Dynamic Programming

Dynamic Programming Jaehyun Park CS 97SI Stanford University

June 29, 2015

Outline

Dynamic Programming 1-dimensional DP 2-dimensional DP Interval DP Tree DP Subset DP

Dynamic Programming

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What is DP?

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Wikipedia definition: “method for solving complex problems by breaking them down into simpler subproblems”

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This definition will make sense once we see some examples – Actually, we’ll only see problem solving examples today

Dynamic Programming

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Steps for Solving DP Problems

1. Define subproblems 2. Write down the recurrence that relates subproblems 3. Recognize and solve the base cases

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Each step is very important!

Dynamic Programming

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Outline

Dynamic Programming 1-dimensional DP 2-dimensional DP Interval DP Tree DP Subset DP

1-dimensional DP

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1-dimensional DP Example

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Problem: given n, find the number of different ways to write n as the sum of 1, 3, 4

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Example: for n = 5, the answer is 6 5 = 1+1+1+1+1 = 1+1+3 = 1+3+1 = 3+1+1 = 1+4 = 4+1

1-dimensional DP

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1-dimensional DP Example

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Define subproblems – Let Dn be the number of ways to write n as the sum of 1, 3, 4

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Find the recurrence – Consider one possible solution n = x1 + x2 + · · · + xm – If xm = 1, the rest of the terms must sum to n − 1 – Thus, the number of sums that end with xm = 1 is equal to Dn−1 – Take other cases into account (xm = 3, xm = 4)

1-dimensional DP

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1-dimensional DP Example

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Recurrence is then Dn = Dn−1 + Dn−3 + Dn−4

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Solve the base cases – D0 = 1 – Dn = 0 for all negative n – Alternatively, can set: D0 = D1 = D2 = 1, and D3 = 2

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We’re basically done!

1-dimensional DP

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Implementation

D[0] = D[1] = D[2] = 1; D[3] = 2; for(i = 4; i