R. Dial. Algorithm 360: Shortest path forest with topological ordering. Commerische Mathematik, 1:269--271, 1959.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to be an interesting area in the study of shortest path problems. Many of the techniques focus on time dependent versions of the classical shortest path problem. Interestingly, apart from a few early attempts on this variant of the problem in 1966 by Cooke and Halsey [8] and subsequently by Dial [11], there are almost no references in the literature until the 1980s when there was a renewed interest in this variant of the problem. Another interesting aspect of shortest path problems in transportation is that the presence of non additive and non linear path costs. The approach we take is to ....

....2.2.2 The Minimum Cost Dynamic Path Problem. The Minimum Cost Dynamic Path Problem looks for a path from a node r to every other node i 6= r, leaving at a time t in a dynamic graph G. Computing a shortest path tree with topological ordering in R using a bucket list B = fB 1 ; B 2 ; B q g [11] to eciently implement the chronological visit of R, where B h denotes nodes to be visited at time t h ; h 2 [1; q] 32 yields the Chrono SPT algorithm. Initially, if departing from a node r at time t = t p then B p = frg and the other buckets are empty. The algorithm terminates when all the ....

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R. Dial. Algorithm 360: Shortest-path forest with topological ordering. Communications of the ACM, 12(11):632-633, 11 1969.

....to this direction. The method has to be user supervised and restarted, if it leaves the vessel of interest, it estimates also the vessel diameter. Path tracking from a seed point to given end point(s) 47] uses the principle of the Dijkstra s shortest (minimal cost) path search in the graph [24, 25]. The 3D volume is taken as a graph with nodes in places of voxels and links connecting the neighboring ones. The graph links are assigned a cost, e.g. the absolute difference of the node values, and a monotone increasing function for computation of the cost along the path is defined. The ....

R.B. Dial. Algorithm 360: Shortest path forest with topological ordering. Communications of the ACM, 12:632-633, 1969.

....path, and for each such edge, the approximate length is at most more than the actual length. Thus we know that ( dist(s,t) 2n = dist(s,t) Further, since each shortest path length is an integer multiple oi and no more than (P) we can use Dial s implementation of Dijkstra s algorithm [13] to compute disti(s, t) in O(m ne ) time. Implementing FINDPATH with this approximate shortest path computation directly im proves the time required by a deterministic implementation of REDUCE. The randomized imple mentation of FINDPATH with approximate shortest path computation requires ....

R. Dial. Algorithm 360: Shortest path forest with topological ordering. Communications of the ACM, 12:632 633, 1969.

....sorting is an upper bound for the number of layers. Hence, the number of vertices in the routing graph is O(jV j ) and, since each vertex has constant degree, the total size of the routing graph is O(jV j ) Because the maximum cost of an edge is 1, we can use Dials shortest path algorithm [6] which has linear running time in this case. The insertion of the edge can clearly be done in linear time. The following theorem summarizes the lemmas above. Theorem 8. Algorithm edge insertion inserts one edge in an embedded upward planar s t graph G = V; E) in time O(jV j ) without ....

R. Dial,Algorithm 360: Shortest path forest with topological ordering, Communications of ACM, 12, 1969, pp. 632-633

.... algorithm is very similar to the well known Dijkstra s shortest path algorithm [6] and is valid for any path cost using a non decreasing function of the arc weights (See this proof in [7] Dial proposed in 1969 the rst implementation of the Moore s shortest path forest using an ordered queue [5]. We present next the IFT shortest path forest algorithm which uses an ordered queue to nd the catchment basins of the watershed based on the de nition just presented. An ordered, hierarchical, or priority queue, with a FIFO restriction is a data structure very popular in some morphological ....

R. Dial. Algorithm 360: Shortest-path forest with topological ordering. Comm. Assoc. Comput. Mach., 1969.

....the sorted list L is maintained. Usually sorting requires at least O(n log n) complexity, where n is the number of nodes of the graph. Fortunately this limit can be reduced to O(n) for integer local costs with an upper bound. In this case it is possible to use a so called bucket list for sorting [3, 4]. Each bucket contains nodes with a particular total cost value only. The length of the bucket list can be delimited to the upper bound of the local costs, if the buckets are arranged in a cyclic array. In our case we use a single byte to define local costs. Thus only 256 buckets are needed, plus ....

Dial, R. B., Algorithm 360: Shortest Path Forest with Topological Ordering, Comm. ACM 12 (1969) 632-633.

