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Formatted question description: https://leetcode.ca/all/1237.html

1237. Find Positive Integer Solution for a Given Equation (Easy)

Given a function  f(x, y) and a value z, return all positive integer pairs x and y where f(x,y) == z.

The function is constantly 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 positive integer f(x, y) for any given positive integer x and y.
  int f(int x, int y);
};

For custom testing purposes you're given an integer function_id and a target z as input, where function_id represent one function from an secret internal list, on the examples you'll know only two functions from the list.  

You may return the solutions in any order.

 

Example 1:

Input: function_id = 1, z = 5
Output: [[1,4],[2,3],[3,2],[4,1]]
Explanation: function_id = 1 means that f(x, y) = x + y

Example 2:

Input: function_id = 2, z = 5
Output: [[1,5],[5,1]]
Explanation: function_id = 2 means that f(x, y) = x * y

 

Constraints:

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

Related Topics:
Math, Binary Search

Solution 1. Brute Force

// OJ: https://leetcode.com/problems/find-positive-integer-solution-for-a-given-equation/
// Time: O(N^2)
// Space: O(1)
class Solution {
public:
    vector<vector<int>> findSolution(CustomFunction& func, int z) {
        vector<vector<int>> ans;
        for (int x = 1; x <= 1000; ++x) {
            for (int y = 1; y <= 1000; ++y) {
                if (func.f(x, y) == z) ans.push_back({ x, y });
            }
        }
        return ans;
    }
};

// OJ: https://leetcode.com/problems/find-positive-integer-solution-for-a-given-equation/
// Time: O(NlogN)
// Space: O(1)
class Solution {
public:
    vector<vector<int>> findSolution(CustomFunction& func, int z) {
        vector<vector<int>> ans;
        for (int x = 1; x <= 1000; ++x) {
            int L = 1, R = 1000;
            while (L <= R) {
                int y = (L + R) / 2, val = func.f(x, y);
                if (val == z) {
                    ans.push_back({ x, y });
                    break;
                } else if (val < z) L = y + 1;
                else R = y - 1;
            }
        }
        return ans;
    }
};

Solution 3. Two Pointers

// OJ: https://leetcode.com/problems/find-positive-integer-solution-for-a-given-equation/
// Time: O(N)
// Space: O(1)
class Solution {
public:
    vector<vector<int>> findSolution(CustomFunction& func, int z) {
        vector<vector<int>> ans;
        for (int x = 1, y = 1000; x <= 1000 && y >= 1; ++x) {
            while (y >= 1 && func.f(x, y) > z) --y;
            if (y >= 1 && func.f(x, y) == z) ans.push_back({ x, y });
        }
        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)
 *     public int f(int x, int y);
 * };
 */
class Solution {
    public List<List<Integer>> findSolution(CustomFunction customfunction, int z) {
        List<List<Integer>> solutions = new ArrayList<List<Integer>>();
        for (int x = 1; x <= 1000; x++) {
            for (int y = 1; y <= 1000; y++) {
                int function = customfunction.f(x, y);
                if (function >= z) {
                    if (function == z) {
                        List<Integer> solution = new ArrayList<Integer>();
                        solution.add(x);
                        solution.add(y);
                        solutions.add(solution);
                    }
                    break;
                }
            }
        }
        return solutions;
    }
}

############

/*
 * // 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<>();
        int x = 1, y = 1000;
        while (x <= 1000 && y > 0) {
            int t = customfunction.f(x, y);
            if (t < z) {
                x++;
            } else if (t > z) {
                y--;
            } else {
                ans.add(Arrays.asList(x++, y--));
            }
        }
        return ans;
    }
}

  • // OJ: https://leetcode.com/problems/find-positive-integer-solution-for-a-given-equation/
// Time: O(N^2)
// Space: O(1)
class Solution {
public:
    vector<vector<int>> findSolution(CustomFunction& func, int z) {
        vector<vector<int>> ans;
        for (int x = 1; x <= 1000; ++x) {
            for (int y = 1; y <= 1000; ++y) {
                if (func.f(x, y) == z) ans.push_back({ x, y });
            }
        }
        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]]:
        res = []
        for x in range(1, 1001):
            left, right = 1, 1000
            while left < right:
                mid = (left + right) >> 1
                if customfunction.f(x, mid) >= z:
                    right = mid
                else:
                    left = mid + 1
            if customfunction.f(x, left) == z:
                res.append([x, left])
        return res

############

# 1237. Find Positive Integer Solution for a Given Equation
# https://leetcode.com/problems/find-positive-integer-solution-for-a-given-equation/

"""
   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]]:
        y = 1000
        res = []        
        for x in range(1,1000):
            while y > 1 and customfunction.f(x,y) > z:
                y -= 1
            
            if customfunction.f(x,y) == z:
                res.append([x,y])
        
        return res

  • /**
 * 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) {
	x, y := 1, 1000
	for x <= 1000 && y > 0 {
		t := customFunction(x, y)
		if t < z {
			x++
		} else if t > z {
			y--
		} else {
			ans = append(ans, []int{x, y})
			x, y = x+1, y-1
		}
	}
	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[][] {
    let x = 1;
    let y = 1000;
    const ans: number[][] = [];
    while (x <= 1000 && y) {
        const t = customfunction.f(x, y);
        if (t < z) {
            ++x;
        } else if (t > z) {
            --y;
        } else {
            ans.push([x--, y--]);
        }
    }
    return ans;
}

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