减少算法的时间复杂度

Question

This is a task taken from codility, which requires O(n) complexity. Although I've solved the task that it gives correct results, the time complexity is O(n*n). Would appreciate any help in explaining how to reduce that complexity to O(n).

Task description:

A non-empty zero-indexed array A consisting of N integers is given. Array A represents numbers on a tape.

Any integer P, such that 0 < P < N, splits this tape into two non-empty parts: A[0], A[1], ..., A[P − 1] and A[P], A[P + 1], ..., A[N − 1].

The difference between the two parts is the value of: |(A[0] + A[1] + ... + A[P − 1]) − (A[P] + A[P + 1] + ... + A[N − 1])|

In other words, it is the absolute difference between the sum of the first part and the sum of the second part.

For example, consider array A such that:

A[0] = 3 A[1] = 1 A[2] = 2 A[3] = 4 A[4] = 3

We can split this tape in four places:

  P = 1, difference = |3 − 10| = 7 P = 2, difference = |4 − 9| = 5 P = 3, difference = |6 − 7| = 1 P = 4, difference = |10 − 3| = 7 

Write a function:

 int solution(vector<int> &A); 

that, given a non-empty zero-indexed array A of N integers, returns the minimal difference that can be achieved.

For example, given:

A[0] = 3 A[1] = 1 A[2] = 2 A[3] = 4 A[4] = 3

the function should return 1, as explained above.

Assume that:

  N is an integer within the range [2..100,000]; each element of array A is an integer within the range [−1,000..1,000]. 

Complexity:

  expected worst-case time complexity is O(N); expected worst-case space complexity is O(N), beyond input storage (not counting the storage required for input arguments). 

Elements of input arrays can be modified.

My solution:  
int solution(vector<int>& A)
{
    // write your code in C++11
    int result = std::numeric_limits<int>::max();
    int part_1 = 0;
    int part_2 = 0;
    for (auto beg = A.cbegin(), end = A.cend(); beg != prev(end); ++beg)
    { 
        part_1 = accumulate(A.cbegin(),beg + 1,0);
        part_2 = accumulate(beg + 1,end,0);
        int current_result = abs(part_1 - part_2);
        if (current_result < result)
        {
            result = current_result;
        }

    }

    return result;
}

Answer 1

Let S[i] = sum of the first i elements . You can compute this with a single for loop:

S[0] = A[0]
for (int i = 1; i < n; ++i)
  S[i] = S[i - 1] + A[i]

Then, for each index 0 < i < n , the sum up until i - 1 is simply S[i - 1] . The sum from i to the end of the array is S[n - 1] - S[i - 1] :

S[n - 1] = A[0] + ... + A[n - 1]
S[i - 1] = A[0] + ... + A[i - 1]

0 < i < n

S[n - 1] - S[i - 1] =  A[0] + ... + A[i - 1] + A[i] + ... + A[n - 1] -
                      (A[0] + ... + A[i - 1])
                    = A[i] + ... + A[n - 1]

After computing S , run another loop and check the differences between the two sums, which are computed in O(1) like I described:

for (int i = 1; i < n; ++i)
{
  sum_left = S[i - 1]
  sum_right = S[n - 1] - S[i - 1]

  // see if |sum_left - sum_right| is better than what you have so far
}

The time complexity is O(n) because you only run two for loops in which you only do O(1) operations at each iteration. The memory complexity is O(n) because you have to store S , which is the same size as your input.

Declaring S as int[n] should be fine judging by the assumptions we're allowed to make.