Write an API that generates fancy sequences using the append, addAll, and multAll operations.
Implement the Fancy class:
Fancy() Initializes the object with an empty sequence.void append(val) Appends an integer val to the end of the sequence.void addAll(inc) Increments all existing values in the sequence by an integer inc.void multAll(m) Multiplies all existing values in the sequence by an integer m.int getIndex(idx) Gets the current value at index idx (0-indexed) of the sequence modulo 109 + 7. If the index is greater or equal than the length of the sequence, return -1.
Example 1:
Input ["Fancy", "append", "addAll", "append", "multAll", "getIndex", "addAll", "append", "multAll", "getIndex", "getIndex", "getIndex"] [[], [2], [3], [7], [2], [0], [3], [10], [2], [0], [1], [2]] Output [null, null, null, null, null, 10, null, null, null, 26, 34, 20] Explanation Fancy fancy = new Fancy(); fancy.append(2); // fancy sequence: [2] fancy.addAll(3); // fancy sequence: [2+3] -> [5] fancy.append(7); // fancy sequence: [5, 7] fancy.multAll(2); // fancy sequence: [5*2, 7*2] -> [10, 14] fancy.getIndex(0); // return 10 fancy.addAll(3); // fancy sequence: [10+3, 14+3] -> [13, 17] fancy.append(10); // fancy sequence: [13, 17, 10] fancy.multAll(2); // fancy sequence: [13*2, 17*2, 10*2] -> [26, 34, 20] fancy.getIndex(0); // return 26 fancy.getIndex(1); // return 34 fancy.getIndex(2); // return 20
Constraints:
1 <= val, inc, m <= 1000 <= idx <= 105105 calls total will be made to append, addAll, multAll, and getIndex.Overview: The main idea is to avoid updating every element directly for each operation. Instead, maintain a cumulative add and mult factor that gets applied on retrieval.
Explanation: By maintaining separate variables for the total add and mult operations performed on the sequence, we can apply them whenever a specific index is requested, rather than updating every element in the sequence each time an operation is made.
The append operation appends a transformed value to the sequence by reversing the current operations effect using modular arithmetic inverses. addAll and multAll update cumulative add and mult factors respectively. getIndex applies stored operations to retrieve the correct value.
Time complexity for append is O(1) with efficient modular inverse calculation. Other operations are O(1).
Space complexity is O(n) for maintaining the sequence.
Overview: Use mathematical properties of addition and multiplication with modular arithmetic for maintaining sequence properties efficiently without updating the sequence on every operation.
Explanation: We can keep track of transformations applied to the entire sequence using lazy evaluations. When retrieving an element by index, compute its final value by applying all saved transformations.
Similar to the Python approach, JavaScript implementation utilizes a lazy evaluation mechanism and mathematical inverses to apply effects during retrieval rather than upon each operation. Inverse calculations are handled via modular exponentiation for efficiency.
Time complexity for each operation remains O(1), leveraging fast modular arithmetic.
Space complexity is O(n) as it maintains the sequence.
This approach optimizes by tracking the cumulative effects of the operations using incremental values and coefficients. Instead of applying operations directly on the sequence, we maintain an additive increment totalAdd and a multiplicative factor totalMult.
The Python solution uses a modulo operation to efficiently manage the applied transformations.
append: Inserts the value by reversing current transformations.addAll: Updates the ongoing cumulative addition effect.multAll: Updates cumulative multiplication effect and adjusts previous additive impact.getIndex: Retrieves an adjusted value by applying stored transformations.C++
Time Complexity: O(1) per operation.
Space Complexity: O(n), where n is the number of values appended.
In this strategy, we store the sequence as-is and apply cumulative transformations only when necessary. This involves calculating the effects just-in-time when accessing elements, thereby delaying computation and optimizing space allocation and access costs.
The Java implementation uses a list to maintain the sequence as originally appended. The operations modify a cumulative set of transformation variables.
append captures values adjusted by existing transformation effects reversed using a modular inverse.addAll increments a global additive transformation.multAll scales both totalAdd and totalMult.getIndex lazily calculates the required result for the index given the cumulative operations.C#
Time Complexity: O(1) per operation.
Space Complexity: O(n), where n is the number of values appended.
| Approach | Complexity |
|---|---|
| Lazy Propagation Using Differential Storage | Time complexity for |
| Efficient Use of Lazy Evaluation and Modular Arithmetic | Time complexity for each operation remains O(1), leveraging fast modular arithmetic. |
| Efficient Operations Tracking | Time Complexity: O(1) per operation. Space Complexity: O(n), where n is the number of values appended. |
| Delayed Application with Lazy Propagation | Time Complexity: O(1) per operation. Space Complexity: O(n), where n is the number of values appended. |
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