1237.FindPositiveIntegerSolutionforaGivenEquation.py

'''
    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:
    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:
    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
'''
#Difficulty: Easy
#45 / 45 test cases passed.
#Runtime: 40 ms
#Memory Usage: 14.1 MB

#Runtime: 40 ms, faster than 71.52% of Python3 online submissions for Find Positive Integer Solution for a Given Equation.
#Memory Usage: 14.1 MB, less than 75.13% of Python3 online submissions for 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]]:
        result = []
        for x in range(1, z+1):
            y = self.binarySearch(customfunction, x, z)
            if y:
                result.append([x, y])
        return result

    def binarySearch(self, customfunction: 'CustomFunction', x: int, z: int) -> int:
        lo = 1
        hi = 1000
        while lo <= hi:
            mid = (lo+hi) // 2
            if customfunction.f(x, mid) == z:
                return mid
            elif customfunction.f(x, mid) > z:
                hi = mid - 1
            else:
                lo = mid + 1