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Lecture 4: Order Statistics Instructor: Saravanan Thirumuruganathan CSE 5311 Saravanan Thirumuruganathan Outline 1 Order Statistics Min, Max k th -smallest and largest Median Mode and Majority CSE 5311 Saravanan Thirumuruganathan In-Class

  1. Lecture 4: Order Statistics Instructor: Saravanan Thirumuruganathan CSE 5311 Saravanan Thirumuruganathan

  2. Outline 1 Order Statistics Min, Max k th -smallest and largest Median Mode and Majority CSE 5311 Saravanan Thirumuruganathan

  3. In-Class Quizzes URL: http://m.socrative.com/ Room Name: 4f2bb99e CSE 5311 Saravanan Thirumuruganathan

  4. Order Statistics i th Order Statistic of a set of n elements is the i th smallest element Selection Problem Input: A set A of n (distinct) numbers and an integer i with 1 ≤ i ≤ n Output: i th smallest element in A The element x ∈ A that is larger than exactly i − 1 other elements of A Select element with rank i CSE 5311 Saravanan Thirumuruganathan

  5. Popular Order Statistics i = 1 i = n i = ⌊ n +1 2 ⌋ and i = ⌈ n +1 2 ⌉ CSE 5311 Saravanan Thirumuruganathan

  6. Popular Order Statistics Minimum: i = 1 Maximum: i = n Median: i = ⌊ n +1 2 ⌋ (lower) and i = ⌈ n +1 2 ⌉ (upper) CSE 5311 Saravanan Thirumuruganathan

  7. Selection Problem Input: A set A of n (distinct) numbers and an integer i with 1 ≤ i ≤ n Output: i th smallest element in A Naive Solution? CSE 5311 Saravanan Thirumuruganathan

  8. Selection Problem Input: A set A of n (distinct) numbers and an integer i with 1 ≤ i ≤ n Output: i th smallest element in A Naive Solution? Sort A and pick A [ i ] Time Complexity: O ( n log n ) CSE 5311 Saravanan Thirumuruganathan

  9. Finding the Minimum CSE 5311 Saravanan Thirumuruganathan

  10. Finding the Minimum Minimum(A): min = A[1] for i = 2 to A.length if min > A[i] min = A[i] return min Analysis: CSE 5311 Saravanan Thirumuruganathan

  11. Finding the Minimum Minimum(A): min = A[1] for i = 2 to A.length if min > A[i] min = A[i] return min Analysis: Complexity Measure: Number of Comparisons CSE 5311 Saravanan Thirumuruganathan

  12. Finding the Minimum Minimum(A): min = A[1] for i = 2 to A.length if min > A[i] min = A[i] return min Analysis: Complexity Measure: Number of Comparisons Number of Comparisons: n − 1 Time Complexity: O ( n ) CSE 5311 Saravanan Thirumuruganathan

  13. Finding the Maximum Maximum(A): max = A[1] for i = 2 to A.length if max < A[i] max = A[i] return max Analysis: Complexity Measure: Number of Comparisons Number of Comparisons: n − 1 Time Complexity: O ( n ) CSE 5311 Saravanan Thirumuruganathan

  14. Recursive Maximum Idea: Use Divide and Conquer to find Maximum CSE 5311 Saravanan Thirumuruganathan

  15. Recursive Maximum Idea: Use Divide and Conquer to find Maximum Analysis: CSE 5311 Saravanan Thirumuruganathan

  16. Recursive Maximum Idea: Use Divide and Conquer to find Maximum Analysis: Recurrence Relation: T ( n ) = 2 T ( n 2 ) + 1 = O ( n ) Number of Comparisons: n − 1 (Intuition) CSE 5311 Saravanan Thirumuruganathan

  17. Simultaneous Maximum and Minimum Aim: Find the maximum and minimum of array A CSE 5311 Saravanan Thirumuruganathan

  18. Simultaneous Maximum and Minimum Aim: Find the maximum and minimum of array A Minimum-Maximum(A): min = Minimum(A) max = Maximum(A) return min, max Analysis: CSE 5311 Saravanan Thirumuruganathan

  19. Simultaneous Maximum and Minimum Aim: Find the maximum and minimum of array A Minimum-Maximum(A): min = Minimum(A) max = Maximum(A) return min, max Analysis: Number of Comparisons: ( n − 1) + ( n − 1) = 2 n − 2 CSE 5311 Saravanan Thirumuruganathan

  20. Simultaneous Maximum and Minimum Aim: Find the maximum and minimum of array A Minimum-Maximum(A): min = Minimum(A) max = Maximum(A) return min, max Analysis: Number of Comparisons: ( n − 1) + ( n − 1) = 2 n − 2 Slightly better: ( n − 1) + ( n − 2) = 2 n − 3 (for e.g., by swapping min with first element of array) CSE 5311 Saravanan Thirumuruganathan

  21. Simultaneous Maximum and Minimum - Visualization CSE 5311 Saravanan Thirumuruganathan

  22. Simultaneous Maximum and Minimum - Better Algorithm CSE 5311 Saravanan Thirumuruganathan

  23. Simultaneous Maximum and Minimum Analysis: CSE 5311 Saravanan Thirumuruganathan

  24. Simultaneous Maximum and Minimum Analysis: Number of Comparisons (approximate): Pairwise + Min of Mins + Max of Maxs ( n 2) + ( n 2) + ( n 2) = 3 n 2 CSE 5311 Saravanan Thirumuruganathan

  25. Finding Second Largest Element - Naive Method CSE 5311 Saravanan Thirumuruganathan

  26. Finding Second Largest Element - Naive Method Find-Second-Largest(A): max = Maximum(A) Swap A[n] with max secondMax = Maximum(A[1:n-1]) return secondMax Analysis: CSE 5311 Saravanan Thirumuruganathan

