The FastMap Algorithm for Shortest Path Computations

We present a new preprocessing algorithm for embedding the nodes of a given edge-weighted undirected graph into a Euclidean space. The Euclidean distance between any two nodes in this space approximates the length of the shortest path between them in the given graph. Later, at runtime, a shortest path between any two nodes can be computed with A* search using the Euclidean distances as heuristic. Our preprocessing algorithm, called FastMap, is inspired by the data mining algorithm of the same name and runs in near-linear time. Hence, FastMap is orders of magnitude faster than competing approaches that produce a Euclidean embedding using Semidefinite Programming. FastMap also produces admissible and consistent heuristics and therefore guarantees the generation of shortest paths. Moreover, FastMap applies to general undirected graphs for which many traditional heuristics, such as the Manhattan Distance heuristic, are not well defined. Empirically, we demonstrate that A* search using the FastMap heuristic is competitive with A* search using other state-of-the-art heuristics, such as the Differential heuristic

Embedding Directed Graphs in Potential Fields Using FastMap-D

Embedding undirected graphs in a Euclidean space has many computational benefits. FastMap is an efficient embedding algorithm that facilitates a geometric interpretation of problems posed on undirected graphs. However, Euclidean distances are inherently symmetric and, thus, Euclidean embeddings cannot be used for directed graphs. In this paper, we present FastMap-D, an efficient generalization of FastMap to directed graphs. FastMap-D embeds vertices using a potential field to capture the asymmetry between the pairwise distances in directed graphs. FastMap-D learns a potential function to define the potential field using a machine learning module. In experiments on various kinds of directed graphs, we demonstrate the advantage of FastMap-D over other approaches.Comment: 9 pages, Published in Symposium on Combinatorial Search(SoCS-2020). Erratum with updated Result

A fast and tight heuristic for A∗ in road networks

We study exact, efficient and practical algorithms for route planning in large road networks. Routing applications often require integrating the current traffic situation, planning ahead with traffic predictions for the future, respecting forbidden turns, and many other features depending on the exact application. While Dijkstra’s algorithm can be used to solve these problems, it is too slow for many applications. A* is a classical approach to accelerate Dijkstra’s algorithm. A* can support many extended scenarios without much additional implementation complexity. However, A*’s performance depends on the availability of a good heuristic that estimates distances. Computing tight distance estimates is a challenge on its own. On road networks, shortest paths can also be quickly computed using hierarchical speedup techniques. They achieve speed and exactness but sacrifice A*’s flexibility. Extending them to certain practical applications can be hard. In this paper, we present an algorithm to efficiently extract distance estimates for A* from Contraction Hierarchies (CH), a hierarchical technique. We call our heuristic CH-Potentials. Our approach allows decoupling the supported extensions from the hierarchical speed-up technique. Additionally, we describe A* optimizations to accelerate the processing of low degree nodes, which often occur in road networks

Efficient time series matching by wavelets.

by Chan, Kin Pong.Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.Includes bibliographical references (leaves 100-105).Abstracts in English and Chinese.Acknowledgments --- p.iiAbstract --- p.iiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Wavelet Transform --- p.4Chapter 1.2 --- Time Warping --- p.5Chapter 1.3 --- Outline of the Thesis --- p.6Chapter 2 --- Related Work --- p.8Chapter 2.1 --- Similarity Models for Time Series --- p.8Chapter 2.2 --- Dimensionality Reduction --- p.11Chapter 2.3 --- Wavelet Transform --- p.15Chapter 2.4 --- Similarity Search under Time Warping --- p.16Chapter 3 --- Dimension Reduction by Wavelets --- p.21Chapter 3.1 --- The Proposed Approach --- p.21Chapter 3.1.1 --- Haar Wavelets --- p.23Chapter 3.1.2 --- DFT versus Haar Transform --- p.27Chapter 3.1.3 --- Guarantee of no False Dismissal --- p.29Chapter 3.2 --- The Overall Strategy --- p.34Chapter 3.2.1 --- Pre-processing --- p.35Chapter 3.2.2 --- Range Query --- p.35Chapter 3.2.3 --- Nearest Neighbor Query --- p.36Chapter 3.3 --- Performance Evaluation --- p.39Chapter 3.3.1 --- Stock Data --- p.39Chapter 3.3.2 --- Synthetic Random Walk Data --- p.45Chapter 3.3.3 --- Scalability Test --- p.51Chapter 3.3.4 --- Other Wavelets --- p.52Chapter 4 --- Time Warping --- p.55Chapter 4.1 --- Similarity Search based on K-L Transform --- p.60Chapter 4.2 --- Low Resolution Time Warping --- p.63Chapter 4.2.1 --- Resolution Reduction of Sequences --- p.63Chapter 4.2.2 --- Distance Compensation --- p.67Chapter 4.2.3 --- Time Complexity --- p.73Chapter 4.3 --- Adaptive Time Warping --- p.77Chapter 4.3.1 --- Time Complexity --- p.79Chapter 4.4 --- Performance Evaluation --- p.80Chapter 4.4.1 --- Accuracy versus Runtime --- p.80Chapter 4.4.2 --- Precision versus Recall --- p.85Chapter 4.4.3 --- Overall Runtime --- p.91Chapter 4.4.4 --- Starting Up Evaluation --- p.93Chapter 5 --- Conclusion and Future Work --- p.95Chapter 5.1 --- Conclusion --- p.95Chapter 5.2 --- Future Work --- p.96Chapter 5.2.1 --- Application of Wavelets on Biomedical Signals --- p.96Chapter 5.2.2 --- Moving Average Similarity --- p.98Chapter 5.2.3 --- Clusters-based Matching in Time Warping --- p.98Bibliography --- p.9