There is a restaurant with a single chef. You are given an array customers, where customers[i] = [arrival_i, time_i]:

  • arrival_i is the arrival time of the ith customer. The arrival times are sorted in non-decreasing order.
  • time_i is the time needed to prepare the order of the ith customer.

When a customer arrives, he gives his order to the chef, and the chef starts preparing it once he is idle. The customer waits until his order is prepared. The chef does not prepare food for more than one customer at a time. The chef prepares food for customers in the order they were given in the input.

Return the average waiting time of all customers. Solutions within 10^-5 from the actual answer are considered accepted.

Examples

Example 1:

Input: customers = [[1,2],[2,5],[4,3]]
Output: 5.00000
Explanation:
1) The first customer arrives at time 1, the chef takes his order and starts preparing it immediately at time 1, and finishes at time 3, so the waiting time of the first customer is 3 - 1 = 2.
2) The second customer arrives at time 2, the chef takes his order and starts preparing it at time 3, and finishes at time 8, so the waiting time of the second customer is 8 - 2 = 6.
3) The third customer arrives at time 4, the chef takes his order and starts preparing it at time 8, and finishes at time 11, so the waiting time of the third customer is 11 - 4 = 7.
So the average waiting time = (2 + 6 + 7) / 3 = 5.00000.

Example 2:

Input: customers = [[5,2],[5,4],[10,3],[20,2]]
Output: 3.25000
Explanation:
1) The first customer arrives at time 5, the chef takes his order and starts preparing it immediately at time 5, and finishes at time 7, so the waiting time of the first customer is 7 - 5 = 2.
2) The second customer arrives at time 5, the chef takes his order and starts preparing it at time 7, and finishes at time 11, so the waiting time of the second customer is 11 - 5 = 6.
3) The third customer arrives at time 10, the chef takes his order and starts preparing it at time 11, and finishes at time 14, so the waiting time of the third customer is 14 - 10 = 4.
4) The fourth customer arrives at time 20, the chef takes his order and starts preparing it immediately at time 20, and finishes at time 22, so the waiting time of the fourth customer is 22 - 20 = 2.
So the average waiting time = (2 + 6 + 4 + 2) / 4 = 3.25000.

Constraints

  • 1 <= customers.length <= 10^5
  • 1 <= arrival_i, time_i <= 10^4
  • arrival_i <= arrival_{i+1} (arrival times are sorted in non-decreasing order)

Thinking Process

  1. Single Server Queue: Classic queueing theory problem
  • Greedy works when local optimal choices lead to global optimum.
  • Often sort first to make the greedy choice obvious.
  • Prove or sanity-check: would swapping two choices ever help?
Greedy choice pick locally best after sorting

Common Approaches

Typical techniques for this pattern:

Approach Time Space Notes
Sort + greedy (this problem) O(n log n) O(1) Interval scheduling, assignment
Local greedy choice O(n) O(1) Jump game, gas station
Greedy + heap O(n log n) O(n) Merge streams, room allocation
Exchange argument O(n) O(1) Prove greedy choice is safe

Solution

class Solution {
public:
    double averageWaitingTime(vector<vector<int>>& customers) {
        long long t = 0, totalTime = 0;
        for(auto& c: customers) {
            int arrival = c[0], order = c[1];
            if(t > arrival) {
                totalTime += t - arrival;
            } else {
                t = arrival;
            }
            totalTime += order;
            t += order;
        }
        return (double) totalTime / customers.size();
    }
};

Solution Explanation

Approach: Sort + greedy (this problem)

Key idea: 1. Single Server Queue: Classic queueing theory problem

How the code works:

  1. Single Server Queue: Classic queueing theory problem
    • Greedy works when local optimal choices lead to global optimum.
    • Often sort first to make the greedy choice obvious.
    • Prove or sanity-check: would swapping two choices ever help?

Walkthrough — input customers = [[1,2],[2,5],[4,3]], expected output 5.00000:

1) The first customer arrives at time 1, the chef takes his order and starts preparing it immediately at time 1, and finishes at time 3, so the waiting time of the first customer is 3 - 1 = 2. 2) The second customer arrives at time 2, the chef takes his order and starts preparing it at time 3, and finishes at time 8, so the waiting time of the second customer is 8 - 2 = 6. 3) The third customer arrives at time 4, the chef takes his order and starts preparing it at time 8, and finishes at time 11, so the waiting time of the third customer is 11 - 4 = 7. So the average waiting time = (2 + 6 + 7) / 3 = 5.00000.

Comparison of Solutions

Solution Code Length Readability Logic Clarity
Solution 1 Longer More explicit Clear if-else logic
Solution 2 Shorter More concise Elegant max() usage

Common Mistakes

  1. All customers arrive before chef finishes: Chef always busy
    • customers = [[1,10],[2,5],[3,3]]
    • Each customer waits for previous to finish
  2. Chef always idle: Customers arrive after chef finishes
    • customers = [[1,2],[5,3],[10,1]]
    • No waiting time, only order preparation time
  3. Single customer: customers = [[1,5]]
    • Waiting time = order time = 5
  4. Simultaneous arrivals: Multiple customers arrive at same time
    • customers = [[5,2],[5,4],[5,3]]
    • Processed sequentially, later ones wait longer
  5. Wrong waiting time calculation: Using start_time - arrival instead of finish_time - arrival
  6. Not handling chef idle case: Assuming chef is always busy
  7. Integer overflow: Not using long long for large sums
  8. Wrong order processing: Processing orders out of sequence
  9. Precision issues: Not using double for division

Key Takeaways

  1. Single Server Queue: Classic queueing theory problem
  2. Sequential Processing: Orders processed in arrival order
  3. Waiting Time Formula: finish_time - arrival_time
  4. Chef Availability: start_time = max(chef_free_time, arrival_time)
  5. Finish Time: finish_time = start_time + order_time

References

Template Reference