Lecture+9+-+Dynamic+Programming+I.pdf

Lecture+9+-+Dynamic+Programming+I.pdf

• 1.
  CIST212-Data Structures and Algorithms Lecture9 Dynamic Programming I Prof. Boxiang Dong https://msuweb.montclair.edu/~dongb/ Office: CCIS-227E Email: dongb@montclair.edu
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  Review • Stack, Queue,Linked List • traverse: O(n) • search: O(n) • insertion: (1) • deletion: O(1) • Binary Search Tree • traverse: inorder, preorder, postorder O(n) • search: O(h) • minimum/maximum: O(h) • successor: O(h) • insertion: O(h) • deletion: O(h)
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  Dynamic Programming • Dynamicprogramming is a general strategy to solve computational problems, like divide-and-conquer.
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  Dynamic Programming • Divide-and-conquerparadigm: • Break the problem into several disjoint subproblems that are similar to the original one but smaller in size. • Solve the subproblems recursively. • Combine these solutions to create a solution to the original problem. • Dynamic programming: • Break the problem into several overlapping subproblems that are similar to the original one but smaller in size. • Solve the subproblems recursively only once. • Combine these solutions to create a solution to the original problem.
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  Dynamic Programming
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  Dynamic Programming • Letslook at the rod-cutting problem. • Given a rod of length n inches and a table of prices pi for i=1,2,…,n, determine the maximum revenue rn obtainable by cutting up the rod and selling the prices.
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  Dynamic Programming
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  Dynamic Programming pn: therevenue from no cut at all ri+rn-i: the revenue from cutting at the i-th place This is called the optimal substructure property: Optimal solutions to a problem incorporate optimal solutions to related subproblems, which we may solve independently.
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  Dynamic Programming • Tomake it simpler, we have Only the right-hand side of a cut at the i–th place will be cut further.
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  Dynamic Programming • Basedon the previous formula, we derive the following recursive algorithm:
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  Dynamic Programming Time complexity:O(2n)
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  Dynamic Programming • Wedo not want to repeatedly calculate r2, r3, …, rn-1. • In dynamic programming, we only need to calculate them once. • The first time we calculate ri, we store it in a place. • The next time we need to use ri, we just retrieve it directly.
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  Dynamic Programming
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  Dynamic Programming Key idea: 1.We use r[0,…,n] to store the maximum revenue. 2. In line 1 of MEMOIZED-CUT-ROD-AUX(p, n, r), If we find that rn is calculated before, we just return it directly; Otherwise, calculate and store it.
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  Dynamic Programming • MEMOIZED-CUT-ROD-AUX(p,n, r) calculates rn in a top-down fashion by recursion. • Most often, we want a bottom-up solution for two reasons: • Logically, it is easy to follow. • Faster, as it avoids recursion.
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  Dynamic Programming Time complexity:O(n2) Space complexity: O(n) Dynamic programming is an approach to trading space for efficiency.
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  Dynamic Programming
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  Your Task • Read:Ch 15.1 (Must-be)