One of Maximum Slice Problems in Codility’s Lessons. [Link]
This is a problem in beginner’s level, and the final answer is really simple. However, I post it as a reminder since I took a long to solve it.
A non-empty array A consisting of N integers is given. A pair of integers (P, Q), such that 0 ≤ P ≤ Q < N, is called a slice of array A. The sum of a slice (P, Q) is the total of A[P] + A[P+1] + … + A[Q].
Write a function:
that, given an array A consisting of N integers, returns the maximum sum of any slice of A.
For example, given array A such that:
A = 3 A = 2 A = -6 A = 4 A = 0
the function should return 5 because:
- (3, 4) is a slice of A that has sum 4,
- (2, 2) is a slice of A that has sum −6,
- (0, 1) is a slice of A that has sum 5,
- no other slice of A has sum greater than (0, 1).
Write an efficient algorithm for the following assumptions:
- N is an integer within the range [1..1,000,000];
- each element of array A is an integer within the range [−1,000,000..1,000,000];
- the result will be an integer within the range [−2,147,483,648..2,147,483,647].
- Detected Time Complexity: O(N)
It uses two variables: ‘max’ to represent maximum sum and ‘acc’ for storing cumulative sum. If acc is negative at the current step, it restarts cumulative sum by assigning 0; that is because the next sum would be negatively affected.
def solution(A): max = A acc = 0 for e in A: acc += e if acc > max: max = acc if acc < 0: acc = 0 return max
We can get the same solution by using 2 max functions. The first max gets cumulative sum for the current slice, and the second max keeps or updates the max slice sum.
def solution(A): max_sum = sub_sum = A for i in range(1, len(A)): sub_sum = max(sub_sum + A[i], A[i]) max_sum = max(max_sum, sub_sum) return max_sum