One example of an NP-hard problem is the Traveling Salesman Problem (TSP), which involves finding the shortest possible route that visits every city in a given set exactly once and returns to the starting city. The TSP is NP-hard because it is not known to have a polynomial-time algorithm to solve it exactly for all instances, meaning that as the size of the problem increases, the time required to solve it grows exponentially.

On the other hand, an example of a P problem is finding the maximum or minimum value in an array of n numbers. This problem can be solved in linear time using a simple algorithm that compares each element of the array to the current maximum or minimum value and updates the value if necessary. This algorithm has a time complexity of O(n), which is considered a polynomial time.

To illustrate the difference in difficulty between these two types of problems, consider the following steps to solve each:

Traveling Salesman Problem (NP-hard)

Generate all possible permutations of the cities in the given set.
For each permutation, calculate the total distance of the route and store the shortest one found so far.
Return the shortest route found.
The TSP algorithm above has a time complexity of O(n!), which grows very quickly as the size of the problem increases, making it infeasible to solve for large instances.

Finding the maximum or minimum value in an array (P)

Set the current maximum or minimum value to the first element of the array.
Iterate over the remaining elements of the array, comparing each element to the current maximum or minimum value and updating it if necessary.
Return the maximum or minimum value found.
The algorithm to find the maximum or minimum value in an array above has a time complexity of O(n), which is a polynomial time, making it much easier to solve than the TSP.
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