Efficiently Finding Increasing Subsequences of Length K in Arrays

In array processing and sequence analysis, finding the number of increasing subsequences of length K in an array of N integers is a common problem. This task is significant in various applications such as bioinformatics, sequence alignment, and data analysis. In this article, we will explore a detailed method, combining dynamic programming with binary indexed trees (BIT) or segment trees, to solve this problem efficiently. We will also explore the underlying steps and provide a pseudocode implementation.

Steps to Solve the Problem

The solution involves several key steps, including the definition of a dynamic programming (DP) table and its initialization, recurrence relation for updating the DP table, and the use of BIT for optimization. Let's delve into each step.

Dynamic Programming Table

The dynamic programming table, dp[i][j], is defined as follows:

i represents the length of the subsequence from 1 to K.

j represents the index in the array from 0 to N-1.

dp[i][j] represents the number of increasing subsequences of length i that end at index j.

For the base case, if K is greater than N, there are no increasing subsequences of length K, so the function returns 0.

Base Case

The base case is straightforward: for subsequences of length 1, every element can be a subsequence of length 1. This is represented as:

dp[1][j] 1 for all j (each element of the array).

Recurrence Relation

The recurrence relation for lengths i from 2 to K is as follows:

For each index j, you sum up all dp[i-1][m] where m j and array[m] array[j]. This means that for each index j, you look at all previous indices m to see if they can form an increasing subsequence when extended by array[j].

Using a Binary Indexed Tree for Optimization

Instead of a simple nested loop for summing the values, you can use a BIT to keep track of the counts of subsequences efficiently. When calculating dp[i][j], use the BIT to query the sum of dp[i-1][m] for all valid m.

Final Count

After filling the DP table, the total number of increasing subsequences of length K can be obtained by summing up dp[K][j] for all j.

Pseudocode

def count_increasing_subsequences(arr, K):
    N  len(arr)
    if K  N:
        return 0
    # Initialize DP table
    dp  [[0] * N for _ in range(K)]
    # Base case: subsequences of length 1
    for j in range(N):
        dp[1][j]  1
    # Fill the DP table
    for i in range(2, K):
        BIT  [0] * (N   1)  # Fenwick Tree for counting
        for j in range(N):
            # Query the BIT for the sum of dp[i-1][m] where m  j and arr[m]  arr[j]
            count  query_BIT(BIT, arr[j] - 1)
            dp[i][j]  count
            # Update BIT with the current dp[i-1][j]
            update_BIT(BIT, arr[j], dp[i-1][j])
    # Sum up all dp[K][j]
    total_count  sum(dp[K-1][j] for j in range(N))
    return total_count

Complexity

The time complexity of this approach is O(N · K · log N) due to the BIT operations. The space complexity is O(N · K) for the DP table.

This method is efficient for moderate values of N and K. Adjustments may be necessary depending on the specific value ranges in the input array.