Gas Station
There are n gas stations in a circle. You have gas[i] fuel at station i and it costs cost[i] fuel to travel to the next station.
Return the starting station index if you can travel around the circuit once clockwise, or -1 if impossible.
If a solution exists, it is guaranteed to be unique.
Example:
1,2,3,4,5 3,4,5,1,2
3
- We start by calculating the total amount of gas available: gastotal​=1+2+3+4+5=15 and the total cost to travel around the circuit: costtotal​=3+4+5+1+2=15.
- Since gastotal​=costtotal​, it is possible to travel around the circuit once.
- We then calculate the cumulative gas level at each station:
- Station 0: 1−3=−2
- Station 1: −2+2−4=−4
- Station 2: −4+3−5=−6
- Station 3: −6+4−1=−3
- Station 4: −3+5−2=0
- The starting station index is the one where the cumulative gas level starts to be non-negative, which is station 3.
- Therefore, the output is 3​.
Constraints:
- 1 <= n <= 10^5
- 0 <= gas[i], cost[i] <= 10^4
Background Knowledge
The "Gas Station" problem is a classic example of a greedy algorithm problem. Greedy algorithms are used to solve optimization problems by making the locally optimal choice at each step, with the hope that these local choices will lead to a globally optimal solution. In this problem, we need to find the starting station index that allows us to travel around the circuit once clockwise. The key concept here is to understand how to use the given information about the amount of fuel at each station and the cost of traveling to the next station to make decisions about which station to start at.
The problem also involves interval concepts, as we are dealing with a circular arrangement of gas stations. This means that we need to consider the fact that the last station is connected to the first station, and we need to make sure that we have enough fuel to complete the circuit. The cumulative sum concept is also relevant here, as we need to keep track of the total amount of fuel we have and the total cost of traveling to each station.
To solve this problem, we need to have a good understanding of how to analyze the given data and make decisions based on that analysis. We need to consider the total fuel available and the total cost of traveling around the circuit, as well as the net fuel available at each station. By analyzing these factors, we can determine whether it is possible to travel around the circuit and, if so, which station we should start at.
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