nested for loop optimisation - c++

Question

I have this expression
for (size_t i = 0; i < expression.size(); i++){
for (size_t j = i + 1; j < expression.size(); j++){
result += (expression.at(j) - expression.at(i));
}
result += (g - expression.at(i));
}
return result;
in the vector expression we have for example [1,2,3]. I am trying to get something like:
f1=[(2-1)+(3-1)]
r1 = g-1
h1 = r1+f1
f2=[3-2]
r2 = g-2
h2 = r2+f2
f3 = 0
r3 = g-3
h3 = r3+f3
then h1+h2+h3
What i am doing right now is in Θ(n^2). Is there a way to make it faster even without for loops?

Answer 1

Addition is commutative and associative so the operations can be reordered and grouped without changing the final result. (Note: not taking into account possible overflows in the intermediate calculations, which may be affected by the order and grouping of operations.)
In pseudo-code with n = expression.size() and x[k] = expression.at(k) the original code can be broken down as follows, with the intermediate results indicated in comments.
a = b = c = d = 0
for i = 0 to (n-1)
for j = (i+1) to (n-1)
a += x[j]
// == x[i+1] + x[i+2] + ... x[n-1]
// a == 0 * x[0] + 1 * x[1] + 2 * x[2] + 3 * x[3] + ... + (n-1) * x[n-1]
for i = 0 to (n-1)
for j = (i+1) to (n-1)
b += x[i];
// == (n-i-1) * x[i]
// b == (n-1) * x[0] + (n-2) * x[1] + ... + 2 * x[n-3] + 1 * x[n-2]
for i = 0 to (n-1)
c += g
// c == n * g
for i = 0 to (n-1)
d += expression.at(i))
// d == x[0] + x[1] + ... + x[n-1]
result = c + a - b - d
= n * g
+ (0 - (n-1) - 1) * x[0]
+ (1 - (n-2) - 1) * x[1]
+ ...
+ ((n-2) - 1 - 1) * x[n-2]
+ ((n-1) - 0 - 1) * x[n-1]
The latter result can be calculated directly from that formula, with one single O(n) loop.