Water Jug Problem using Memoization

Given two jugs with the maximum capacity of m and n liters respectively. The jugs don’t have markings on them which can help us to measure smaller quantities. The task is to measure d liters of water using these two jugs. Hence our goal is to reach from initial state (m, n) to final state (0, d) or (d, 0).


Input: 4 3 2
Output: (0, 0) –> (4, 0) –> (4, 3) –> (0, 3) –> (3, 0) –> (3, 3) –> (4, 2) –> (0, 2)

Input: 5 2 4
Output: (0, 0) –> (5, 0) –> (5, 2) –> (0, 2) –> (2, 0) –> (2, 2) –> (4, 0)

Approach: An approach using BFS has been discussed in the previous post. In this post an approach using memoization and recursion has been discussed. At any point, there can be a total of six possibilities:

  • Empty the first jug completely
  • Empty the second jug completely
  • Fill the first jug
  • Fill the second jug
  • Fill the water from the second jug into the first jug until the first jug is full or the second jug has no water left
  • Fill the water from the first jug into the second jug until the second jug is full or the first jug has no water left

Approach: Using Recursion, visit all the six possible moves one by one until one of them returns True. Since there can be repetitions of same recursive calls, hence every return value is stored using memoization to avoid calling the recursive function again and returning the stored value.

Below is the implementation of the above approach:





# This function is used to initialize the 
# dictionary elements with a default value.
from collections import defaultdict
# jug1 and jug2 contain the value 
# for max capacity in respective jugs 
# and aim is the amount of water to be measured. 
jug1, jug2, aim = 4, 3, 2
# Initialize dictionary with 
# default value as false.
visited = defaultdict(lambda: False)
# Recursive function which prints the 
# intermediate steps to reach the final 
# solution and return boolean value 
# (True if solution is possible, otherwise False).
# amt1 and amt2 are the amount of water present 
# in both jugs at a certain point of time.
def waterJugSolver(amt1, amt2): 
    # Checks for our goal and 
    # returns true if achieved.
    if (amt1 == aim and amt2 == 0) or (amt2 == aim and amt1 == 0):
        print(amt1, amt2)
        return True
    # Checks if we have already visited the
    # combination or not. If not, then it proceeds further.
    if visited[(amt1, amt2)] == False:
        print(amt1, amt2)
        # Changes the boolean value of
        # the combination as it is visited. 
        visited[(amt1, amt2)] = True
        # Check for all the 6 possibilities and 
        # see if a solution is found in any one of them.
        return (waterJugSolver(0, amt2) or
                waterJugSolver(amt1, 0) or
                waterJugSolver(jug1, amt2) or
                waterJugSolver(amt1, jug2) or
                waterJugSolver(amt1 + min(amt2, (jug1-amt1)),
                amt2 - min(amt2, (jug1-amt1))) or
                waterJugSolver(amt1 - min(amt1, (jug2-amt2)),
                amt2 + min(amt1, (jug2-amt2))))
    # Return False if the combination is 
    # already visited to avoid repetition otherwise
    # recursion will enter an infinite loop.
        return False
print("Steps: ")
# Call the function and pass the
# initial amount of water present in both jugs.
waterJugSolver(0, 0)



0 0
4 0
4 3
0 3
3 0
3 3
4 2
0 2

Time complexity: O(M * N)
Auxiliary Space: O(M * N)

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