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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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