K-shortest path

In step 1, a k-shortest path algorithm (Minieka 1978) fork = 2 is used to obtain all the OD pairs that have unique feasible paths. The packages of these OD pairs are routed tilong their single paths and removed from the problem, updating till the ptuameters. 

The application research of K shortest path algorithm in avoiding mine disaster... 

Minkowski metric with k = 1. The shortest path from a to b walking on a co-ordinate axis. Named for the streets of Manhattan, which resemble a co-ordinate axis... 

For aqrclic graphs, connectivity indices for path subgraphs can be calculated by the Wiener operator applied to the product of the path-% matrix and the geodesic matrix B whose elements are all equal to zero except for those corresponding to the shortest paths i-j of length k that are equal to 1 ... 

Gupta, S. K., Kececioglu, J. D., Schaffer, A. A. (1995) Improving the practical space and time efficiency of the shortest-path approach to sum-of-pairs multiple sequence alignment, J. Comp. Biol. 2 459-472. 

Table 12.5. The molar stiffness function K calculated by group contributions, the number N p of atoms in the shortest path across the backbone of the repeat unit, the number of hydrogen atoms Nr in the repeat unit, the quantity used in correlation for K, and fitted or predicted (fit/pre) values of K, for 120 polymers. For 26 of these polymers, K cannot be calculated via group contributions, but can be calculated by using the new correlation. N and °%v, which are also used in the correlation, are listed in Table 2.2. K is in units of grams°-25-cm1-5/mole0-75.

To get higher orders of distances we repeat the process. We look for an element that is still zero, a, = 0 and inspect whether aij > 1 and aj k > 1 for y = 1,..., n. We insert into a k the minimal nonzero sum of Uij + aj k- This is justified, because in a cyclic molecule, say cyclohexane, atom 1 is connected to atom 6 either directly or via 2, 3,4, and 5. So we are fixing the shortest path. For completeness, we give the final distance matrix that has eventually a maximum order of 3 ... 

In order to solve the shortest path problem for the double weighting network, a new weight for graph G is constructed with the target weight and limit weight and which is expressed as w = aw, -H (1 - a) w and Wj = (k,W2 + fcj (0 < a < 1). If there is a feasible path p from the specified source node s to the destination node t, a function can be established as follows ... 

R. Dorairaj and G. Lakhani. A VLSI implementation of all-pair shortest path problem. In ICPP 81 pages 207-209. S. K. Sahni, editor. The Pennsylvania State University Press, 1987. 

Figure 6.18 Top Chemical wave in the BZ reaction propagating through a membrane labyrinth. The wave was initiated at the lower left corner of the 3.2 x 3.2 cm maze. Bottom Velocity field showing local propagation direction obtained from analysis of fifty images obtained at 50-s intervals. Lines show shortest paths between five points and the starting point S. (Reprinted with permission from Steinbock, O. Toth, A. Showalter, K. 1995. Navigating Complex Labyrinths Optimal Paths from Chemical Waves, Science 267, 868 871.

AKC 10] Akca Z., Ralphs T.K., Berger R.T., Solution methods for the multi-trip elementary shortest path problem with resource constraints . Optimization Online the Mathematical Programming Society, 2010... 

R.K. Ahuja, K. Mehlhom, J.B. Orlin, R.E. Tarjan, Faster algorithms for the shortest path problem, J. Asso. Comput. Mach., 1990. 

K.L. Clarkson, Approximation algorithms for shortest path motion planning, proc. 19th Ann. Symp. on Theory of Computing, 1987, 56-65. 

The final step of the Isomap algorithm is the eigendecomposition of the shortest path matrix, as with MDS the computational cost of this step is 0 n ). Therefore, the overall complexity of the Isomap algorithm is 0(D log(A )n log(n)) -I- 0 n k + log(n))) + 0 n ) and since only the most computationally expensive term is being considered the complexity of Isomap can be thought of being 0 n ). 

Narvaez, P, Siu, K.Y., Tzeng, H.Y. New dynamic algorithms for shortest path tree computation. lEEE/ACM Transactions on Networking 8(6), 734-746 (2000)... 

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