Practice Prelim 2 Solutions

# Practice Prelim 2 Solutions - ORIE 3310/5310 Practice...

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ORIE 3310/5310 Practice Prelim 2 (Spr2009) Solutions Spring 2010 1. (a) See class notes on Bellman’s algorithm (for finding a shortest (1 , n )-path in a directed acyclic graph). (b) Now suppose we wish to find a shortest (1 , n )-path which contains a specific edge , say edge ( i, j ). How would you use the basic procedure to achieve this? Solution. There are two ways to solve this problem. The first method: Find the shortest path from 1 to i using the basic procedure (Bellman), then find the shortest path from j to n . Then, the two solutions together with edge ( i, j ) is a shortest path from 1 to n that contains edge ( i, j ). Note that it is possible that there is no solution to one of these problems (i.e. no path from 1 to i or no path from j to n ), in which case we know that there is no path from 1 to n that contains ( i, j ). The second method: change the length of edge ( i, j ) to c ij - M , where M is a sufficiently large positive number (for example, you could choose M to be the sum of the absolute value of all other edge costs). Then, since we wish to minimize total path length, the

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