The Water Jug Problem is a classic puzzle that has intrigued mathematicians and computer scientists for years. It involves measuring a specific amount of water using two jugs of different capacities. Despite its simplicity, the problem offers deep insights into problem-solving strategies and has become a popular example in artificial intelligence (AI) for teaching concepts like search algorithms and state space representation.

In this blog, we will explore the Water Jug Problem in detail. We’ll cover the problem statement, various approaches like BFS and DFS, its significance in AI, practical applications, and recent advancements. By the end, you’ll have a thorough understanding of how this problem fits into the broader field of AI.

Problem Statement

You have two jugs, one with a capacity of X litres and the other with a capacity of Y litres. Your objective is to measure exactly Z litres of water by:

Filling either jug.

Emptying either jug.

Pour water from one jug to the other until one jug is full or the other is empty.

Justification: You can achieve 4 litres by filling the 5-litre jug and then pouring water into the 3-litre jug until it’s full. This leaves exactly 2 litres in the 5-litre jug. Empty the 3-litre jug and pour the remaining 2 litres into it. Then, refill the 5-litre jug and pour water into the 3-litre jug until it’s full again, which leaves exactly 4 litres in the 5-litre jug.

Justification: There is no way to measure exactly 5 litres using these jugs. The capacities do not allow for the exact measurement of 5 litres, as the operations only allow you to measure 2, 4, 6, and 8 litres, but not 5.

Justification: You can measure 8 litres by filling the 9-litre jug and then pouring water into the 7-litre jug until it’s full. This leaves exactly 2 litres in the 9-litre jug. Then, empty the 7-litre jug and pour the remaining 2 litres into it. Finally, refill the 9-litre jug and pour water into the 7-litre jug until it’s full, leaving exactly 4 litres in the 9-litre jug.

We can solve the problem using 2 approaches: BFS (Breadth-first search) and DFS (Depth-first search). Let’s learn each approach one by one.

Problem Constraints

jug1Capacity and jug2Capacity must be positive integers, where 1 ≤ X, Y ≤ 1000.

targetAmount (Z) is a non-negative integer, where 0 ≤ Z ≤ X + Y.

Both jugs are unmarked, meaning there are no measurement lines to guide partial fills.

You can perform the operations (fill, empty, pour) any number of times.

Water cannot be split in any other way except by transferring between the two jugs.

Using the BFS Approach

The Breadth-First Search (BFS) approach is used to solve the Water Jug Problem by exploring all possible states of the two jugs systematically. The key idea is to treat each state, representing the amount of water in each jug, as a node in a graph. BFS ensures that the shortest sequence of operations is explored first, making it ideal for this problem where we want to find the fastest way to measure the target amount of water.

In this approach, we start with both jugs empty and use a queue to keep track of all possible actions that can be taken. Each action (filling, emptying, or pouring between jugs) generates a new state, which is added to the queue if it hasn’t been visited before. We continue this process until either the target amount of water is measured or all possible states have been explored. This method guarantees that if a solution exists, it will be found in the shortest number of operations.

Step-by-Step Algorithm

Initialization:

If the targetAmount exceeds the combined capacity of both jugs, return False, as it’s impossible to measure that amount.

Initialise a queue with the starting state where both jugs are empty: (0, 0).

Create a set called visited to track the states that have already been explored.

BFS Loop:

While the queue is not empty, pop the first state (jug1, jug2) from the queue.

If either jug contains the targetAmount or their combined water equals the target, return True.

Generate New States:

For the current state (jug1, jug2), generate all possible new states by performing the following actions:

Fill jug1.

Fill jug2.

Empty jug1.

Empty jug2.

Pour water from jug2 into jug1 until jug1 is full or jug2 is empty.

Pour water from jug1 into jug2 until jug2 is full or jug1 is empty.

Check and Add New States:

For each new state (new_jug1, new_jug2) generated in the previous step, check if it has already been visited.

If it hasn’t been visited, add it to the queue and mark it as visited by adding it to the visited set.

