Reducing the time complexity of an algorithm with nested loops

问题

I have the following algorithm which I want to rewrite so it has time complexity O(n). I am new to algorithms but from my understanding since the two for loops both do a multiple of n iterations, the complexity will always be O(n²). Is it even possible to reduce the complexity of this?

Algorithm example(ArrayA, ArrayB, n)                                           
Input: 2 arrays of integers, ArrayA and ArrayB, both length n          
Output: integer

value <- 0                                                    1 operation
for i <- 0 to n-1                                             n-1 operations
    for j <- 0 to n-1                                         (n-1)^2 operations
        value <- value + (ArrayA[i] * ArrayB[j])              3(n-1)^2 operations
return value                                                  1 operation

Total primitive operations: n² + 2n - 1, giving it a time complexity of O(n²).

回答1:

By applying a bit of algebra:

So here is an algorithm which computes the same result in O(n) time:

sum_A ← 0
for i ← 0 to n-1
    sum_A ← sum_A + ArrayA[i]

sum_B ← 0
for j ← 0 to n-1
    sum_B ← sum_B + ArrayB[j]

return sum_A * sum_B

Generally speaking, an algorithm with nested loops cannot always be changed to reduce the time complexity; but in some cases you can do it, if you can identify something specific about the computation which means it can be done in a different way.

For sums like this, it's sometimes possible to compute the result more efficiently by writing something algebraically equivalent. So, put your mathematician's hat on when faced with such a problem.

回答2:

This type of operation is going to only ever run in n2 time. The reason being is that you have to compare each element of i, to each element of j. For example:

i*j, i*j+1,...,i*j+(n-1)
(i+1)*j, (i+1)*(j+1),...,(i+1)*(j+n-1)
.
.
.
(i+n-1)*j, (i+n-1)*(j+1),...,(i+n-1)*(j+n-1)

There's just no way to reduce the complexity.

来源：https://stackoverflow.com/questions/60551353/reducing-the-time-complexity-of-an-algorithm-with-nested-loops