# Exercise 8.3.5

$\star$ In the first card-sorting algorithm in this section, exactly how many sorting passes are needed to sort $d$-digit decimal numbers in the worst case? How many piles of cards would an operator need to keep track of in the worst case?

The algorithm becomes exponential. Here's a simple breakdown for sorting n-digits numbers in base 3.

- sort all on first digit into 3 piles
- sort pile with first digit = 0 into 3 piles on second digit
- sort pile with second digit = 0 on third digit
...
- sort pile with second digit = 1 on third digit
...
- sort pile with third digit = 1 on third digit
...
- sort pile with first digit = 1 into 3 piles on second digit
..
...


This is exponential. We need to perform $\Theta(k^d)$ passes. Furthermore, we need to keep track of $\Theta(nk)$ piles.

Those correspond to space and time.