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Searching Methods in Data Structures

Explore the concepts of linear and binary search methods in data structures, along with their differences and applications. Learn about sorting algorithms like Bubble Sort, Selection Sort, Insertion Sort, and Quick Sort with examples and implementations.

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Searching Methods in Data Structures

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  1. Data Structure Chapter 3

  2. Searching Searching is a process of finding out a particular element in a list of ‘n’ elements Developed By :- Misha Ann Alexander Data Structures

  3. Methods of Searching • Linear Search (Sequential Search) • Binary Search Developed By :- Misha Ann Alexander Data Structures

  4. Linear Search(Sequential Search) In a linear search, we start at the top of the list, and compare the element there with the key If we have a match, the search terminates and the index number is returned Developed By :- Misha Ann Alexander Data Structures

  5. Linear Search Comparison = 05 Developed By :- Misha Ann Alexander Data Structures

  6. Linear Search No of Comparisons = 11 Developed By :- Misha Ann Alexander Data Structures

  7. Binary Search • A binary search is a divide and conquer algorithm In a binary search, we look for the key in the middle of the list If we get a match, the search is over If the key is greater than the thing in the middle of the list, we search the top half If the key is smaller, we search the bottom half Developed By :- Misha Ann Alexander Data Structures

  8. Binary Search Beg = 0 End = 6 Mid = 0+6/2 Search Element = 25 If(key==a[mid] 0 1 2 3 4 5 6 6 13 14 25 33 43 51 Developed By :- Misha Ann Alexander Data Structures

  9. Binary Search Beg = 0 End = 6 Mid = 0+6/2 Search Element = 13 0 1 2 3 4 5 6 6 13 14 25 33 43 51 If 13 < a[mid] then search in the left side. Beg = 0 End = mid – 1 Mid = 0+2/2 Developed By :- Misha Ann Alexander Data Structures

  10. Binary Search Beg = 0 End = 6 Mid = 0+6/2 0 1 2 3 4 5 6 6 13 14 25 33 43 51 If key > a[mid] Beg = mid+1 End = 6 Mid = 4+6/2 Developed By :- Misha Ann Alexander Data Structures

  11. Difference between linear and binary search • Linear Search 1. Data can be in any order. 2. Multidimensional array also can be used. 3.TimeComplexity:- O( n) 4. Not an efficient method to be used if there is a large list. • Binary Search • Data should be in a sorted order. • Only single dimensional array is used. • Time Complexity:- O(log n 2) • Efficient for large inputs also. Developed By :- Misha Ann Alexander Data Structures

  12. Application of Searching Developed By :- Misha Ann Alexander Data Structures

  13. Sorting (Bubble Sort) 5 4 4 4 4 4 5 3 3 3 3 3 5 2 2 2 2 2 5 1 1 5 1 1 1 1 2 3 4 5 Developed By :- Misha Ann Alexander Data Structures

  14. Selection Sort Developed By :- Misha Ann Alexander Data Structures

  15. 77 11 33 2 6 1 99 77 11 33 2 6 1 99 1 11 33 2 6 77 99 1 2 33 11 6 77 99 Developed By :- Misha Ann Alexander Data Structures 77 , 11 , 33 , 2 , 6 , 1 , 99

  16. 1 2 6 11 33 77 99 1 2 6 11 33 77 99 1 2 6 11 33 77 99 Developed By :- Misha Ann Alexander Data Structures

  17. Insertion Sort 1. Insertion sort keeps making the left side of the array sorted until the whole array is sorted. It sorts the values seen far away and repeatedly inserts unseen values in the array into the left sorted array. 2. It is the simplest of all sorting algorithms. Although it has the same complexity as Bubble Sort, the insertion sort is a little over twice as efficient as the bubble sort. Developed By :- Misha Ann Alexander Data Structures

  18. Insertion Sort Developed By :- Misha Ann Alexander Data Structures

  19. 77 11 33 2 6 1 99 -99 77 -99 11 77 -99 11 33 77 -99 2 11 33 77 -99 2 6 11 33 77 -99 1 2 6 11 33 77 -99 1 2 6 11 33 77 99 Developed By :- Misha Ann Alexander Data Structures

  20. Quick Sort Given an array of n elements (e.g., integers): • If array only contains one element, return • Else • pick one element to use as pivot. • Partition elements into two sub-arrays: • Elements less than or equal to pivot • Elements greater than pivot • Quicksort two sub-arrays • Return results Developed By :- Misha Ann Alexander Data Structures

  21. Example of Quick Sort 40 20 10 80 60 50 7 30 100 • Choose an element as the pivot element. • Given a pivot, partition the elements of the array such • that the resulting array consists of: • 1. One sub-array that contains elements >= pivot • 2. Another sub-array that contains elements < pivot Developed By :- Misha Ann Alexander Data Structures

  22. Example 40 20 10 80 60 50 7 30 100 30 20 10 80 60 50 7 40 100 30 20 10 40 60 50 7 80 100 30 20 10 7 60 50 40 80 100 Developed By :- Misha Ann Alexander Data Structures

