Bitonic Sequence: A sequence is called Bitonic if it is first increasing, then decreasing. In other words, an array arr[0..n-i] is Bitonic if there exists an index i where 0<=i<=n-1 such that
x0 <= x1 …..<= xi and xi >= xi+1….. >= xn-1
- A sequence, sorted in increasing order is considered Bitonic with the decreasing part as empty. Similarly, decreasing order sequence is considered Bitonic with the increasing part as empty.
- A rotation of Bitonic Sequence is also bitonic.
Bitonic Sorting: It mainly involves two steps.
- Forming a bitonic sequence (discussed above in detail). After this step we reach the fourth stage in below diagram, i.e., the array becomes {3, 4, 7, 8, 6, 5, 2, 1}
- Creating one sorted sequence from bitonic sequence: After first step, first half is sorted in increasing order and second half in decreasing order.
We compare first element of first half with first element of second half, then second element of first half with second element of second, and so on. We exchange elements if an element of first half is smaller.
After above compare and exchange steps, we get two bitonic sequences in array. See fifth stage in below diagram. In the fifth stage, we have {3, 4, 2, 1, 6, 5, 7, 8}. If we take a closer look at the elements, we can notice that there are two bitonic sequences of length n/2 such that all elements in first bitonic sequence {3, 4, 2, 1} are smaller than all elements of second bitonic sequence {6, 5, 7, 8}.
We repeat the same process within two bitonic sequences and we get four bitonic sequences of length n/4 such that all elements of leftmost bitonic sequence are smaller and all elements of rightmost. See sixth stage in below diagram, arrays is {2, 1, 3, 4, 6, 5, 7, 8}.
If we repeat this process one more time we get 8 bitonic sequences of size n/8 which is 1. Since all these bitonic sequence are sorted and every bitonic sequence has one element, we get the sorted array.
Example
Java
/* Java program for Bitonic Sort. Note that this program works only when size of input is a power of 2. */public class BitonicSort { /* The parameter dir indicates the sorting direction, ASCENDING or DESCENDING; if (a[i] > a[j]) agrees with the direction, then a[i] and a[j] are interchanged. */ void compAndSwap(int a[], int i, int j, int dir) { if ((a[i] > a[j] && dir == 1) || (a[i] < a[j] && dir == 0)) { // Swapping elements int temp = a[i]; a[i] = a[j]; a[j] = temp; } } /* It recursively sorts a bitonic sequence in ascending order, if dir = 1, and in descending order otherwise (means dir=0). The sequence to be sorted starts at index position low, the parameter cnt is the number of elements to be sorted.*/ void bitonicMerge(int a[], int low, int cnt, int dir) { if (cnt > 1) { int k = cnt / 2; for (int i = low; i < low + k; i++) compAndSwap(a, i, i + k, dir); bitonicMerge(a, low, k, dir); bitonicMerge(a, low + k, k, dir); } } /* This function first produces a bitonic sequence by recursively sorting its two halves in opposite sorting orders, and then calls bitonicMerge to make them in the same order */ void bitonicSort(int a[], int low, int cnt, int dir) { if (cnt > 1) { int k = cnt / 2; // sort in ascending order since dir here is 1 bitonicSort(a, low, k, 1); // sort in descending order since dir here is 0 bitonicSort(a, low + k, k, 0); // Will merge whole sequence in ascending order // since dir=1. bitonicMerge(a, low, cnt, dir); } } /*Caller of bitonicSort for sorting the entire array of length N in ASCENDING order */ void sort(int a[], int N, int up) { bitonicSort(a, 0, N, up); } /* A utility function to print array of size n */ static void printArray(int arr[]) { int n = arr.length; for (int i = 0; i < n; ++i) System.out.print(arr[i] + " "); System.out.println(); } // Driver method public static void main(String args[]) { int a[] = { 3, 7, 4, 8, 6, 2, 1, 5 }; int up = 1; BitonicSort ob = new BitonicSort(); ob.sort(a, a.length, up); System.out.println("\nSorted array"); printArray(a); }} |
Output:
Sorted array 1 2 3 4 5 6 7 8
Time Complexity: O(n*log2(n))
Auxiliary Space: O(1)
Please refer complete article on Bitonic Sort for more details!
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