#3021

Medium

auto_awesomeMath-driven solution strategy

LeetCode Problem Workspace

Alice and Bob Playing Flower Game

Alice and Bob play a turn-based game on a circular field with flowers, where players must choose pairs that satisfy parity conditions.

math

Problem Statement

Alice and Bob are playing a turn-based game on a circular field surrounded by flowers. The field is arranged such that there are two integers, n and m, where n represents the number of flowers on the circle, and m represents the number of flowers between Alice and Bob. There are x flowers between Alice and Bob in the clockwise direction and y flowers in the anti-clockwise direction. The goal is to compute the number of valid pairs (x, y) that satisfy certain conditions.

The valid pairs must satisfy the following condition: (x, y) is valid if and only if x and y have different parities. You are tasked with calculating the number of such pairs that can exist, given the values of n and m, where 1 <= n, m <= 10^5. The problem can be solved by applying a math-driven approach that considers the parities of x and y.

Examples

Example 1

Input: n = 3, m = 2

Output: 3

The following pairs satisfy conditions described in the statement: (1,2), (3,2), (2,1).

Example 2

Input: n = 1, m = 1

Output: 0

No pairs satisfy the conditions described in the statement.

Constraints

1 <= n, m <= 105

Solution Approach

Parity Analysis

The key to solving this problem efficiently is recognizing that valid pairs (x, y) must have different parities. This means that if x is odd, y must be even, and if x is even, y must be odd. This observation significantly reduces the number of possible pairs.

Counting Valid Pairs

To count valid pairs, we can iterate through the possible values of x and y while ensuring that the difference in their parities is satisfied. For each possible x, calculate the corresponding y values that maintain the parity condition. The final result will be the total count of such pairs.

Efficient Computation

To avoid brute-forcing through all combinations, an efficient approach would involve leveraging modular arithmetic or precomputing the number of odd and even flowers based on the input values n and m. This allows for a more optimized solution that scales well with large input sizes.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the final approach used. A brute-force method would have a time complexity of O(n*m), but by leveraging parity analysis and modular arithmetic, we can reduce the time complexity significantly. The space complexity also depends on whether we use auxiliary arrays to store intermediate results or not.

What Interviewers Usually Probe

Problem-solving involves quick analysis of mathematical patterns.
Candidate demonstrates proficiency in math-driven algorithms.
Approach to parity analysis shows efficiency and optimization.

Common Pitfalls or Variants

Common pitfalls

Overlooking the parity condition can lead to incorrect pair counting.
Failing to optimize the counting approach could result in inefficient solutions for large inputs.
Not understanding the circular nature of the field might complicate boundary conditions.

Follow-up variants

Consider different sizes of n and m to evaluate solution performance.
Explore edge cases such as the smallest values for n and m.
Examine how changes in the circular structure affect the solution for n and m.

FAQ

What is the key insight behind solving the Alice and Bob Playing Flower Game problem?

The key insight is recognizing that valid pairs (x, y) must have different parities, meaning if x is odd, y must be even, and vice versa.

How can I optimize the solution for large values of n and m?

To optimize for larger inputs, consider using modular arithmetic or precomputing the number of odd and even flowers, reducing unnecessary computations.

What does the 'circle' in the Alice and Bob Playing Flower Game problem represent?

The circle represents a circular field, meaning that the flower positions are considered cyclic, and the calculation of x and y must account for the circular nature.

Why are parities important in solving this problem?

The parities are crucial because the game condition requires that the number of flowers between Alice and Bob, in both directions, must have different parities to be valid.

What is the time complexity of the optimal solution for this problem?

The time complexity of the optimal solution depends on the approach used, but with parity analysis and modular arithmetic, it can be reduced to an efficient solution with time complexity close to O(n).

terminal

Solution

Solution 1: Mathematics

According to the problem description, in each move, the player will choose to move in a clockwise or counterclockwise direction and then pick a flower. Since Alice moves first, when $x + y$ is odd, Alice will definitely win the game.

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class Solution:
    def flowerGame(self, n: int, m: int) -> int:
        a1 = (n + 1) // 2
        b1 = (m + 1) // 2
        a2 = n // 2
        b2 = m // 2
        return a1 * b2 + a2 * b1

Solution 2: Mathematics (Optimized)

The result obtained from Solution 1 is $\lfloor \frac{n + 1}{2} \rfloor \times \lfloor \frac{m}{2} \rfloor + \lfloor \frac{n}{2} \rfloor \times \lfloor \frac{m + 1}{2} \rfloor$.

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class Solution:
    def flowerGame(self, n: int, m: int) -> int:
        a1 = (n + 1) // 2
        b1 = (m + 1) // 2
        a2 = n // 2
        b2 = m // 2
        return a1 * b2 + a2 * b1

Continue Practicing

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