Find Positive Integer Solution for a Given Equation

Find Positive Integer Solution for a Given Equation

What is this problem about?

The Find Positive Integer Solution for a Given Equation interview question is an interactive search problem. You are given a function $f(x, y)$ that is monotonically increasing in both $x$ and $y$. You are also given a target value $z$. Your goal is to find all pairs of positive integers $(x, y)$ such that $f(x, y) = z$. You don't know the implementation of $f$, but you can call it.

Why is this asked in interviews?

Companies like Google and TikTok use this to test your ability to optimize searches in a 2D space. Since the function is monotonic, a brute-force $O(N^2)$ search is sub-optimal. It evaluates your knowledge of Two Pointers interview patterns or Binary Search in a specialized coordinate system.

Algorithmic pattern used

This problem is best solved using the Two Pointers pattern on a monotonic grid.

1. Initialize: Start at the top-left corner $(x=1, y=1000)$ or top-right. Let's use x = 1 and y = 1000 (assuming 1000 is the upper bound).
2. Move Pointers:
   • If $f(x, y) == z$, add $(x, y)$ to result. Since $f$ increases with $x$ and $y$, the next $x$ must be larger and next $y$ must be smaller. x++, y--.
   • If $f(x, y) < z$, we need a larger result. Since $f$ increases with $x$, move x++.
   • If $f(x, y) > z$, we need a smaller result. Since $f$ increases with $y$, move y--.
3. Termination: Stop when x > 1000 or y < 1.

Example explanation

Target $z=5$. Function $f(x,y) = x+y$.

1. Start at $(1, 1000)$. $f(1, 1000) = 1001 > 5$. y--. ...
2. At $(1, 4)$. $f(1, 4) = 5$. Match! x++, y--.
3. At $(2, 3)$. $f(2, 3) = 5$. Match! x++, y--.
4. At $(3, 2)$. $f(3, 2) = 5$. Match! x++, y--.
5. At $(4, 1)$. $f(4, 1) = 5$. Match! x++, y--. Final result: [[1,4], [2,3], [3,2], [4,1]].

Common mistakes candidates make

• Brute Force: Using two nested loops from 1 to 1000, which results in $10^6$ function calls. The two-pointer approach only uses $2000$.
• Incorrect Movement: Moving the pointers in a way that skips potential solutions (e.g., incrementing $x$ when the value is already too large).
• Bounds: Forgetting that $x$ and $y$ must be positive integers ($\geq 1$).

Interview preparation tip

This is identical to searching in a sorted 2D matrix. Whenever you have a 2D space where values increase along rows and columns, the Two Pointers approach from the corners is the standard $O(N+M)$ solution.