Though the smartest solution to this problem is the two-pass solution, I have come up with a stack-based solution which requires O(n) space. It can be further optimized to eliminate the stack structure.

The idea is as follows: assign as few candies as possible. First assign 1 candy to the first kid, then go through the kid one by one. If we encounter kids in increasing rating value, we simply increment the candy number for each kid and we are done. But if we encounter kids in decreasing rating value, we need to fix previous candy values to avoid zero or negative candies to the current kid. Thus I use a stack to keep track of the kids with decreasing ratings, so that we can give more candies to them when the 2nd requirement is violated. I use a subroutine `clearStack`

to count the actual number of candies required for these kids. For each kid, there are three cases:

- current kid has higher rating than previous kid: in this case, we assign one more candy to this kid. Also pop all previous kids out (and count candies for them), and push current kid's candy number
- current kid has equal rating to previous kid: we just assign 1 candy to this kid. Use the same stack operation as 1.
- current kid has less rating to previous kid: we also assign only 1 candy to this kid temporarily, and push it to stack (to be fixed later)

I also use a sentinel in the end of ratings to trigger `clearStack`

at the last element.

Code is as follows:

```
class Solution {
public:
int candy(vector<int>& ratings) {
ratings.insert(ratings.end(), INT_MAX); //sentinel at the end
int n = ratings.size();
if(n <= 1) return n; //corner cases
int sum(0); //total number of candies
int curr(0); //number of candies for current kid
stack<int> s;
s.push(1); //candy for the first kid
for(int k = 1; k < n; ++k) {
if(ratings[k] >= ratings[k-1]) { //current higher or equal than previous
if(ratings[k] == ratings[k-1]) curr = 1;
else curr = s.top()+1;
sum += clearStack(s); //count all previous
s.push(curr); //push current candy number in stack (don't count yet)
} else { //lower than previous
s.push(1); //just give one candy to this kid, don't count yet
}
}
return sum;
}
//count the actual number of candies of kids in the stack
int clearStack(stack<int>& s) {
int sum = 0;
int curr(0), last(0);
while(!s.empty()) {
int curr = s.top(); s.pop();
if(curr <= last) curr = last+1;
sum += curr;
last = curr;
}
return sum;
}
};
```