Fancy Sequence

Hard

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Arrays

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 <= 100
0 <= idx <= 10⁵
At most 10⁵ calls total will be made to append, addAll, multAll, and getIndex.

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the constraints on the integer values used for `add`, `mult`, and the initial `append` values? Can they be negative, zero, or have a maximum value?
How large can the sequence grow, and how many operations (add, mult, append, getIndex) will be performed? I'm thinking about potential overflow issues and the choice of data structures.
For the `getIndex` operation, what should I return if the index is out of bounds (less than 0 or greater than or equal to the current sequence length)? Should I throw an exception or return a specific value like -1?
Are there any specific constraints on the number of calls to each method? For example, will `getIndex` be called far more frequently than `append`?
What data type should be used to store the sequence elements to prevent integer overflow during `add` and `mult` operations? Should I use `long` or handle potential overflow by using modulo arithmetic with a prime number?

Brute Force Solution

Approach

The brute force strategy for the fancy sequence is to perform each operation one at a time directly as requested. This means applying the add, multiply, or get operation without trying to be clever or optimize anything. It's like following the instructions exactly as they are written, no shortcuts.

Here's how the algorithm would work step-by-step:

When we 'add' a number, we go through the entire existing sequence and add that number to each element.
When we 'multiply' by a number, we go through the entire existing sequence and multiply each element by that number.
When we're asked to 'get' the element at a certain position, we simply perform all the add and multiply operations that have happened up to that point, in order, on the initial value(s) that existed at that position.
We don't remember or optimize anything from past operations; each 'get' request re-calculates from scratch using the history of operations.

Code Implementation

class FancySequence:

    def __init__(self):
        self.sequence = []
        self.add_operations = []
        self.multiply_operations = []

    def append(self, value):
        self.sequence.append(value)

    def addAll(self, increment):
        self.add_operations.append(increment)

    def multAll(self, multiplier):
        self.multiply_operations.append(multiplier)

    def getIndex(self, index_to_get):
        if index_to_get >= len(self.sequence):
            return -1

        value_at_index = self.sequence[index_to_get]

        # Need to apply all operations in order
        for multiplier in self.multiply_operations:

            value_at_index = (value_at_index * multiplier) % 1000000007

        # Need to apply all operations in order
        for increment in self.add_operations:

            value_at_index = (value_at_index + increment) % 1000000007

        return value_at_index

Big(O) Analysis

Time Complexity

O(n*m) – Let n be the number of elements in the sequence and m be the total number of add and multiply operations performed. Each 'add' or 'multiply' operation iterates through all n elements of the sequence. The 'get' operation also iterates through all operations (up to m) that modified the sequence up to the index being retrieved. Therefore, in the worst case, the 'get' operation takes O(m) time, where m is the number of modifications. Since 'add' and 'multiply' also take O(n) time and each operation potentially recalculates the whole array up to a given index, and we might call the get method for each element, the time complexity is O(n*m).

Space Complexity

O(1) – The provided solution, based on the plain English explanation, doesn't use any auxiliary data structures. It operates directly on the sequence by modifying elements in place or recalculating values on demand. Therefore, no extra memory is allocated to store intermediate results, operation history, or visited locations. The space complexity is constant, independent of the number of operations or the size of the sequence.

Optimal Solution

Approach

The problem asks us to efficiently perform a sequence of operations (addition, multiplication, and setting values) on a list of numbers. The key is to avoid actually updating all the numbers every time an operation is performed. Instead, we track how the operations transform the numbers indirectly using multiplicative and additive changes.

Here's how the algorithm would work step-by-step:

Keep track of two running values: one for the cumulative multiplication factor and one for the cumulative addition factor applied to all numbers so far.
When adding a number to the sequence, don't add the number directly. Instead, first multiply it by the inverse of the current multiplicative factor and then subtract the current additive factor. This effectively 'undoes' the previous operations so the new number is the 'raw' value before previous transformations.
When multiplying the sequence, update the multiplicative factor by multiplying it by the given number. When adding to the sequence, update the additive factor by multiplying it by the same number.
When adding to the sequence, simply update the additive factor by adding the given number to it.
When getting a value, apply the cumulative multiplication and then the cumulative addition to the 'raw' number (the number originally added after the 'undoing' process) to get the final result.
Since we're multiplying, we also need to divide at some point. So, remember to calculate the inverse of any numbers when multiplying or dividing. This needs to be done in modulo arithmetic.
By deferring the actual calculations until a value is requested, and only doing them on a single number when needed, we avoid repeated operations on the entire sequence.

