Longest Increasing Subsequence:

We have discussed Overlapping Subproblems and Optimal Substructure properties respectively.

Let us discuss Longest Increasing Subsequence (LIS) problem as an example problem that can be solved using Dynamic Programming.
The Longest Increasing Subsequence (LIS) problem is to find the length of the longest subsequence of a given sequence such that all elements of the subsequence are sorted in increasing order. For example, the length of LIS for {10, 22, 9, 33, 21, 50, 41, 60, 80} is 6 and LIS is {10, 22, 33, 50, 60, 80}

Longest-Increasing-Subsequence

More Examples:

Input  : arr[] = {3, 10, 2, 1, 20}
Output : Length of LIS = 3
The longest increasing subsequence is 3, 10, 20

Input  : arr[] = {3, 2}
Output : Length of LIS = 1
The longest increasing subsequences are {3} and {2}

Input : arr[] = {50, 3, 10, 7, 40, 80}
Output : Length of LIS = 4
The longest increasing subsequence is {3, 7, 40, 80}
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Optimal Substructure:

Let arr[0..n-1] be the input array and L(i) be the length of the LIS ending at index i such that arr[i] is the last element of the LIS.
Then, L(i) can be recursively written as:
L(i) = 1 + max( L(j) ) where 0 < j < i and arr[j] < arr[i]; or
L(i) = 1, if no such j exists.
To find the LIS for a given array, we need to return max(L(i)) where 0 < i < n.
Thus, we see the LIS problem satisfies the optimal substructure property as the main problem can be solved using solutions to subproblems.

Following is a simple recursive implementation of the LIS problem. It follows the recursive structure discussed above.

Python Programming

Python 
# A naive Python based recursive implementation of LIS problem
 
global max_lis_length # stores the final LIS
 
# Recursive implementation for calculating the LIS
def _lis(arr, n):
    # Following declaration is needed to allow modification
    # of the global copy of max_lis_length in _lis()
    global max_lis_length
 
    # Base Case
    if n == 1:
        return 1
 
    current_lis_length = 1
 
    for i in xrange(0, n-1):
        # Recursively calculate the length of the LIS
        # ending at arr[i]
        subproblem_lis_length = _lis(arr, i)
 
        # Check if appending arr[n-1] to the LIS
        # ending at arr[i] gives us an LIS ending at
        # arr[n-1] which is longer than the previously
        # calculated LIS ending at arr[n-1]
        if arr[i] < arr[n-1] and \
            current_lis_length < (1+subproblem_lis_length):
            current_lis_length = (1+subproblem_lis_length)
 
    # Check if currently calculated LIS ending at
    # arr[n-1] is longer than the previously calculated
    # LIS and update max_lis_length accordingly
    if (max_lis_length < current_lis_length):
        max_lis_length = current_lis_length
 
    return current_lis_length
 
# The wrapper function for _lis()
def lis(arr, n):
 
    # Following declaration is needed to allow modification
    # of the global copy of max_lis_length in lis()
    global max_lis_length
 
    max_lis_length = 1 # stores the final LIS
 
    # max_lis_length is declared global at the top
    # so that it can maintain its value
    # between the recursive calls of _lis()
    _lis(arr , n)
 
    return max_lis_length
 
# Driver program to test the functions above
def main():
    arr = [10, 22, 9, 33, 21, 50, 41, 60]
    n = len(arr)
    print "Length of LIS is", lis(arr, n)
 
if __name__=="__main__":
    main()

Output :

Length of LIS is 5

Overlapping Subproblems :

Considering the above implementation, following is recursion tree for an array of size 4. lis(n) gives us the length of LIS for arr[].

              lis(4)
        /        |     \
      lis(3)    lis(2)   lis(1)
     /   \        /
   lis(2) lis(1) lis(1)
   /
lis(1)

We can see that there are many subproblems which are solved again and again. So this problem has Overlapping Substructure property and recomputation of same subproblems can be avoided by either using Memorization or Tabulation. Following is a tabluated implementation for the LIS problem.

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Python Programming

Python 
# Dynamic programming Python implementation of LIS problem
 
# lis returns length of the longest increasing subsequence
# in arr of size n
def lis(arr):
    n = len(arr)
 
    # Declare the list (array) for LIS and initialize LIS
    # values for all indexes
    lis = [1]*n
 
    # Compute optimized LIS values in bottom up manner
    for i in range (1 , n):
        for j in range(0 , i):
            if arr[i] > arr[j] and lis[i]< lis[j] + 1 :
                lis[i] = lis[j]+1
 
    # Initialize maximum to 0 to get the maximum of all
    # LIS
    maximum = 0
 
    # Pick maximum of all LIS values
    for i in range(n):
        maximum = max(maximum , lis[i])
 
    return maximum
# end of lis function
 
# Driver program to test above function
arr = [10, 22, 9, 33, 21, 50, 41, 60]
print "Length of lis is", lis(arr)

Output :

Length of lis is 5

Note that the time complexity of the above Dynamic Programming (DP) solution is O(n^2) and there is a O(nLogn) solution for the LIS problem.

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