Formatted question description: https://leetcode.ca/all/1606.html

# 1606. Find Servers That Handled Most Number of Requests (Hard)

You have k servers numbered from 0 to k-1 that are being used to handle multiple requests simultaneously. Each server has infinite computational capacity but cannot handle more than one request at a time. The requests are assigned to servers according to a specific algorithm:

• The ith (0-indexed) request arrives.
• If all servers are busy, the request is dropped (not handled at all).
• If the (i % k)th server is available, assign the request to that server.
• Otherwise, assign the request to the next available server (wrapping around the list of servers and starting from 0 if necessary). For example, if the ith server is busy, try to assign the request to the (i+1)th server, then the (i+2)th server, and so on.

You are given a strictly increasing array arrival of positive integers, where arrival[i] represents the arrival time of the ith request, and another array load, where load[i] represents the load of the ith request (the time it takes to complete). Your goal is to find the busiest server(s). A server is considered busiest if it handled the most number of requests successfully among all the servers.

Return a list containing the IDs (0-indexed) of the busiest server(s). You may return the IDs in any order.

Example 1:

Input: k = 3, arrival = [1,2,3,4,5], load = [5,2,3,3,3]
Output: [1]
Explanation:
All of the servers start out available.
The first 3 requests are handled by the first 3 servers in order.
Request 3 comes in. Server 0 is busy, so it's assigned to the next available server, which is 1.
Request 4 comes in. It cannot be handled since all servers are busy, so it is dropped.
Servers 0 and 2 handled one request each, while server 1 handled two requests. Hence server 1 is the busiest server.


Example 2:

Input: k = 3, arrival = [1,2,3,4], load = [1,2,1,2]
Output: [0]
Explanation:
The first 3 requests are handled by first 3 servers.
Request 3 comes in. It is handled by server 0 since the server is available.
Server 0 handled two requests, while servers 1 and 2 handled one request each. Hence server 0 is the busiest server.


Example 3:

Input: k = 3, arrival = [1,2,3], load = [10,12,11]
Output: [0,1,2]
Explanation: Each server handles a single request, so they are all considered the busiest.


Example 4:

Input: k = 3, arrival = [1,2,3,4,8,9,10], load = [5,2,10,3,1,2,2]
Output: [1]


Example 5:

Input: k = 1, arrival = [1], load = [1]
Output: [0]


Constraints:

• 1 <= k <= 105
• 1 <= arrival.length, load.length <= 105
• arrival.length == load.length
• 1 <= arrival[i], load[i] <= 109
• arrival is strictly increasing.

Related Topics:
Ordered Map

## Solution 1. Set + Heap

set<int> free contains the index of available servers. busy is a min-heap each item of which is a pair of { endTime, serverIndex }.

When we find an available server, we erase it from free, and put { endTime, serverIndex } into a min-heap busy.

For each arrival[i], we first free all those servers from busy whose endTime is smaller than or equal to arrival[i], then find the first available server in free whose index is greater than i % k in circular order.

About time complexity, iterating the N elements in A and L takes O(N), within each iteration, the amortized time complexity is O(logK). It’s because the sizes of the free and busy are at most K, so each push and pop operation takes O(logK) time.

Note that for the loop popping busy, since at most we can pop busy K times and each pop takes O(logK) time, it looks like it’s O(KlogK) and the busy’s popping operation takes O(NKlogK) overall. But since we at most pop busy N - K times through out the entire function, so the busy’s popping operation takes at most (N - k)logK time. Thus the funtion’s overall time complexity is still O(NlogK).

// OJ: https://leetcode.com/problems/find-servers-that-handled-most-number-of-requests/

