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An algorithmic puzzle

I posted the following problem on Twitter yesterday: I have intervals \(L_1, \ldots, L_m\) which are represented as half-open intervals \(L_k = [u_k, v_k)\). I know that the intervals are all distinct and that for any two intervals \(L_i, L_j\) with \(i \neq j\) either one is contained in the other or two are disjoint.

I want to sample uniformly from all pairs \(\{(i, j) : L_i \cap L_j = \emptyset \}\), or determine that there are no such pairs. How?

A solution now follows. It might not be the optimal solution, but it’s pretty close.

The two observations that make the solution work are:

  1. If every interval is disjoint from at least one other interval, then rejection sampling runs in expected time at most \(O(m)\), because the probability of accepting a random pair is at least the lowest probability of selecting \(j\) disjoint from \(i\) over all \(i\).
  2. A root interval (one containing everything else) is contained in no disjoint pair and thus can be removed from the list without changing the distribution. If there is no root interval then every interval is disjoint from at least one other.

To unpack the second observation: If there is no root interval, take some \(L_i\). Then \(L_i\) is contained in some maximal length interval. This is not a root interval, so there is some \(L_j\) that is not contained in it (and thus is disjoint from it). Such an \(L_j\) is necessarily disjoint from \(L_i\).

It’s easy to find a root interval in \(O(m)\), but removing it might introduce a new root-interval (in the degenerate case the whole list could be a chain of the form \(L_i = [i, m)\)).

Here’s how to simultaneously remove all root intervals in \(O(m \log(m))\):

First note that a root interval is necessarily of maximum length. Thus if we sort the intervals in order of decreasing length (that’s the \(O(m \log(m) \) bit. You can do it in \(O(m)\) if the interval boundaries are not-too-large intervals using bucket sort), the root intervals must form some initial prefix of the list. By renumbering where necessary lets assume that the \(L_i\) are sorted in that order.

Now calculate the intervals \(B_i = [\min\limits_{k \geq i} u_i, \max\limits_{k \geq i} v_i) \). These are the bounding intervals of the tail sets and can easily be worked out in \(O(m)\) by just iterating backwards over the list.

\(L_i\) contains all \(k \geq i\) if and only if it contains \(U_i\). So now we find the first index \(k\) that is not a root index just by iterating forward through the \(L_i\) until \(U_i\) is not contained in \(L_i\).

We now take all intervals from that point and discard everything earlier. If the set is now empty, we raise an error because there are no disjoint pairs. If it’s not empty, we can now run the rejection sampling algorithm on it as desired and have it terminate in \(O(m)\).

think there’s another solution that may sometimes be better involving explicitly tracking the tree structure and counting the number of disjoint pairs that live in each sub-interval, but the solution fell apart the more I looked at it and I wasn’t convinced it was ever actually better.

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The Clarke Gap

I coined a term on Twitter earlier and almost immediately found it useful, so I thought I’d register it somewhere a bit more permanent.

The “Clarke Gap” of a technology is the degree to which it is distinguishable from magic.

(The term of course comes Clarke’s third law that any significantly advanced technology is indistinguishable from magic, or more specifically from the corollary to it that any piece of technology distinguishable from magic is insufficiently advanced).

The Clarke gap is characterised mostly by how likely something is to go wrong in a way that requires you to understand what the system is actually doing in order to fix it. The more likely it is, the larger the Clarke gap is. If something always works and you never need to understand it, it might as well be magic. If you have to open the hood every five minutes and poke the bits that have got misaligned, it’s obviously technology. Most things are somewhere in between.

It’s not that things with a low Clarke gap don’t do wrong, but when they go wrong they tend to do it in a way that is either easy or extremely hard to fix – e.g. a computer has a fairly low Clarke gap – when it goes seriously wrong you need a lot of technical expertise to figure out what’s happened and why – but computers go wrong all the time and mostly you can just turn it off and on again (either the whole computer or the misbehaving program) and it will work.

A low Clarke gap is neither wholly good or wholly bad. It’s mostly about which cases you’re optimising for: Systems with a low Clarke gap tend to be much easier to learn at a casual usage level, but much harder to acquire expertise in.

For any given class of technology, the Clarke gap tends to go down over time. If you look at search engines now versus search engines twenty years ago, the Clarke gap has gone way down – a modern search engine is very much trying to guess your intent and thinks it knows better than you, while one from twenty years ago is basically a boolean query engine with maybe some spelling correction if you’re lucky. I suspect this has a correspondingly negative impact on people’s abilities to pick up search skills – we’re so used to search being magically good, we don’t pick up the necessarily knowledge about what to do when it’s not.

