Construction of Longest Increasing Subsequence (N log N)

In my previous post, I have explained about longest increasing sub-sequence (LIS) problem in detail. However, the post only covered code related to querying size of LIS, but not the construction of LIS. I left it as an exercise. If you have solved, cheers. If not, you are not alone, here is code.

If you have not read my previous post, read here. Note that the below code prints LIS in reverse order. We can modify print order using a stack (explicit or system stack). I am leaving explanation as an exercise (easy).

// C++ implementation to find longest increasing subsequence
// in O(n Log n) time.
#include <bits/stdc++.h>
using namespace std;

// Binary search
int GetCeilIndex(int arr[], vector<int> &T, int l, int r,
                 int key)
{
    while (r - l > 1)
    {
        int m = l + (r - l)/2;
        if (arr[T[m]] >= key)
            r = m;
        else
            l = m;
    }

    return r;
}

int LongestIncreasingSubsequence(int arr[], int n)
{
    // Add boundary case, when array n is zero
    // Depend on smart pointers

    vector<int> tailIndices(n, 0); // Initialized with 0 
    vector<int> prevIndices(n, -1); // initialized with -1

    int len = 1; // it will always point to empty location
    for (int i = 1; i < n; i++)
    {
        if (arr[i] < arr[tailIndices[0]])
        {
            // new smallest value
            tailIndices[0] = i;
        }
        else if (arr[i] > arr[tailIndices[len-1]])
        {
            // arr[i] wants to extend largest subsequence
            prevIndices[i] = tailIndices[len-1];
            tailIndices[len++] = i;
        }
        else
        {
            // arr[i] wants to be a potential condidate of
            // future subsequence
            // It will replace ceil value in tailIndices
            int pos = GetCeilIndex(arr, tailIndices, -1,
                                   len-1, arr[i]);

            prevIndices[i] = tailIndices[pos-1];
            tailIndices[pos] = i;
        }
    }

    cout << "LIS of given input" << endl;
    for (int i = tailIndices[len-1]; i >= 0; i = prevIndices[i])
        cout << arr[i] << " ";
    cout << endl;

    return len;
}

int main()
{
    int arr[] = { 2, 5, 3, 7, 11, 8, 10, 13, 6 };
    int n = sizeof(arr)/sizeof(arr[0]);

    printf("LIS size %d\n", LongestIncreasingSubsequence(arr, n));

    return 0;
}

Exercises:

1. You know Kadane‘s algorithm to find maximum sum sub-array. Modify Kadane’s algorithm to trace starting and ending location of maximum sum sub-array.

2. Modify Kadane‘s algorithm to find maximum sum sub-array in a circular array. Refer GFG forum for many comments on the question.

3. Given two integers A and B as input. Find number of Fibonacci numbers existing in between these two numbers (including A and B). For example, A = 3 and B = 18, there are 4 Fibonacci numbers in between {3, 5, 8, 13}. Do it in O(log K) time, where K is max(A, B). What is your observation?

— Venki. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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