From 0ade4b2efb5965c2c5176a59e9b8bbb5da119db0 Mon Sep 17 00:00:00 2001 From: Danila Fedorin Date: Tue, 25 Oct 2022 22:46:05 -0700 Subject: [PATCH] Add a tiny section about the tropical semiring to polynomial post --- content/blog/search_polynomials.md | 56 ++++++++++++++++++++++++++++-- 1 file changed, 54 insertions(+), 2 deletions(-) diff --git a/content/blog/search_polynomials.md b/content/blog/search_polynomials.md index a4177aa..c6ec303 100644 --- a/content/blog/search_polynomials.md +++ b/content/blog/search_polynomials.md @@ -202,7 +202,7 @@ way, polynomials like this give us _less_ information than our original ones we're just warming up - there are more interesting semirings for us to consider! -#### Polynomials over Sets of Paths, \\(\\mathcal{P}(\\Pi)\\) +#### The Semiring of Sets of Paths, \\(\\mathcal{P}(\\Pi)\\) Until now, we explicitly said that "all paths of the same length are equivalent". If we're giving directions, though, we might benefit from knowing not just that there _is_ a way, but what roads that @@ -267,7 +267,7 @@ consisting of each path from \\(A\\) concatenated with each path from \\(B\\): {{< latex >}} -A \times B = \{ a \rightarrow b\ |\ a \in A, b \in B \} +A \times B \triangleq \{ a \rightarrow b\ |\ a \in A, b \in B \} {{< /latex >}} The fact that this definition of multiplication on sets is associative @@ -325,3 +325,55 @@ Multiplying them gives: This resulting polynomial gives us all the paths from city A to city C, grouped by their length! + +#### The Tropical Semiring, \\(\\mathbb{R}\\) +I only have one last semiring left to show you before we move on to something +other than paths between cities. It's a fun semiring though, as even its name +might suggest: we'll take a look at a _tropical semiring_. + +In this semiring, we go back to numbers; particularly, real numbers (e.g., +\\(1.34\\), \\(163\\), \\(e\\), that kind of thing). We even use addition -- +sort of. In the tropical semiring, addition serves as the _multiplicative_ +operation! This is even confusing to write, so I'm going to switch up notation; +in the rest of this section, I'll use \\(\\otimes\\) to represent the +multiplicative operation in semirings, and \\(\\oplus\\) to represent the +additive one. The symbols \\(\\times\\) and \\(+\\) will be used to represent +the regular operations on real numbers. With that, the operations on our +tropical semiring over real numbers are defined as follows: + +{{< latex >}} +\begin{array}{rcl} +x \otimes y & \triangleq & x + y\\ +x \oplus y & \triangleq & \min(x,y) +\end{array} +{{< /latex >}} + +What is this new semiring good for? How about this: suppose that in addition to +the duration of the trip, you'd like to track the distance you must travel for +each route (shorter routes do sometimes have more traffic!). Let's watch what +happens when we add and multiply polynomials over this semiring. +When we add terms with the same power but different coefficients, like +\\(ax\oplus bx\\), we end up with a term \\(\min(a,b)x\\). In other words, +for each trip duration, we pick the shortest length. When we multiply two +polynomials, like \\(ax\otimes bx\\), we get \\((a+b)x\\); in other words, +when sequencing two trips, we add up the distances to get the combined +distance, just like we'd expect. + +We can, of course, come up with a polynomial to match our initial example. +Say that the trips from A to B are represented by \\(2.0x^2\oplus1.5x\\\) (the +shortest two-hour trip is \\(2\\) units of distance long, and the one-hour +trip is \\(1.5\\) units long), and that the trips from B to C are represented +by \\(4.0x^3\oplus1.0x^2\\). Multiplying the two polynomials out gives: + +{{< latex >}} +\begin{array}{rcl} +(2.0x^2\oplus1.5x)(4.0x^3\oplus1.0x^2) & = & 6.0x^5 \oplus \min(2.0+1.0, 1.5+4.0)x^4 \oplus 2.5x^3 \\ + & = & 6.0x^5 \oplus 3.0x^4 \oplus 2.5x^3 +\end{array} +{{< /latex >}} + +The only time we used the additive operation in this case was to pick between +two trips of equal druation but different length (two-hour trip from A to B +followed by a two-hour trip from B to C, or one-hour trip from A to C followed +by a three-hour trip from B to C). The first trip wins out, since it requires +only \\(3.0\\) units of distance.