In figure 56 we illustrate the way the dynamic

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Unformatted text preview: rithm Thus, we can compute N0,n−1 with an algorithm that consists primarily of three nested for-loops. The outside loop is executed n times. The loop inside is executed at most n times. And the inner-most loop is also executed at most n times. Therefore, the total running time of this algorithm is O(n3 ). Theorem 5.15: Given a chain-product of n two-dimensional matrices, we can compute a parenthesization of this chain that achieves the minimum number of scalar multiplications in O(n3 ) time. Proof: We have shown above how we can compute the optimal number of scalar multiplications. But how do we recover the actual parenthesization? The method for computing the parenthesization itself is is actually quite straightforward. We modify the algorithm for computing Ni, j values so that any time we find a new minimum value for Ni, j , we store, with Ni, j , the index k that allowed us to achieve this minimum. In Figure 5.6, we illustrate the way the dynamic programming solution to the matrix chain-product problem fills in the array N . j N i,j i i,k + didk+1dj+1 k+1,j Figure 5.6: Illustration of the way the matrix chain-product dynamic-programming algorithm fills in the array N . Now that we have worked through a complete example of the use of the dynamic programming method, let us discuss the general aspects of the dynamic programming technique as it can be applied to other problems. Chapter 5. Fundamental Techniques 278 5.3.2 The General Technique The dynamic programming technique is used primarily for optimization problems, where we wish to find the “best” way of doing something. Often the number of different ways of doing that “something” is exponential, so a brute-force search for the best is computationally infeasible for all but the smallest problem sizes. We can apply the dynamic programming technique in such situations, however, if the problem has a certain amount of structure that we can exploit. This structure involves the following three components: Simple Subproblems: There has to be some way of breaking the global optimization problem into subproblems, each having a similar structure to the original problem. Moreover, there should be a simple way of defining subproblems with just a few indices, like i, j, k, and so on. Subproblem Optimality: An optimal solution to the global problem must be a composition of optimal subproblem solutions, using a relatively simple combining operation. We should not be able to find a globally optimal solution that contains suboptimal subproblems. Subproblem Overlap: Optimal solutions to unrelated subproblems can contain subproblems in common. Indeed, such overlap improves the efficiency of a dynamic programming algorithm that stores solutions to subproblems. Now that we have given the general components of a dynamic programming algorithm, we next give another example of its use. 5.3.3 The 0-1 Knapsack Problem Suppose a hiker is about to go on a trek through a rain forest carrying a single knapsack. Suppose further that she knows the maximum total weight W that she can carry, and she has a set S of n different useful items that she can potentially take with her, such as a folding chair, a tent, and a copy of this book. Let us assume that each item i has an integer weight wi and a benefit value bi , which is the utility value that our hiker assigns to item i. Her problem, of course, is to optimize the total value of the set T of items that she takes with her, without going over the weight limit W . That is, she has the following objective: maximize ∑ bi i∈T subject to ∑ wi ≤ W . i∈T Her problem is an instance of the 0-1 knapsack problem. This problem is called a “0-1” problem, because each item must be entirely accepted or rejected. We consider the fractional version of this problem in Section 5.1.1, and we study how knapsack problems arise in the context of Internet auctions in Exercise R-5.12. 5.3. Dynamic Programming 279 A First Attempt at Characterizing Subproblems We can easily solve the 0-1 knapsack problem in Θ(2n ) time, of course, by enumerating all subsets of S and selecting the one that has highest total benefit from among all those with total weight not exceeding W . This would be an inefficient algorithm, however. Fortunately, we can derive a dynamic programming algorithm for the 0-1 knapsack problem that runs much faster than this in most cases. As with many dynamic programming problems, one of the hardest parts of designing such an algorithm for the 0-1 knapsack problem is to find a nice characterization for subproblems (so that we satisfy the three properties of a dynamic programming algorithm). To simplify the discussion, number the items in S as 1, 2, . . . , n and define, for each k ∈ {1, 2, . . . , n}, the subset Sk = {items in S labeled 1, 2, . . . , k}. One possibility is for us to define subproblems by using a parameter k so that subproblem k is the best way to fill the knapsack using only items from the set Sk . This is a va...
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