Most problems in computer science can be reduced in some part to one of a few hundred or so classic problems. There are the Dining Philosophers, the Chinese Restaurant, the Traveling Salesman, the Byzantine Generals, and so on. Last week I ran into a fun one I hadn’t seen in a while — the Stable Marriage Problem — so I thought I’d write a bit about it.

The problem that needed solving was re-ordering the pages of out-of-order legal contracts so that we can compare their content to the original. The solution turns out to be very simple and based on the rough metaphor of a cast of singles trying to find love.

### Problem setup: Imagine you’re a busy lawyer..

You send people 40-page contracts all day. Usually they send these contracts back signed and unchanged, but sometimes somebody crosses out a line or adds a margin note before signing. The contract they signed is now different than the one you sent them! To detect this, you’re stuck re-reading 100% of incoming contracts to see which have changes and which don’t.

You like your billing rate, but this is not how you want to spend your time. So you write a computer program to automatically analyze pairs of pages, `<outgoing_page, incoming_page>`

, to detect any signatures and pen marks that have been made on the incoming page.

You run the program. It works wonderfully! But it also fails catastrophically sometimes, producing bizarre results. Inspecting these errors, you realize that some of these contracts haven’t just been marked up with a pen, they’ve also been shuffled around and had pages lost. The errors result from your algorithm trying to compare `outgoing_page 2`

with some badly matched `incoming_page 4`

.

### Ways a document can be “out of order”

Let’s frame our problem as being about two documents we’d like to compare:

`A`

, the outgoing contract, and`B`

, the incoming contract

Document `A`

has pages `a_1, a_2, ... a_n`

and document `B`

has pages `b_1, b_2, ... b_m`

. We know `A`

and `B`

are the same documents, but `b_1 .. b_m`

deviates from `A`

in a few inconvenient ways:

- Pages have been removed
- Pages have been added
- Pages have been reordered

What we’d like to find is a **match** between `A`

and `B`

. A match is a set of pairings between the elements of `A`

and `B`

, like this: `[ <a_1,b_1>, <a_2,b_2> … <a_n,b_n> ]`

. Specifically, we want the match which results in `<a_i, b_j>`

pairs that are most similar to each other.

That’s simple to say but complex to do: what constitutes the set of most similar matches? Should we optimize for the average match strength (which skews for passion)? Median match strength (which skews for settling)? Should we always make perfect matches when possible (might leave some elements woefully undermatched)?

### Stable Marriage Matching

A stable marriage match is a great matching framework for finding matches between populations of different size. (Note: If you follow that hyperlink, Wikipedia will tell you stable marriage matching is for populations of equal size. Just ignore that.)

It optimizes for the following global property, which we call a “stable match”:

```
No element in a match will have a better viable option.
```

Here’s that sentence expressed in terms of a test you can apply to some match **(a, b)** to see if it’s stable:

```
for all mates (a,b):
if a would rather be with c than b:
if c would rather be with a than c's current match:
(a,b) is unstable! (a,c) deserve each other!
same rule for b
```

The counterintuitive observation here is that match **(a, b)** is considered stable *even if *a* and

**both prefer other mates…as long as those secret crushes don’t crush back*. Elements **a **and **b **are each other’s best viable compromise when factoring in everyone else’s preferences. This strategy also results in the guarantee that if there does exist some mutually in-love Romeo and Juliet, they will be matched together.**

*b*

The other great part of Stable Marriage Matching is that the solution is simple and well understood — just look up the Gale–Shapley algorithm. You can implement it in an hour and be on your way.

OK. So let’s assume that we now have some function **match **that produces the following output:

```
match(A, B) -> { matches: [<a,b>], additions: [b], deletions: [a] }
```

The match function takes an outgoing document `A`

and an incoming document `B`

and produces a set of page-to-page matches, along with a list of added and deleted pages. If `|B| > |A|`

we’ll call any unmatched `b`

added, and if `|A| > |B|`

we’ll call any unmatched `a`

deleted. This isn’t addition/deletion strategy isn’t perfect — there are better strategies involving match strength — but it will do fine for many cases.

The only thing left is to define some measure of attraction that helps us decide which pages `a`

get married to which pages `b`

.

### “Chunky Jaccard” as a measure of document attraction

Attraction really means similarity here. And just like matching, document similarity is simple in concept but more nuanced to actually define. Are two things similar if they have the same letters? The same words? The same words in the same order? The same words in the same order and font styling?

In general, you can define a spectrum of text similarity measures that go from coarse (Jaccard) to strict (labeled “Extremist” below). In the below figure, `A`

and `B`

are the sets of tokens in two pages, and `a_i`

and `b_i`

are the i^th tokens of those pages.

- Jaccard Similarity is a coarse measure of document similarity. It looks at two documents as bags of words and returns the measure of how many words were the same divided by how many existed in total. If the documents are perfectly equal,
`J(A,B) = 1.0`

, and if they are perfectly disjoint,`J(A,B) = 0.0`

- What I labeled as **Extremist **would be to take the opposite approach from “bag of words” and compare the i^th word of each document to each other. On the one hand, I understand the extremism — after all, documents aren’t bags of words. Word order matters! But on the other hand, this is extremely brittle in the presence of errors. What if the OCR of a document inserted an extra word? That single, off-by-one index shift would cause the similarity score to plummet by 0.5 on average.
**“Chunky” Jaccard Similarity**represents a balance between the bag-of-words model and the extremist variant. We take some`chunk_size`

, which is expressed as a percentage, and then carve up the document into N segments, each one`chunk_size`

percent of the document. When`chunk_size`

approaches zero, Chunky Jaccard is the same as the extremist measure. When`chunk_size = 1`

, Chunky Jaccard is the same as the bag of words version.

So that’s really it. Pick one of the two Jaccard variants, implement it, and use that as your measure of attraction in the Gale-Shapley algorithm.

### Matching made simple

Once we implemented the above, our document analysis errors disappeared completely. On top of that, we had a more complete analysis to show our users, including pages added and removed in addition to page-level modifications.

It turns out that in our dataset, the document shuffling was almost entirely caused by Page 1 of the initial document going missing. The cover letter that accompanied the initial contract had been omitted in the signed response.

### Strong visuals make things easy to remember

This is one of those problems where if you had to come up with a solution yourself, it would take a while to devise an approach that was straightforward to implement and also lacked any nasty edge cases.

I think the colorful metaphors for classic algorithms are so useful because they make it easy to remember when you can pull a standard solution out of your bag of tricks. In general, humans are terrible at remembering numbers but fantastic at remembering stories.

So don’t dismiss the stories behind these algorithms as metaphor abuse. Ham them up! Visualize ridiculous re-enactments of them in your mind so that they’ll be easier to recall when you need them most.

#### PS: Come solve interesting problems at Instabase

If solving high-impact problems with real world data sounds like a good way to spend your day, reach out and say hello. Instabase is hiring strong CS generalists, as well as specialists in systems, machine learning, NLP, and web front-end.