L. Ford. Network Flow Theory. Technical Report P-932, The Rand Corporation, 1956.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithm, all stored arcs form a rooted tree with this property. There is also a trivial min max relation characterizing the minimum length of an s t path. To this end, call a subset A # of A an s (U) for some subset U of V satisfying s and t U . Then the following was observed by Robacker [1956]: Theorem 1.2. The minimum length of an s t path is equal to the maximum number of pairwise disjoint s t cuts. Proof. Trivially, the minimum is at least the maximum, since each s t path intersects each s cut in an arc. The fact that the minimum is equal to the maximum follows by ....

....than Q. This contradicts the assumption that Q has a minimum number of arcs. In particular, it follows that, if there are no directed circuits of negative length, there is a shortest path traversing 1 arcs. 15) Also in this case there is an easy algorithm, the Bellman Ford method (Bellman [1958] Ford [1956]) determining a shortest r s path. Let n : V . The algorithm calculates functions f 0 , f 1 , f 2 , f n : V successively by the following rule: i) Put f 0 (r) 0 and f 0 (v) r . ii) For k n, if f k has been found, put f k 1 (v) min f k (v) min (u,v)#A (f ....

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L.R. Ford, Jr, Network Flow Theory, Paper P-923, The RAND Corporation, Santa Monica, California,

....for the general problem are called label correcting. Label correcting SSSP algorithms may select arbitrary labelled vertex to be scanned next, hence generally to re insert the vertices for scanning until they are finally settled (d(v) #(s; v) The well known Bellman Ford algorithm [Bel58, For56] uses FIFO strategy to select vertices. The next vertex to be scanned is removed from the head of the queue; a vertex that becomes labelled is added to the tail of the queue if it is not already in the queue. The algorithm runs in O(mn) time in the worst case. For graphs where negative cycles ....

L.R. Ford. Network flow theory. Rand Corp., Santa Monica, Cal., 1956.

....described by Viterbi ( 119] which is essentially the dynamic programming approach to solving a minimum path problem with non negative costs. A more general description was earlier given by Bellman ( 20] Shortest path algorithms has been described by Dijkstra ( 37] and by Bell man and Ford. [42, 21]) The viterbi algorithm is a mainstay of estimation in Markov Chains and the related Hidden Markov Models. The reason viterbi algorithm can be used in this minimization is the following. Suppose we have a sequence of blanks following some known letters: t e . The dynamic programming ....

L. R. Ford Jr., "Network Flow Theory", Paper P-923, RAND Corporation, Santa Monica, California, 1956.

....at a time. The network is in fact running the sequential algorithm with extended memory. At the same time we wanted to compare our algorithm with similar algorithm [BCKP01] which does not employ reverse search (DSP Q) but which traverses graph using the queue (classical Bellman Ford approach [For56,Bel58]) The distribution of DSP Q algorithm is the very same as the distribution of DSP R algorithm. Our aim was to document, that avoiding the additional data structure (such as the queue in this case) does not impact the running time. In the implementation of the DSP R and the DSP Q algorithms we ....

L.R. Ford. Network flow theory. Rand Corp., Santa Monica, Cal., 1956.

....node by means of different strategies are known as label correcting or as list search algorithms. The queue is a very natural choice when implementing the set Q. The algorithm based on the corresponding selection rule, i.e. the FIFO rule, is named L queue, and it is credited to Bellman (1958) Ford (1956) and Moore (1958) It is very simple to implement, and its time complexity is O(mn) whatever the sign of the arc costs. A list which combines the properties of the data structures queue and stack is the deque, in which addition and deletion are possible at either end of the list. A deque is used ....

L. R. Ford Jr. (1956), Network flow theory, Rand Co., P-293.

....shortest path itself from s to v can be traced on termination by a method, explained in Section 3. Although we will prove termination, there are problem instances, even for a deterministic algorithm as the SE algorithm, which take an exponential number of updates. The Arc algorithm originates from Ford [Ford] A termination proof for the case of integer arc distances can be found in [Ahuja] Our termination proof, which does not assume that the arc distances are integer, has not been explicitly stated in literature. In the Arc algorithm the choice of the arc (v; w) with g(w) g(v) d(v; w) ....

....path itself from s to v can be traced on termination by a method, explained in Section 3. Although we will prove termination, there are problem instances, even for a deterministic algorithm as the SE algorithm, which take an exponential number of updates. The Arc algorithm originates from Ford [Ford] A termination proof for the case of integer arc distances can be found in [Ahuja] Our termination proof, which does not assume that the arc distances are integer, has not been explicitly stated in literature. In the Arc algorithm the choice of the arc (v; w) with g(w) g(v) d(v; w) is ....

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L.R. Ford jr., Network Flow Theory, Report P923, Rand Corp., Santa Monica, CA.

....means that at least 5 minutes must pass between the end of A and the beginning of B. A special time point O is often defined to represent the origin of time: O 60 start(A) means that A cannot start before date 60; end(A) Gamma 120 O means that A cannot end after date 120. Ford s algorithm [Ford, 1956; Gondran Minoux, 1984] which computes the length of the longest paths from a node N 0 to the other nodes N 1 . N n of a valued oriented graph, can be used to compute the earliest start and end times of activities. Informally, this algorithm can be described as follows. 1. Let (N 0 ) 0 ....

Ford, Jr., L.R. [1956], Network Flow Theory, Technical report, Rand Corporation.

