Algorithm for truck moving around a circle of gas stations

Question

You have a truck moving around a circular track with gas stations spaced out around the circle. Each station has a finite amount of gas. The gas tank on the truck is infinit

Answer 1

Yes O(n) is possible. Definitely not TSP.

Let x_i be the amount of gas available at station i minus the amount of gas required to go to next station.

A requirement is Σ x_i ≥ 0 (enough gas to complete a full circle).

Consider S_i = x₁ + x₂ + ... + x_i

Note that S_n ≥ 0.

Now pick the smallest (or even largest will do, making it easier to write code for) k such that S_k is the least and start at the station next to it.

Now for k < j ≤ n, we have the gas in tank = S_j - S_k ≥ 0.

for 1 ≤ j ≤ k, we have gas in tank = x_k+1 + .. + x_n + x₁ + x₂ + .. + x_j = S_n - S_k + S_j ≥ 0.

Thus starting at k+1 will ensure there is enough gas accumulated at each station to get to the next station.

// C++ code. gas[i] is the gas at station i, cost[i] is the cost from station i to (i+1)%n
int circ(vector &gas, vector &cost) {
    int min_S=INT_MAX, S=0, position=0;
    for(int i=0;i=0)
        return position;
    else
        return -1;
}