IOI13E - Editorial






Heap, Sorting, Binary Search


As is commonly the case for problems that ask how long some agents need to achieve a goal, the answer can be found by a binary search on the answer. So we need to decide whether the robots can clear the floor in S seconds.
We can simplify the problem slightly by noting that for each toy, we only need to know how many weak and how many small robots are able to move it, which can be found by binary searching the two lists (after sorting them). Of course, if a toy cannot be moved by any robot, return ­ -1.
Let’s first decide the actions of the weak robots, starting from the weakest. There will be some set of toys it can handle. Since they effectively differ only in size, the weakest robot should work from the largest downwards, so as to make the job of the small robots easier. Also, there is never any reason for it to move fewer than S toys, unless it runs out. Now consider the 2nd-weakest robot. There will be extra toys it can handle, plus any light toys that the weakest robot didn’t have time for. Since the weakest robot is finished, the difference in weights are irrelevant, and the 2nd-weakest robot should again work in decreasing order of size amongst the toys it can handle. The same process can be continued for the remaining weak robots.
Now consider the small robots, from largest to smallest. These can again take up to S toys, starting from the largest remaining one. If a robot is unable to handle the largest remaining toy, then S was too small.
Implementation can be done using a priority queue, implemented as a binary heap, representing toys that are light enough to be handled by the current robot and ordered with the largest at the head. The toys are initially sorted by weight. Each time a new weak robot is considered, new elements are added from the list of toys to the priority queue, and the robot then removes items starting from the head of the queue.
Assuming that T is larger than A, B, the running time will be O(T (log T)^2): one log T for the binary search, the other for the priority queue operations. I think that it may be possible to reduce this to something like O(T·log T·log(max(A, B))) using the right sort of data structure for the priority queue (to allow S items to be removed in log time): something like an interval tree for the number of toys of each size.

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