  27. Finding Second Largest Element - Naive Method Find-Second-Largest(A): max = Maximum(A) Swap A[n] with max secondMax = Maximum(A[1:n-1]) return secondMax Analysis: n − 1: for finding maximum n − 2: for finding 2nd maximum 2 n − 3: total CSE 5311 Saravanan Thirumuruganathan

  28. Finding Second Largest Element - Tournament Method CSE 5311 Saravanan Thirumuruganathan

  29. Finding Second Largest Element - Tournament Method Observation: In a tournament, second best person could have only be defeated by the best person. It is not necessarily the other element in the final “match” Find-Second-Largest(A): max = Recursive-Maximum(A) candidates = list of all elements of A that were directly compared with max secondMax = Maximum(candidates) return secondMax CSE 5311 Saravanan Thirumuruganathan

  30. Finding Second Largest Element - Tournament Method Analysis: CSE 5311 Saravanan Thirumuruganathan

  31. Finding Second Largest Element - Tournament Method Analysis: Number of Comparisons: ( n − 1) + ( ⌈ lg n ⌉ − 1) = n + ⌈ lg n ⌉ − 2 CSE 5311 Saravanan Thirumuruganathan

  32. Selection Problem Input: A set A of n (distinct) numbers and an integer i with 1 ≤ i ≤ n Output: i th smallest element in A Naive Solution? Sort A and pick A [ i ] Time Complexity: O ( n log n ) Surprising Result : Can be solved in O ( n ) time! CSE 5311 Saravanan Thirumuruganathan

  33. QuickSelect Divide and Conquer Strategy - Ideas? Called QuickSelect or Randomized-Select Invented by Tony Hoare Works excellent in practice CSE 5311 Saravanan Thirumuruganathan

  34. QuickSelect - Case 1 CSE 5311 Saravanan Thirumuruganathan

  35. QuickSelect - Case 2 CSE 5311 Saravanan Thirumuruganathan

  36. QuickSelect - Case 3 CSE 5311 Saravanan Thirumuruganathan

  37. QuickSelect PseudoCode Randomized-Select(A, p, r, i) if p == r: return A[p] q = Randomized-Partition(A, p, r) k = q - p + 1 if i == k return A[q] elseif i < k return Randomized-Select(A, p, q-1, i) else return Randomized-Select(A, q+1, r, i-k) CSE 5311 Saravanan Thirumuruganathan

  38. QuickSelect - Intuition CSE 5311 Saravanan Thirumuruganathan

  39. QuickSelect - Analysis Recurrence Relation: CSE 5311 Saravanan Thirumuruganathan

  40. QuickSelect - Analysis Recurrence Relation: T ( n ) = T ( | L | ) + n or T ( n ) = T ( | R | ) + n Best Case: CSE 5311 Saravanan Thirumuruganathan

  41. QuickSelect - Analysis Recurrence Relation: T ( n ) = T ( | L | ) + n or T ( n ) = T ( | R | ) + n Best Case: T ( n ) = T ( n 2 ) + n ⇒ T ( n ) = O ( n ) Worst Case: CSE 5311 Saravanan Thirumuruganathan

  42. QuickSelect - Analysis Recurrence Relation: T ( n ) = T ( | L | ) + n or T ( n ) = T ( | R | ) + n Best Case: T ( n ) = T ( n 2 ) + n ⇒ T ( n ) = O ( n ) Worst Case: T ( n ) = T ( n − 1) + n ⇒ T ( n ) = O ( n 2 ) Worst than sorting ! Lucky Case: (assume a 1:9 split) CSE 5311 Saravanan Thirumuruganathan

  43. QuickSelect - Analysis Recurrence Relation: T ( n ) = T ( | L | ) + n or T ( n ) = T ( | R | ) + n Best Case: T ( n ) = T ( n 2 ) + n ⇒ T ( n ) = O ( n ) Worst Case: T ( n ) = T ( n − 1) + n ⇒ T ( n ) = O ( n 2 ) Worst than sorting ! Lucky Case: (assume a 1:9 split) T ( n ) = T ( 9 n 10 ) + n ⇒ T ( n ) = O ( n ) CSE 5311 Saravanan Thirumuruganathan

  44. QuickSelect and QuickSort Similarities: CSE 5311 Saravanan Thirumuruganathan

  45. QuickSelect and QuickSort Similarities: Both invented by Tony Hoare Both use D&C and randomization Best and Average case behavior is good but has bad worst case behavior (same: O ( n 2 )) Works very well in practice Differences: CSE 5311 Saravanan Thirumuruganathan

  46. QuickSelect and QuickSort Similarities: Both invented by Tony Hoare Both use D&C and randomization Best and Average case behavior is good but has bad worst case behavior (same: O ( n 2 )) Works very well in practice Differences: QuickSelect iterates on one partition only while QuickSort on both Objective: Sorting vs Selection CSE 5311 Saravanan Thirumuruganathan

  47. Median of Median Algorithm QuickSelect works well in Practice - linear expected time Worst case is worse than sorting O ( n 2 ) Can we solve Selection problem in worst case Linear time? CSE 5311 Saravanan Thirumuruganathan

  48. Median of Median Algorithm QuickSelect works well in Practice - linear expected time Worst case is worse than sorting O ( n 2 ) Can we solve Selection problem in worst case Linear time? Yes! Designed by Blum, Floyd, Pratt, Rivest & Tarjan in 1973 Basic Idea: Identify a good pivot so that partition is “balanced” Aka “Median of Median” or “Worst case Linear time Order Statistics” CSE 5311 Saravanan Thirumuruganathan

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