Computing the Shortest Path:

We propose shortest path algorithms that use A* semch in combination with a new graph-theoretic lower-bounding technique based on landmarks and the triangle inequality. Our algorithms compute optimal shortest paths and work on any directed graph. We give experimental results showing that the most efficient of our new algorithms outperforms previous algorithms, in particular A* search with Euclidean bounds, by a wide margin on road networks and on some synthetic problem families. 1 I n t r o d u c t i o n The shortest pa th problem is a fundamental problem with numerous applications. In this paper we study one of the most common variants of the problem, where the goal is to find a point-to-point shortest pa th in a weighted, directed graph. We refer to this problem as the P2P problem. We assume that for the stone underlying network, the problem will be solved repeatedly. Thus, we allow preprocessing, with the only restriction that the additional space used to store precomputed data is limited: linear in the graph size with a small constant factor. Our goal is a fast algorithm for answering point-to-point shortest pa th queries. A natural application of the P2P problem is providing driving directions, for example services like Mapquest, Yahoo! Maps and Microsoft MapPoint, and some GPS devices. One can spend time preprocessing maps for these applications, but the underlying graphs are very large, so memory usage much exceeding the graph size is prohibitive. This motivates the linearspace assmnption. Shortest pa th problems have been extensively studied. The P2P problem with no preprocessing has been addressed, for example, in [21, 27, 29, 36]. While no nontrivial theoretical results are known for the general P2P problem, there has been work on the special case of undirected planar graphs with slightly superlinear pre-~rosoft Research, 1065 La Avenida, Mountain View, CA 94062. Email: goldberg0m:icrosoft, corn; URL: h t tp ://www. avglab, com/andrew/index, html. tComputer Science Division, UC Berkeley. Part of this work was done while the author was visiting Microsoft Research. Emaih chrishtrOcs, berkeley, edu. processing space. The best bound in this context (see [10]) is superlinear in the output pa th size unless the pa th is very long. Preprocessing using geometric information and hierarchical decomposition is discussed in [19, 28, 34]. Other related work includes algorithms for the single-source shortest pa th problem, such as [1, 4, 6, 7, 13, 14, 16, 17, 18, 22, 25, 32, 35], and algorithms for approximate shortest paths that use preprocessing [3, 23, a3]. Usually one can solve the P2P problem while searching only a small portion of the graph; the algori thm's running t ime then depends only on the number of visited vertices. This motivates an output-sensitive complexity measure tha t we adopt. We measure algorithm performance as a function of the number of vertices on the output path. Note that this measure has the additional benefit of being machine-independent. In Artificial Intelligence settings, one often needs to find a solution in a huge search space. The classical A search (also known as heuristic search) [8, 20] technique often finds a solution while searching a small subspace. A search uses estimates on distances to the destination to guide vertex selection in a search from the source. Pohl [27] studied the relationship between A search and Dijkstra 's algorithm in the context of the P2P problem. He observed that if the bounds used in A search are feasible (as defined in section 2), A search is equivalent to Dijkstra 's algorithm on a graph with nonnegative arc lengths and therefore finds the optimal path. In classical applications of A search to the P2P problem, distance bounds are implicit in the domain description, with no preprocessing required. For example, for Euclidean graphs, the Euclidean distance between two vertices gives a lower bound on the distance between them. Our first contribution is a new preprocessing-based technique for computing distance bounds. The preprocessing entails carefiflly choosing a small (constant) number of landmarks, then computing and storing shortest pa th distances between all vertices and each of these landmarks. Lower bounds are computed in constant t ime using these distances in combination with the triangle inequality. These lower bounds yield a new class of algorithms, which we call ALT algorithms since they are based on A search, landmarks, and the triangle inequality. Here we are talking about the triangle