Real-world example of exponential time complexity

Question

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 for 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?

Answer 1

• 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.

Answer 2

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!