# The Universe of Discourse

Sat, 11 Nov 2017

A little while back I thought of a perfect metaphor for explaining what a randomized algorithm is. It's so perfect I'm sure it must have thought of many times before, but it's new to me.

Suppose you have a lake, and you want to know if there are fish in the lake. You dig some worms, pick a spot, bait the hook, and wait. At the end of the day, if you have caught a fish, you have your answer: there are fish in the lake.[1]

But what if you don't catch a fish? Then you still don't know. Perhaps you used the wrong bait, or fished in the wrong spot. Perhaps you did everything right and the fish happened not to be biting that day. Or perhaps you did everything right except there are no fish in the lake.

But you can try again. Pick a different spot, try a different bait, and fish for another day. And if you catch a fish, you know the answer: the lake does contain fish. But if not, you can go fishing again tomorrow.

Suppose you go fishing every day for a month and you catch nothing. You still don't know why. But you have a pretty good idea: most likely, it is because there are no fish to catch. It could be that you have just been very unlucky, but that much bad luck is unlikely.

But perhaps you're not sure enough. You can keep fishing. If, after a year, you have not caught any fish, you can be almost certain that there were no fish in the lake at all. Because a year-long run of bad luck is extremely unlikely. But if you are still not convinced, you can keep on fishing. You will never be 100% certain, but if you keep at it long enough you can become 99.99999% certain with as many nines as you like.

That is a randomized algorithm, for finding out of there are fish in a lake! It might tell you definitively that there are, by producing a fish. Or it might fail, and then you still don't know. But as long as it keeps failing, the chance that there are any fish rapidly becomes very small, exponentially so, and can be made as small as you like.

For not-metaphorical examples, see:

• The Miller-Rabin primality test: Given an odd number K, is K composite? If it is, the Miller-Rabin test will tell you so 75% of the time. If not, you can go fishing again the next day. After n trials, you are either !!100\%!! certain that K is composite, or !!100\%-\frac1{2^{2n}}!! certain that it is prime.

• Frievalds’ algorithm: given three square matrices !!A, B, !! and !!C!!, is !!C!! the product !!A×B!!? Actually multiplying !!A×B!! could be slow. But if !!A×B!! is not equal to !!C!!, Frievald's algorithm will quickly tell you that it isn't—half the time. If not, you can go fishing again. After n trials, you are either !!100\%!! certain that !!C!! is not the correct product, or !!100\%-\frac1{2^n}!! certain that it is.

[1] Let us ignore mathematicians’ pettifoggery about lakes that contain exactly one fish. This is just a metaphor. If you are really concerned, you can catch-and-release.