There are a number of factors that tend to push it this way. If users commonly experience a problem, the natural thing to do is to try to make that problem non-existent or easy to solve. This is good development practice! You’re making people’s lives easier. But by doing this you’re also removing a lot of intermediate ways in which the system can go wrong, and making the system more complex and thus increasing the number of more subtle ways it has to fail. This lowers the Clarke gap.

I think this is a useful phenomenon to pay attention to, and consequently to have a term for, so I’m probably going to keep using the term until a better one comes around.

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The Great Asshole Filter

(Content note: I try to keep my blog significantly more profanity light than my normal vocabulary, but “asshole” is being used as a term of art in one of the links this post is based on, and I will follow that usage).

As you’ve doubtless heard by now, Patreon are currently tilling their users into the soil with a major change that largely screws over people who rely on the $1-2/month donations (which is largely how I donate on Patreon).

Christie Koehler offers an extremely lucid explanation for why they’re doing this: It’s to avoid financial regulations that would cause them to be classed as a “money transmitter”, which would be significantly more onerous. They are not explaining this because if they admit to knowing it then it potentially increases their liability.

How did they get away with being classed as a money services business so far? Well, basically by ignoring the problem and hoping the problem would ignore them. It’s the standard startup thing: Ignore the regulations until you can’t, then either comply with or attempt to change the regulations.

I’ve been cooling a bit on regulation over the last couple of years. I still fundamentally believe that we need something to fetter the runaway optimisation process that is the free market (though I’m definitely also not on team full communism now), but I think it’s increasingly clear (at least in the UK and the USA. I’m less clued in on politics elsewhere, though even where it seems better I’m not sure it seems better enough) that whatever it is we’re doing right now isn’t working, and the current generation of governments is fundamentally untrustworthy to create regulation (see: Attempts to regulate crypto by mandating back doors).

Anyway, all these thoughts were bouncing around in my head and they latched on to something seemingly unrelated that I read a while ago (and very strongly agree with): The idea of an asshole filter (Ironically, I support the author on Patreon at a $2 level, to publish on Live Journal, another platform with a history of tilling their users into the soil).

The idea of the asshole filter is this: If you have strongly defined boundaries which you erratically enforce, then you will mostly be surrounded by people who are willing to violate those boundaries.

And it occurred to me that regulation and startup culture are currently interacting in exactly this way, especially in anything remotely fintech or employment related: There are large regulations which it is profitable to ignore, and we are erratically enforcing them. Thus many of the startups who succeed are the ones who are prepared to violate boundaries for as long as they can get away with it.

This isn’t a Fully General Theory Of Why Startups Are Mostly Assholes – e.g. I’m not currently aware of any regulations Twitter is skirting the boundary of (though I suspect that a lot of news agencies are probably using it to do an end run around various social norms of reporting if not regulations – it’s basically an environment where they can lie as much as they do in the headlines and get away with it because character limit. Maybe 280 will help with that? Note: 280 won’t help with that) – but I suspect it accounts for a lot of the worst behaviour. If we create an environment where only assholes can win, we probably shouldn’t be surprised when assholes take advantage of that to win.

I’m not sure what to do about this. In many ways the fact that current regulations are not enforced on small businesses is a feature – requiring full regulatory compliance from day one basically means you’re never going to successfully start a business due to the level of overhead required to comply with it. Sometimes this is even formalised in terms of size requirements on businesses, which would at least help make regulation less of an asshole filter though wouldn’t do anything to stop users getting tilled into the soil as soon as the regulation kicks in.

In theory this could be fixed with better regulation (maybe with strictly less regulation, but it’s not like I trust the banks to behave themselves if unregulated either), but right now I don’t trust our governments to do anything that is obviously in people’s best interests, let alone things that require a complex argument based on social dynamics to even notice might be an improvement. I also have literally no control over what our government does despite ostensibly living in a democracy, so even if this were a solution there wasn’t anything I could do about it.

So the  best I can offer is a word of warning: That hot new startup that is doing something that nobody else is offering despite there being many players in a similar space? It’s probably because they’re assholes, and even if they’re not right now, success is probably going to force them to till you into the soil.

That doesn’t mean you can’t use them, but maybe you shouldn’t unless you want to look forward to becoming fertiliser.

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Non-fiction

I’m going to be experimenting with a thing this December: I will be (mostly) giving up consuming fiction.

What this means:

  • I’m not going to read any fiction books
  • I’m not going to watch any TV or movies
  • I’m not going to play any video games

I’m going to allow myself the following exceptions:

  • I have a couple of updating serials I read (webcomics, a few web novels) that I will continue to keep up with as they come out (but won’t e.g. go back and read archives of)
  • I will probably watch the Dr Who Christmas Special.

