Fancy Sequence

Fancy Sequence

What is this problem about?

The Fancy Sequence coding problem asks you to design a data structure that supports four operations on a dynamic sequence of integers:

1. append(val): Add a number to the end.
2. addAll(inc): Add a value to every element currently in the sequence.
3. multAll(m): Multiply every element currently in the sequence by a value.
4. getIndex(idx): Return the value at a specific index modulo $10^9 + 7$. The challenge is that addAll and multAll affect potentially thousands of elements at once, so you cannot iterate through the sequence for every operation.

Why is this asked in interviews?

Google uses this Hard difficulty Design problem to test your knowledge of Modular Arithmetic and algebraic transformations. It evaluates if you can identify a mathematical invariant that allows you to perform range updates in O(1) time. It requires understanding modular inverses and how to represent a sequence of linear transformations (ax + b) as a single operation.

Algorithmic pattern used

The problem uses a Linear Transformation State (val * multiplier + increment).

1. Maintain a global m (current total multiplier) and a (current total increment).
2. When append(x) is called:
   • You need to store a "raw" value such that when the current global m and a are applied, it equals x.
   • Stored value = (x - a) / m (performed using modular inverse).
3. addAll(inc): a = (a + inc) % MOD.
4. multAll(m_new): m = (m * m_new) % MOD, a = (a * m_new) % MOD.
5. getIndex(idx): result = (stored_val[idx] * m + a) % MOD.

Example explanation

1. append(2): Global m=1, a=0. Store 2.
2. addAll(3): Global m=1, a=3.
3. append(1): We want (stored * 1 + 3) % MOD = 1.
   • stored = (1 - 3) * inv(1) = -2.
4. multAll(2): Global m = 12 = 2. Global a = 32 = 6.
5. getIndex(0): stored(2) * 2 + 6 = 10.
6. getIndex(1): stored(-2) * 2 + 6 = 2.

Common mistakes candidates make

• Linear Update: Iterating through the array for addAll or multAll, leading to O(N) per update.
• Precision: Forgetting the modulo operator at every intermediate step.
• Modular Inverse: Not realizing that division in modular arithmetic must be handled using Fermat's Little Theorem (for prime MOD).

Interview preparation tip

Familiarize yourself with the property of linear functions: f(g(x)) where f(x) = ax + b and g(x) = cx + d is still a linear function acx + (ad + b). This allows you to chain infinitely many additions and multiplications into just two variables.