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Lecture 4 Dynamic Programming

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  1. Lecture 4 Dynamic Programming

  2. Basic Algorithm Design Techniques • Divide and conquer • Dynamic Programming • Greedy • Common Theme: To solve a large, complicated problem, break it into many smaller sub-problems.

  3. Dynamic Programming • Idea: Break the problem into many closely relatedsub-problems, memorize the result of the sub-problems to avoid repeated computation.

  4. Warmup Example: Fibonacci Numbers f(n) = f(n-1) + f(n-2), f(1) = f(2) = 1

  5. Naïve solution • Follow the recursion directly f(7) f(6) f(5) f(3) f(4) f(5) f(4) f(3) f(4) … …

  6. Naïve solution • Follow the recursion directly f(7) f(6) f(5) f(4) f(5) f(3) f(4) … … f(3) f(4)

  7. Memoization f(n) IF n < 3 THENRETURN 1. IF f(n) has been computed before THEN RETURN stored result. res = f(n-1) + f(n-2) Mark f(n) as computed, Store f(n) = res RETURN res Dynamic Programming Table

  8. Filling in the table iteratively • Observation: f(n) is always computed in the order of n = 1, 2, 3, … • No need to recurse f(n) IF n < 3 THENRETURN 1. a[1] = a[2] = 1 FORi = 3 TO n a[i] = a[i-1] + a[i-2] RETURN a[n]

  9. Basic Steps in Designing Dynamic Programming Algorithms • Relate the problem recursively to smaller sub-problems. (Transition function, f(n) = f(n-1)+f(n-2)) • Organize all sub-problems as a dynamic programming table. (A table for all values of n) • Fill in values in the table in an appropriate order.(n = 1, 2, 3, …)

  10. Example 1: Longest Increasing Subsequence • Input: Array of numbersa[] = {4, 2, 5, 3, 9, 7, 8, 10, 6} • Subsequence: list of numbers appearing in the same order, but may not be consecutiveExample: {4, 2, 5}, {4, 3, 8}, {2, 5, 7, 8, 10} • Subsequence b[] is increasing if b[i+1] > b[i] for all i. • Output: Length of the longest increasing subsequence.Example: 5, the subsequence is {2, 5, 7, 8, 10}or {2, 3, 7, 8, 10} (both have length 5)

  11. Example 2 Knapsack Problem • There is a knapsack that can hold items of total weight at most W. There are now n items with weights w1,w2,…, wn. Each item also has a value v1,v2,…,vn. • Goal: Select some items to put into knapsack1. Total weight is at most W.2. Total value is as large as possible.

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