From eab5ae9359feea12234421c3775f991c121bdc47 Mon Sep 17 00:00:00 2001
From: "Brady J. Garvin" <bgarvin@cse.unl.edu>
Date: Mon, 30 Oct 2023 13:29:11 -0500
Subject: [PATCH] Recorded work from Monday.
---
monoid-design.md | 28 ++++++++++++++--------------
1 file changed, 14 insertions(+), 14 deletions(-)
diff --git a/monoid-design.md b/monoid-design.md
index f5cf156..9877814 100644
--- a/monoid-design.md
+++ b/monoid-design.md
@@ -270,32 +270,32 @@ function isBalanced(parentheses) {
## Quotienting a Free Monoid by Relations
-* The sequence `…` cancels to the identity.
-* If we simplify any input by repeatedly removing occurrences of `…`, we eventually end up with a string of the form `…` where `…` is ….
+* The sequence `()` cancels to the identity.
+* If we simplify any input by repeatedly removing occurrences of `()`, we eventually end up with a string of the form `)*(*` where `*` is **the Kleene star**.
## Parentheses Monoid
* Monoid:
- * `P = (…, ∙, …)` where `(a, b) ∙ (c, d) = (…, …)`
+ * `P = (𝐍 × 𝐍, ∙, (0, 0))` where `(a, b) ∙ (c, d) = (a + c - min(b, c), b + d - min(b, c))`
## Divide-and-Conquer Design
```
In: '()(())'
- Out:
+ Out: 0, 0
/ \
/ \
/ \
In: '()(' In: '())'
- Out: Out:
+ Out: 0, 1 Out: 1, 0
/ \ / \
/ \ / \
In: '(' In: ')(' In: '(' In: '))'
- Out: Out: Out: Out:
+ Out: 0, 1 Out: 1, 1 Out: 0, 1 Out: 2, 0
/ \ / \
/ \ / \
In: ')' In: '(' In: ')' In: ')'
- Out: Out: Out: Out:
+ Out: 1, 0 Out: 0, 1 Out: 1, 0 Out: 1, 0
```
## New Iterative Solution
@@ -303,22 +303,22 @@ function isBalanced(parentheses) {
```js
function encode(character) {
if (character === '(') {
- return […, …];
+ return [0, 1];
}
if (character === ')') {
- return […, …];
+ return [1, 0];
}
return [0, 0];
}
function combine([leftCloses, leftOpens], [rightCloses, rightOpens]) {
- const cancellations = …;
- return […, …];
+ const cancellations = Math.min(leftOpens, rightCloses);
+ return [leftCloses + rightCloses - cancellations, leftOpens + rightOpens - cancellations];
}
function iterativeReduce(parentheses) {
- let result = …;
- for (const element of …) {
+ let result = [0, 0];
+ for (const element of [...parentheses].map(encode)) {
result = combine(result, element);
}
return result;
@@ -337,7 +337,7 @@ async function parallelReduce(parentheses, threadCount) {
jobs.push(job);
}
const [closes, opens] = iterativeReduce(await Promise.all(jobs));
- return …;
+ return closes === 0 && opens === 0;
}
```
--
GitLab