Question

There are N patients (numbered from 0 to

n-1) who want to visit the doctor. The doctor
has S possible appointment slots, numbered
from 1 to S. Each of the patients has two
preferences. Patient K would like to visit the
doctor during either slot A[k] or slot B[k]. The
doctor can treat only one patient during each
slot.
Is it possible to assign every patient to one of their preferred slots so that there will be at most one patient assigned to each slot?
Input: Two arrays A and B, both of N integers, and an integer S
1 <= N <= 10⁶
1 <= S <= 2*10⁶
1 <= A[i] <= S
1 <= B[i] <= S
no patient has two preference for the same slot i.e. A[i] != B[i]
Output: print "true" if it is possible to assign every patient to one of their preferred slots, one patient to one slot, and "false" otherwise.

Answer 1

The task involves assigning patients to preferred appointment slots without overlap, akin to a bipartite graph matching problem, where the outcome 'true' or 'false' signifies whether each patient can be scheduled in one of their chosen slots without any slot overlap.

Step-by-step explanation:

When determining whether it is possible to assign each patient to one of their preferred appointment slots without any overlaps, we are dealing with a problem that can be modeled using graph theory or combinatorics. This problem is similar to finding a matching in a bipartite graph where each vertex on one side represents a patient and vertices on the other side represent available slots. Edges would connect patients to their preferred slots. The goal is to find a matching that covers all patient vertices.

With the given inputs:

1. Array A represents the first choice of slots for patients.
2. Array B represents the second choice of slots for patients.
3. S represents the total number of slots available.

We can iterate over the patients and try to assign slots based on their preferences. This requires ensuring that a slot is not assigned to more than one patient. If we manage to find assignments for all patients within the constraints, we print 'true'; otherwise, we print 'false'.

For the problem given in Solution 3.1, the approach is to check if there are patients whose preferred slots overlap with others and if there are enough alternative slots to accommodate each patient. The calculation of probabilities regards a different problem, related to the duration between patients' arrivals at a facility, which requires knowledge of exponential distribution.