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Searching

Searching. CS 105 See Section 14.6 of Horstmann text. The Search Problem. Given an array A storing n numbers, and a target number v, locate the position in A (if it exists) where A[i] = v Example Input: A = {3,8,2,15,99,52}, v = 99 Output: position 4 Notes

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Searching

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  1. Searching CS 105 See Section 14.6 of Horstmann text

  2. The Search Problem • Given an array A storing n numbers, and a target number v, locate the position in A (if it exists) where A[i] = v • Example • Input: A = {3,8,2,15,99,52}, v = 99 • Output: position 4 • Notes • Array positions (indexes) go from 0..n-1 • When the target does not exist in the array, return an undefined position, such as -1

  3. Algorithm 1: Linear Search Algorithm LinearSearch(A,v): Input: An array A of n numbers, search target v Output: position i where A[i]=v, -1 if not found for i  0 to n - 1 do if A[i] = v then return i return -1

  4. Linear Search time complexity • Worst-case: O( n ) • If v does not exist or if v is the last element in the array • Best-case: O( 1 ) • When the target element v is the first element in the array • Average-case: O( n ) • e.g., if the target element is somewhere in the middle of the array, the for-loop will iterate n/2 times. Still O( n )

  5. Searching in sorted arrays • Suppose the array is arranged in increasing or non-decreasing order • Linear search on a sorted array still yields the same analysis • O( n ) worst-case time complexity • Can exploit sorted structure by performing binary search • Strategy: inspect middle of the search list so that half of the list is discarded at every step

  6. Algorithm 2: Binary Search Algorithm BinarySearch(A,v): Input: A SORTED array A of n numbers, search target v Output: position i where A[i]=v, -1 if not found low  0, high  n-1 while low <= high do mid  (low+high)/2 if A[mid]= v then return mid else if A[mid] < v then low  mid+1 else high  mid-1 return -1

  7. Binary Search time complexity • Time complexity analysis: how many iterations of the while loop?(assume worst-case) • Observe values for low and high • Note that before loop entry,size of search space = n = high-low+1. • On each suceeding iteration, search space is cut in half • Loop terminates when search space collapses to 0 or 1

  8. Binary Search time complexity • search space size iteration numbern 1n/2 2n/4 3… …1 x • x = number of iterations • Observe 2x = n, log 2x = log n,x = log n iterations carried out • Binary Search worst-case time complexity:O( log n )

  9. Binary Search time complexity • Worst-case: O( log n ) • If v does not exist • Best-case: O( 1 ) • When the target element v happens to be in the middle of the array • Average-case: O( log n ) • Technically, some statistical analysis needed here (beyond the scope of this course)

  10. Binary Search (version 2) • Can use recursion instead of iteration • BinSearch(A,v): return BinSearchHelper(A,0,n-1,v) • BinSearchHelper(A,low,high,v): if low > high then return -1 mid  (low+high)/2 if A[mid] = v then return mid else if A[mid] < v then return BinSearchHelper(A,mid+1,high,v) else return BinSearchHelper(A,low,mid-1,v)

  11. Summary • Linear search on an array takes O( n ) time in the worst-case • If the array is sorted, can perform binary search in O( log n ) time • Iterative and recursive versions • Both algorithms run in O( 1 ) time in the best-case • Case where the first element inspected is the target element

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