APPLICATIONS OFTEN CALL for a graph model where we associate *weights* or *costs* with each edge. In an airline map where edges represent flight routes, these weights might represent distances or fares. In an electric circuit where edges represent wires, the weights might represent the length of the wire, its cost, or the time that it takes a signal to propagate through it. In a job-scheduling problem, weights might represent time or the cost of performing tasks or of waiting for tasks to be performed.

Questions that entail cost minimization naturally arise for such situations. We examine algorithms for two such problems: (*i*) find the lowest-cost way to connect all of the points, and (*ii*) find the lowest-cost ...

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