Code Implementation

class FancySequence:
    def __init__(self):
        self.sequence = []
        self.multiplicativeFactor = 1
        self.additiveFactor = 0
        self.modulo = 10**9 + 7

    def append(self, value):
        # Adjust the value to 'undo' previous operations.
        adjustedValue = (value - self.additiveFactor) * self.self_inverse(self.multiplicativeFactor, self.modulo) % self.modulo
        self.sequence.append(adjustedValue)

    def addAll(self, increment):
        # Update additive factor.
        self.additiveFactor = (self.additiveFactor + increment) % self.modulo

    def multAll(self, multiplier):
        # Update both factors when multiplying.
        self.multiplicativeFactor = (self.multiplicativeFactor * multiplier) % self.modulo
        self.additiveFactor = (self.additiveFactor * multiplier) % self.modulo

    def getIndex(self, indexToGet):
        if indexToGet >= len(self.sequence):
            return -1

        # Apply cumulative operations to get final value.
        rawValue = self.sequence[indexToGet]
        return (rawValue * self.multiplicativeFactor + self.additiveFactor) % self.modulo

    def self_inverse(self, number, modulo):
        # Calculates modular inverse using Fermat's Little Theorem.
        return pow(number, modulo - 2, modulo)

Big(O) Analysis

Time Complexity

O(1) – The Fancy Sequence operations (append, addAll, multAll, getIndex) each take constant time. Appending involves calculating a modular inverse which can be done in O(1) using exponentiation by squaring. Applying modular arithmetic to compute multiplicative and additive changes also take constant time. Accessing an element at a specific index also requires constant time. Thus, each operation has a time complexity of O(1).

Space Complexity

O(N) – The solution stores the 'raw' values in a list. The size of this list increases as more numbers are added to the sequence. Therefore, the auxiliary space used is proportional to the number of add operations performed, which can be at most N, where N is the number of elements added to the sequence. The multiplicative and additive factors require constant space, but the list of 'raw' values dominates, resulting in O(N) space complexity.

Edge Cases

Case	How to Handle
Initializing with a zero-length sequence.	Handle the empty sequence by initializing the sequence with an empty list and setting the multiplier and adder to default values (e.g., 1 and 0).
Adding a very large number of elements to the sequence, exceeding memory limits.	Consider using a modular arithmetic approach and pre-allocate memory if the sequence size is known in advance or implement a dynamic resizing strategy with checks to prevent memory overflow.
Multiplying by zero.	Reset the multiplier to 1 to avoid making all subsequent elements zero.
Adding an extremely large or small number, leading to integer overflow/underflow in getIndex.	Use modular arithmetic with a large prime number during calculations in `add` and `getIndex` operations to avoid exceeding the maximum integer value.
Repeatedly multiplying without adding, leading to extremely large multiplier value.	Apply modular arithmetic on the multiplier itself to prevent overflow.
Calling getIndex with an index that is too small before any operations have been performed.	Return 0 if the index is out of the current bounds of the sequence, consistent with initializing the sequence with zeros.
Alternating add and multiply operations causing significant precision loss in the multiplier or adder (if using floats).	Stick to integer arithmetic and modular operations if floating-point precision is a concern for a specific application.
Modular inverse does not exist during division operation when reversing the operations for getIndex because the multiplier is not coprime to mod.	Implement a mechanism to detect when the modular inverse does not exist and signal an error or fallback to a slower, but more robust calculation.

Initializing with a zero-length sequence.

How to Handle:

Handle the empty sequence by initializing the sequence with an empty list and setting the multiplier and adder to default values (e.g., 1 and 0).

Adding a very large number of elements to the sequence, exceeding memory limits.

How to Handle:

Consider using a modular arithmetic approach and pre-allocate memory if the sequence size is known in advance or implement a dynamic resizing strategy with checks to prevent memory overflow.

Multiplying by zero.

How to Handle:

Reset the multiplier to 1 to avoid making all subsequent elements zero.

Adding an extremely large or small number, leading to integer overflow/underflow in getIndex.

How to Handle:

Use modular arithmetic with a large prime number during calculations in `add` and `getIndex` operations to avoid exceeding the maximum integer value.

Repeatedly multiplying without adding, leading to extremely large multiplier value.

How to Handle:

Apply modular arithmetic on the multiplier itself to prevent overflow.

Calling getIndex with an index that is too small before any operations have been performed.

How to Handle:

Return 0 if the index is out of the current bounds of the sequence, consistent with initializing the sequence with zeros.

Alternating add and multiply operations causing significant precision loss in the multiplier or adder (if using floats).

How to Handle:

Stick to integer arithmetic and modular operations if floating-point precision is a concern for a specific application.

Modular inverse does not exist during division operation when reversing the operations for getIndex because the multiplier is not coprime to mod.

How to Handle:

Implement a mechanism to detect when the modular inverse does not exist and signal an error or fallback to a slower, but more robust calculation.