Solving the UVa 514 Rails Problem: A Comprehensive Guide

Welcome to this in-depth guide on how to solve the UVa 514 problem, also known as the Rails Problem. This tutorial covers the problem statement, the necessary operations, and step-by-step instructions on how to simulate the process using a stack.

Problem Statement

The UVa 514 problem involves arranging train cars in a specific order as they leave a yard. You are given a sequence of train cars represented by integers, and you need to determine if it's possible to achieve the desired output sequence using the operations of pushing cars onto the stack and popping cars from the stack.

Input and Operations

Input: The input consists of a list of integers representing the order of the train cars.

Operations: Trains can enter the yard by pushing onto the stack. Trains can leave the yard by popping from the stack.

Output: Determine if it is possible to achieve the desired output sequence given the constraints of the stack.

Steps to Solve the Problem

Simulate the Process

The key to solving this problem is to use a stack to simulate the entry and exit of train cars. The objective is to match the desired output sequence with the order of cars in the stack.

Maintain Order

Follow these steps for each car in the desired output sequence:

If the next car is already on top of the stack, pop it from the stack. If it is not on top of the stack, use the input sequence to push additional cars onto the stack until you either find the next car or exhaust the input sequence.

Check Feasibility

If you can successfully pop all cars from the stack in the desired order, then the output is feasible. If not, it indicates that the desired order cannot be achieved.

Example

Let's take an example where the input sequence is 1 2 3 4 5 and the desired output sequence is 4 5 3 2 1.

Push 1, 2, 3, 4 onto the stack. Pop 4 (desired next car). Push 5 and pop 5. Continue this process until the stack is empty or you exhaust the input sequence.

Pseudocode

Here’s a simple pseudocode outline for the solution:

initialize stack
set next_input_index to 0
for each car in desired_output:
    while stack

is empty or top of stack is not car:

    if next_input_index  total_cars:
        push input_sequence[next_input_index] onto stack
        next_input_index   1
    else:
        break
    if top of stack is car:
        pop from stack
    else:
        print "Not possible"
        return

print "Possible"

Complexity

The time complexity of this solution is O(n), where n is the number of train cars, since each car is pushed and popped at most once. This makes the algorithm efficient for large input sizes.

Conclusion

By following this simulation approach, you can determine whether the desired order of train cars can be achieved using a stack-based system. This method not only solves the problem but also provides a clear understanding of the underlying logic and operations involved.

If you have any questions or need further clarification, feel free to reach out! Happy coding!