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1237. Find Positive Integer Solution for a Given Equation

Description

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.

While the exact formula is hidden, the function is monotonically increasing, i.e.:

  • f(x, y) < f(x + 1, y)
  • f(x, y) < f(x, y + 1)

The function interface is defined like this:

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);
};

We will judge your solution as follows:

  • The judge has a list of 9 hidden implementations of CustomFunction, along with a way to generate an answer key of all valid pairs for a specific z.
  • The judge will receive two inputs: a function_id (to determine which implementation to test your code with), and the target z.
  • The judge will call your findSolution and compare your results with the answer key.
  • If your results match the answer key, your solution will be Accepted.

 

Example 1:

Input: function_id = 1, z = 5
Output: [[1,4],[2,3],[3,2],[4,1]]
Explanation: The hidden formula for function_id = 1 is f(x, y) = x + y.
The following positive integer values of x and y make f(x, y) equal to 5:
x=1, y=4 -> f(1, 4) = 1 + 4 = 5.
x=2, y=3 -> f(2, 3) = 2 + 3 = 5.
x=3, y=2 -> f(3, 2) = 3 + 2 = 5.
x=4, y=1 -> f(4, 1) = 4 + 1 = 5.

Example 2:

Input: function_id = 2, z = 5
Output: [[1,5],[5,1]]
Explanation: The hidden formula for function_id = 2 is f(x, y) = x * y.
The following positive integer values of x and y make f(x, y) equal to 5:
x=1, y=5 -> f(1, 5) = 1 * 5 = 5.
x=5, y=1 -> f(5, 1) = 5 * 1 = 5.

 

Constraints:

  • 1 <= function_id <= 9
  • 1 <= z <= 100
  • It is guaranteed that the solutions of f(x, y) == z will be in the range 1 <= x, y <= 1000.
  • It is also guaranteed that f(x, y) will fit in 32 bit signed integer if 1 <= x, y <= 1000.

Solutions

Solution 1: Enumeration + Binary Search

According to the problem, we know that the function $f(x, y)$ is a monotonically increasing function. Therefore, we can enumerate $x$, and then binary search $y$ in $[1,…z]$ to make $f(x, y) = z$. If found, add $(x, y)$ to the answer.

The time complexity is $O(n \log n)$, where $n$ is the value of $z$, and the space complexity is $O(1)$.

Solution 2: Two Pointers

We can define two pointers $x$ and $y$, initially $x = 1$, $y = z$.

  • If $f(x, y) = z$, we add $(x, y)$ to the answer, then $x \leftarrow x + 1$, $y \leftarrow y - 1$;
  • If $f(x, y) \lt z$, at this time for any $y’ \lt y$, we have $f(x, y’) \lt f(x, y) \lt z$, so we cannot decrease $y$, we can only increase $x$, so $x \leftarrow x + 1$;
  • If $f(x, y) \gt z$, at this time for any $x’ \gt x$, we have $f(x’, y) \gt f(x, y) \gt z$, so we cannot increase $x$, we can only decrease $y$, so $y \leftarrow y - 1$.

After the loop ends, return the answer.

The time complexity is $O(n)$, where $n$ is the value of $z$, and the space complexity is $O(1)$.

  • /*
     * // 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)
     *     public int f(int x, int y);
     * };
     */
    
    class Solution {
        public List<List<Integer>> findSolution(CustomFunction customfunction, int z) {
            List<List<Integer>> ans = new ArrayList<>();
            for (int x = 1; x <= 1000; ++x) {
                int l = 1, r = 1000;
                while (l < r) {
                    int mid = (l + r) >> 1;
                    if (customfunction.f(x, mid) >= z) {
                        r = mid;
                    } else {
                        l = mid + 1;
                    }
                }
                if (customfunction.f(x, l) == z) {
                    ans.add(Arrays.asList(x, l));
                }
            }
            return ans;
        }
    }
    
  • /*
     * // This is the custom function interface.
     * // You should not implement it, or speculate about its implementation
     * class CustomFunction {
     * public:
     *     // 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)
     *     int f(int x, int y);
     * };
     */
    
    class Solution {
    public:
        vector<vector<int>> findSolution(CustomFunction& customfunction, int z) {
            vector<vector<int>> ans;
            for (int x = 1; x <= 1000; ++x) {
                int l = 1, r = 1000;
                while (l < r) {
                    int mid = (l + r) >> 1;
                    if (customfunction.f(x, mid) >= z) {
                        r = mid;
                    } else {
                        l = mid + 1;
                    }
                }
                if (customfunction.f(x, l) == z) {
                    ans.push_back({x, l});
                }
            }
            return ans;
        }
    };
    
  • """
       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
    
    
  • /**
     * This is the declaration of customFunction API.
     * @param  x    int
     * @param  x    int
     * @return 	    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)
     */
    
    func findSolution(customFunction func(int, int) int, z int) (ans [][]int) {
    	for x := 1; x <= 1000; x++ {
    		y := 1 + sort.Search(999, func(y int) bool { return customFunction(x, y+1) >= z })
    		if customFunction(x, y) == z {
    			ans = append(ans, []int{x, y})
    		}
    	}
    	return
    }
    
  • /**
     * // This is the CustomFunction's API interface.
     * // You should not implement it, or speculate about its implementation
     * class CustomFunction {
     *      f(x: number, y: number): number {}
     * }
     */
    
    function findSolution(customfunction: CustomFunction, z: number): number[][] {
        const ans: number[][] = [];
        for (let x = 1; x <= 1000; ++x) {
            let l = 1;
            let r = 1000;
            while (l < r) {
                const mid = (l + r) >> 1;
                if (customfunction.f(x, mid) >= z) {
                    r = mid;
                } else {
                    l = mid + 1;
                }
            }
            if (customfunction.f(x, l) == z) {
                ans.push([x, l]);
            }
        }
        return ans;
    }
    
    

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