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Upcoming Contests

Upcoming Contests. TopCoder Marathon Match 49 (currently running – ends Feb. 18 th ). TopCoder Open 2009. Algorithms qualifying rounds: Tuesday, February 24 th @ 7:00 AM Saturday, February 28 th @ 12:00 PM Wednesday, March 4 th @ 9:00 PM. TopCoder Open 2009. Marathon Matches:

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Upcoming Contests

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  1. Upcoming Contests • TopCoder Marathon Match 49 (currently running – ends Feb. 18th)

  2. TopCoder Open 2009 • Algorithms qualifying rounds: • Tuesday, February 24th @ 7:00 AM • Saturday, February 28th @ 12:00 PM • Wednesday, March 4th @ 9:00 PM

  3. TopCoder Open 2009 • Marathon Matches: • Wednesday, February 25th @ 12:00 PM • Wednesday, March 11th @ 12:00 PM • Wednesday, March 25th @ 12:00 PM

  4. TopCoder Open 2009 • Other contests as well • Visit http://www.topcoder.com/tco09 for more information

  5. TopCoderEvents Calendar (Feb.)

  6. Events Calendar (March)

  7. Dynamic Programming Programming Puzzles and Competitions CIS 4900 / 5920 Spring 2009

  8. Lecture Outline • Order Notation • Solution to Text Segmentation • Dynamic Programming • Text Segmentation w/ Memoization • Practice

  9. Algorithmic Complexity and Order Notation • We would like to be able to describe how long a program takes to run

  10. Algorithmic Complexity and Order Notation • We would like to be able to describe how long a program takes to run • We should express the runtime in terms of the input size

  11. Algorithmic Complexity and Order Notation • Absolute time measurements are not helpful, as computers get faster all the time • Also, runtime is clearly dependent on the input (and input size)

  12. Algorithmic Complexity and Order Notation • We define O(*) as follows

  13. Text Segmentation • Some languages are written without spaces between the words

  14. Text Segmentation • Difficult for search engines to decipher the meaning of a search

  15. Text Segmentation • Difficult for search engines to decipher the meaning of a search • Difficult to return relevant results

  16. Text Segmentation • Given a string without spaces, what is the best segmentation of the string?

  17. Text Segmentation • Examples: “upordown”  “up or down”

  18. Text Segmentation • Examples: “upordown”  “up or down” “upsidedown”  “upside down”

  19. Text Segmentation • Examples: “upordown”  “up or down” “upsidedown”  “upside down” “haveaniceday”  “have a nice day”

  20. Text Segmentation • Ambiguities are possible, e.g.: “theyouthevent”  ?

  21. Text Segmentation • At least three ways to segment “theyouthevent” into valid words: 1) “they out he vent” 2) “the you the vent” 3) “the youth event”

  22. Text Segmentation • At least three ways to segment “theyouthevent” into valid words: 1) “they out he vent” 2) “the you the vent” 3) “the youth event” (most likely)

  23. Text Segmentation:Some ambiguities • whorepresents.com • therapistfinder.com • speedofart.com • expertsexchange.com • penisland.com These examples are borrowed from Peter Norvig.

  24. Text Segmentation • How can this be done?

  25. Text Segmentation:Solution • Recursion: Pmax(y) = maxi P0(y[0:i]) x Pmax(y[i:n]); Pmax(“”) = 1; where n = length(y)

  26. Text Segmentation:Solution • Recursion: Pmax(y) = maxi P0(y[0:i]) x Pmax(y[i:n]); Pmax(“”) = 1; where n = length(y) “base case”

  27. Text Segmentation:Solution Pmax(“theyouthevent”) = max( P0(“t”) x Pmax(“heyouthevent”), P0(“th”) x Pmax(“eyouthevent”), … P0(“theyouthevent”) x Pmax(“”) );

  28. Text Segmentation:Solution double get_Pmax(const string &s) { if(s.length() == 0) return get_P0(s); double max = 0.0; for(int i = 1; i <= s.length(); ++i) { double temp = get_P0(s.substr(0, i - 0)) * get_Pmax(s.substr(i)); if(temp > max) max = temp; } return max; }

  29. Text Segmentation:Solution double get_Pmax(const string &s) { if(s.length() == 0) return get_P0(s); double max = 0.0; for(int i = 1; i <= s.length(); ++i) { double temp = get_P0(s.substr(0, i - 0)) * get_Pmax(s.substr(i)); if(temp > max) max = temp; } return max; } P0(“”)x Pmax(“”)

