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53
Computing the shortest path: A* search meets graph theory
, 2005
"... We study the problem of finding a shortest path between two vertices in a directed graph. This is an important problem with many applications, including that of computing driving directions. We allow preprocessing the graph using a linear amount of extra space to store auxiliary information, and usi ..."
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Cited by 98 (4 self)
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We study the problem of finding a shortest path between two vertices in a directed graph. This is an important problem with many applications, including that of computing driving directions. We allow preprocessing the graph using a linear amount of extra space to store auxiliary information, and using this information to answer shortest path queries quickly. Our approach uses A ∗ search in combination with a new graphtheoretic lowerbounding technique based on landmarks and the triangle inequality. We also develop new bidirectional variants of A ∗ search and investigate several variants of the new algorithms to find those that are most efficient in practice. 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. We also experiment with several synthetic graph families.
A New Approach to Dynamic All Pairs Shortest Paths
, 2002
"... We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with nonnegative realvalued edge weights that supports any sequence of operatio ..."
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Cited by 68 (8 self)
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We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with nonnegative realvalued edge weights that supports any sequence of operations in e O(n amortized time per update and unit worstcase time per distance query, where n is the number of vertices. We can also report shortest paths in optimal worstcase time. These bounds improve substantially over previous results and solve a longstanding open problem. Our algorithm is deterministic and uses simple data structures.
Reach for A∗: Efficient pointtopoint shortest path algorithms
 IN WORKSHOP ON ALGORITHM ENGINEERING & EXPERIMENTS
, 2006
"... We study the pointtopoint shortest path problem in a setting where preprocessing is allowed. We improve the reachbased approach of Gutman [16] in several ways. In particular, we introduce a bidirectional version of the algorithm that uses implicit lower bounds and we add shortcut arcs which reduc ..."
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Cited by 59 (5 self)
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We study the pointtopoint shortest path problem in a setting where preprocessing is allowed. We improve the reachbased approach of Gutman [16] in several ways. In particular, we introduce a bidirectional version of the algorithm that uses implicit lower bounds and we add shortcut arcs which reduce vertex reaches. Our modifications greatly reduce both preprocessing and query times. The resulting algorithm is as fast as the best previous method, due to Sanders and Schultes [27]. However, our algorithm is simpler and combines in a natural way with A∗ search, which yields significantly better query times.
Finding shortest nonseparating and noncontractible cycles for topologically embedded graphs
 Discrete Comput. Geom
, 2005
"... We present an algorithm for finding shortest surface nonseparating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3−ε), this represents a considerable improvement over p ..."
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Cited by 38 (8 self)
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We present an algorithm for finding shortest surface nonseparating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3−ε), this represents a considerable improvement over previous results by Thomassen, and Erickson and HarPeled. We also give algorithms to find a shortest noncontractible cycle in O(g O(g) V 3/2) time, which improves previous results for fixed genus. This result can be applied for computing the (nonseparating) facewidth of embedded graphs. Using similar ideas we provide the first nearlinear running time algorithm for computing the facewidth of a graph embedded on the projective plane, and an algorithm to find the facewidth of embedded toroidal graphs in O(V 5/4 log V) time. 1
Fully Dynamic All Pairs Shortest Paths with Real Edge Weights
 In IEEE Symposium on Foundations of Computer Science
, 2001
"... We present the first fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with realvalued edge weights. Given a dynamic directed graph G such that each edge can assume at most S di#erent real values, we show how to support updates in O(n amortized time and que ..."
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Cited by 35 (10 self)
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We present the first fully dynamic algorithm for maintaining all pairs shortest paths in directed graphs with realvalued edge weights. Given a dynamic directed graph G such that each edge can assume at most S di#erent real values, we show how to support updates in O(n amortized time and queries in optimal worstcase time. No previous fully dynamic algorithm was known for this problem. In the special case where edge weights can only be increased, we give a randomized algorithm with onesided error which supports updates faster in O(S We also show how to obtain query/update tradeo#s for this problem, by introducing two new families of algorithms. Algorithms in the first family achieve an update bound of O(n/k), and improve over the best known update bounds for k in the . Algorithms in the second family achieve an update bound of ), and are competitive with the best known update bounds (first family included) for k in the range (n/S) # Work partially supported by the IST Programme of the EU under contract n. IST199914. 186 (ALCOMFT) and by CNR, the Italian National Research Council, under contract n. 01.00690.CT26. Portions of this work have been presented at the 42nd Annual Symp. on Foundations of Computer Science (FOCS 2001) [8] and at the 29th International Colloquium on Automata, Languages, and Programming (ICALP'02) [9].
An O(n log n) algorithm for maximum stflow in a directed planar graph
"... We give the first correct O(n log n) algorithm for finding a maximum stflow in a directed planar graph. After a preprocessing step that consists in finding singlesource shortestpath distances in the dual, the algorithm consists of repeatedly saturating the leftmost residual stot path. ..."
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Cited by 27 (0 self)
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We give the first correct O(n log n) algorithm for finding a maximum stflow in a directed planar graph. After a preprocessing step that consists in finding singlesource shortestpath distances in the dual, the algorithm consists of repeatedly saturating the leftmost residual stot path.
Engineering Route Planning Algorithms
 ALGORITHMICS OF LARGE AND COMPLEX NETWORKS. LECTURE NOTES IN COMPUTER SCIENCE
, 2009
"... Algorithms for route planning in transportation networks have recently undergone a rapid development, leading to methods that are up to three million times faster than Dijkstra’s algorithm. We give an overview of the techniques enabling this development and point out frontiers of ongoing research on ..."
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Cited by 26 (15 self)
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Algorithms for route planning in transportation networks have recently undergone a rapid development, leading to methods that are up to three million times faster than Dijkstra’s algorithm. We give an overview of the techniques enabling this development and point out frontiers of ongoing research on more challenging variants of the problem that include dynamically changing networks, timedependent routing, and flexible objective functions.
Shortest paths in directed planar graphs with negative lengths: A linearspace O(n log² n)time algorithm
 PROC. 20TH ANN. ACMSIAM SYMP. DISCRETE ALGORITHMS
, 2009
"... We give an O(n log² n)time, linearspace algorithm that, given a directed planar graph with positive and negative arclengths, and given a node s, finds the distances from s to all nodes. ..."
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Cited by 21 (5 self)
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We give an O(n log² n)time, linearspace algorithm that, given a directed planar graph with positive and negative arclengths, and given a node s, finds the distances from s to all nodes.
Dynamic highwaynode routing
 In Proc. 6th Workshop on Experimental and Efficient Algorithms. LNCS
, 2007
"... Abstract. We introduce a dynamic technique for fast route planning in large road networks. For the first time, it is possible to handle the practically relevant scenarios that arise in presentday navigation systems: When an edge weight changes (e.g., due to a traffic jam), we can update the preproc ..."
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Cited by 20 (5 self)
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Abstract. We introduce a dynamic technique for fast route planning in large road networks. For the first time, it is possible to handle the practically relevant scenarios that arise in presentday navigation systems: When an edge weight changes (e.g., due to a traffic jam), we can update the preprocessed information in 2–40 ms allowing subsequent fast queries in about one millisecond on average. When we want to perform only a single query, we can skip the comparatively expensive update step and directly perform a prudent query that automatically takes the changed situation into account. If the overall cost function changes (e.g., due to a different vehicle type), recomputing the preprocessed information takes typically less than two minutes. The foundation of our dynamic method is a new static approach that generalises and combines several previous speedup techniques. It has outstandingly low memory requirements of only a few bytes per node. 1
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the spec ..."
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Cited by 17 (7 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.