Real-world example of exponential time complexity

I'm looking for an intuitive, real-world example of a problem that takes (worst case) exponential time complexity to solve for a talk I am giving.

Here are examples fo开发者_开发问答r other time complexities I have come up with (many of them taken from this SO question):

• O(1) - determining if a number is odd or even
• O(log N) - finding a word in the dictionary (using binary search)
• O(N) - reading a book
• O(N log N) - sorting a deck of playing cards (using merge sort)
• O(N^2) - checking if you have everything on your shopping list in your trolley
• O(infinity) - tossing a coin until it lands on heads

Any ideas?

• O(10^N): trying to break a password by testing every possible combination (assuming numerical password of length N)

p.s. why is your last example is of complexity O(infinity) ? it's linear search O(N) .. there are less than 7 billion people in the world.

A pizza restaurant has several toppings to choose from

• Pepperoni
• Chilli peppers
• Pineapple (don't knock it until you've tried it!)

Customers may choose any combination of toppings or none at all for their pizza. Now consider an algorithm that finds every possible unique combination of toppings. This is an exponential algorithm with time complexity O(2^n).

Look how the possible combinations grow (exponentially) when you add a new topping to the menu:

0 toppings: 1 combination (no toppings at all)
1 toppings: 2 combinations (none, a)
2 toppings: 4 combinations (none, a, b, ab)
3 toppings: 8 combinations (none, a, b, c, ab, ac, bc, abc)
...
...
10 toppings: 1,024 combinations
20 toppings: 1,048,576 combinations

So with just 20 types of toppings, there are over 1 million possible combinations!

A brute-force and naive n-queens problem's solution.

You have to place n queens on a n*n board without them to be taken by others.

while there are untried configs, go to next solution and test it

Assuming every queen is on a given row, there are n possibilities for the queen to be placed and n for the (n-1) other queens (because duplicate rows are not checked).

Therefore, you've got a O(n^n) complexity

The brute force solution of the traveling salesman problem is O(n!) which is approximately O(N^N)

What about finding a subset of integers within a set such that their sum is a designated value X?

I believe this has complexity O(2^(n/2))