Path, shortest

Table 6.3. Sample molecules acetone and isobutene described by atom pair (ap) descriptors.

The distance matrix D of a graph G with n vertices is a square n x n symmetric matrix as represented by Eq. (13), where is the distance between the vertices Vi and Vj in the graph (i.e., the number of edges on the shortest path). 

In graph theory, the conversion of the adjacency matrix into the distance matrix is known as the "all pairs shortest path problem",... 

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

We denote the topological distance between atoms i and j (i.e., the number of bonds for the shortest path in the structure diagram) dy, and the properties for atoms i and j are referred to as pi and pj, respectively. The value of the autocorrelation function a d) for a certain topological distance d results from summation over all products of a property p of atoms i and j having the required distance d. 

It is strongly recommended to keep the source between the exhaust and the person. When the source is quite close to the person and between the person and the exhaust, the airflow around the person could generate a wake that includes the source or the generated contaminant and thereby the person s exposure could increase (see Fig. 10.3). In these cases, it is better to use a side exhaust, where the person is situated beside the shortest path from source to exhaust. 

Shortest path and the alternative paths between any two components... 

Direct interactions, shortest path, common targets or common regulators... 

Pajek http //vlado.fmf.uni-lj.si/pub/networks/pajek/ Network building software Random networks, shortest paths. Many options for network building 106... 

In a simple (nonweighted) connected graph, the graph distance dy between a pair of vertices V and Vj is equal to the length of the shortest path cormecting the two vertices, i.e. the number of edges on the shortest path. The distance between two adjacent vertices is 1. The distance matrix D(G) of a simple graph G with N vertices is the square NxN symmetric matrix in which [D],j=cl,j [9, 10]. 

To understand the method of steepest accent, consider a hiker who wishes to reach the summit of a volcanic island (assumed unimodal) by taking the shortest path. The shortest path is the steepest. Suppose our hiker is a mathematician, and rather than use his eyesight he decides to determine mathematically the best direction in which to proceed. Further, he knows that if he goes 20 ft to the north of his current position he will rise 20 ft and if he goes 20 ft to the east he will fall 10 ft. If the surface were a plane, he could approximate it by the following equation ... 

Hence the shortest path from r to d in G must go r-m-p-d and thus since r dominates p, by Lemma 4.2, r dominates m. Thus r chain... 

In the present example, the only fire detector is in the fire room. When it alarms, the fire room occupant is awakened immediately, but occupants in the suite can hardly hear it. They are slow to respond. The program EXITT knows where all persons are, decides mother will get baby, seeks the shortest path to get there and calculates the time at 1.3 m/sec. Mother and baby then follow the shortest route to the corridor door. The times required for these actions are given in Table II. 

In the first step a Shortest Path-Algorithm (Dijkstra or A ) calculates one or more shortest paths between two given nodes of the plant graph [5],... 

If no conflict free shortest path can be found, the conflict resolution tries to solve the conflict by waiting, speed reduction or evasion of the currently considered AGV. The applied priority rule is a First-Come-First-Serve heuristic. By using a FCFS-priority rule, paths that were once scheduled stay conflict free in the future [6, 7]. If the detected conflict can be solved, the resulting path is again analyzed for conflicts because new conflicts can emerge from the resolution of an earlier conflict. 

Cherkassky, B., Goldberg, A. and Radzik, T. (1994) Shortest paths algorithms Theory and experimental evaluation. SODA ACM-SIAM Symposium on Discrete Algorithms (A Conference on Theoretical and Experimental Analysis of Discrete Algorithms). 

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