Fancy Sequence - Solution & Explanation

Problem Overview: You design a sequence that supports four operations: append(val), addAll(inc), multAll(m), and getIndex(i). Each modification affects every element already in the sequence. The challenge is handling up to 10^5 operations efficiently while applying additions and multiplications under modulo 1e9+7.

Approach 1: Direct Simulation (O(n) per update, O(1) query)

The simplest idea stores the sequence in an array and directly applies addAll or multAll by iterating through every element. Each operation modifies the entire list using a loop. While getIndex becomes a constant-time lookup, update operations cost O(n) time. With up to 10^5 operations, this approach quickly becomes too slow and exceeds time limits. Space complexity is O(n) for storing the sequence.

Approach 2: Lazy Global Transformation (Affine Function) (O(1) per operation)

Every operation applied to the sequence can be modeled as an affine transformation: x → a * x + b. Instead of updating each element, maintain global multipliers a and b. When you append a value, store it in a normalized form that cancels the current transformation using modular inverse. Later, when retrieving with getIndex, reconstruct the real value by applying the current transformation. All operations run in O(1) time with O(n) space. This technique relies on modular arithmetic and inverse computation, a common trick in math-heavy design problems.

Approach 3: Operations Tracking with Differential Storage (O(1) amortized)

Another design tracks the sequence values along with cumulative operations applied after insertion. Instead of modifying stored elements, maintain operation history such as total multipliers and additions. Each element remembers the operation state at the time it was appended. When retrieving an index, compute the difference between the stored state and the current state to derive the final value. This avoids repeated updates and keeps operations constant time. The technique works well for problems involving chained transformations and falls under advanced design patterns.

Approach 4: Segment Tree with Lazy Propagation (O(log n))

A more general solution uses a segment tree with lazy propagation. Each node stores values for a range and defers updates using lazy tags representing multiplication and addition. When addAll or multAll is called, the update propagates lazily across the tree. Queries push pending updates down the path before reading the value. Time complexity becomes O(log n) per operation with O(n) space. This approach is more flexible but heavier than the affine transformation trick.

Recommended for interviews: The affine transformation approach with lazy global parameters is the expected optimal solution. It achieves O(1) operations while handling both multiplication and addition efficiently. Discussing the brute-force simulation first shows understanding of the problem constraints, while the optimized math-based design demonstrates strong algorithmic thinking.