## Skew-cost coding

### Problem 219

Published on Saturday, 29th November 2008, 05:00 am; Solved by 1021; Difficulty rating: 70%Let A and B be bit strings (sequences of 0's and 1's).

If A is equal to the leftmost length(A) bits of B, then A is said to be a prefix of B.

For example, 00110 is a prefix of 001101001, but not of 00111 or 100110.

A prefix-free code of size `n` is a collection of `n` distinct bit strings such that no string is a prefix of any other. For example, this is a prefix-free code of size 6:

0000, 0001, 001, 01, 10, 11

Now suppose that it costs one penny to transmit a '0' bit, but four pence to transmit a '1'.

Then the total cost of the prefix-free code shown above is 35 pence, which happens to be the cheapest possible for the skewed pricing scheme in question.

In short, we write Cost(6) = 35.

What is Cost(10^{9}) ?