Simple DP #1

Problem Statement

The Longest Increasing Subsequence problem is to find the longest increasing subsequence of a given sequence. Given a sequence S= {a1 , a2 , a3, a4, ............., an-1, an } we have to find a longest subset such that for all j and i, j<i in the subset aj<ai.

First of all we have to find the value of the longest subsequences(LSi) at every index i with last element of sequence being ai. Then largest LSi would be the longest subsequence in the given sequence. To begin LSi is assigned to be one since ai is element of the sequence(Last element). Then for all j such that j<i and aj<ai ,we find Largest LSj and add it to LSi. Then algorithm take O(n2) time.

Pseudo-code for finding the length of the longest increasing subsequence:

This algorithms complexity could be reduced by using better data structure rather than array. Storing predecessor array and variable like largest_sequences_so_far and its index would save a lot time.

Similar concept could be applied in finding longest path in Directed acyclic graph.

Pseudo code

for i=0 to n-1          

LS[i]=1            

for j=0 to i-1                      

if(a[i] >  a[j] and LS[i]<=LS[j])                      

LS[i] = LS[j]+1 for i=0 to n-1            

if(largest < LS[i])                        

largest = LS[i]

Solution

To Be Uploaded Soon...