# The Universe of Discourse

Wed, 19 Oct 2022

Consider this problem:

Input: A set !!S!! of items, of which an unknown subset, !!S_{\text{bad}}!!, are ‘bad’, and a function, !!\mathcal B!!, which takes a subset !!S'!! of the items and returns true if !!S'!! contains at least one bad item:

$$\mathcal B(S') = \begin{cases} \mathbf{false}, & \text{if S'\cap S_{\text{bad}} = \emptyset} \\ \mathbf{true}, & \text{otherwise} \\ \end{cases}$$

Think of a boxful of electronic components, some of which are defective. You can test any subset of components simultaneously, and if the test succeeds you know that each of those components is good. But if the test fails all you know is that at least one of the components was bad, not how many or which ones.

The obvious method is simply to test the components one at a time:

$$S_{\text{bad}} = \{ x\in S \mid \mathcal B(\{x\}) \}$$

This requires exactly !!|S|!! calls to !!\mathcal B!!.

But if we expect there to be relatively few bad items, we may be able to do better:

• Call !!\mathcal B(S)!!. That is, test all the components at once. If none is bad, we are done.
• Otherwise, partition !!S!! into (roughly-equal) halves !!S_1!! and !!S_2!!, and recurse.

In the worst case this takes (nearly) twice as many calls as just calling !!\mathcal B!! on the singletons. But if !!k!! items are bad it requires only !!O(k\log |S|)!! calls to !!\mathcal B!!, a big win if !!k!! is small compared with !!|S|!!.

My question is: does this technique have a name? If I wanted to tell someone to use it, what would I say?

It's tempting to say "binary search" but it's not very much like binary search. Binary search finds a target value in a sorted array. If !!S!! were an array sorted by badness we could use something like binary search to locate the first bad item, which would solve this problem. But !!S!! is not a sorted array, and we are not really looking for a target value.

Is the history of this algorithm lost in time, or do we know who first invented it, or at least wrote it down? I think it sometimes pops up in connection with coin-weighing puzzles.

[ Addendum 20221023: this is the pure binary-splitting variation of adaptive group testing. I wrote a followup. ]