3404. Count Special Subsequences - Detailed Explanation

Problem Statement

Given an integer array, a "special subsequence" is defined as a subsequence that follows a fixed pattern in order. In this problem, the fixed pattern is:

[4, 8, 15, 16, 23, 42]

A subsequence is obtained by deleting some (or no) elements from the array without changing the order of the remaining elements. A special subsequence is one in which the numbers 4, 8, 15, 16, 23, and 42 appear in that exact order (they do not have to be contiguous, but each must appear at least once and in order).

The task is to count the number of special subsequences in the given array and return the result modulo 10⁹ + 7.

Example Inputs, Outputs, and Explanations

Example 1

Input:

arr = [4, 8, 15, 16, 23, 42]

Output:

1

Explanation:
The array itself is the only special subsequence that exactly follows the pattern.

Example 2

Input:

arr = [4, 4, 8, 15, 16, 8, 23, 42, 15, 16, 23, 42]

Output:

?

Explanation:
There are multiple ways to form the special subsequence [4, 8, 15, 16, 23, 42] by choosing different occurrences of the same number while preserving order. The answer is the total count of such valid ways modulo 10⁹+7.

Example 3

Input:

arr = [4, 8, 4, 15, 8, 16, 23, 15, 16, 42, 23, 42]

Output:

?

Explanation:
Even when the array contains multiple copies of some numbers, each valid special subsequence must follow the order 4 → 8 → 15 → 16 → 23 → 42. The task is to count all such subsequences.

Hints for the Approach

1. Dynamic Programming for Counting Subsequences:
   We can solve this problem using a DP approach that counts the number of ways to form a subsequence matching the pattern up to each stage. Since the pattern has 6 numbers, we maintain an array dp of length 6 where:

   • dp[0] counts the ways to form a subsequence ending in 4.
   • dp[1] counts the ways to form a subsequence [4, 8].
   • … and so on, until dp[5] which counts the ways to form the complete sequence [4, 8, 15, 16, 23, 42].

2. Iterative Updates:
   As you iterate through the input array, when you encounter a number that matches a pattern element:

   • If it’s the first element (4), start a new subsequence.
   • For any later element (say 8, 15, etc.), extend any subsequence that is waiting for that number by adding the count from the previous stage.

3. Reverse Order Update:
   To avoid using updated values in the same iteration, update the dp array from the last index to the first index when processing each number.

4. Modulo Arithmetic:
   Since the answer can be very large, take the modulo 10⁹+7 at every update.

Detailed Explanation of the Approach

DP State and Transition

• State:
  Let pattern = [4, 8, 15, 16, 23, 42].
  Define an array dp of length 6, where dp[i] represents the number of ways to form a subsequence that matches pattern[0] to pattern[i] (inclusive).

• Transition:
  For each number num in the input array:

  • Iterate i from 5 down to 0.
  • If num == pattern[i]:
    • If i == 0, update:

      dp[0] = (dp[0] + 1) % mod
      

      because a new subsequence starting with 4 can be started.
    • Otherwise, update:

      dp[i] = (dp[i] + dp[i - 1]) % mod
      

      meaning that each way to form a subsequence ending with pattern[i-1] can now be extended by num to form a subsequence ending with pattern[i].

Final Answer

After processing the entire array, dp[5] holds the number of complete special subsequences that match the pattern. Return dp[5].

Complexity Analysis

• Time Complexity: O(n), where n is the length of the input array, since we process each element once.
• Space Complexity: O(1), as we only use an array of fixed size (6 elements).

Code Implementation

Python Code

Edge Cases

1. Empty Array:

   • If the input array is empty, there are no subsequences, so the output should be 0.

2. Array Missing Pattern Elements:

   • If the array does not contain one or more of the numbers from the pattern [4, 8, 15, 16, 23, 42], then it is impossible to form a complete special subsequence. In such cases, the count should be 0.

3. Partial Pattern Occurrence:

   • When the array contains only some initial part of the pattern (for example, only 4s or 4 followed by 8s), the DP should correctly count subsequences that are incomplete. However, since the task is to count complete special subsequences, the final answer must be 0.

4. All Elements are the Same:

   • If the array consists entirely of a single number (for example, only 4s), then only the first element of the pattern is present repeatedly, and no complete special subsequence can be formed. The result should be 0.

5. Large Array with Many Duplicates:

   • When the array is very large and contains many duplicate values of the pattern elements, the count of subsequences can grow very large. It is crucial to perform modulo arithmetic at each update to avoid integer overflow.

Common Mistakes

1. Incorrect Update Order:

   • A common mistake is updating the DP array in forward order. This may lead to using the updated value for the current iteration and overcounting subsequences. The correct approach is to update the DP array in reverse order (from pattern index 5 down to 0).

2. Not Using Modulo at Every Step:

   • Forgetting to apply the modulo operation during each update may result in overflow or incorrect results when the count becomes very large. Always take modulo 10⁹+7 when updating DP values.

3. Off-By-One Errors:

   • Errors can occur while iterating through the array or pattern indices. Ensure that the loops correctly cover the entire range of the array and the pattern.

4. Overcounting Incomplete Subsequences:

   • Some solutions may mistakenly count partial matches (e.g., subsequences that match [4, 8] but never complete the entire pattern). The final result should only consider fully formed special subsequences corresponding to [4, 8, 15, 16, 23, 42].

5. Type Conversion and Overflow Issues (in Java):

   • In Java, if the DP array is not defined as a long array, the intermediate counts might overflow an int. Use long for intermediate DP values and then cast to int after applying modulo.

6. Mismatched Pattern Check:

   • When checking if a number from the array matches a pattern element, ensure that the equality check is done properly. A slight typo or misinterpretation of the pattern order could lead to a completely wrong count.

By carefully handling these edge cases and avoiding the common pitfalls, you can implement a robust solution that accurately counts the number of special subsequences.

Related LeetCode Problems

• Longest Increasing Subsequence (LeetCode 300)

• Count Ways to Form a Target String Given a Dictionary (LeetCode 1639)