Shortest Path Applications
 The most obvious applications arise in transportation or communications,
such as finding the best route to drive between Chicago and Phoenix or figuring
how to direct packets to a destination across a network.
 Consider the problem of image segmentation, that is, separating two
characters in a scanned, bitmapped image of printed text. We need to find the
separating line between two points that cuts through the fewest number of black
pixels. This grid of pixels can be modeled as a graph, with any edge across a
black pixel given a high cost. The shortest path from top to bottom defines the
best separation between left and right.
 A major problem in speech recognition is distinguishing between words that
sound alike (homophones), such as to, two, and too. We can construct a graph
whose vertices correspond to possible words, with an edge between possible
neighboring words. If the weight of each edge measures the likelihood of
transition, the shortest path across the graph defines the best interpretation
of a sentence.
 Suppose we want to draw an informative picture of a graph. The center of the
page should coorespond to the ``center'' of the graph, whatever that means. A
good definition of the center is the vertex that minimizes the maximum distance
to any other vertex in the graph. Finding this center point requires knowing the
distance (i.e. shortest path) between all pairs of vertices.

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