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Algorithm

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  1. Getting Started with Algorithm
    What is an Algorithm?
  2. Characteristics of Algorithm
    1 Topic
  3. Analysis Framework
  4. Performance Analysis
    3 Topics
  5. Mathematical Analysis
    2 Topics
  6. Sorting Algorithm
    Sorting Algorithm
    10 Topics
  7. Searching Algorithm
    6 Topics
  8. Fundamental of Data Structures
    Stacks
  9. Queues
  10. Graphs
  11. Trees
  12. Sets
  13. Dictionaries
  14. Divide and Conquer
    General Method
  15. Binary Search
  16. Recurrence Equation for Divide and Conquer
  17. Finding the Maximum and Minimum
  18. Merge Sort
  19. Quick Sort
  20. Stassen’s Matrix Multiplication
  21. Advantages and Disadvantages of Divide and Conquer
  22. Decrease and Conquer
    Insertion Sort
  23. Topological Sort
  24. Greedy Method
    General Method
  25. Coin Change Problem
  26. Knapsack Problem
  27. Job Sequencing with Deadlines
  28. Minimum Cost Spanning Trees
    2 Topics
  29. Single Source Shortest Paths
    1 Topic
  30. Optimal Tree Problem
    1 Topic
  31. Transform and Conquer Approach
    1 Topic
  32. Dynamic Programming
    General Method with Examples
  33. Multistage Graphs
  34. Transitive Closure
    1 Topic
  35. All Pairs Shortest Paths
    6 Topics
  36. Backtracking
    General Method
  37. N-Queens Problem
  38. Sum of Subsets problem
  39. Graph Coloring
  40. Hamiltonian Cycles
  41. Branch and Bound
    2 Topics
  42. 0/1 Knapsack problem
    2 Topics
  43. NP-Complete and NP-Hard Problems
    1 Topic
Lesson 7, Topic 5
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Ternary Search

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Like the binary search, it also separates the lists into sub-lists. This procedure divides the list into three parts using two intermediate mid values. As the lists are divided into more subdivisions, so it reduces the time to search a key value.

We can divide the array into three parts by taking mid1 and mid2 which can be calculated as shown below. Initially, l and r will be equal to 0 and n-1 respectively, where n is the length of the array.

mid1 = l + (r-l)/3
mid2 = r – (r-l)/3

Note: Array needs to be sorted to perform ternary search on it.

Steps to perform Ternary Search:

  1. First, we compare the key with the element at mid1. If found equal, we return mid1.
  2. If not, then we compare the key with the element at mid2. If found equal, we return mid2.
  3. If not, then we check whether the key is less than the element at mid1. If yes, then recur to the first part.
  4. If not, then we check whether the key is greater than the element at mid2. If yes, then recur to the third part.
  5. If not, then we recur to the second (middle) part.
KodNest Capture15

The complexity of Ternary Search Technique

  1. Time Complexity: O(log3 n)
  2. Space Complexity: O(1)

Input and Output

Input:
A sorted list of data: 12 25 48 52 67 79 88 93
The search key 52
Output:
Item found at location: 3

Algorithm

ternarySearch(array, start, end, key)

Input: An sorted array, start and end location, and the search key

Output: location of the key (if found), otherwise wrong location.

Begin
   if start <= end then
      midFirst := start + (end - start) /3
      midSecond := midFirst + (end - start) / 3
      if array[midFirst] = key then
         return midFirst
      if array[midSecond] = key then
         return midSecond
      if key < array[midFirst] then
         call ternarySearch(array, start, midFirst-1, key)
      if key > array[midSecond] then
         call ternarySearch(array, midFirst+1, end, key)
      else
         call ternarySearch(array, midFirst+1, midSecond-1, key)
   else
      return invalid location
End

Ternary Search Program in C

// C program to illustrate 
// recursive approach to ternary search 
  
#include <stdio.h> 
  
// Function to perform Ternary Search 
int ternarySearch(int l, int r, int key, int ar[]) 
{ 
    if (r >= l) { 
  
