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how to make barcode in c#.net Figure 45 Edge lengths often matter in Software
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Breadth rst search treats all edges as having the same length This is rarely true in applications where shortest paths are to be found For instance, suppose you are driving from San Francisco to Las Vegas, and want to nd the quickest route Figure 45 shows the major highways you might conceivably use Picking the right combination of them is a shortestpath problem in which the length of each edge (each stretch of highway) is important For the remainder of this chapter, we will deal with this more general scenario, annotating every edge e E with a length le If e = (u, v), we will sometimes also write l(u, v) or l uv These le s do not have to correspond to physical lengths They could denote time (driving time between cities) or money (cost of taking a bus), or any other quantity that we would like to conserve In fact, there are cases in which we need to use negative lengths, but we will brie y overlook this particular complication 44 Dijkstra s algorithm
441 An adaptation of breadth rst search
Breadth rst search nds shortest paths in any graph whose edges have unit length Can we adapt it to a more general graph G = (V, E) whose edge lengths l e are positive integers A more convenient graph Here is a simple trick for converting G into something BFS can handle: break G s long edges into unitlength pieces, by introducing dummy nodes Figure 46 shows an example of this 108 Figure 46 Breaking edges into unitlength pieces
B A C
transformation To construct the new graph G , For any edge e = (u, v) of E, replace it by l e edges of length 1, by adding le 1 dummy nodes between u and v Graph G contains all the vertices V that interest us, and the distances between them are exactly the same as in G Most importantly, the edges of G all have unit length Therefore, we can compute distances in G by running BFS on G Alarm clocks If ef ciency were not an issue, we could stop here But when G has very long edges, the G it engenders is thickly populated with dummy nodes, and the BFS spends most of its time diligently computing distances to these nodes that we don t care about at all To see this more concretely, consider the graphs G and G of Figure 47, and imagine that the BFS, started at node s of G , advances by one unit of distance per minute For the rst 99 minutes it tediously progresses along S A and S B, an endless desert of dummy nodes Is there some way we can snooze through these boring phases and have an alarm wake us up whenever something interesting is happening speci cally, whenever one of the real nodes (from the original graph G) is reached We do this by setting two alarms at the outset, one for node A, set to go off at time T = 100, and one for B, at time T = 200 These are estimated times of arrival, based upon the edges currently being traversed We doze off and awake at T = 100 to nd A has been discovered At this point, the estimated time of arrival for B is adjusted to T = 150 and we change its alarm accordingly Figure 47 BFS on G is mostly uneventful The dotted lines show some early wavefronts G:

