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

Answer 1

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!