Longest Bitonic Subsequence

A sequence is said to be bitonic if it is first increasing and then decreasing. In this problem, an array of all positive integers is given. We have to find a subsequence which is increasing first and then decreasing.

To solve this problem, we will define two subsequences, they are the Longest Increasing Subsequence and the Longest Decreasing Subsequence. The LIS array will hold the length of increasing subsequence ending with array[i]. The LDS array will store the length of decreasing subsequence starting from array[i]. Using these two arrays, we can get the length of longest bitonic subsequence.

Input and Output

Input:
A sequence of numbers. {0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15}
Output:
The longest bitonic subsequence length. Here it is 7.

Algorithm

longBitonicSub(array, size)

Input: The array, the size of an array.

Output − Max length of longest bitonic subsequence.

Begin
   define incSubSeq of size same as the array size
   initially fill all entries to 1 for incSubSeq

   for i := 1 to size -1, do
      for j := 0 to i-1, do
         if array[i] > array[j] and incSubSeq[i] < incSubSum[j] + 1, then incSubSum[i] := incSubSum[j] + 1
      done
   done

   define decSubSeq of size same as the array size.
   initially fill all entries to 1 for incSubSeq

   for i := size - 2 down to 0, do
      for j := size - 1 down to i+1, do
         if array[i] > array[j] and decSubSeq[i] < decSubSum[j] + 1, then decSubSeq [i] := decSubSeq [j] + 1
      done
   done

   max := incSubSeq[0] + decSubSeq[0] – 1
   for i := 1 to size, do
      if incSubSeq[i] + decSubSeq[i] – 1 > max, then max := incSubSeq[i] + decSubSeq[i] – 1
   done

   return max
End

Example

#include<iostream>
using namespace std;

int longBitonicSub( int arr[], int size ) {
   int *increasingSubSeq = new int[size];          //create increasing sub sequence array
   for (int i = 0; i < size; i++)
      increasingSubSeq[i] = 1;              //set all values to 1

   for (int i = 1; i < size; i++)           //compute values from left ot right
      for (int j = 0; j < i; j++)
         if (arr[i] > arr[j] && increasingSubSeq[i] < increasingSubSeq[j] + 1)
            increasingSubSeq[i] = increasingSubSeq[j] + 1;

   int *decreasingSubSeq = new int [size];       //create decreasing sub sequence array
   for (int i = 0; i < size; i++)
      decreasingSubSeq[i] = 1;              //set all values to 1

   for (int i = size-2; i >= 0; i--)          //compute values from left ot right
      for (int j = size-1; j > i; j--)
         if (arr[i] > arr[j] && decreasingSubSeq[i] < decreasingSubSeq[j] + 1)
            decreasingSubSeq[i] = decreasingSubSeq[j] + 1;

   int max = increasingSubSeq[0] + decreasingSubSeq[0] - 1;
   for (int i = 1; i < size; i++) //find max length
      if (increasingSubSeq[i] + decreasingSubSeq[i] - 1 > max)
         max = increasingSubSeq[i] + decreasingSubSeq[i] - 1;
   return max;
}

int main() {
   int arr[] = {0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15};
   int n = 16;
   cout << "Length of longest bitonic subsequence is " << longBitonicSub(arr, n);
}

Output

Length of longest bitonic subsequence is 7