End of Search:

If the queue is empty and no solution has been found, return False, as all possible states have been explored without reaching the targetAmount.

from collections import deque
class Solution:
def canMeasureWater(self, jug1Capacity: int, jug2Capacity: int, targetAmount: int) -> bool:
# Special case: If targetAmount is greater than the total capacity, return False
if targetAmount > jug1Capacity + jug2Capacity:
return False
# Initialise a queue for BFS and a set to track visited states
queue = deque([(0, 0)]) # (amount in jug1, amount in jug2)
visited = set((0, 0)) # to avoid revisiting the same state
# Perform BFS
while queue:
jug1, jug2 = queue.popleft()
# If either jug contains the exact target amount, return True
if jug1 == targetAmount or jug2 == targetAmount or jug1 + jug2 == targetAmount:
return True
# List of all possible actions (fill, empty, transfer)
possible_actions = [
(jug1Capacity, jug2), # fill jug1
(jug1, jug2Capacity), # fill jug2
(0, jug2), # empty jug1
(jug1, 0), # empty jug2
(min(jug1 + jug2, jug1Capacity), jug2 - (min(jug1 + jug2, jug1Capacity) - jug1)), # transfer jug2 -> jug1
(jug1 - (min(jug1 + jug2, jug2Capacity) - jug2), min(jug1 + jug2, jug2Capacity)) # transfer jug1 -> jug2
]
# Check each possible action
for new_jug1, new_jug2 in possible_actions:
if (new_jug1, new_jug2) not in visited:
visited.add((new_jug1, new_jug2)) # mark state as visited
queue.append((new_jug1, new_jug2)) # add new state to the queue
# If all possible states are exhausted and no solution is found, return False
return False
def main():
solution = Solution()
# Example 1
jug1Capacity = 3
jug2Capacity = 5
targetAmount = 4
result = solution.canMeasureWater(jug1Capacity, jug2Capacity, targetAmount)
print(f"Example 1: Can we measure {targetAmount} litres? {result}")
# Example 2
jug1Capacity = 2
jug2Capacity = 6
targetAmount = 5
result = solution.canMeasureWater(jug1Capacity, jug2Capacity, targetAmount)
print(f"Example 2: Can we measure {targetAmount} litres? {result}")
if __name__ == "__main__":
main()

Java Code

import java.util.*;
public class Solution {
// Method to check if we can measure the targetAmount using two jugs
public boolean canMeasureWater(int jug1Capacity, int jug2Capacity, int targetAmount) {
// If target amount exceeds the total capacity of both jugs combined, it's impossible
if (targetAmount > jug1Capacity + jug2Capacity) {
return false;
}
// Queue to perform BFS, starts with both jugs empty
Queue<int[]> queue = new LinkedList<>();
queue.add(new int[]{0, 0}); // (amount in jug1, amount in jug2)
// Set to keep track of visited states to prevent revisiting
Set<String> visited = new HashSet<>();
visited.add("0,0"); // Initially, both jugs are empty
// BFS loop
while (!queue.isEmpty()) {
int[] current = queue.poll();
int jug1 = current[0];
int jug2 = current[1];
// If either jug contains the exact target amount, or combined, return true
if (jug1 == targetAmount || jug2 == targetAmount || jug1 + jug2 == targetAmount) {
return true;
}
// List of possible actions
List<int[]> nextStates = Arrays.asList(
new int[]{jug1Capacity, jug2}, // Fill jug1
new int[]{jug1, jug2Capacity}, // Fill jug2
new int[]{0, jug2}, // Empty jug1
new int[]{jug1, 0}, // Empty jug2
new int[]{Math.min(jug1 + jug2, jug1Capacity), jug2 - (Math.min(jug1 + jug2, jug1Capacity) - jug1)}, // Pour jug2 -> jug1
new int[]{jug1 - (Math.min(jug1 + jug2, jug2Capacity) - jug2), Math.min(jug1 + jug2, jug2Capacity)} // Pour jug1 -> jug2
);
// Explore all possible states
for (int[] state : nextStates) {
String stateStr = state[0] + "," + state[1];
if (!visited.contains(stateStr)) {
visited.add(stateStr);
queue.add(state);
}
}
}
// If BFS completes without finding a solution, return false
return false;
}
public static void main(String[] args) {
Solution solution = new Solution();
// Example 1
int jug1Capacity1 = 3;
int jug2Capacity1 = 5;
int targetAmount1 = 4;
System.out.println("Example 1: Can measure " + targetAmount1 + " litres? " + solution.canMeasureWater(jug1Capacity1, jug2Capacity1, targetAmount1));
// Example 2
int jug1Capacity2 = 2;
int jug2Capacity2 = 6;
int targetAmount2 = 5;
System.out.println("Example 2: Can measure " + targetAmount2 + " litres? " + solution.canMeasureWater(jug1Capacity2, jug2Capacity2, targetAmount2));
}
}

