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32
An Incremental Algorithm for a Generalization of the ShortestPath Problem
, 1992
"... The grammar problem, a generalization of the singlesource shortestpath problem introduced by Knuth, is to compute the minimumcost derivation of a terminal string from each nonterminal of a given contextfree grammar, with the cost of a derivation being suitably defined. This problem also subsume ..."
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Cited by 115 (1 self)
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The grammar problem, a generalization of the singlesource shortestpath problem introduced by Knuth, is to compute the minimumcost derivation of a terminal string from each nonterminal of a given contextfree grammar, with the cost of a derivation being suitably defined. This problem also subsumes the problem of finding optimal hyperpaths in directed hypergraphs (under varying optimization criteria) that has received attention recently. In this paper we present an incremental algorithm for a version of the grammar problem. As a special case of this algorithm we obtain an efficient incremental algorithm for the singlesource shortestpath problem with positive edge lengths. The aspect of our work that distinguishes it from other work on the dynamic shortestpath problem is its ability to handle "multiple heterogeneous modifications": between updates, the input graph is allowed to be restructured by an arbitrary mixture of edge insertions, edge deletions, and edgelength changes.
New Dynamic Algorithms for Shortest Path Tree Computation
 IEEE/ACM Transactions on Networking
, 2000
"... The OSPF and ISIS routing protocols widely used in today's Internet compute a shortest path tree (SPT) from each router to other routers in a routing area. Many existing commercial routers recompute an SPT from scratch following changes in the link states of the network. Such recomputation of an en ..."
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Cited by 55 (1 self)
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The OSPF and ISIS routing protocols widely used in today's Internet compute a shortest path tree (SPT) from each router to other routers in a routing area. Many existing commercial routers recompute an SPT from scratch following changes in the link states of the network. Such recomputation of an entire SPT is inecient and may consume a considerable amount of CPU time. Moreover, as there may coexist multiple SPTs in a network with a set of given link states, recomputation from scratch causes frequent unnecessary changes in the topology of an existing SPT and may lead to routing instability. In this paper, we present new dynamic SPT algorithms that make use of the structure of the previously computed SPT. Besides efficiency, our algorithm design objective is to achieve routing stability by making minimum changes to the topology of an existing SPT (while maintaining shortest path property) when some link states in the network have changed. We establish an algorithmic framework that allows ...
Randomized Fully Dynamic Graph Algorithms with Polylogarithmic Time per Operation
 JOURNAL OF THE ACM
, 1999
"... This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new d ..."
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Cited by 52 (0 self)
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This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique which combines a novel graph decomposition with randomization. They are LasVegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of \Omega\Gamma m 0 ) operations, where m 0 is the number of edges in the initial graph, the expected time for p updates is O(p log 3 n) 1 for connectivity and bipartiteness. The worstcase time for one query is O(log n= log log n). For the kedge witness problem ("Does the removal of k given edges disconnect the graph?") the expected time for p updates is O(p log 3 n) and expected time for q queries is O(qk log 3 n). Given a grap...
On the Computational Complexity of Dynamic Graph Problems
 THEORETICAL COMPUTER SCIENCE
, 1996
"... ..."
Lifelong Planning A*
, 2005
"... Heuristic search methods promise to find shortest paths for pathplanning problems faster than uninformed search methods. Incremental search methods, on the other hand, promise to find shortest paths for series of similar pathplanning problems faster than is possible by solving each pathplanning p ..."
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Cited by 28 (3 self)
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Heuristic search methods promise to find shortest paths for pathplanning problems faster than uninformed search methods. Incremental search methods, on the other hand, promise to find shortest paths for series of similar pathplanning problems faster than is possible by solving each pathplanning problem from scratch. In this article, we develop Lifelong Planning A * (LPA*), an incremental version of A * that combines ideas from the artificial intelligence and the algorithms literature. It repeatedly finds shortest paths from a given start vertex to a given goal vertex while the edge costs of a graph change or vertices are added or deleted. Its first search is the same as that of a version of A * that breaks ties in favor of vertices with smaller gvalues but many of the subsequent searches are potentially faster because it reuses those parts of the previous search tree that are identical to the new one. We present analytical results that demonstrate its similarity to A * and experimental results that demonstrate its potential advantage in two different domains if the pathplanning problems change only slightly and the changes are close to the goal.
