how to make barcode in c#.net Figure 410 Single-edge extensions of known shortest paths in Software

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Figure 410 Single-edge extensions of known shortest paths
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Known region R
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Here s a plan for computing shortest paths: expand outward from the starting point s, steadily growing the region of the graph to which distances and shortest paths are known This growth should be orderly, rst incorporating the closest nodes and then moving on to those further away More precisely, when the known region is some subset of vertices R that includes s, the next addition to it should be the node outside R that is closest to s Let us call this node v; the question is: how do we identify it To answer, consider u, the node just before v in the shortest path from s to v:
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Since we are assuming that all edge lengths are positive, u must be closer to s than v is This means that u is in R otherwise it would contradict v s status as the closest node to s outside R So, the shortest path from s to v is simply a known shortest path extended by a single edge But there will typically be many single-edge extensions of the currently known shortest paths (Figure 410); which of these identi es v The answer is, the shortest of these extended paths Because, if an even shorter single-edge-extended path existed, this would once more contradict v s status as the node outside R closest to s So, it s easy to nd v: it is the node outside R for which the smallest value of distance(s, u) + l(u, v) is attained, as u ranges over R In other words, try all single-edge extensions of the currently known shortest paths, nd the shortest such extended path, and proclaim its endpoint to be the next node of R We now have an algorithm for growing R by looking at extensions of the current set of shortest paths Some extra ef ciency comes from noticing that on any given iteration, the only new extensions are those involving the node most recently added to region R All other extensions will have been assessed previously and do not need to be recomputed In the following pseudocode, dist(v) is the length of the currently shortest single-edge-extended path leading to v; it is for nodes not adjacent to R 113
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Initialize dist(s) to 0, other dist( ) values to R = { } (the known region ) while R = V : Pick the node v R with smallest dist( ) Add v to R for all edges (v, z) E: if dist(z) > dist(v) + l(v, z): dist(z) = dist(v) + l(v, z) Incorporating priority queue operations gives us back Dijkstra s algorithm (Figure 48) To justify this algorithm formally, we would use a proof by induction, as with breadth- rst search Here s an appropriate inductive hypothesis At the end of each iteration of the while loop, the following conditions hold: (1) there is a value d such that all nodes in R are at distance d from s and all nodes outside R are at distance d from s, and (2) for every node u, the value dist(u) is the length of the shortest path from s to u whose intermediate nodes are constrained to be in R (if no such path exists, the value is )
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The base case is straightforward (with d = 0), and the details of the inductive step can be lled in from the preceding discussion
Running time
At the level of abstraction of Figure 48, Dijkstra s algorithm is structurally identical to breadth- rst search However, it is slower because the priority queue primitives are computationally more demanding than the constant-time eject s and inject s of BFS Since makequeue takes at most as long as |V | insert operations, we get a total of |V | deletemin and |V | + |E| insert/decreasekey operations The time needed for these varies by implementation; for instance, a binary heap gives an overall running time of O((|V | + |E|) log |V |)
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