midsamp206-sol2

midsamp206-sol2 - ECS 222A: Algorithm Design and Analysis...

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Unformatted text preview: ECS 222A: Algorithm Design and Analysis Handout ?? UC Davis Charles Martel Nov. 2006 Samplel Exam Solutions 1 Dynamic Programming (a) T (1 , , 0) = 0 T (1 ,w i ,s i ) = v i else, T (1 ,j,k ) =- (b) We can either use item i or not. If we use item i , we look in the cell already computed ( i- 1 ,j- w i ,k- s i ) in order to find the optimal solution with the weight and size left. This leads to the following forumla: T ( i,j,k ) = max [ T ( i- 1 ,j,k ) , T ( i- 1 ,j- w i ,k- s i ) + v i ] (c) cleary, our table is of size n * W * S and we do (1) work at each cell. Then, overall work is ( nWS ) (d) If we have the entire table around, we can simply do backtracking. We must first find the optimal solution which lay in the bottom plane, in some cell T ( n,w * ,s * ). We can find this cell by brute force in worst case time of ( SW ) time. Once found, we follow this method. Starting with i = n can see if T ( i,w * ,s * ) = T ( i- 1 ,w * ,s * ). If so, item i was not included. If T ( i,w * ,s * ) = T ( i- 1 ,w * - w i ,s * - s i ) + v i , then we know item i was included and we continue from cell T ( i- 1 ,w *- w i ,s *- s i ). We then repeat this procedure on whichever cell gave us our value. We stop when we reach an ( i, , 0) cell, or i = 1 so we know no more items are to be added. 2 Network Flow [10 points] Suppose you are given a maximum flow f in a network G=(V,E) with integer capacities. a) Suppose that we increase the capacity of a single edge by 1. Describe how to find a new maximum flow in O( | E | + | V | ) time. Form the new residual graph G f and look for a path from s to t in O( | E | + | V | ) time using BFS or DFS. If found, augment the flow (by one since integer capacities). The flow can go up by at most one since the min-cut increased by at most one from the change....
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midsamp206-sol2 - ECS 222A: Algorithm Design and Analysis...

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