### Computer Science: The Travelling Salesman Problem

I was thinking about the Travelling Salesman problem this morning. I came up with an algorithm that permits a few nice optimizations. My guess is that Knuth probably already came up with this algorithm, formally analyzed it, and then came up with 15 others that were much better. Nonetheless, I figured I'd jot it down.

Warning: This is a rough draft of an algorithm. You shouldn't bother reading this if you want real rigor. I'm just writing it down to get it off my chest.

Rather than picking the first segment of the journey and working my way to the end, I want to pick a middle segment of the journey and work my way outwards recursively.

Given a list of distances (or times, or costs, or whatever) between cities, sort the list from shortest distance to longest distance regardless of which cities are involved. Now, loop over this list and use each element of the list as a middle segment of your journey. This gives us a list of "first steps" (or rather, first middle steps). Looping over the list from shortest distance to longest distance is an important optimization because it increases the likelihood that we'll find an optimal path early, allowing us to skip over a lot of potential paths later.

Also sort the list of distances so that for each city, we have a list of other cities in ascending distance order. By sorting all of the city pairs in order of (first city, distance), you can use one sort for all of the pairs.

Now, here comes the recursion. We have a bunch of middle parts of the journey. Use recursion to extend the journey either by adding to the beginning of the journey or by adding to the end of the journey. Keep recursing until you have a complete path or a partial path that is already longer than the best path seen so far. Now, we can either extend the journey at the beginning or at the end. Recursively try extending the journey by either adding to the beginning or the end. However, do it in order so that you try adding the shorter distances first. There's an implicit merge sort going on in this algorithm. This, again, is an optimization to allow you to skip work later.

While we were recursing, we had a function that took two things, a middle chunk of the journey and a set of cities not yet visited. Apply memoization so that anytime we get the same parameters, we return the same answer (by using a cache, of course). This is an important optimization.

Last of all, using the above algorithm, we'll quickly come up with the first complete path that has a decently minimal distance. Keep track of this as the best complete path seen so far. Anytime we are recursing and we come up with a partial path that is longer than the best complete path seen so far, we can stop recursing, give up, and try something else. This is an important optimization to save work "at the leaves of the tree".

I can't even wrap my head around the big O of this algorithm. I suspect it's one of those cases where you use words like "amortized cost".

This algorithm does have a weakness. If every distance between cities is exactly the same, it'll try every possibility. Similarly, if every distance between cities is exactly the same except one pair of cities which has a longer distance, it'll still try every possibility. I'm not sure of the degree to which the memoization fixes this problem. It'd be good to extend the algorithm somehow to recognize this situation and short circuit, i.e. automatically throw out paths of duplicate length.

denlunev said…
Nice article! I implemented Travelling Salesman Problem solution using genetic algorithm.
It's an heuristic algorithm in some cases doesn't produce 100% right solution, but very effective on big amount of data.

Demo - http://app.mozgoweb.com/, source code - https://github.com/denlunev/TSP
jjinux said…
> Nice article! I implemented Travelling Salesman Problem solution using genetic algorithm.

Nice :-D

### Ubuntu 20.04 on a 2015 15" MacBook Pro

I decided to give Ubuntu 20.04 a try on my 2015 15" MacBook Pro. I didn't actually install it; I just live booted from a USB thumb drive which was enough to try out everything I wanted. In summary, it's not perfect, and issues with my camera would prevent me from switching, but given the right hardware, I think it's a really viable option. The first thing I wanted to try was what would happen if I plugged in a non-HiDPI screen given that my laptop has a HiDPI screen. Without sub-pixel scaling, whatever scale rate I picked for one screen would apply to the other. However, once I turned on sub-pixel scaling, I was able to pick different scale rates for the internal and external displays. That looked ok. I tried plugging in and unplugging multiple times, and it didn't crash. I doubt it'd work with my Thunderbolt display at work, but it worked fine for my HDMI displays at home. I even plugged it into my TV, and it stuck to the 100% scaling I picked for the othe

### Drawing Sierpinski's Triangle in Minecraft Using Python

In his keynote at PyCon, Eben Upton, the Executive Director of the Rasberry Pi Foundation, mentioned that not only has Minecraft been ported to the Rasberry Pi, but you can even control it with Python . Since four of my kids are avid Minecraft fans, I figured this might be a good time to teach them to program using Python. So I started yesterday with the goal of programming something cool for Minecraft and then showing it off at the San Francisco Python Meetup in the evening. The first problem that I faced was that I didn't have a Rasberry Pi. You can't hack Minecraft by just installing the Minecraft client. Speaking of which, I didn't have the Minecraft client installed either ;) My kids always play it on their Nexus 7s. I found an open source Minecraft server called Bukkit that "provides the means to extend the popular Minecraft multiplayer server." Then I found a plugin called RaspberryJuice that implements a subset of the Minecraft Pi modding API for B

### ERNOS: Erlang Networked Operating System

I've been reading Dreaming in Code lately, and I really like it. If you're not a dreamer, you may safely skip the rest of this post ;) In Chapter 10, "Engineers and Artists", Alan Kay, John Backus, and Jaron Lanier really got me thinking. I've also been thinking a lot about Minix 3 , Erlang , and the original Lisp machine . The ideas are beginning to synthesize into something cohesive--more than just the sum of their parts. Now, I'm sure that many of these ideas have already been envisioned within Tunes.org , LLVM , Microsoft's Singularity project, or in some other place that I haven't managed to discover or fully read, but I'm going to blog them anyway. Rather than wax philosophical, let me just dump out some ideas: Start with Minix 3. It's a new microkernel, and it's meant for real use, unlike the original Minix. "This new OS is extremely small, with the part that runs in kernel mode under 4000 lines of executable code.&quo