Using register allocation algorithms for determining table layout Tue, 12 Jun 2007
S and I decided to assign tables for our guests during the wedding reception. There were a bunch of good reasons for doing this which I'm not going to go into here. However, assigning 100 or so guests to 10 or so tables while maximizing "goodness" and minimizing "badness" isn't trivial to do on paper by hand. At some point during wedding planning, I decided we could do table layout with a modified register allocation algorithm. This is a quick summary of translating register allocation into table layout along with some commentary on how this works nicely and also where it doesn't quite work.
More after the jump....BREAK
Register allocation is typically done using graph-coloring with some additional bits that turn the NP-complete graph-coloring algorithm into a linear-time one that results in a good approximation. Andrew Appel talks about register allocation in his book on compilers entitled Modern Compiler Implementation in ML in chapter 11. It involves a graph of nodes representing temps connected by edges that represent interference between the temps. There's also a list of move-related nodes and nodes related by move-edges are coalesced if possible leading the connected temps to be colored with the same register.
In table layout, there are groups of people who can't sit at the same table with other groups of people. This could be due to family issues, differences in politics, past history, ... Additionally, there are groups of people you want to sit at the same table with other groups if possible.
Let's do some substitution. We'll substitute registers for tables, edges representing people who can't sit at the same table as interference, edges representing people who would be good sitting together as move-related, and temps as people and we have mapped the register allocation problem into a table layout problem.
I'm not going to go into the details of register allocation--Appel talks about the George and Appel algorithm from "Iterated register coalescing" (1996) for 25 pages and it'd be hard to summarize that into a blog entry.
There are a few things that don't translate well from register allocation to table layout like spilling. In register allocation if you can't get it to work you spill a temp into memory and then start over. I suppose you could uninvite particularly ornery people, that's one possible mapping for spilled temps. Another possibility is that you look for the least-worst pairing and remove that edge from the graph. A third possibility is to break up a larger table into two smaller tables giving you an extra color to work with.
Speaking of tables, there's one big difference between tables and registers: a register can be assigned to an unlimited number of temps so long as they don't interfere with one another. A table has a limited number of seats. So if you were to use a register allocation algorithm, you have the additional constraint that a limited number of people can be assigned to each table. If the register allocation implementation that you use is deterministic, this constraint could cause your situation to be unsolvable without backtracking. It would be a good idea to introduce a random element that causes assignments to occur in different orders between iterations.
The move-related edges (which represent people who should be sitting together) could be prioritized and that priority could be used for selecting moves to coalesce. For example, you might want to keep couples together and perhaps families, too. Perhaps you want to put all the children at a single table that you can put close to the bathroom.
Theoretically, this sounds like it would work pretty well for many cases, at a minimum returning a layout that can be tinkered with. We never used this method--by the time I had finished grad school, S was already past most of the table layout issues.
It's possible the constraint on people per table handicaps the register allocation algorithm to such a degree that it would be better to use a more general purpose constraint satisfaction problem solver instead. This requires further study.