E has a real-valued cost cost(e). A minimum spanning tree of a connected, undirected planar graph G is a spanning tree of minimum total edge cost. If G is directed and r is a vertex from which all other vertices are reachable, then a shortest path tree from r is a spanning tree that contains a minimum-cost path from r to every...

In this thesis we develop a discriminative learning method for dependency parsing using online large-margin training combined with spanning tree inference algorithms. We will show that this method provides state-of-the-art accuracy, is extensible through the feature set and can be implemented efficiently. Furthermore, we display the language independent nature of the method by evaluating it on over a dozen diverse languages as well as show its practical applicability through integration into a sentence compression system. We start by presenting an online large-margin learning framework that is a generaliza- tion of the work of Crammer and Singer [34, 37] to structured outputs, such as sequences and parse trees. This will lead to the heart of this thesis – discriminative dependency pars- ing. Here we will formulate dependency parsing in a spanning tree framework, yielding efficient parsing algorithms for both projective and non-projective tree structures. We will then extend the parsing algorithm to incorporate features over larger substructures with- out an increase in computational complexity for the projective case. Unfortunately, the non-projective problem then becomes NP-hard so we provide structurally motivated ap- proximate algorithms. Having defined a set of parsing algorithms, we will also define a rich feature set and train various parsers using the online large-margin learning framework. We then compare our trained dependency parsers to other state-of-the-art parsers on 14 diverse languages: Arabic, Bulgarian, Chinese, Czech, Danish, Dutch, English, German, Japanese, Portuguese, Slovene, Spanish, Swedish and Turkish. Having built an efficient and accurate discriminative dependency parser, this thesis will then turn to improving and applying the parser. First we will show how additional re- sources can provide useful features to increase parsing accuracy and to adapt parsers to new domains. We will also argue that the robustness of discriminative inference-based learning algorithms lend themselves well to dependency parsing when feature representa- tions or structural constraints do not allow for tractable parsing algorithms. Finally, we integrate our parsing models into a state-of-the-art sentence compression system to show its applicability to a real world problem.

Untyped dependency parsing can be viewed as the problem of finding maximum spanning trees (MSTs) in directed graphs. Using this representation, the Eisner (1996) parsing algorithm is sufficient for searching the space of projective trees. More importantly, the representation is extended naturally to non-projective parsing using Chu-Liu-Edmonds (Chu and Liu, 1965; Edmonds, 1967) MST algorithm. These efficient parse search methods support large-margin discriminative training methods for learning dependency parsers. We evaluate these methods experimentally on the English and Czech treebanks. 1

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Dissertation présentée en vue de l’obtention du titre de Docteur en

A precursor is a compound which is transformed to a class of functional molecules within short steps. It is an important process in the production of natural drugs to decide whether a given compound is a precursor or not. We present two strategies to select precursor compounds in the secondary metabolism of terpenoids: one is to find the packing of basic molecules in the given cyclic structure, and the other is to find the synthetic map of the given set of compounds. Both strategies play important roles in reproducing tracer experiments on a computer. 1 Introduction Finding the biosynthetic pathway of a hormone or a natural drug is the key issue for its industrial production. Even a limited increase in its production rate may lead to a drastic change in the synthetic scheme at a commercial level, because of its very low yield from raw materials. For example, the average yield of paclitaxel (a recently approved anticancer drug) from the yew bark is in the range of 0.014--0.017%. ...