Sum of Mutated Array Closest to Target — Python

# Time:  O(nlogn)
# Space: O(1)

class Solution(object):
    def findBestValue(self, arr, target):
        """
        :type arr: List[int]
        :type target: int
        :rtype: int
        """
        arr.sort(reverse=True)
        max_arr = arr[0]
        while arr and arr[-1]*len(arr) <= target:
            target -= arr.pop()
        # let x = ceil(t/n)-1
        # (1) (t/n-1/2) <= x:
        #    return x, which is equal to ceil(t/n)-1 = ceil(t/n-1/2) = (2t+n-1)//2n
        # (2) (t/n-1/2) > x:
        #    return x+1, which is equal to ceil(t/n) = ceil(t/n-1/2) = (2t+n-1)//2n
        # (1) + (2) => both return (2t+n-1)//2n
        return max_arr if not arr else (2*target+len(arr)-1)//(2*len(arr))


# Time:  O(nlogn)
# Space: O(1)
class Solution2(object):
    def findBestValue(self, arr, target):
        """
        :type arr: List[int]
        :type target: int
        :rtype: int
        """
        arr.sort(reverse=True)
        max_arr = arr[0]
        while arr and arr[-1]*len(arr) <= target:
            target -= arr.pop()
        if not arr:
            return max_arr
        x = (target-1)//len(arr)
        return x if target-x*len(arr) <= (x+1)*len(arr)-target else x+1


# Time:  O(nlogm), m is the max of arr, which may be larger than n
# Space: O(1)
class Solution3(object):
    def findBestValue(self, arr, target):
        """
        :type arr: List[int]
        :type target: int
        :rtype: int
        """
        def total(arr, v):
            result = 0
            for x in arr:
                result += min(v, x)
            return result

        def check(arr, v, target):
            return total(arr, v) >= target
        
        left, right = 1, max(arr)
        while left <= right:
            mid = left + (right-left)//2
            if check(arr, mid, target):
                right = mid-1
            else:
                left = mid+1
        return left-1 if target-total(arr, left-1) <= total(arr, left)-target else left