Also board games are fine, as they’re fundamentally a social activity rather than novelty on tap like the above.

This isn’t a firm commitment. I might give up if I find it too hellish. It’s just an experiment. I’m writing about it partly for general interest and partly to make myself slightly more likely to stick to it.

I’m not doing this because I think there’s anything wrong with these things. I’m certainly not doing it with an intent to make this a permanent change, yikes.

The reason I’m doing this is that these are things that I can spend an unbounded amount of time on: If I’m not quite sure what to do, I’ll read some fiction. Sometimes I’ll watch TV. This means that it’s easy to put things off that I don’t want to do because there’s always something I can procrastinate with.

For example, I have been intending for literally six months to start meditating. I have books on it and everything. Have I started? Ha, no. I’ve dug my heels in so hard.

And one of the reasons for that is that it’s always something I can do later, that I don’t want to do now. Maybe not having something to fill my time with will bore me into submission make me more likely to try it.

I’d also like to get back to writing more. Despite this being my second blog post today, I’ve let the blog lapse a bit since starting my PhD. Some of that is because I now have other writing commitments, but I’d still like to fix that.

In general I would like to see what I fill my time with when I do not have an automatic source of things to reach for. I don’t expect the answer will be especially high value – I suspect it’s mostly going to be Twitter and annoying the cats – but it will be interesting to see.

Update: This turned out to be very poorly timed. I’ve given up on this for now, but will attempt to resume at some point in the new year. It seemed to work very well for a few days before disruption started, but I think needs better planning.

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The politics of “I don’t know”

My politics these days are increasingly vague, because I’m increasingly uncomfortable with proposing solutions when I don’t understand the problems.

If I had actual political power, that would be one thing – I would try things and see what worked – but I don’t, so I’m left with angry shouting about why everyone fails to see that my obviously correct solution is obviously correct as my main socially acceptable option for political expression.

And I don’t know what the correct solution is.

I’m not quite at the Socrates level of knowing that I know nothing. I know a few things. Unfortunately most of the things I know are “Wow this problem is hard, huh?”

(I also know that the current brand of politics in both my countries is very bad and should be opposed. This post is more about how I’d like things to work once everything isn’t on fire)

Unfortunately “This problem is hard” seems to be literally the only thing peoples’ politics are willing to universally unite against. If a politician admits that a problem is hard and they don’t know what to do about it, they might as well start writing their resignation letter now and save the papers the effort of raking them over the coals and you the time of voting them out of office.

Here are two problems I know to be hard beyond our current ability to deal with:

  1. Centralised planning of a complex system
  2. Decentralised coordination in response to large-scale problems

If you’ll forgive the horrendous oversimplification of politics in a post complaining about the horrendous oversimplification of politics, economic left vs right wing political opinion seems to be largely a split on which one of those two things we want to deny is hard.

Either we should let the market solve everything including the things that the market can’t solve because they require individuals to act against their best interests in order to achieve a collectively better result, or we nationalise everything and the people who are an inconveniently tiny fraction of the population to worry about in our centralised planning get crushed by the system.

How should we solve both of these problems simultaneously? I don’t know.

I sometimes refer to the problem of how you get a large group of people to coordinate for their mutual benefit as the fundamental problem of civilization. Many (most?) ideologies think they have an answer to this problem, but all of the answers I’ve seen seem to be pretty bad.

Which is not to say the current compromise system is good either. In many ways we’re suffering from a blend of the worst problems of each – we can’t coordinate properly, but the giant machine of society still crushes people. To some extent we’ve compromised by each side adopting the worst excesses of the other. Massive accumulation of capital in individuals results in people happily doing their own version of centralised planning, while states generally totally fail to solve coordination problems because politicians are more concerned with soundbites that will get them reelected than the good of the country or world.

Can we do better? I don’t know. Probably. Certainly I’m going to choose to believe we can. But I think as long as we deny hard problems are hard, we’re going to keep failing to solve them.

Does this make me a centrist? I guess this makes me a centrist, so I’ve probably now outed myself to my friends (who are mostly left-wing, as am I on a lot of issues) as literally/figuratively worse than Hitler. I can’t say I’ve found most centrist politicians any more appealing than the run of the mill on either side, but maybe the problem there is politicians and the system they work in, rather than their specific politics.

Does this mean I think better things aren’t possible? No. But I do think that whatever your current politics are, if you don’t admit “I don’t know” as an answer, the way you want us to be going probably won’t get us there.

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