A New Approach to AllPairs Shortest Paths on RealWeighted Graphs
 Theoretical Computer Science
, 2003
"... We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Her ..."
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Cited by 27 (2 self)
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We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Here m and n are the number of edges and vertices, respectively.
New Dynamic SPT Algorithm based on a BallandString Model
, 1999
"... A key functionality in today's widely used interior gateway routing protocols such as OSPF and ISIS involves the computation of a shortest path tree (SPT). In many existing commercial routers, the computation of an SPT is done from scratch following changes in the link states of the network. As the ..."
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Cited by 25 (0 self)
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A key functionality in today's widely used interior gateway routing protocols such as OSPF and ISIS involves the computation of a shortest path tree (SPT). In many existing commercial routers, the computation of an SPT is done from scratch following changes in the link states of the network. As there may coexist multiple SPTs in a network with a set of given link states, such recomputation of an entire SPT not only is inefficient but also causes frequent unnecessary changes in the topology of an existing SPT and creates routing instability. This paper presents a new dynamic SPT algorithm that makes use of the structure of the previously computed SPT. Our algorithm is derived by recasting the SPT problem into an optimization problem in a dual linear programming framework, which can also be interpreted using a ballandstring model. In this model, the increase (or decrease) of an edge weight in the tree corresponds to the lengthening (or shortening) of a string. By stretching the strings...
Parallel RealTime Optimization: Beyond Speedup
 PARALLEL PROCESSING LETTERS
, 1999
"... Traditionally, interest in parallel computation centered around the speedup provided by parallel algorithms over their sequential counterparts. In this paper, we ask a different type of question: Can parallel computers, due to their speed, do more than simply speed up the solution to a problem? ..."
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Cited by 25 (23 self)
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Traditionally, interest in parallel computation centered around the speedup provided by parallel algorithms over their sequential counterparts. In this paper, we ask a different type of question: Can parallel computers, due to their speed, do more than simply speed up the solution to a problem? We show that for realtime optimization problems, a parallel computer can obtain a solution that is better than that obtained by a sequential one. Specifically, a sequential and a parallel algorithm are exhibited for the problem of computing the bestpossible approximation to the minimumweight spanning tree of a connected, undirected and weighted graph whose vertices and edges are not all available at the outset, but instead arrive in real time. While the parallel algorithm succeeds in computing the exact minimumweight spanning tree, the sequential algorithm can only manage to obtain an approximate solution. In the worst case, the ratio of the weight of the solution obtained seque...
Fast replanning for navigation in unknown terrain
 Transactions on Robotics
"... Abstract—Mobile robots often operate in domains that are only incompletely known, for example, when they have to move from given start coordinates to given goal coordinates in unknown terrain. In this case, they need to be able to replan quickly as their knowledge of the terrain changes. Stentz ’ Fo ..."
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Cited by 22 (7 self)
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Abstract—Mobile robots often operate in domains that are only incompletely known, for example, when they have to move from given start coordinates to given goal coordinates in unknown terrain. In this case, they need to be able to replan quickly as their knowledge of the terrain changes. Stentz ’ Focussed Dynamic A (D) is a heuristic search method that repeatedly determines a shortest path from the current robot coordinates to the goal coordinates while the robot moves along the path. It is able to replan faster than planning from scratch since it modifies its previous search results locally. Consequently, it has been extensively used in mobile robotics. In this article, we introduce an alternative to D that determines the same paths and thus moves the robot in the same way but is algorithmically different. D Lite is simple, can be rigorously analyzed, extendible in multiple ways, and is at least as efficient as D. We believe that our results will make Dlike replanning methods even more popular and enable robotics researchers to adapt them to additional applications. Index Terms—A, D (Dynamic A), navigation in unknown terrain, planning with the freespace assumption, replanning, search, sensorbased path planning. I.
Offline Algorithms for Dynamic Minimum Spanning Tree Problems
, 1994
"... We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modificat ..."
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Cited by 17 (9 self)
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We describe an efficient algorithm for maintaining a minimum spanning tree (MST) in a graph subject to a sequence of edge weight modifications. The sequence of minimum spanning trees is computed offline, after the sequence of modifications is known. The algorithm performs O(log n) work per modification, where n is the number of vertices in the graph. We use our techniques to solve the offline geometric MST problem for a planar point set subject to insertions and deletions; our algorithm for this problem performs O(log 2 n) work per modification. No previous dynamic geometric MST algorithm was known.