....line scheduling, as well as in intelligent autonomous machines. Over the years, This work was supported by the Hong Kong Research Grant Council under Grant No. CUHK 58 93E 2 there have been many fundamental graph algorithms developed, including the significant results due to Bellman[4] Ford[11], Dijkstra[9] Floyd[10] and others[2] An interesting observation due to Bellman [4] is that, given an N node directed or undirected graph and a single destination path problem with a destination node (arbitrarily chosen to be node 1 without loss of generality) the shortest path length D i (i = ....

L. R. Ford. `Network Flow Theory'. Technical Report P-923, Rand Corp., Santa Monica, CA, 1956.

....node by means of different strategies are known as label correcting or as list search algorithms. The queue is a very natural choice when implementing the set Q. The algorithm based on the corresponding selection rule, i.e. the FIFO rule, is named L queue, and it is credited to Bellman [1958] Ford [1956], and Moore [1959] It is very simple to implement, and its time complexity is O(mn) whatever the sign of the arc costs. A list which combines the properties of the data structures queue and stack is the deque, in which addition and deletion are possible at either end of the list. A deque is used ....

L. R. Ford Jr. (1956). Network flow theory. Rand Co., P-293.

....such as visiting intermediate vertices before reaching destination. Our problem then refers to the one pair shortest path problem or, more generally, to the single source problem. Algorithms solving that problem are usually based on the methods proposed by Bellman [5] Dijkstra [34] Ford [44], and Moore [85] Methods solving all pairs problems such as that of Floyd [43] are consequently inappropriate. The problem of determining k th shortest paths do not apply practically to our case. But disposing of second shortest path costs is of theoretical importance in Chapter 6. 2.2.2 The ....

....next sections are devoted to review some shortest path tree algorithms that use above search strategies. Computational complexities are mentioned to allow comparisons between these algorithms. 2. 4 Label correcting algorithms The first well known label correcting algorithm was introduced by Ford [44], and then detailed by Bellman [5] and Moore [85] Their method recursively solves Bellman s equations (2.4) and has a computational complexity of O(mn) Proofs and comments about this method can be found in [57, 89] We present three variants of this prototype algorithm that have been suggested ....

L.R. Ford, "Network flow theory", Report P-923, The Rand Corporation, Santa Monica, CA, 1956.

....for the negative cycle problem combine a shortest path algorithm and a cycle detection strategy. We study combinations of shortest path algorithms and cycle detection strategies to determine the best combination. The shortest path algorithms we study are based on the labeling method of Ford [6, 7]. Most cycle detection strategies for the labeling method look for cycles in the graph of parent pointers maintained by the method. The facts that these cycles correspond to negative cycles in the input graph and that if the input graph has a negative cycle then eventually the parent pointer graph ....

....w) is admissible. If d(v) d(w) 1, we define d (v; w) v; w) A shortest path tree of G is a spanning tree rooted at s such that for any v 2 V , the s to v path in the tree is a shortest path from s to v. 3 Labeling Method In this section we briefly outline the general labeling method [6, 7] for solving the shortest path problem. See e.g. 3, 8, 22] for more detail. Most shortest path algorithms, and all those which we study in this paper, are based on the labeling method. For every vertex v, the method maintains its distance label d(v) and parent p(v) Initially for every vertex ....

L. Ford. Network Flow Theory. Technical Report P-932, The Rand Corporation, 1956.

.... For every edge, we define a cost as the minimum of the potentials of the cliques containing the two points : x i ; x i 1 ) min c i 2Cn jx i ;x i 1 2c i OE(c i ; f c i ) Then, h) P p Gamma1 i=0 (x i ; x i 1 ) Thus, our first algorithm [Mer 91] is an adaptation of Ford s algorithm [For 56] For 62] to image processing. There are still important differences : ffl our potential is general and adapted to image processing ffl our graph is planar and each vertex has the same degree (except on the border of the image) ffl we exploit this special structure of the graph (planar and with ....

L.R., Jr., Ford, Network Flow Theory, The Rand Corp., 1956, August, P-923.

....de la Paix B 5000 Namur, Belgium Keywords : computational complexity, shortest paths, inverse problems, traffic modelling. 1 Introduction Shortest paths problems form a fundamental part of transportation research, because they naturally arise in a large number of related optimization problems [9]. Shortest path techniques are applied in fields as diverse as traffic modelling [8] and computerized tomography [16] Very efficient algorithms have been proposed during the last three decades to solve the shortest paths problem (see for example [1, 7, 15, 12, 2] for the single source problem) ....

L.R. Ford, "Network flow theory", Report P-923, Rand Corp., Santa Monica, CA, 1956.

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L. Ford. Network Flow Theory. Technical Report P-932, The Rand Corporation, 1956.

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L. R. Ford, Jr. Network flow theory. Technical Report P-923, The Rand Corporation, Santa Monica, CA, August 1956.

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Jr. L. R. Ford. Network Flow Theory. Technical Report P-932, The Rand Corporation, 1956.

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L.R. Ford. Network flow theory. Rand Corp., Santa Monica, Cal., 1956.

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L. R. Ford, Jr. Network flow theory. Technical Report P-923, The Rand Corporation, Santa Monica, CA, August 1956.

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L. Ford. Network Flow Theory. Technical Report P-932, The Rand Corporation, 1956.

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L. Ford. Network Flow Theory. Technical Report P-932, The Rand Corporation, 1956.

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L. Ford, Network flow theory, Paper p-923, RAND Corporation, Santa Monica, California (1956).

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Lester R. Ford Jr. Network Flow Theory. Rand Corporation, 1956.

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L. R. Ford Jr. [1956], Network Flow Theory, Technical Report, Rand Corporation.

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