# [tahoe-dev] "servers of happiness"

David-Sarah Hopwood david-sarah at jacaranda.org
Wed Oct 14 04:58:16 UTC 2009

```Zooko Wilcox-O'Hearn wrote:
> I asked my wife Amber for help formalizing my intuition about what
> sort of share placement makes me happy.  We came up with this:
>
> First of all, let's call a set of servers "sufficient" if you can
> download the file from that set (i.e. if at least K distinct shares
> are hosted in that set of servers).

Call it K-sufficient.

> Now consider the largest set of servers such that every K-sized
> subset of it is sufficient.
> Let's call the size of that largest set S.

This largest set isn't necessarily unique. I think you mean:

A set U is K-happy iff every K-sized subset of U is K-sufficient.
Let S be the largest integer such that there exists a K-happy set
of size S.

Then S is unique, even though there may be more than one K-happy set
of that size.

> "Happyness" is that I configure a Happyness number H, and if an
> upload results in an S >= H then I'm happy.

Note that this only makes sense for H >= K. If H < K then consider
an arbitrary set U of size H: it is vacuously K-happy, because it has
no K-sized subsets. Therefore S >= H, because there must exist K-happy
sets at least as large as U. This is despite the fact there the file

Anyway, your intuition, if it is correct, is equivalent to the
simpler statement:

I am happy iff there exists a K-happy set of size H.

because if there exists a K-happy set of any size S >= H, then
there exists a K-happy set of size H.

(This follows from the fact that if U' is a subset of U, then
the set of K-sized subsets of U' is a subset of the set of K-sized
subsets of U. Therefore, if there is a K-happy set of size S then
all subsets of it are K-happy, and therefore there are K-happy
sets of sizes 0..S-1.)

--
David-Sarah Hopwood  ⚥  http://davidsarah.livejournal.com

```