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MIT1_204S10_lec13

# MIT1_204S10_lec13 - 1.204 Lecture 13 Dynamic programming...

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1.204 Lecture 13 Dynamic programming: Method Resource allocation Introduction Divide and con quer starts with the entire p roblem, divides it into subproblems and then combines them into a solution This is a top-down approach Dynamic programming starts with the smallest, simplest subproblems and combines them in stages to obtain solutions to larger subproblems until we get the solution to the original problem This is a bottom-up approach Dynamic programming is used much more than divide and conquer It is more flexible and controllable It is more efficient on most problems since it must consider far fewer combinations 1

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Principle of optimality “Princi ple of op timalit y”: In an optimal sequence of decisions or choices, each subsequence must also be optimal For some problems, an optimal sequence may be found by making decisions one at a time and never making a mistake True for greedy algorithms (except label correctors) For many problems it’s not possible to make stepwise decisions based only on local information so that the sequence of decisions is optimal sequence of decisions is optimal. One way to solve such problems is to enumerate all possible decision sequences and choose the best Dynamic programming can drastically reduce the amount of computation by avoiding sequences that cannot be optimal by the “principle of optimality” Project selection example Supp ose we have: \$4 million budget 3 possible projects (e.g. flood control) Each funded at \$1 million increments from \$0 to \$4 million Each increment produces a different marginal benefit Dynamic programming problems are usually discrete, not continuous We want to find the plan that produces the maximum benefit Stages are the number of decisions to be made We have 3 stages, since we have 3 projects States are the number of distinct possibilities At each stage there are 5 states (\$0, 1, 2, 3, 4 million) 2
3 Project selection formulation We build a multistage graph to represent this problem: Source node at start of graph, representing ‘null’ initial stage Set of nodes at each stage for each state Sink node at end of graph, which is a collapsed representation of the final state Each node characterized by V(i,j): V(i,j) is value (benefit) obtained up to (but not including) stage i by committing j resources Each node also stores its predecessor node in P(i) Each arc is characterized by E(m,n): E(m,n) is value obtained by spending n resources on project m Project selection data Investment Benefit Investment Benefit Investment Benefit Project 0 Project 1 Project 2 In theory, projects could have dependencies, but in practice it’s an improbable model. In the example above: Project 1’s benefits could depend on project 0 investment 0 0 0 0 0 0 1 6 1 5 1 1 2 8 2 11 2 4 3 8 3 16 3 5 4 10 4 17 4 6 Project 1 s benefits could depend on project 0 investment But not on project 2 investment

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