Complexity Analysis

Time Complexity:

In the worst case, the number of states explored is limited by the total number of unique combinations of water in both jugs. Each jug can have an amount between 0 and its maximum capacity.

Therefore, the time complexity is O(X * Y), where X is the capacity of jug1, and Y is the capacity of jug2. This accounts for all possible states of the two jugs.

Space Complexity:

The space complexity is also O(X * Y) due to the storage required for the queue and the visited set, which can potentially hold all possible states of the two jugs.

The Depth-First Search (DFS) approach for the Water Jug Problem explores all possible configurations of water in the two jugs by recursively diving into each potential action (fill, empty, or pour) until it either finds a solution or exhausts all options. DFS focuses on fully exploring one path before backtracking to try other possibilities. It checks each state to see if the target amount of water is reached and uses a set to track visited states, preventing infinite loops.

At each state, DFS performs the allowable actions: filling one of the jugs, emptying one of the jugs, or transferring water between the jugs. It continues to explore all possible actions until it either reaches the target amount or confirms that the target is unreachable. The recursive nature of DFS means that it may not always find the shortest solution, but it will still return the correct answer if a solution exists. By tracking visited states, the algorithm avoids revisiting the same configurations and reduces redundant work.

Step-by-Step Algorithm

Initial Check:

If the targetAmount is greater than the combined capacity of the two jugs, return False, since it’s impossible to measure that amount.

DFS Setup:

Use a helper function dfs() to recursively explore all possible states of water in both jugs.

Track visited states with a set to prevent reprocessing the same jug configurations.

Base Case in DFS:

At each recursive step, check if either jug or the sum of both jugs contains the targetAmount. If so, return True.

If the current state (amount in both jugs) has already been visited, return False to avoid loops.

Perform Actions:

For the current state of the jugs (jug1, jug2), generate all possible actions:

Fill jug1.

Fill jug2.

Empty jug1.

Empty jug2.

Transfer water from jug2 to jug1 until one of them is full or empty.

Transfer water from jug1 to jug2 until one of them is full or empty.

Recursively call dfs() for each new state generated by these actions.

Return Result:

If any of the recursive calls return True, then a solution has been found, and the function returns True. Otherwise, once all possible states have been explored, return False.