  23. 30 20 10 7 40 50 60 80 100 30 20 10 7 40 50 60 80 100 Subpart 1 Subpart 2 30 20 10 7 50 60 80 100 7 20 10 30 Subpart 4 Subpart 3 Developed By :- Misha Ann Alexander Data Structures

  24. 60 80 100 7 20 10 Subpart 5 20 10 10 20 Sorted Elements 7 10 20 30 40 50 60 80 100 Developed By :- Misha Ann Alexander Data Structures

  25. Algorithm of Quick Sort • //Sort an array A of n elements. • Create global integer array A[1:n+1]; • //The elements to be sorted are in A[1]…….A[n] • //Step 1: Put a very large element in A[n+1]; • //Find maximum of the n elements. • //Store a large element at the end of the sequence in A. • void max(1,n) • { • int max; • max = A[1]; • for (int I=2; I<=n; I++} Developed By :- Misha Ann Alexander Data Structures

  26. if (max < A[I]) max = A[I]; • A[n+1] = ++max; • return(0); • } • //The Quicksort algorithm • void Quicksort(int left, int right) • { • if (left >= right) return(0); • int pivot = Partition(left, right + 1); • Quicksort(left, pivot – 1); • Quicksort(pivot + 1,right); • return(0); • } Developed By :- Misha Ann Alexander Data Structures

  27. //Partititon returns position of the pivot element • //The pivot element is assumes to be the leftmost one • int Partititon(int left, int right) • { • int pivot_element = A[left]; • int left_to_right = ++left; • int right_to_left = --right; • while(A[left_to_right] < pivot_element) ++left_to_right; • while(A[right_to_left] > pivot_element) ++right_to_left; • //posititon for pivot element is at right_to_left • A[left] = A[right_to_left], A[right_to_left] = pivot_element; • return (right_to_left); • } Developed By :- Misha Ann Alexander Data Structures

  28. Merge Sort • Merge-sort is a sorting algorithm based on the divide-and-conquer paradigm . • Divide: divide the input data S in two disjoint subsets S1and S2 • Recur: solve the subproblems associated with S1and S2 • Conquer: combine the solutions for S1and S2 into a solution for S Developed By :- Misha Ann Alexander Data Structures

  29. 40 20 10 80 60 50 7 30 20 40 10 80 50 60 7 30 10 20 40 80 7 30 50 60 7 10 20 30 40 50 60 80 Developed By :- Misha Ann Alexander Data Structures

  30. Algorithm for Merge Sort • MERGE (A, R, LBA, S, LBB, C, LBC) • Set NA:= LBA, NB := LBB, PTR := LBC, UBA:= LBA+R-1, UBB := LBB+S-1. • Return. • MERGEPASS (A, N, L, B) • Set Q: = INT(N/(2*L)), S := 2*L*Q and R := N-S. • Repeat for J= 1, 2, ....,Q:a) Set LB: = 1+(2*J-2)*L. • b) Call MERGE(A, L, LB, A, L, LB+L, B, LB) • [End of loop] Developed By :- Misha Ann Alexander Data Structures

  31. 3. If R<=L, then:Repeat for J=1, 2... R: • Set B(S+J) := A(S+J) • [End of loop] • Else: • Call MERGE(A, L, S+1, A, R, L+S+1, B, S+1) • Return Developed By :- Misha Ann Alexander Data Structures

  32. MERGESORT (A, N) • Set L := 1. • Repeat Steps 3 to 6 while L<N: • Call MERGEPASS(A, N, L, B) • Call MERGEPASS(B, N, 2*L, A) • Set L := 4*L[End of step 2 loop] • Exit Developed By :- Misha Ann Alexander Data Structures

  33. Radix Sort(Pass 1) Developed By :- Misha Ann Alexander Data Structures

  34. Pass 2 Developed By :- Misha Ann Alexander Data Structures

  35. Pass 3 Developed By :- Misha Ann Alexander Data Structures

  36. Radix Sort • Sorted Elements • 12 , 45 , 121 Developed By :- Misha Ann Alexander Data Structures

  37. Shell Sort • Founded by Donald Shell and named the sorting algorithm after himself in 1959. • 1st algorithm to break the quadratic time barrier but few years later, a sub quadratic time bound was proven • Shellsort works by comparing elements that are distant rather than adjacent elements in an array or list where adjacent elements are compared. Developed By :- Misha Ann Alexander Data Structures

  38. Shell Sort • Shellsort makes multiple passes through a list and sorts a number of equally sized sets using the insertion sort. Developed By :- Misha Ann Alexander Data Structures

  39. Shell Sort • Sort: 18 32 12 5 38 33 16 2 • D1 = n/2 (8/2 = 4) • 18 16 12 5 38 33 32 2 Developed By :- Misha Ann Alexander Data Structures

  40. 2 32 12 5 38 33 16 18 D2 = d1/2 (4/2 = 2) Developed By :- Misha Ann Alexander Data Structures

  41. Shell Sort Advantage • Advantage of Shellsort is that its only efficient for medium size lists. For bigger lists, the algorithm is not the best choice. Fastest of all O(N^2) sorting algorithms. • 5 times faster than the bubble sort and a little over twice as fast as the insertion sort, its closest competitor. Developed By :- Misha Ann Alexander Data Structures

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