ch08 The Greedy Approach

Optimal substructure
A problem exhibits optimal substructure if an optimal solution to the problem contains within it optimal solutions to subproblems. In the 0-1 knapsack problem, when we consider an item for inclusion in the knapsack, we must compare the solution to the subproblem in which the item is included with the solution to the subproblem in which the item is excluded before we can make the choice. The problem formulated in this way gives rise to many overlapping subproblems--a hallmark of dynamic programming.
Chapter 8
The Greedy Approach
Introduction
Greedy algorithms are usually designed to solve optimization problems. They typically consist of an iterative procedure that tries to find a local optimal solution. A greedy algorithm makes a correct guess on the basis of little calculation without worrying about the future. It builds a solution step by step. Each step increases the size of the partial solution and is based on local optimization. The choice made is that which produces the largest immediate gain while maintaining feasibility. The algorithms are typically efficient. The hard part is proving that the algorithm does indeed solve the problem.
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Algorithm 8.1 DIJKSTRA Input: A weighted directed graph G=(V,E), where V={1,...,n} Output: The distance from vertex 1 to every other vertex in G 1. X={1}; YV-{1}; [1]0 2. for y2 to n 3. if y is adjacent to 1 then [y]length[1,y] 4. else [y] 5. end if 6. end for 7. for j1 to n-1 8. Let yY be such that [y] is minimum 9. XX{y} {add vertex y to X} 10. YY-{y} {delete vertex y from Y} 11. for each edge (y,w) 12. if wY and [y]+length[y,w]<[w] then 13. [w][y]+length[y,w] 14. end for 15. end for
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Representation: adjacency lists Lemma 8.1 (correctness): In Algorithm DIJKSTRA, when a vertex y is chosen in Step 8, if its label [y] is finite then [y]=[y] Theorem 8.1 (time complexity): Given a directed graph G with nonnegative weights on its edges and a source vertex s, Algorithm DIJKSTRA finds the length of the distance from s to every other vertex in (n^2) time.
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Fractional knapsack
Given n items of sizes s1,...,sn, and values v1,...,vn and size C, the knapsack capacity, the objective is to find nonnegative real numbers x1,...,xn that
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The shortest path problem
Let G=(V,E) be a directed graph in which each edge has a nonnegative length, and a distinguished vertex s called the source. Then determine the distance from s to every other vertex in V, where the distance from vertex s to vertex x is defined as the length of a shortest path from s to x.
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Greedy-choice property
A globally optimal solution can be arrived at by making a locally optimal (greedy) choice. When we are considering which choice to make, we make the choice that looks best in the current problem and then solve the subproblem arising after the choice is made. The choice may depends on choices so far, but it cannot depend on any future choices or on the solutions to subproblems. It usually progresses in a top-down fashion, making one greedy choice after another, reducing each given problem instance to a smaller one. In dynamic programming, we make a choice at each step, but the choice usually depends on the solutions to subproblems. Consequently we solve dynamicprogramming problems in a bottom-up manner, progressing from smaller subproblems to large subproblems. But we must prove that a greedy choice at each step yields a globally optimal solution. 8
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File compression
Suppose we are given a file, which is a string of characters. Compress the file as much as possible in such a way that the original file can be reconstructed. Let the set of characters in the file be C={c1,c2,...,cn}. Let also f(ci), 1in, be the frequency of character ci in the file. Since the frequency of some characters may be much larger than others, it is reasonable to use variable length encodings. When the encodings vary in length, we stipulate that the encoding of one character must not be the prefix of the encoding of another character; such codes are called prefix codes.
max xi v i
i 1 n n
s.t. x i si C
Greedy strategy: For each item compute yi=vi/si. Sort the items by decreasing ratio,and fill the knapsack with as much as possible from the first item, then the second, and so forth. Characteristics: The algorithm consists of a simple iterative procedure that selects that item which produces the largest 3 immediate gain while maintaining feasibility.