Leetcode #1237: Find Positive Integer Solution for a Given Equation
In this guide, we solve Leetcode #1237 Find Positive Integer Solution for a Given Equation in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
Given a callable function f(x, y) with a hidden formula and a value z, reverse engineer the formula and return all positive integer pairs x and y where f(x,y) == z. You may return the pairs in any order.
Quick Facts
- Difficulty: Medium
- Premium: No
- Tags: Math, Two Pointers, Binary Search, Interactive
Intuition
The constraints hint that we can reason about two ends of the data at once, which is perfect for a two-pointer scan.
Moving one pointer at a time keeps the invariant intact and avoids nested loops.
Approach
Place pointers at the left and right ends and move them based on the comparison or target condition.
This yields a clean linear pass after any required sorting.
Steps:
- Set left and right pointers.
- Move a pointer based on the condition.
- Update the best answer while scanning.
Example
interface CustomFunction {
public:
// Returns some positive integer f(x, y) for two positive integers x and y based on a formula.
int f(int x, int y);
};
Python Solution
"""
This is the custom function interface.
You should not implement it, or speculate about its implementation
class CustomFunction:
# Returns f(x, y) for any given positive integers x and y.
# Note that f(x, y) is increasing with respect to both x and y.
# i.e. f(x, y) < f(x + 1, y), f(x, y) < f(x, y + 1)
def f(self, x, y):
"""
class Solution:
def findSolution(self, customfunction: "CustomFunction", z: int) -> List[List[int]]:
ans = []
for x in range(1, z + 1):
y = 1 + bisect_left(
range(1, z + 1), z, key=lambda y: customfunction.f(x, y)
)
if customfunction.f(x, y) == z:
ans.append([x, y])
return ans
Complexity
The time complexity is , where is the value of , and the space complexity is . The space complexity is .
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.