Python Code

class Solution:
def canMeasureWater(self, jug1Capacity: int, jug2Capacity: int, targetAmount: int) -> bool:
# If target amount exceeds the total capacity of both jugs, it's impossible
if targetAmount > jug1Capacity + jug2Capacity:
return False
# Set to keep track of visited states to prevent cycles
visited = set()
# DFS helper function
def dfs(jug1, jug2):
# If we have reached the target amount, return True
if jug1 == targetAmount or jug2 == targetAmount or jug1 + jug2 == targetAmount:
return True
# If this state has already been visited, return False
if (jug1, jug2) in visited:
return False
# Mark this state as visited
visited.add((jug1, jug2))
# Try all possible actions (fill, empty, pour)
return (dfs(jug1Capacity, jug2) or # fill jug1
dfs(jug1, jug2Capacity) or # fill jug2
dfs(0, jug2) or # empty jug1
dfs(jug1, 0) or # empty jug2
dfs(min(jug1 + jug2, jug1Capacity), jug2 - (min(jug1 + jug2, jug1Capacity) - jug1)) or # pour jug2 -> jug1
dfs(jug1 - (min(jug1 + jug2, jug2Capacity) - jug2), min(jug1 + jug2, jug2Capacity))) # pour jug1 -> jug2
# Start DFS with both jugs empty
return dfs(0, 0)
# Example usage
if __name__ == "__main__":
solution = Solution()
# Example 1
jug1Capacity = 3
jug2Capacity = 5
targetAmount = 4
print(f"Example 1: Can we measure {targetAmount} litres? {solution.canMeasureWater(jug1Capacity, jug2Capacity, targetAmount)}")
# Example 2
jug1Capacity = 2
jug2Capacity = 6
targetAmount = 5
print(f"Example 2: Can we measure {targetAmount} litres? {solution.canMeasureWater(jug1Capacity, jug2Capacity, targetAmount)}")

Java Code

import java.util.HashSet;
import java.util.Set;
public class Solution {
// Method to check if we can measure the target amount using two jugs
public boolean canMeasureWater(int jug1Capacity, int jug2Capacity, int targetAmount) {
// If target amount exceeds the total capacity of both jugs, it's impossible
if (targetAmount > jug1Capacity + jug2Capacity) {
return false;
}
// Set to keep track of visited states to avoid cycles
Set<String> visited = new HashSet<>();
// Call the DFS function with both jugs initially empty
return dfs(jug1Capacity, jug2Capacity, targetAmount, 0, 0, visited);
}
// DFS function to explore all possible states
private boolean dfs(int jug1Capacity, int jug2Capacity, int targetAmount, int jug1, int jug2, Set<String> visited) {
// If we reach the target amount, return true
if (jug1 == targetAmount || jug2 == targetAmount || jug1 + jug2 == targetAmount) {
return true;
}
// If this state has been visited before, return false
String state = jug1 + "," + jug2;
if (visited.contains(state)) {
return false;
}
// Mark this state as visited
visited.add(state);
// Try all possible actions (fill, empty, pour)
return dfs(jug1Capacity, jug2Capacity, targetAmount, jug1Capacity, jug2, visited) || // fill jug1
dfs(jug1Capacity, jug2Capacity, targetAmount, jug1, jug2Capacity, visited) || // fill jug2
dfs(jug1Capacity, jug2Capacity, targetAmount, 0, jug2, visited) || // empty jug1
dfs(jug1Capacity, jug2Capacity, targetAmount, jug1, 0, visited) || // empty jug2
dfs(jug1Capacity, jug2Capacity, targetAmount, Math.min(jug1 + jug2, jug1Capacity), jug2 - (Math.min(jug1 + jug2, jug1Capacity) - jug1), visited) || // pour jug2 -> jug1
dfs(jug1Capacity, jug2Capacity, targetAmount, jug1 - (Math.min(jug1 + jug2, jug2Capacity) - jug2), Math.min(jug1 + jug2, jug2Capacity), visited); // pour jug1 -> jug2
}
public static void main(String[] args) {
Solution solution = new Solution();
// Example 1
int jug1Capacity1 = 3;
int jug2Capacity1 = 5;
int targetAmount1 = 4;
System.out.println("Example 1: Can we measure " + targetAmount1 + " litres? " + solution.canMeasureWater(jug1Capacity1, jug2Capacity1, targetAmount1));
// Example 2
int jug1Capacity2 = 2;
int jug2Capacity2 = 6;
int targetAmount2 = 5;
System.out.println("Example 2: Can we measure " + targetAmount2 + " litres? " + solution.canMeasureWater(jug1Capacity2, jug2Capacity2, targetAmount2));
}
}

Complexity Analysis

Time Complexity:

The time complexity of the DFS approach is O(X * Y), where X is the capacity of jug1 and Y is the capacity of jug2. This is because each unique combination of water in the two jugs can form a possible state, and the DFS explores all of them.

