longest increasing subsequence in O(n) code example
Example 1: longest common subsequence
int maxSubsequenceSubstring(char x[], char y[],
int n, int m)
{
int dp[MAX][MAX];
// Initialize the dp[][] to 0.
for (int i = 0; i <= m; i++)
for (int j = 0; j <= n; j++)
dp[i][j] = 0;
// Calculating value for each element.
for (int i = 1; i <= m; i++) {
for (int j = 1; j <= n; j++) {
// If alphabet of string X and Y are
// equal make dp[i][j] = 1 + dp[i-1][j-1]
if (x[j - 1] == y[i - 1])
dp[i][j] = 1 + dp[i - 1][j - 1];
// Else copy the previous value in the
// row i.e dp[i-1][j-1]
else
dp[i][j] = dp[i][j - 1];
}
}
// Finding the maximum length.
int ans = 0;
for (int i = 1; i <= m; i++)
ans = max(ans, dp[i][n]);
return ans;
}
Example 2: longest increasing subsequence when elements hae duplicates
using namespace std;
// define maximum possible length of X and Y
// lookup[i][j] stores the length of LCS of subarray X[0..i-1], Y[0..j-1]
int lookup[N][N];
// Function to find LCS of array X[0..m-1] and Y[0..n-1]
void LCS(int X[], int Y[], int m, int n)
{
// return if we have reached the end of either array
if (m == 0 || n == 0)
return;
// if last element of X and Y matches
if (X[m - 1] == Y[n - 1])
{
LCS(X, Y, m - 1, n - 1);
cout << X[m - 1] << " ";
return;
}
// else when the last element of X and Y are different
if (lookup[m - 1][n] > lookup[m][n - 1])
LCS(X, Y, m - 1, n);
else
LCS(X, Y, m, n - 1);
}
// Function to find length of Longest Common Subsequence of
// array X[0..m-1] and Y[0..n-1]
void findLCS(int X[], int Y[], int m, int n)
{
// first row and first column of the lookup table
// are already 0 as lookup[][] is globally declared
// fill the lookup table in bottom-up manner
for (int i = 1; i <= m; i++)
{
for (int j = 1; j <= n; j++)
{
// if current element of X and Y matches
if (X[i - 1] == Y[j - 1])
lookup[i][j] = lookup[i - 1][j - 1] + 1;
// else if current element of X and Y don't match
else
lookup[i][j] = max(lookup[i - 1][j], lookup[i][j - 1]);
}
}
// find longest common sequence
LCS(X, Y, m, n);
}
// Function to remove duplicates from a sorted array
int removeDuplicates(int X[], int n)
{
int k = 0;
for (int i = 1; i < n; i++)
if (X[i] != X[k])
X[++k] = X[i];
// return length of sub-array containing all distinct characters
return k + 1;
}
// Iterative function to find length of longest increasing subsequence (LIS)
// of given array using longest common subsequence (LCS)
void findLIS(int X[], int n)
{
// create a copy of the original array
int Y[n];
for (int i = 0; i < n; i++)
Y[i] = X[i];
// sort the copy of the original array
sort(Y, Y + n);
// remove all the duplicates from Y
int m = removeDuplicates (Y, n);
// perform LCS of both
findLCS(X, Y, n, m);
}
// Longest Increasing Subsequence using LCS
int main()
{
int X[] = { 100,1,100,1,100 };
int n = sizeof(X)/sizeof(X[0]);
cout << "The LIS is ";
findLIS(X, n);
return 0;
}