Problem 11

I’m using problems in The Advent of Code 2016 to motivate my study of Common Lisp though, sadly, I’ve not been devoting time to it recently. Luckily several of my colleagues are also doing the challenge and one of them came by and described what sounded like a particularly interesting problem – number 11 – , and I decided to try my hand at it in Lisp. It promises to be engaging and instructive.

I’m not going to repeat the problem here (but some of the stuff I’ll write will only make sense in the context of the details of the problem); in the abstract it is a path finding problem: The problem world is in some initial state, there are set of intricate rules by which you can change the world state, and there is an end goal state. The task is to find the shortest number of steps to that end goal.

For me this is a rich problem both in terms of algorithms and language. I think I’ll try and make short posts about the steps I take. My course of action will be (and these may change, of course):

1. Figure out what algorithms apply to this problem
2. Think of data structures that will help
3. Implement this in Common Lisp, from scratch where it looks interesting
1. I’ve had at least one case so far where I had a naive way of coding things and then I discovered a different Lisp idiom that made it better. I hope to show these examples too.

There are several tutorials and expositions on path finding. The one I like the best is from Red Blob Games. It has interactive visualizations of the algorithms which are very motivating and the tutorial itself starts slow and easy and makes nice progress to end at the A* algorithm. If you want a refresher, please refer to that. If you’ve read Problem 11 you’ll note that it looks different, but with the proper abstraction, this is what it is.

Now let’s take up where the tutorial leaves off. The A* algorithm seems to be the ticket and we want to figure out some implementation details. The interesting things the A* algorithm requires are a priority queue and a heuristic for determining how far a given state is from the end state.

Popular lore has it that, for this application, a Fibonacci heap is a good implementation for the priority queue for this application, but The algorithm design manual points us to rank-paired heaps as something that is better in practice. I found this paper on rank-pairing heaps. The authors have written the paper very well: They start with a nice systematic introduction to heaps and then slowly introduce modifications they make to existing structures and algorithms.

To start with however, I’m going to implement a priority queue using a simple binary heap (there is a good illustrated guide here). At a later stage, when the whole thing is worked out, I’m going to come back and implement rank-pairing heaps.

The data structure for holding the world state can be separated out from all this priority queue business, but it does interface with the A* algorithm in two places 1) The world state has to be amenable to hashing, such that we can put it in a map-like structure and 2) The implementation of rules for going from state to state i.e. the manner in which we find child nodes of a given parent node can, we intuit, be made considerably easy or difficult based on our choice of world state representation.

This whole thing is going to go down as a series of blog posts and I’ll link to each one (creating a linked list) but also try and set them out here, so that this post serves as a table of contents.

1. Binary heap priority queue