Since the set of visited states ensures that no state is visited more than once, the time complexity is proportional to the total number of possible states.

Space Complexity:

The space complexity is also O(X * Y) due to the storage required for the visited set, which keeps track of all the states that have been explored.

Additionally, DFS is recursive, so the call stack adds extra space overhead. In the worst case, the recursion depth can reach X * Y, leading to O(X * Y) space complexity as well.

Significance in AI

The Water Jug Problem is a fundamental example in AI that demonstrates problem-solving techniques using search algorithms. Its simplicity helps in understanding key concepts in AI.

Search Algorithms: It introduces the concepts of BFS and DFS, which are essential in navigating state spaces in AI.

State Space Representation: The problem models real-world issues where a series of decisions lead to a specific goal, a common scenario in AI problem-solving.

Optimization: By exploring different paths to reach a solution, the problem highlights the trade-offs between efficiency and thoroughness in AI search techniques.

Applications in Planning: The problem-solving approach used here is similar to AI planning systems where steps must be carefully chosen to reach a desired outcome.

Applications of the Water Jug Problem

The Water Jug Problem, though simple, has several real-world applications and is used to illustrate important concepts in computer science and AI. Here are some key applications:

Puzzle Solving: It is often used to demonstrate problem-solving techniques in algorithms, helping students and professionals understand BFS and DFS in action.

AI Pathfinding: The Water Jug Problem can model situations where AI systems need to reach a goal through a series of valid operations, similar to pathfinding algorithms in AI.

Robotics: Robots use similar decision-making processes when navigating environments or performing tasks with limited resources, such as water distribution or energy management.

Optimization Problems: This problem is analogous to resource allocation issues, where a system needs to manage resources efficiently to achieve a specific target.

Game Development: In games where characters need to solve puzzles, the logic behind the Water Jug Problem can be applied to design interactive problem-solving scenarios.

Challenges and Advancements

The Water Jug Problem, while simple, presents several challenges and opportunities for improvement, especially when applied to more complex systems.

State Explosion: As the number of operations and jug sizes increase, the number of possible states grows rapidly, making the problem computationally expensive.

Inefficient Solutions: DFS may not always provide the most efficient solution path, highlighting the challenge of finding the shortest sequence of operations.

Advanced Search Algorithms: Techniques like A* search and dynamic programming can be applied to optimise the solution-finding process, reducing the number of explored states.

Real-World Adaptation: Scaling the problem to real-world applications, such as robotics or resource management, requires integrating more advanced AI techniques to handle large state spaces efficiently.

Conclusion

The Water Jug Problem is an excellent example for understanding how basic AI search algorithms like BFS and DFS work. It simplifies complex decision-making processes, making it an ideal teaching tool for beginners in computer science and AI. Through this problem, we can explore how different algorithms handle problem-solving in structured environments.

Beyond its academic use, the Water Jug Problem has practical applications in fields such as robotics, resource allocation, and game development. It highlights key AI concepts such as state space representation, optimization, and pathfinding. As advancements in AI continue, more efficient algorithms will further improve the ability to solve resource management challenges based on the principles of this problem. Passionate about learning Artificial Intelligence? Try the Certificate Program in Artificial Intelligence and Machine Learning by Hero Vired.

FAQs

What is the Water Jug Problem?

It involves measuring a specific amount of water using two jugs with different capacities.

What are the key operations allowed in the Water Jug Problem?

You can fill, empty, or transfer water between the two jugs.

Which search algorithms can solve the Water Jug Problem?

Both BFS (Breadth-First Search) and DFS (Depth-First Search) can solve it.

Is the Water Jug Problem relevant in AI?

Yes, it demonstrates key concepts like state space and problem-solving using search algorithms.

What is the time complexity of solving the Water Jug Problem?

The time complexity is O(X * Y), where X and Y are the jug capacities.

Can the problem always be solved?

No, it can only be solved if the target amount is achievable using the jug capacities.

What are the real-world applications of the Water Jug Problem?

It applies to resource management, robotics, and game development.

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