....matching case, our algorithm for the assignment problem has an efficient sequential version. This version runs in O( p nm log(nC) time i.e. as fast as the algorithm of Gabow and Tarjan [12] The sequential version is obtained by setting l to 1, k to p n, and using shortest path algorithms of [7, 34] to solve the shortest path subproblems in linear time, as described in [18] A somewhat different sequential algorithm that combines the push relabel and augmenting path methods and achieves the same time bound has been proposed independently by Orlin and Ahuja [26] 5.3 Zero One Minimum Cost ....

R. B. Dial. Algorithm 360: Shortest path forest with topological ordering. Comm. ACM, 12:632--633, 1969.

....algorithms whp. While this problem is interesting in its own right it also stresses the asymptotic superiority of our new algorithm. 2 Adaptive Bucket Splitting 2.1 Preliminaries Our algorithm is based on keeping nodes in buckets. This technique has already been used in Dial s implementation [7] of Dijkstra s algorithm for integer weights in f0; Cg: a queued node v is stored in the bucket B i with index i = tent(v) In each iteration the algorithm removes a node v from the rst nonempty bucket 1 For a problem of size n, we say that an event occurs with high probability (whp) if ....

R. B. Dial. Algorithm 360: Shortest-path forest with topological ordering. Communications of the ACM, 12(11):632-633, 1969.

....priority queues. Unless mentioned otherwise, we refer to priority queues whose operation time bounds depend only on the number of elements on the queue as heaps. The fastest implementations of heaps are described in [4, 14, 19] Alternative implementations of priority queues use buckets (e.g. [2, 7, 11, 12]) Operation times for bucketbased implementations depend on the maximum event duration C, defined in Section 2. See [3] for a related data structure. Heaps are particularly efficient when the number of elements on the heap is small. Bucket based priority queues are particularly efficient when the ....

R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632--633, 1969.

....priority queues. Unless mentioned otherwise, we refer to priority queues whose operation time bounds depend only on the number of elements on the queue as heaps. The fastest implementations of heaps are described in [4, 12, 15] Alternative implementations of priority queues use buckets (e.g. [2, 6, 9, 10]) Operation times for bucket based implementations depend on the maximum event duration C, defined in Section 2. See [3] for a related data structure. Heaps are more efficient when the number of elements on the heap is small. Bucket based priority queues are more efficient when the maximum event ....

R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632--633, 1969.

....each node price so that there is a path in GA from every excess to some deficit (a node v with e f (v) 0) and every node reachable in GA from a node with excess lies on such a path. This amounts to a modified shortest paths computation, and can be done in O(m) time using ideas from Dial s work [3]. We perform a global update every time Gamma max has increased by at least ffl since the last global update. We developed global updates from an implementation heuristic for the minimum cost circulation problem [11] but in retrospect, they prove similar in the assignment context to the ....

R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632-- 633, 1969.

....all priority queue operations take O(n) time, so the binary heap is an s heap. Operation bounds may depend on parameters other than the number of elements. The fastest implementations of s heaps are described in [4, 12, 17] Alternative implementations of priority queues use buckets (e.g. [2, 7, 9, 10]) Operation times for bucket based implementations depend on the maximum event duration C, defined in Section 2, and are not very sensitive to the number of elements. See [3] for a related data structure. s heaps are particularly efficient when the number of elements on the s heap is small. ....

R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632--633, 1969.

....Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Appendix 1 Appendix 2 Figure 1.2. Suggested reading order Chapter 2 Shortest Path Algorithms Due to the importance of finding shortest paths through networks, a significant amount of research has gone into this area[7, 5, 2, 3, 8, 9, 10, 14]. This chapter aims to provide a brief overview of some of the major shortest path algorithms and justifies my choice of Dijkstra s algorithm. Additionally, a description of the method I used in implementing the algorithm is given, including the handling of multiple source nodes. 2.1 Comparison ....

....are empty. The next node to be scanned is taken from the bucket indexed by L, as the labelled nodes with minimum potential are stored in this bucket. If the bucket indexed by L is empty, L is incremented. If all buckets are empty, the algorithm terminates. This implementation was proposed by Dial[7]. Bucket operations will take linear time, and there are no more than n Theta C buckets to be examined, so the implementation is linear with O(m n Theta C) This algorithm is shown in algorithm 2.4. 2.4. The Selection Algorithms 15 Dijkstra Bucket: Inputs: V , the set of nodes in the ....

R.B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Communications of the ACM, 12(11), November 1969.

.... Eva Tardos. 1 1. Introduction The shortest paths problem is one of the most fundamental network optimization problems. This problem comes up in practice and arises as a subproblem in many network optimization algorithms. Algorithms for this problem have been studied for a long time. See e.g. [2, 5, 6, 7, 18, 19, 21]. However, advances in the theory of shortest paths algorithms are still being made. See e.g. 1, 9, 13] A good description of the classical algorithms and their implementations appears in [10] On a network with negative length arcs, the best currently known time bound of O(nm) is achieved by ....