  30. Text Segmentation:Solution double get_Pmax(const string &s) { if(s.length() == 0) return get_P0(s); double max = 0.0; for(int i = 1; i <= s.length(); ++i) { double temp = get_P0(s.substr(0, i - 0)) * get_Pmax(s.substr(i)); if(temp > max) max = temp; } return max; } max(…) P0(“”)x Pmax(“”)

  31. Text Segmentation:Solution double get_Pmax(const string &s) { if(s.length() == 0) return get_P0(s); double max = 0.0; for(int i = 1; i <= s.length(); ++i) { double temp = get_P0(s.substr(0, i - 0)) * get_Pmax(s.substr(i)); if(temp > max) max = temp; } return max; } base case max(…) P0(“”)x Pmax(“”)

  32. Text Segmentation:Solution double get_Pmax(const string &s) { if(s.length() == 0) return get_P0(s); double max = 0.0; for(int i = 1; i <= s.length(); ++i) { double temp = get_P0(s.substr(0, i - 0)) * get_Pmax(s.substr(i)); if(temp > max) max = temp; } return max; } base case max(…) P0(“”)x Pmax(“”) Pmax(y) = maxiP0(y[0:i]) x Pmax(y[i:n])

  33. Text Segmentation:Solution • Recursion: Pmax(y) = maxiP0(y[0:i]) x Pmax(y[i:n]); Pmax(“”) = 1; where n = length(y) base case

  34. Text Segmentation:Solution • Note that this algorithm is (extremely) inefficient

  35. Text Segmentation:Computing Pmax(“ent”) Pmax(“ent”) P0(“e”) x Pmax(“nt”) P0(“en”) x Pmax(“t”) P0(“ent”) x Pmax(“”) P0(“n”) x Pmax(“t”) P0(“nt”) x Pmax(“”) P0(“t”) x Pmax(“”) P0(“t”) x Pmax(“”)

  36. Text Segmentation:Computing Pmax(“ent”) Unnecessary Call Pmax(“ent”) P0(“e”) x Pmax(“nt”) P0(“en”) x Pmax(“t”) P0(“ent”) x Pmax(“”) P0(“n”) x Pmax(“t”) P0(“nt”) x Pmax(“”) P0(“t”) x Pmax(“”) P0(“t”) x Pmax(“”)

  37. Fibonacci Numbers • F0 = 0 • F1 = 1 • Fn = Fn-1 + Fn-2 for n > 1

  38. Fibonacci Numbers • F0 = 0 • F1 = 1 • Fn = Fn-1 + Fn-2 for n > 1 • 0, 1, 1, 2, 3, 5, 8, 13, …

  39. Fibonacci Numbers • How would we write code for this?

  40. Fibonacci Numbers • How would we write code for this? • F0 = 0 • F1 = 1 • Fn = Fn-1 + Fn-2 for n > 1

  41. Fibonacci Numbers int fibonacci(int n) { if(n == 0 || n == 1) return n; else return fibonacci(n-1) + fibonacci(n-2); }

  42. Fibonacci Numbers: Computing F(5) F(5) F(4) F(3) F(3) F(2) F(1) F(2) F(1) F(0) F(1) F(0) F(2) F(1) F(1) F(0)

  43. Dynamic Programming • Same algorithm as before, but compute each value only once

  44. Dynamic Programming • Same algorithm as before, but compute each value only once • This significantly reduces the runtime

  45. Dynamic Programming • Can be done in one of two ways • iteration • recursion w/ memoization

  46. Dynamic Programming • Can be done in one of two ways • iteration (bottom-up) • recursion w/ memoization (top-down)

  47. Dynamic Programming • Can be done in one of two ways • iteration (bottom-up) • recursion w/ memoization (top-down) • We will focus on the second of these (for now)

  48. Dynamic Programming:Recursion w/ Memoization • “Memoize”* previously computed function evaluations *the word “memoize” stems from the word “memo”; it is not a misspelling of the word “memorize”

  49. Dynamic Programming:Recursion w/ Memoization • “Memoize”* previously computed function evaluations • There is no reason to run the same computation twice *the word “memoize” stems from the word “memo”; it is not a misspelling of the word “memorize”

  50. Fibonacci Numbers:F(5) revisited F(5) F(4) F(3) F(3) F(2) F(1) F(2) F(1) F(0) F(1) F(0) F(2) F(1) F(1) F(0)

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