We present an improvement of the multi-level bucket shortest path algorithm of Denardo and Fox [9] and justify this improvement, both theoretically and experimentally. We prove that if the input arc lengths come from a natural probability distribution, the new algorithm runs in linear average time while the original algorithm does not. We also describe an implementation of the new algorithm. Our experimental data suggests that the new algorithm is preferable to the original one in practice. Furthermore, for integral arc lengths that fit into a word of today's computers, the performance is close to that of breadth-first search, suggesting limitations on further practical improvements.

A method for lossy compression of genus-0 surfaces is presented. Geometry, texture and other surface attributes are incorporated in a unified manner. The input surfaces are represented by surfels (surface elements), i.e., by a set of disks with attributes. Each surfel, with its attribute vector, is optimally mapped onto a sphere in the sense of geodesic distance preservation. The resulting spherical vector-valued function is resampled. Its components are decorrelated by the Karhunen-Loève transform, represented by spherical wavelets and encoded using the zerotree algorithm. Methods for geodesic distance computation on surfel-based surfaces are considered. A novel efficient approach to dense surface flattening/mapping, using rectangular distance matrices, is employed. The distance between each surfel and a set of key-surfels is optimally preserved, leading to greatly improved resolution and eliminating the need for interpolation, that complicates and slows down existing surface unfolding methods. Experimental surfel-based surface compression results demonstrate successful compression at very low bit rates. Key words: textured surface compression, spherical mapping, geodesic paths, surfels, spherical wavelets ⋆ This research was supported by the Kurt Lion Foundation. At Tel-Aviv University, it was supported by the Ministry of Science. At Konstanz University, it was supported by the DFG Graduiertenkolleg “Explorative Analysis and Visualization

Abstract—Computing the shortest path in a graph is an important problem and it is very useful in various applications. The standard shortest path problem has been studied extensively and intensively, but it can’t handle the situation when there is a switch cost between arcs. For example, in a train transportation network, the switch cost between arcs contains waiting time in stations, times of transfer and so on. Obviously, the switch cost is an important factor for users to make decisions. Taking into consideration of the switch cost between arcs, we extend the standard shortest path problem and propose an algorithm and its optimized version to solve the extended single source shortest path problem. Test results show that the proposed algorithms can give reasonable and acceptable results for users.

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This paper considersthe problem of Multi-GeographyRoute Planning (MGRP) where the geographical information may be spread over multiple heterogeneous interconnected maps. Wefirstdesignaflexibleandscalablerepresentationtomodel individual geographies and their interconnections. Given such a representation, we develop an algorithm that exploits precomputation and caching of geographical data for path planning. A utility-based approach is adopted to decide which paths to precompute and store. To validate the proposed approach we test the algorithm over the workload of a campus level evacuation simulation that plans evacuation routes over multiple geographies: indoor CAD maps, outdoor maps, pedestrian and transportation networks, etc. The empirical results indicate that the MGRP algorithm withtheproposedutilitybasedcachingstrategysignificantly outperforms the state of the art solutions when applied to a large university campus data under varying conditions. 1.

Dijkstra’s algorithm solves the single-source shortest path problem on any directed graph in O(m + n log n) worst-case time when a Fibonacci heap is used as the frontier set data structure. Here n is the number of vertices and m is the number of edges in the graph. If the graph is nearly acyclic, then other algorithms can achieve a time complexity lower than that of Dijkstra’s algorithm. Abuaiadh and Kingston gave a single source shortest path algorithm for nearly acyclic graphs with O(m + n log t) worst-case time complexity, where the new parameter t is the number of delete-min operations performed in priority queue manipulation. For nearly acyclic graphs, the value of t is expected to be small, allowing the algorithm to outperform Dijkstra’s algorithm. Takaoka, using a different definition for acyclicity, gave an algorithm with O(m + n log k) worstcase time complexity. In this algorithm, the new parameter k is the maximum cardinality of the strongly connected components in the graph. This thesis presents several new shortest path algorithms that define trigger