""" This is a python implementation for questions involving task assignments between people. Here Bitmasking and DP are used for solving this. Question :- We have N tasks and M people. Each person in M can do only certain of these tasks. Also a person can do only one task and a task is performed only by one person. Find the total no of ways in which the tasks can be distributed. """ from __future__ import print_function from collections import defaultdict class AssignmentUsingBitmask: def __init__(self,task_performed,total): self.total_tasks = total #total no of tasks (N) # DP table will have a dimension of (2^M)*N # initially all values are set to -1 self.dp = [[-1 for i in range(total+1)] for j in range(2**len(task_performed))] self.task = defaultdict(list) #stores the list of persons for each task #finalmask is used to check if all persons are included by setting all bits to 1 self.finalmask = (1<<len(task_performed)) - 1 def CountWaysUtil(self,mask,taskno): # if mask == self.finalmask all persons are distributed tasks, return 1 if mask == self.finalmask: return 1 #if not everyone gets the task and no more tasks are available, return 0 if taskno > self.total_tasks: return 0 #if case already considered if self.dp[mask][taskno]!=-1: return self.dp[mask][taskno] # Number of ways when we dont this task in the arrangement total_ways_util = self.CountWaysUtil(mask,taskno+1) # now assign the tasks one by one to all possible persons and recursively assign for the remaining tasks. if taskno in self.task: for p in self.task[taskno]: # if p is already given a task if mask & (1<<p): continue # assign this task to p and change the mask value. And recursively assign tasks with the new mask value. total_ways_util+=self.CountWaysUtil(mask|(1<<p),taskno+1) # save the value. self.dp[mask][taskno] = total_ways_util return self.dp[mask][taskno] def countNoOfWays(self,task_performed): # Store the list of persons for each task for i in range(len(task_performed)): for j in task_performed[i]: self.task[j].append(i) # call the function to fill the DP table, final answer is stored in dp return self.CountWaysUtil(0,1) if __name__ == '__main__': total_tasks = 5 #total no of tasks (the value of N) #the list of tasks that can be done by M persons. task_performed = [ [ 1 , 3 , 4 ], [ 1 , 2 , 5 ], [ 3 , 4 ] ] print(AssignmentUsingBitmask(task_performed,total_tasks).countNoOfWays(task_performed)) """ For the particular example the tasks can be distributed as (1,2,3), (1,2,4), (1,5,3), (1,5,4), (3,1,4), (3,2,4), (3,5,4), (4,1,3), (4,2,3), (4,5,3) total 10 """