new post: min-plus with apl

_posts/2024-04-04-apl-min-plus.md

@@ -0,0 +1,112 @@
+---
+title: "Tropical Semiring in Dyalog APL"
+layout: post
+use_katex: true
+---
+
+# A few of my favorite things
+
+The [Tropical (min-plus)
+semiring](https://en.wikipedia.org/wiki/Tropical_semiring) is one of my
+favorite examples of how changing one's perspective can make difficult problems
+much simpler.[^1]
+In this semiring, instead of using the usual addition and multiplication,
+we replace them with minimum and addition, so, for instance, \\( 1 \oplus 2 = 1
+\\) and \\( 3 \otimes 2 = 5 \\).
+I've [written]({% post_url 2023-05-11-kompreni %}) and [presented]({% post_url
+2018-02-20-algebraic-structure %}) about [this]({% post_url
+2022-09-13-ascb-in-rust %}) before.
+
+# Links from the farm
+
+Recently, someone on the [APL Farm](https://aplwiki.com/wiki/APL_Farm) posted
+[an excellent
+article](http://blog.sigfpe.com/2009/05/three-projections-of-doctor-futamura.html)
+by `sigfpe` (AKA [Dan](https://mathstodon.xyz/@dpiponi)
+[Piponi](https://twitter.com/sigfpe)), which prompted me to post a
+[different](http://blog.sigfpe.com/2007/06/how-to-write-tolerably-efficient.html)
+article where they work with the min-+ semiring.
+I also posted [this article](https://r6.ca/blog/20110808T035622Z.html) by
+Russell O'Connor, which is one of my all-time favorites.
+In it, there's an example where the distances between nodes in a graph gets
+turned into a Tropical matrix, and algorithms for computing shortest distances
+become iterated linear algebra over this different semiring.
+And then I wanted to work in APL instead of Haskell...
+
+# An open dialogue
+
+After a bit of head scratching, I realized that [Dyalog
+APL](https://www.dyalog.com/) is really good for manipulating matrices over the
+min-+ semiring.
+Taking `exampleGraph2` from O'Connor's blog post, and letting \\( 10^{20} \\)
+stand in for \\( \infty \\)[^2]:
+
+```apl
+inf←1e20
+dist←6 6⍴0 7 9 inf inf 14 7 0 10 15 inf inf 9 10 0 11 inf 2 inf 15 11 0 6 inf inf inf inf 6 0 9 14 inf 2 inf 9 0
+```
+
+we have the following distance matrix
+
+```apl
+0.0E0  7.0E0  9.0E0  1.0E20 1E20 1.4E1
+7.0E0  0.0E0  1.0E1  1.5E1  1E20 1.0E20
+9.0E0  1.0E1  0.0E0  1.1E1  1E20 2.0E0
+1.0E20 1.5E1  1.1E1  0.0E0  6E0  1.0E20
+1.0E20 1.0E20 1.0E20 6.0E0  0E0  9.0E0
+1.4E1  1.0E20 2.0E0  1.0E20 9E0  0.0E0
+```
+
+With this matrix, representing distances between vertices in our graph, we can
+use the APL matrix product operator `.` _with different operations_ to perform
+our calculations; instead of `+` and `×` for addition and multiplication, we
+turn to `⌊` for min and `+`; then `dist ⌊.+ dist` gives us the two-hop distances:
+
+```apl
+      dist ⌊.+ dist
+ 0  7  9 20 23 11
+ 7  0 10 15 21 12
+ 9 10  0 11 11  2
+20 15 11  0  6 13
+23 21 11  6  0  9
+11 12  2 13  9  0
+```
+
+We can make this made more succinct with `⍨`, telling the interpreter to apply our
+function with `dist` as both arguments:
+
+```apl
+      ⌊.+⍨ dist
+ 0  7  9 20 23 11
+ 7  0 10 15 21 12
+ 9 10  0 11 11  2
+20 15 11  0  6 13
+23 21 11  6  0  9
+11 12  2 13  9  0
+```
+
+I was very excited that I could write down the steps of the
+Gauss-Jordan-Floyd-Warshall-Kleene algorithm in so few characters!
+Moreover, iterating that step until convergence is similarly succinct using the
+power operator `⍣`; we can keep running `⌊.+` until the output matches (`≡`) the input:
+
+```apl
+      ⌊.+⍨⍣≡dist
+ 0  7  9 20 20 11
+ 7  0 10 15 21 12
+ 9 10  0 11 11  2
+20 15 11  0  6 13
+20 21 11  6  0  9
+11 12  2 13  9  0
+```
+
+I can't help but feel that this was the type of thing Iverson meant by
+[notation as a tool of thought](https://www.jsoftware.com/papers/tot.htm).
+
+
+[^1]: As Alan Kay [says](https://quoteinvestigator.com/2018/05/29/pov/), point
+    of view is worth 80 IQ points.
+
+[^2]: I tried this in [BQN](https://mlochbaum.github.io/BQN/) as well, since it
+    has \\( \infty \\), but the linear algebra and fixed point iteration are
+    nicer in APL.

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