CareerCup

I believe O(n) is possible since they are sorted by start time.
Keep auxiliary variables storing the current maximum end time of the processes seen so far and the time the current set of consecutive processes started.

Go through the entries in the log in order. If the current process starts at a time larger than the previous maximum time the processor was busy, then this is a break (it was idle) and update the current start time.

// for simplicity, consider the log as an array with N entrys each with start and end times.
// in reality this would be just fetching from the log file - no big difference.
int GetMaxBusyTime(Entry* entries, int N) {
     int max_end = 0, res = 0, cur_begin = 0;
     for (int i = 0 ; i < N ; i++) {
          if (entries[i].start > max_end) { // it was idle before this one
              res = max(res , max_end - cur_begin + 1);
              cur_begin = entries[i].start;
              max_end = entries[i].end;
          }
          else {
              max_end = max(max_end, entries[i].end);
          }
     }
     return max(res , max_end - cur_begin + 1); // also check the final one
}