COMP 171 Data Structures and Algorithms

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COMP 171Data Structures and Algorithms

• Tutorial 4
• In-Class Exercises Algorithm Design

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Exercises 2.3-6 (p.37)

• Observe that the while loop of the Insertion-sort
  procedure uses a linear search to scan (backward)
  through the sorted subarray A0..i-1. Can we use
  a binary search instead to improve the overall
  worst-case running time of insertion sort to T
  (n?n)?

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• InsertionSort(A)
• for i ? 1 to size(A)-1
• key ? Ai
• j ? i 1
• while j ? 0 and Aj gt key
• Aj 1 ? Aj
• j ? j 1
• end while
• Aj 1 ? key
• end for i
• end InsertionSort

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• BinarySearch(A, left, right, s_value)
• if left ? right then
• middle ? ?(left right) / 2?
• if Amiddle s_value then
• return middle
• else if Amiddle gt s_value then
• BinarySearch(A, left, middle-1, s_value)
• else
• BinarySearch(A, middle1, right, s_value)
• end if
• else
• return 1
• end if
• end BinarySearch

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Exercises 2.3-7 (p.37)

• Describe a T(n?n)-time algorithm that, given a
  set of S of n integers and another integer x,
  determines whether or not there exist two
  elements in S whose sum is exactly x.

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Exercises 2-4 (p.39)

• Let A1..n be an array of n distinct numbers. If
  i lt j and Ai gt Aj, then the pair (i, j) is
  called an inversion of A.
• List the five inversions of the array 2, 3, 8,
  6, 1
• What array with elements from the set 1, 2, ,
  n has the most inversions? How many does it have?

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• What is the relationship between the running time
  of insertion sort and the number of inversions in
  the input array? Justify your answer.
• Given an algorithm that determines the number of
  inversions in any permutation on n elements in
  T(n?n) worst-case time. (Hint Modify merge sort)

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• mergesort(A, left, right)
• if left lt right then
• middle ? ?(left right) / 2?
• mergesort(A, left, middle)
• mergesort(A, middle1, right)
• merge(A, left, middle1, right)
• end if
• end mergesort

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• merge(A, p, q, r)
• n1 ? q p
• n2 ? r q 1
• create array L0..n1, R0..n2
• for i ? 0 to n1-1 do Li ? Api
• for j ? 0 to n2-1 do Rj ? Aqj
• Ln1 ? Rn2 ? 8
• I ? j ? 0
• for k ? p to r
• if LI lt Rj then
• Ak ? Li
• i ? i 1
• else
• Ak ? Rj
• j ? j 1
• end if
• end for k
• end merge