183 research outputs found

    A Sidetrack-Based Algorithm for Finding the k Shortest Simple Paths in a Directed Graph

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    We present an algorithm for the k shortest simple path problem on weighted directed graphs (kSSP) that is based on Eppstein's algorithm for a similar problem in which paths are allowed to contain cycles. In contrast to most other algorithms for kSSP, ours is not based on Yen's algorithm and does not solve replacement path problems. Its worst-case running time is on par with state-of-the-art algorithms for kSSP. Using our algorithm, one may find O(m) simple paths with a single shortest path tree computation and O(n + m) additional time per path in well-behaved cases, where n is the number of nodes and m is the number of edges. Our computational results show that on random graphs and large road networks, these well-behaved cases are quite common and our algorithm is faster than existing algorithms by an order of magnitude. Further, the running time is far better predictable due to very small dispersion

    A fast 0(n) Embedding Algorithm, based on the Hopcroft-Tarjan Planary Test

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    The embedding problem for a planar undirected graph G = (V,E) consists of constructing adjacency lists A(v) for each node v in V, in which all the neighbors of v appear in clockwise order with respect to a planar drawing of G. Such a set of adjacency lists is called a (combinatorial) embedding of G. Chiba presented a linear time algorithm based on the 'vertex-addition' planarity testing algorithm of Lempel, Even and Cederbaum using a PQ-tree. It is very complicated to implement this data structure. He also pointed out that it is fairly complicated to modify the linear 'path-addition' planarity testing algorithm of Hopcroft and Tarjan, such that it produces an embedding. We present a straightforward extension of the Hopcroft and Tarjan planarity testing algorithm which is easy to implement. Our method runs in linear time and performs very efficiently in practice

    Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

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    The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is NP{\sf NP}-hard in general graphs. Confining to trees renders polynomial time algorithms possible and is of fundamental importance for approaches on more general graph classes. Various variants of this problem in trees have been intensively studied. We consider the general case, where trees are neither rooted nor ordered and the isomorphism is maximum w.r.t. a weight function on the mapped vertices and edges. For trees of order nn and maximum degree Δ\Delta our algorithm achieves a running time of O(n2Δ)\mathcal{O}(n^2\Delta) by exploiting the structure of the matching instances arising as subproblems. Thus our algorithm outperforms the best previously known approaches. No faster algorithm is possible for trees of bounded degree and for trees of unbounded degree we show that a further reduction of the running time would directly improve the best known approach to the assignment problem. Combining a polynomial-delay algorithm for the enumeration of all maximum common subtree isomorphisms with central ideas of our new algorithm leads to an improvement of its running time from O(n6+Tn2)\mathcal{O}(n^6+Tn^2) to O(n3+TnΔ)\mathcal{O}(n^3+Tn\Delta), where nn is the order of the larger tree, TT is the number of different solutions, and Δ\Delta is the minimum of the maximum degrees of the input trees. Our theoretical results are supplemented by an experimental evaluation on synthetic and real-world instances

    Automatisches Layout von Diagrammen

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    Wir geben eine EinfĂĽhrung in das Gebiet des automatischen Zeichnens von Diagrammen und zeigen auf, wie mathematische und algorithmische Methoden des Operations Research dazu beitragen, qualitativ hochwertige Layouts praktisch effizient zu erzeugen
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