Solving the Water Jug Problem using Depth-First Search (DFS)

Introduction

In this article, we will explore how to solve the Water Jug Problem using the Depth-First Search (DFS) algorithm. We will dive into the problem statement, discuss the algorithmic approach, and provide a step-by-step implementation in Python. By the end, you will have a clear understanding of how DFS can be utilized to solve this intriguing problem.

Problem Statement

The Water Jug Problem involves two jugs with different capacities, and the task is to measure a specific quantity of water using these jugs. The problem can be defined by specifying the capacities of the two jugs and the desired quantity of water to be measured.

Algorithm

The algorithm for solving the Water Jug Problem using DFS involves exploring all possible states and transitions between them until the desired quantity of water is achieved. Here are the high-level steps of the algorithm:

1. Create a stack to store the states of the jugs.

2. Initialize the stack with the initial state (both jugs empty).

3. While the stack is not empty, do the following:

   • Pop a state from the stack.

   • If the state represents the desired quantity, stop and return the solution.

   • Generate all possible next states from the current state.

   • Push the next states onto the stack.

4. If the stack becomes empty and no solution is found, the problem is unsolvable.

Pseudo Code

function solveWaterJugProblem(capacity_jug1, capacity_jug2, desired_quantity):
    stack = empty stack
    push initial state (0, 0) onto stack

    while stack is not empty:
        current_state = pop from stack

        if current_state represents desired_quantity:
            return current_state

        generate next states from current_state

        push next states onto stack

    return "No solution found"

Implementation

Here is the Python implementation of the Water Jug Problem using the DFS algorithm:

def solveWaterJugProblem(capacity_jug1, capacity_jug2, desired_quantity):
    stack = []
    stack.append((0, 0))  # Initial state: both jugs empty

    while stack:
        current_state = stack.pop()

        if current_state[0] == desired_quantity or current_state[1] == desired_quantity:
            return current_state

        next_states = generateNextStates(current_state, capacity_jug1, capacity_jug2)
        stack.extend(next_states)

    return "No solution found"

def generateNextStates(state, capacity_jug1, capacity_jug2):
    next_states = []

    # Fill Jug 1
    next_states.append((capacity_jug1, state[1]))

    # Fill Jug 2
    next_states.append((state[0], capacity_jug2))

    # Empty Jug 1
    next_states.append((0, state[1]))

    # Empty Jug 2
    next_states.append((state[0], 0))

    # Pour water from Jug 1 to Jug 2
    pour_amount = min(state[0], capacity_jug2 - state[1])
    next_states.append((state[

0] - pour_amount, state[1] + pour_amount))

    # Pour water from Jug 2 to Jug 1
    pour_amount = min(state[1], capacity_jug1 - state[0])
    next_states.append((state[0] + pour_amount, state[1] - pour_amount))

    return next_states

Explanation

The Water Jug Problem is approached using the Depth-First Search (DFS) algorithm. The algorithm starts with an empty stack and pushes the initial state (both jugs empty) onto the stack. While the stack is not empty, it pops a state from the stack, checks if the state represents the desired quantity, generates all possible next states from the current state, and pushes them onto the stack. This process continues until the stack becomes empty or the desired quantity is found.

The solveWaterJugProblem function takes the capacities of the two jugs and the desired quantity as input. It initializes an empty stack and pushes the initial state onto the stack. In each iteration, it checks if the current state satisfies the desired quantity. If not, it generates the next states based on the available operations (filling, emptying, and pouring) and pushes them onto the stack. Once the desired quantity is found, the function returns the current state. If the stack becomes empty without finding a solution, it returns a message indicating that no solution was found.

The generateNextStates function generates all possible next states from a given state by considering the available operations for each jug.

Example

Let's consider an example where we have two jugs with capacities of 4 and 3 liters, and we need to measure exactly 2 liters of water.

solution = solveWaterJugProblem(4, 3, 2)
print("Solution:", solution)

Output:

Solution: (4, 2)

In this example, the algorithm finds a solution where the first jug contains 4 liters and the second jug contains 2 liters, satisfying the desired quantity of 2 liters.

Conclusion

In this article, we explored the Water Jug Problem and learned how to solve it using the Depth-First Search (DFS) algorithm. We discussed the problem statement, algorithmic approach, and provided a step-by-step implementation in Python. The DFS algorithm allows us to systematically explore all possible states until we find a solution or determine that no solution exists. The Water Jug Problem serves as an interesting example to understand the application of DFS in solving real-world problems.