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CS 312: Algorithm Analysis

This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License . CS 312: Algorithm Analysis. Lecture #22: Intro. to Dynamic Programming. Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick. Objectives.

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CS 312: Algorithm Analysis

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  1. This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License. CS 312: Algorithm Analysis Lecture #22: Intro. to Dynamic Programming Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick

  2. Objectives • Motivate Dynamic Programming (DP) • Understand the tabular approach to DP • Use DP to solve some example problems • Extract the composition of a solution from a DP table

  3. Families of Algorithms • Introductory • Importance of algorithm analysis, seen through RSA • Elegant Design Principles • Divide-and-Conquer • Graph Exploration • Greedy • Drawback: only usable on very specific types of problems • Need: “Sledgehammers of thealgorithms craft” • Dynamic Programming • Linear Programming

  4. Divide & Conquer A B C B E F G E G H E F G

  5. A B C B E F G E G H E F G Dynamic Programming start solving sub-problems at the bottom.

  6. Dynamic Programming A E solutionE F solutionF G solutionG B C B E F G E G H E F G Build a table of results as you go

  7. Dynamic Programming A E solutionE F solutionF G solutionG B solutionB B C B E F G E G H E F G Avoid recomputing intermediate results;Consult the table

  8. Dynamic Programming A E solutionE F solutionF G solutionG B solutionB H solutionH B C B E F G E G H E F G Avoid recomputing intermediate results;Consult the table

  9. A B C B E F G E G H E F G Dynamic Programming A C B H E F G

  10. DP Key Idea #1 Not every sub-problem is new. Save time: retain prior results.

  11. DP Key Idea #2 • Devise a minimal description for any problem instance and sub-problem • Use this description as key to a table

  12. Dynamic Programming • Identify sub-problems • Define their minimal description • Perspective #1: Divide & Conquer with Memory Table • Solve the sub-problems one by one, smallest first • Store the solutions to sub-problems in a table and reuse the solutions to solve larger sub-problems • Until the top (original) problem instance is solved • Perspective #2: DAG • Nodes are sub-problems; edges are dependencies between the sub-problems • Linearize! • Solve the sub-problems one by one in the linearized order

  13. Example: Binomial Coefficients How many ways are there to choose k items from a set of n items?

  14. Example: C(5,3) C(5,3) C(4,2) C(4,3) C(3,1) C(3,2) C(3,2) C(3,3) C(2,0) C(2,1) C(2,1) C(2,2) C(2,1) C(2,2) 1 C(1,0) C(1,1) C(1,0) C(1,1) C(1,0) C(1,1) 1 1 1 1 1 1 1 1 1 (C(n,k))

  15. C(5,3): Be Wise C(5,3) C(4,2) C(4,3) C(3,1) C(3,2) C(3,3) C(2,0) C(2,1) C(2,2) C(1,0) C(1,1)

  16. From DAG to Table • What is the minimal description? • Can you embed this DAG in a table? C(5,3) C(4,2) C(4,3) C(3,1) C(3,2) C(3,3) C(2,0) C(2,1) C(2,2) C(1,0) C(1,1) Embed in a table; evaluate in the same order as before Linearize our DAG

  17. C(5, 3) k: n: Does this look familiar?

  18. Pascal’s Triangle Blaise Pascal (1623-1662) Second person to invent the calculator Religious philosopher Mathematician and physicist

  19. “Dynamic Programming” • Very little to do with writing code • Developed in the area of Operations Research • Conceived to optimally plan multi-stage processes • Then, “programming” = “planning” • Solves an optimization problem • Described by recurrence relations among problems and sub-problems • Possibly time-variant • i.e. a dynamic system • In the CS setting, often time-invariant • Coined by Bellman in the 1950s

  20. DP: From Problemto Table to Algorithm • Start with a problem definition • Devise a minimal description for any problem instance and sub-problem • Define recurrence to specify the relationship of problems to sub-problems • i.e., Define the conceptual DAG on sub-problems • Embed the DAG in a table • Use the index variables • 2-D case: index rows and columns • Two possible strategies • Fill in the sub-problem cells, proceeding from the smallest to the largest • Draw the DAG in the table from the top problem down to the smallest sub-problems; solve the relevant sub-problems in their table cells, from smallest to largest. i.e., solve top problem recursively, using the table as a memory function

  21. Making Change using DP • Given , what’s the smallest number of coins required to make change for ? • Example: • m = 4 different denominations of coins • Values d = [1, 2, 4, 7] – Look familiar? • In general, does the greedy algorithm always give the optimal answer? • Greedy can fail. • Example: • d = [1, 10, 18], = 20 • Goal: Solve every case using some other strategy • How about Dynamic Programming?

  22. Making Change using DP • Problem: Given , what’s the smallest number of coins to make change for? • Define sub-problem representation: • Let min. number of coins needed to make change for using coins of type 1 through . • Coins have denominations in the array ; assume • Basic ideas for relationship among sub-problems as a recurrence relation: • If , then take coins • If , either takea coin of value or do not take, and solve the problem (make change) for the balance • Take out a piece of scratch paper and write a mathematical version of the recurrence relation

  23. Making Change using DP • If , then take coins • If , either take a coin of value or do not

  24. Making Change using DP • Design the table to contain the sub-problem results from the recurrence relation • Design the DP algorithm to walk the table properly • Solve an example by running the DP algorithm • Modify the resulting algorithm for space and time, if necessary

  25. Given j, what’s the smallest number of “coins” to make j in change? Example

  26. Making Change j i How much space? How much time?

  27. Algorithm in Pseudo-code function coins(d, N) Input: Array d[1..n] specifies the coinage; N is the number of units for which to make change Output: Minimum number of coins needed to make change for N units using coins from d array c[1..n,0..N] for i=1 to n do c[i,0] = 0 for i=1 to n do for j=1 to N do if i=1 and j<d[i] then c[i,j] = +infinity else if i=1 then c[i,j] = 1+c[1,j-d[1]] else if j<d[i] then c[i,j] = c[i-1,j] else c[i,j] = min(c[i-1,j],1+c[i,j-d[i]]) return c[n,N] Easy translation from table + recurrence to pseudo-code!

  28. Pre-computing vs. on-the-fly • Eager: Pre-computing • Fill the table bottom-up, then extract solution • Lazy: On-demand • Build the DAG (in the table) top-down • Solve only the necessary table entries (again from bottom-up)

  29. Comparison • How does DP algorithm compare to greedy? • speed • space • correctness • simplicity

  30. Assignment • HW #15, #16 • Greedy: Divisible Knapsack • Dynamic Programming: (coin problem) • Project #5 • Gene Sequence Alignment (DP)

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