# Cartesian Tree-Product May 02, 2013

I have to admit that my day-to-day life involves very little algorithmic problems, but here and there I get a chance to think. In this post, I’d like to discuss an interesting problem that I’ve met several times already in my programming career, each time in different settings. When I met it again last week, I decided it’s time to formalize it a little. In lack of a better name, I call it “Cartesian Tree-Product” (not to be confused with Cartesian product of graphs), and here’s how it goes:

Say you’re given an expression tree. To
limit the scope of the problem, we’ll assume the tree is binary, its internal nodes can either be
`AND`

or `OR`

, and its leaves hold “atomic comparators” (which are of no interest to us).
Here’s an example expression:

```
(x=5 OR y=6) AND z=7
```

Which is represented by the following expression tree:

```
AND
/ \
OR \
/ \ \
x=5 y=6 z=7
```

Now the problem is to produce all different sub-trees such that:

- Each sub-tree consists only of
`AND`

internal nodes - Each sub-tree satisfies the original expression (any assignment that satisfies a sub-tree must also satisfy the original tree)
`OR`

-ing together all the sub-trees produces a tree that is mathematically-equivalent to the original one (any assignment that satisfies the original tree must satisfy the reconstructed tree)

In other words, we want to produce all partial expressions of the original expression, which will satisfy it and which can together reconstruct it. Big words aside, here’s what we want for the expression above:

```
x=5 AND z=7
y=6 AND z=7
```

Each expression here satisfies the original one (try it), and if we `OR`

the two, we get a
mathematically-equivalent tree:

```
(x=5 AND z=7) OR (y=6 AND z=7)
OR
/ \
/ \
AND AND
/ \ / \
x=5 z=7 y=6 z=7
```

(To clarify: By *sub-tree* or *partial expression*, I mean it is constructed only from
the *leaves* of the original tree/expression)

If you take a second look, it resembles Cartesian products where some “joints” (nodes) in the tree
duplicate the resulting tree. Intuitively, we “split” the tree for every internal `OR`

and continue
with both copies. For instance, if we take `(x=5 OR y=6) AND (z=7 OR w=8)`

,
we’ll get 4 sub-trees

```
x=5 AND z=7
x=5 AND w=8
y=6 AND z=7
y=6 AND w=8
```

However, it depends on the structure of the tree; `(x=5 OR y=6 OR z=7) AND w=8`

produces 3 sub-trees.
This closely relates to the rules of distributivity
in propositional logic, but it seems to me that the “Cartesian product” notion is a generalization
of the concept.

## How is it Useful?

Since we’re dealing with expression trees, it’s hardly surprising that the two times I had to use this algorithm related to syntax. In the first case, I wrote a fuzz-testing tool for an interactive program, like the MySQL shell. The program accepted commands, conforming to a well-defined syntax, and I wanted to generate commands at random and see that it didn’t crash.

For instance, a command might look like `map-lun <vol-name|vol-id> lun-id`

and we’ll want to try
both variants, i.e., `map-lun vol-name lun-id`

and `map-lun vol-id lun-id`

. Of course the syntax
is generally much more complex, with nested brackets, optional arguments, etc. It gets interesting,
but we can still map it to the problem outlined above.

Another real-life use case is running queries against a huge dataset. In order not to complicate
our query engine (written in C for performance), it can perform only intersections (`AND`

s) of filters.
If you want to query for more complex conditions, you have to run it multiple times with the
partial queries and “sum up” the results. But we don’t want the end user “doing the math” for us,
and waiting for one query to finish before we start the next means we have to load data from the
store multiple times. If we could process it in chunks, we’d benefit from cache locality and shorten
query times.

## The Algorithm

The code is strikingly short, but that’s not to mean it’s easy to follow. The heart of it is two, recursively-nested for-loops:

Let’s try it out on the expression `(x=5 OR y=6) AND (z=7 OR w=8 OR q=9) AND r=10`

:

Does the trick.

Trying to estimate the complexity of this beast may be futile, but it clearly seems to be
doing “more work” than a mere SAT
problem: it doesn’t just find one satisfying assignment, it looks for all satisfying assignments!
This ought to put it in the NP-hard class. In fact, if we generate a binary expression of
alternating `AND`

s and `OR`

s, it can get much worse than exponential complexity!

Here’s a little function that generates an interleaved binary expression:

E.g.,

Now

Notice how each increment either doubles or squares the number of results… that’s because `AND`

s and
`OR`

s are interleaved (`AND`

s double, `OR`

s square). Seems more like a
power tower to me.

## Extension: the Inverse Problem

The inverse problem is also useful. In the inverse problem we’re given a set of expressions and we’re trying to generate the most “compact form”, i.e., “undo the effects” of the distributivity law.

I once wrote a test harness where each test specified its prerequisites declaratively. For instance, a test might need to run after the system had come up from an emergency shutdown, so the framework would bring the system to the required state and then then run the test. Obviously, it may take a while to bring the system to a certain state. It could range from minutes to days. And we have hundreds of tests!

In this case, we are given a list of tests and we want to find the most efficient order to run them, meaning, we want to minimize the setup and teardown times when moving between different system states.

In the example above, we can first bring the system to states A and B (e.g., “running over 10
hours” and “having less than 1 TB of free space”), then setup C, run `FooTest`

, teardown C
and setup state D, run `BarTest`

, teardown B, setup E, and run `SpamTest`

.

In essence, we want to take the requirement lists from each test and reconstruct the most compact tree that represents them, then we follow that tree in BFS order and reduce the overall time.

```
A
/ \
B \
/ \ \
C D E
```

It took me a while to realize it’s basically the inverse of the original problem. It is much simpler, in terms of complexity, so perhaps it’s not strictly the inverse, but the two clearly work in “opposite directions”.

Anyway, it’s funny how I met this problem from different angles three times already. Just thought I’d share.