Sum of Beautiful Subsequences

Sum of Beautiful Subsequences

What is this problem about?

"Sum of Beautiful Subsequences" is a challenging problem that defines "beauty" based on certain mathematical criteria—usually involving properties like arithmetic progressions, divisibility, or GCD—within a subsequence. A subsequence is a sequence derived from an array by deleting zero or more elements without changing the order of the remaining elements. The task is to calculate the beauty of every possible subsequence and return the total sum, often modulo $10^9 + 7$.

Why is this asked in interviews?

This "Hard" problem, often seen in advanced rounds for companies like Infosys, tests high-level skills in Number Theory and Dynamic Programming. It's designed to see if a candidate can handle complex counting problems where the number of states is large. It requires the ability to use advanced data structures like Binary Indexed Trees (Fenwick Trees) or Segment Trees to maintain and query property-based counts as you iterate through the array.

Algorithmic pattern used

This problem typically requires Dynamic Programming optimized with a Binary Indexed Tree (BIT) or Hash Maps. The "beauty" property usually allows you to define a DP state like dp[i][val], representing the sum of beauties of subsequences ending at index i with a certain property val. To avoid O(N²) or O(N * MaxVal) complexity, you use a BIT to quickly fetch the sum of DP values from previous indices that satisfy the "beautiful" criteria (e.g., values that are divisors or within a certain range).

Example explanation

Imagine "beauty" is defined by how many elements in a subsequence are multiples of their predecessor. Array: [2, 4, 8]

• Subsequences: [2], [4], [8], [2, 4], [2, 8], [4, 8], [2, 4, 8]
• Beauties (calculated based on the specific rule):
  • [2, 4] is beautiful because 4 is a multiple of 2.
  • [2, 4, 8] is beautiful because 4 is a multiple of 2 and 8 is a multiple of 4. The algorithm would use a map to keep track of how many beautiful subsequences end in a certain number, updating it as it traverses the array.

Common mistakes candidates make

1. TLE (Time Limit Exceeded): Attempting to generate all $2^n$ subsequences explicitly. This is almost never the intended solution for a Hard problem.
2. Modulo errors: Forgetting to apply the modulo operation at every addition step, leading to integer overflow.
3. Incorrect Property Tracking: Mismanaging the state in the BIT or DP table, such as double-counting subsequences.

Interview preparation tip

Hard problems involving subsequences and mathematical properties are almost always DP problems. Focus on defining what "information" you need to carry forward from the previous elements to make a decision about the current element. Practice using Fenwick Trees for range sum queries, as they are a frequent optimization for these types of problems.