// Time: O(N * logK)
// Space: O(K)
// Ref: https://leetcode.com/problems/find-servers-that-handled-most-number-of-requests/discuss/876793/Java-O(nlogn)-use-both-TreeSet-and-PriorityQueue
class Solution {
public:
vector<int> busiestServers(int k, vector<int>& A, vector<int>& L) {
vector<int> cnt(k);
set<int> free;
for (int i = 0; i < k; ++i) free.insert(i);
priority_queue<pair<int, int>, vector<pair<int, int>>, greater<>> busy; // endTime, serverIndex
for (int i = 0; i < A.size(); ++i) {
int start = A[i], end = start + L[i];
while (busy.size() && busy.top().first <= start) {
int server = busy.top().second;
busy.pop();
free.insert(server);
}
if (free.empty()) continue;
auto it = free.lower_bound(i % k);
if (it == free.end()) it = free.begin();
cnt[*it]++;
busy.emplace(end, *it);
free.erase(*it);
}
vector<int> ans;
int mx = *max_element(begin(cnt), end(cnt));
for (int i = 0; i < k; ++i) {
if (cnt[i] == mx) ans.push_back(i);
}
return ans;
}
};


## Solution 2.

free[t] is the available time of server t. Initially free[t] = 0 for all 0 <= t < k. When a server t takes request i, free[t] = arrival[i] + load[i], i.e. it’s free at the end time of the request.

map<int, int> m is a map from the start time of a request to its load/length. It stores the requests that are not yet handled.

We round-robin visit the servers. For the i-th request, we put it into the request pool m as m[arrival[i]] = load[i]. Its corresponding i % k-th server is free at free[i % k], so we scan (binary search) in the m to find requests that this i % k-th server can handle.

Note that all the requests in m thus far are the leftover requests that can’t be handled by previous servers. So this i % k-th server can just take whatever it can handle. Every time it handles a request, update its free time to be the end time of the request, increment cnt[i % k], remove the request from m, and keep finding the next request that it can handle.

Every time a server successfully handles a request, we mark this round i as the last successful round. If we’ve scanned k servers since the last successful round but still haven’t handled any leftover requests, then no server could handle those leftover requests, we should break now.

// OJ: https://leetcode.com/problems/find-servers-that-handled-most-number-of-requests/

// Time: O(NlogN)
// Space: O(N)
// Ref: https://leetcode.com/problems/find-servers-that-handled-most-number-of-requests/discuss/876998/C%2B%2B-Map
class Solution {
public:
vector<int> busiestServers(int k, vector<int>& A, vector<int>& L) {
vector<int> cnt(k), free(k), ans;
map<int, int> m;
for (int i = 0, last = 0;; ++i) {
if (i < A.size()) m[A[i]] = A[i] + L[i];
else if (i - last > k) break; // If we've scanned k servers since the last time we handle a request, but still haven't handled any leftover requests, then no server could handle those leftover requests, break.
auto it = m.lower_bound(free[i % k]);
while (it != end(m)) { // when there are requests whose start times are greater than or equal to the free time of server i % k
last = i; // update the last successfully handled request.
++cnt[i % k]; // let this i % k-th server handle this request.
free[i % k] = it->second;
m.erase(it); // remove this request
it = m.lower_bound(free[i % k]);
} // all the leftover requests in m are passed over to the next server to handle
}
int mx = *max_element(begin(cnt), end(cnt));
for (int i = 0; i < k; ++i) {
if (cnt[i] == mx) ans.push_back(i);
}
return ans;
}
};


Java

class Solution {
public List<Integer> busiestServers(int k, int[] arrival, int[] load) {
int[] requests = new int[k];
TreeSet<Integer> set = new TreeSet<Integer>();
for (int i = 0; i < k; i++)
set.add(i);
PriorityQueue<int[]> priorityQueue = new PriorityQueue<int[]>(new Comparator<int[]>() {
public int compare(int[] array1, int[] array2) {
return array1[1] - array2[1];
}
});
int length = arrival.length;
for (int i = 0; i < length; i++) {
while (!priorityQueue.isEmpty() && priorityQueue.peek()[1] <= arrival[i])
set.add(priorityQueue.poll()[0]);
int server = i % k;
if (!set.isEmpty()) {
Integer nextServer = set.ceiling(server);
if (nextServer == null)
nextServer = set.first();
requests[nextServer]++;
set.remove(nextServer);
priorityQueue.offer(new int[]{nextServer, arrival[i] + load[i]});
}
}
int maxRequests = requests[0];
for (int i = 1; i < k; i++)
maxRequests = Math.max(maxRequests, requests[i]);
List<Integer> busiest = new ArrayList<Integer>();
for (int i = 0; i < k; i++) {
if (requests[i] == maxRequests)
busiest.add(i);
}
return busiest;
}
}