        // Find the mid1 and mid2 
        int mid1 = l + (r - l) / 3; 
        int mid2 = r - (r - l) / 3; 
  
        // Check if key is present at any mid 
        if (ar[mid1] == key) { 
            return mid1; 
        } 
        if (ar[mid2] == key) { 
            return mid2; 
        } 
  
        // Since key is not present at mid, 
        // check in which region it is present 
        // then repeat the Search operation 
        // in that region 
  
        if (key < ar[mid1]) { 
  
            // The key lies in between l and mid1 
            return ternarySearch(l, mid1 - 1, key, ar); 
        } 
        else if (key > ar[mid2]) { 
  
            // The key lies in between mid2 and r 
            return ternarySearch(mid2 + 1, r, key, ar); 
        } 
        else { 
  
            // The key lies in between mid1 and mid2 
            return ternarySearch(mid1 + 1, mid2 - 1, key, ar); 
        } 
    } 
  
    // Key not found 
    return -1; 
} 
  
// Driver code 
int main() 
{ 
    int l, r, p, key; 
  
    // Get the array 
    // Sort the array if not sorted 
    int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }; 
  
    // Starting index 
    l = 0; 
  
    // length of array 
    r = 9; 
  
    // Checking for 5 
  
    // Key to be searched in the array 
    key = 5; 
  
    // Search the key using ternarySearch 
    p = ternarySearch(l, r, key, ar); 
  
    // Print the result 
    printf("Index of %d is %d\n", key, p); 
  
    // Checking for 50 
  
    // Key to be searched in the array 
    key = 50; 
  
    // Search the key using ternarySearch 
    p = ternarySearch(l, r, key, ar); 
  
    // Print the result 
    printf("Index of %d is %d", key, p); 
} 

Output :

Index of 5 is 4
Index of 50 is -1

Iterative Approach of Ternary Search in C

// C program to illustrate 
// iterative approach to ternary search 
  
#include <stdio.h> 
  
// Function to perform Ternary Search 
int ternarySearch(int l, int r, int key, int ar[]) 
  
{ 
    while (r >= l) { 
  
        // Find the mid1 and mid2 
        int mid1 = l + (r - l) / 3; 
        int mid2 = r - (r - l) / 3; 
  
        // Check if key is present at any mid 
        if (ar[mid1] == key) { 
            return mid1; 
        } 
        if (ar[mid2] == key) { 
            return mid2; 
        } 
  
        // Since key is not present at mid, 
        // check in which region it is present 
        // then repeat the Search operation 
        // in that region 
  
        if (key < ar[mid1]) { 
  
            // The key lies in between l and mid1 
            r = mid1 - 1; 
        } 
        else if (key > ar[mid2]) { 
  
            // The key lies in between mid2 and r 
            l = mid2 + 1; 
        } 
        else { 
  
            // The key lies in between mid1 and mid2 
            l = mid1 + 1; 
            r = mid2 - 1; 
        } 
    } 
  
    // Key not found 
    return -1; 
} 
  
// Driver code 
int main() 
{ 
    int l, r, p, key; 
  
    // Get the array 
    // Sort the array if not sorted 
    int ar[] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }; 
  
    // Starting index 
    l = 0; 
  
    // length of array 
    r = 9; 
  
    // Checking for 5 
  
    // Key to be searched in the array 
    key = 5; 
  
    // Search the key using ternarySearch 
    p = ternarySearch(l, r, key, ar); 
  
    // Print the result 
    printf("Index of %d is %d\n", key, p); 
  
    // Checking for 50 
  
    // Key to be searched in the array 
    key = 50; 
  
    // Search the key using ternarySearch 
    p = ternarySearch(l, r, key, ar); 
  
    // Print the result 
    printf("Index of %d is %d", key, p); 
} 

Output :

Index of 5 is 4
Index of 50 is -1
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