....Section 6) dikq is orders of magnitude slower than dikh, which itself is relatively slow on this problem. See Figure 5. Because of the poor performance, we do not include dikq in our tests. Another way to implement Dijkstra s algorithm is by using the bucket data structure, as proposed by Dial [5]. This implementation maintains an array of buckets, with the i th bucket containing all nodes v with d(v) i. When distance label of a node changes, the node is removed from a bucket corresponding to its old distance label (if the label was finite) and inserted into the bucket corresponding to ....

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R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632-- 633, 1969.

....than the actual length. Thus we know that dist (s; t) dist (s; t) n ffl 0 (P ) 2n = dist (s; t) ffl 0 (P ) 2 Further, since each shortest path length is an integer multiple of ffl 0 (P ) 2n , and no more than (P ) we can use Dial s implementation of Dijkstra s algorithm [2] to compute dist (s; t) in O(m nffl Gamma1 ) time. Implementing FindPath with this approximate shortest path computation directly improves the time required by a deterministic implementation of Reduce. The randomized implementation of FindPath with approximate shortest path computation ....

R. Dial. Algorithm 360: Shortest path forest with topological ordering. Communications of the ACM, 12:632--633, 1969.

....was done while the author was at Computer Science Department, Stanford University, and supported in part by NSF Grant CCR 9307045. 1 Introduction The shortest paths problem is a fundamental network optimization problem. Algorithms for this problem have been studied for a long time. See e.g. [2, 7, 8, 10, 14, 15, 16]. An important special case of the problem occurs when no arc length is negative. In this case, implementations of Dijkstra s algorithm [8] achieve the best time bounds. An implementation of [11] runs in O(m n log n) time. Here n and m denote the number of nodes and arcs in the network, ....

....n p log C) time. In a recent computational study [4, 5] however, a 2 level bucket implementation of Dijkstra s algorithm gave the best overall performance among the codes studied. In particular, the implementation proved to be much more robust than the classical 1 level bucket implementation [7, 9, 18]. In this paper we study relative performance of the multi level bucket implementations of the algorithm. We conduct computational experiments and explain their results. Our study leads to better understanding of the multi level implementations and confirms that the 1 level implementation is much ....

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R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632--633, 1969.

....O(mff(m; n) time. 6 This is more natural and allows the use of 2 Delta instead of 3 Delta in (1) but the resulting algorithm for routing flow through the components is more complicated. Since the arc lengths are 0 and 1, we can compute d in linear time, for example using Dial s algorithm [13]. We compute u f (S k ; T k ) for i = 1; d (s) as follows. For every i, We initialize u f (S k ; T k ) to zero. Then we examine all arcs (i; j) If d(i) d(j) we increase u f (S i ; T i ) by u f (i; j) This procedure takes O(m) time. 6 Time Bounds In this section we derive bounds on ....

R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632-- 633, 1969.

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R. Dial. Algorithm 360: Shortest path forest with topological ordering. Commerische Mathematik, 1:269--271, 1959.

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R. Dial. Algorithm 360: Shortest path forest with topological ordering. Communications of the ACM, 12:632--633, 1969.

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R. Dial. Algorithm 360: Shortest path forest with topological ordering. Comm. ACM, pages 632--633, 1969.

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R. Dial. Algorithm 360: Shortest path forest with topological ordering. Communications of the ACM, 12:632--633, 1969.

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R. Dial. Algorithm 360: Shortest path forest with topological ordering. Comm. ACM, pages 632--633, 1969.

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Dial, R.: Algorithm 360: Shortest path forest with topological ordering. Communications of ACM 12 (1969) 632--633

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R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632-633, 1969.

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R. B. Dial. Algorithm 360: Shortest-path forest with topological ordering. Communications of the ACM, 12(11):632--633, 1969.

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R. B. Dial. Algorithm 360: Shortest-path forest with topological ordering. Communications of the ACM, 12(11):632-633, 1969.

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R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632--633, 1969. 37

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R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632--633, 1969.

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R. Dial. Algorithm 360: Shortest path forest with topological ordering. Communications of ACM, 12:632--633, 1969.

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R. Dial. Algorithm 360: Shortest path forest with topological ordering. Comm. ACM, pages 632--633, 1969.

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R. B. Dial. Algorithm 360: Shortest Path Forest with Topological Ordering. Comm. ACM, 12:632--633, 1969.

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