Counting Solutions to Linear Inequalities: Exploring the Floorsum Technique

Counting Solutions to Linear Inequalities: Exploring the Floorsum Technique

Introduction:

In many mathematical and programming problems, there is a need to count the number of non-negative integer solutions to a linear inequality of the form ax + by ≤ c. The coefficients a, b, and c can be quite large, up to 10^9. In this blog post, we will explore a new technique called "Floorsum" that efficiently calculates the number of solutions to such inequalities.

The Floorsum Function:

The key to solving this problem lies in the implementation of the floorsum function:

def floorsum(n, m, a, b):
    ans = 0
    while True:
        if a >= m:
            ans += n * (n - 1) // 2 * (a // m)
            a %= m
        if b >= m:
            ans += n * (b // m)
            b %= m
        y_max = a * n + b
        if y_max < m:
            break
        n = y_max // m
        b = y_max % m
        m, a = a, m
    return ans

The floorsum function is a clever implementation that calculates the sum of the floor values of a linear expression with changing coefficients. Let's understand how it works:

  1. Initialize the variable ans to 0, which will store the final count of solutions.
  2. Enter a while loop that continues until a certain condition is met (explained later).
  3. If a is greater than or equal to m, add a contribution to ans based on the value of n, a, and m. This step accounts for the terms in the linear expression that have a common multiple of m.
  4. Reduce a modulo m to ensure it is less than m.
  5. If b is greater than or equal to m, add a contribution to ans based on the value of n and b. This step accounts for the terms in the linear expression that are independent of m.
  6. Reduce b modulo m to ensure it is less than m.
  7. Calculate the maximum possible value of y (y_max) that satisfies the inequality ax + by ≤ c. If y_max is less than m, it means there are no further solutions to consider, so the while loop breaks.
  8. Update the values of n and b to continue the calculation with the new coefficients.
  9. Swap the values of m and a to prepare for the next iteration of the while loop.
  10. Return the final count of solutions stored in ans.

The floorsum function cleverly iterates through the linear expression, accounting for terms with common multiples of m and terms independent of m. By adjusting the coefficients and tracking the maximum value of y, it effectively calculates the number of valid solutions.

The Count Function:

The count function builds upon the floorsum technique to provide a convenient way to count the solutions to the given linear inequality:

def count(a, b, c):
    return floorsum(c // a + 1, b, a, c % a)

The count function takes the coefficients a, b, and c as parameters and uses floorsum to calculate the number of non-negative solutions to the inequality ax + by ≤ c

the input by dividing c by a and adding 1 to account for the non-negative constraint. It then calls the floorsum function with the modified parameters, a, b, and c % a, where c % a represents the remainder when c is divided by a.

Application and Efficiency:

The floorsum technique is particularly useful when dealing with large coefficients and constraints. It efficiently calculates the count of non-negative solutions to linear inequalities, providing a valuable tool for various mathematical and programming problems.

The time complexity of the floorsum function depends on the values of a, b, and c. In the worst case, where a is close to m, the while loop may iterate multiple times. However, overall, the floorsum technique offers an efficient approach to counting solutions in linear inequalities.

Conclusion:

The floorsum technique introduces a powerful and efficient way to count the number of non-negative solutions to linear inequalities. By leveraging clever adjustments to the coefficients and carefully tracking the maximum value of y, it provides an elegant solution to a common problem. The count function builds upon the floorsum technique, making it easy to apply in various scenarios. Understanding and implementing this technique can greatly enhance one's ability to tackle problems involving linear inequalities with large coefficients. So, next time you encounter a similar problem, consider using the floorsum technique and witness its effectiveness.

Code snippet:

from collections import Counter

def floorsum(n, m, a, b):
    ans = 0
    while True:
        if a >= m:
            ans += n * (n - 1) // 2 * (a // m)
            a %= m
        if b >= m:
            ans += n * (b // m)
            b %= m
        y_max = a * n + b
        if y_max < m:
            break
        n = y_max // m
        b = y_max % m
        m, a = a, m
    return ans

def count(a, b, c):
    return floorsum(c // a + 1, b, a, c % a)

Example usage


solution_count = count(5, 7, 30)
print(solution_count)  # Output: 16

So, go ahead and apply the floorsum technique to solve counting problems involving linear inequalities, and witness its efficiency in action!

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