From 2663b97c55ce35fb603944ba0c37a111ebea4ddf Mon Sep 17 00:00:00 2001
From: Joshua Potter <jrpotter2112@gmail.com>
Date: Fri, 23 Feb 2024 19:43:29 -0700
Subject: [PATCH] Add merge sort and more notes on floors/ceilings.

---
 .../plugins/obsidian-to-anki-plugin/data.json |  12 +-
 notes/_journal/2024-02-23.md                  |   6 +-
 notes/algebra/floor-ceiling.md                |  32 ++++
 notes/algorithms/images/merge-sort.gif        | Bin 0 -> 7854 bytes
 notes/algorithms/index.md                     |  97 +++++++++++
 notes/algorithms/sorting/merge-sort.md        | 156 ++++++++++++++++++
 6 files changed, 296 insertions(+), 7 deletions(-)
 create mode 100644 notes/algorithms/images/merge-sort.gif
 create mode 100644 notes/algorithms/sorting/merge-sort.md

diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
index a41ce05..e3651d5 100644
--- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
+++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
@@ -79,10 +79,11 @@
     "lattice-path-chessboard.png",
     "lattice-path-chessboard-colored.png",
     "lattice-path-before-recurrence.png",
-    "lattice-path-after-recurrence.png"
+    "lattice-path-after-recurrence.png",
+    "merge-sort.gif"
   ],
   "File Hashes": {
-    "algorithms/index.md": "a5ff7313f71777f1f3536e27dd9894fa",
+    "algorithms/index.md": "cd7c7ba91fb2f961c9f2437777e8e2ac",
     "algorithms/sorting/index.md": "2d5a18a3079d96fa9e3d4289181a8b6c",
     "algorithms/sorting/insertion-sort.md": "00e4edb132d473b0516fde3307ebae30",
     "bash/index.md": "3dfeb538d781e4645e3aaaf32beb1034",
@@ -166,7 +167,7 @@
     "_journal/2024-02/2024-02-14.md": "aa009f9569e175a8104b0537ebcc5520",
     "_journal/2024-02-16.md": "5cc129254afd553829be3364facd23db",
     "_journal/2024-02/2024-02-15.md": "16cb7563d404cb543719b7bb5037aeed",
-    "algebra/floor-ceiling.md": "e9f905f5ec01921e0f19afe8732fb38b",
+    "algebra/floor-ceiling.md": "828b11e642d6803a740b58f27ff047cf",
     "algebra/index.md": "90b842eb694938d87c7c68779a5cacd1",
     "algorithms/binary-search.md": "08cb6dc2dfb204a665d8e8333def20ca",
     "_journal/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048",
@@ -189,12 +190,13 @@
     "algebra/radices.md": "0fcd901c798eaed8075ff1375e2429dd",
     "_journal/2024-02-22.md": "e01f1d4bd2f7ac2a667cdfd500885a2a",
     "_journal/2024-02/2024-02-21.md": "f423137ae550eb958378750d1f5e98c7",
-    "_journal/2024-02-23.md": "75eec3feffa90a219de77151771fe3fe",
+    "_journal/2024-02-23.md": "219ce9ad15a8733edd476c97628b71fd",
     "_journal/2024-02/2024-02-22.md": "312e55d57868026f6e80f7989a889c2b",
     "c17/strings.md": "bbe8983602adbeb38eff214beddedd84",
     "c17/index.md": "78576ee41d0185df82c59999142f4edb",
     "c17/escape-sequences.md": "ebc63c6cdfbe60bbc2708c1b0c8da8bb",
-    "c17/declarations.md": "46b135d583a992991c889d518fec1c0f"
+    "c17/declarations.md": "46b135d583a992991c889d518fec1c0f",
+    "algorithms/sorting/merge-sort.md": "f66f482e5bd551c765fcba564c938d67"
   },
   "fields_dict": {
     "Basic": [
diff --git a/notes/_journal/2024-02-23.md b/notes/_journal/2024-02-23.md
index 0bd53ec..e33e2a1 100644
--- a/notes/_journal/2024-02-23.md
+++ b/notes/_journal/2024-02-23.md
@@ -7,7 +7,7 @@ title: "2024-02-23"
 - [ ] Sheet Music (10 min.)
 - [x] OGS (1 Life & Death Problem)
 - [ ] Korean (Read 1 Story)
-- [ ] Interview Prep (1 Practice Problem)
+- [x] Interview Prep (1 Practice Problem)
 - [x] Log Work Hours (Max 3 hours)
 
 * 101weiqi (serial numbers)
@@ -19,4 +19,6 @@ title: "2024-02-23"
 	* Q-9107
 * Read about extension and truncation of integral values using unsigned and two's-complement encoding.
 * Read through last sections of "Discrete Mathematics: An Open Introduction"'s first chapter.
-	* I took a few little notes but I didn't pay as close attention to these sections as others.
\ No newline at end of file
+	* I took a few notes but didn't pay as close attention to these sections as others.
+* Leetcode [Decode Ways](https://leetcode.com/problems/decode-ways/solutions/).
+* Practiced implementing heap sort and quicksort.
\ No newline at end of file
diff --git a/notes/algebra/floor-ceiling.md b/notes/algebra/floor-ceiling.md
index 520d0b9..02c6b73 100644
--- a/notes/algebra/floor-ceiling.md
+++ b/notes/algebra/floor-ceiling.md
@@ -240,6 +240,22 @@ Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete M
 <!--ID: 1708115683351-->
 END%%
 
+%%ANKI
+Basic
+Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, what is the size of `A[p..r]` in terms of $n = q - p + 1$?
+Back: $\lceil n / 2 \rceil$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467192-->
+END%%
+
+%%ANKI
+Basic
+Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, what is the size of `A[r+1..q]` in terms of $n = q - p + 1$?
+Back: $\lfloor n / 2 \rfloor$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467198-->
+END%%
+
 %%ANKI
 Basic
 Given `A[p..q]` and $r = \lceil (p + q) / 2 \rceil$, how does the size of `A[p..r-1]` compare to `A[r..q]`?
@@ -248,6 +264,22 @@ Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete M
 <!--ID: 1708115683354-->
 END%%
 
+%%ANKI
+Basic
+Given `A[p..q]` and $r = \lceil (p + q) / 2 \rceil$, what is the size of `A[r..q]` in terms of $n = q - p + 1$?
+Back: $\lceil n / 2 \rceil$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467202-->
+END%%
+
+%%ANKI
+Basic
+Given `A[p..q]` and $r = \lceil (p + q) / 2 \rceil$, what is the size of `A[p..r-1]` in terms of $n = q - p + 1$?
+Back: $\lfloor n / 2 \rfloor$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467207-->
+END%%
+
 %%ANKI
 Basic
 Given `A[p..q]` and $r = \lfloor (p + q) / 2 \rfloor$, how does the size of `A[p..r-1]` compare to `A[r..q]`?
diff --git a/notes/algorithms/images/merge-sort.gif b/notes/algorithms/images/merge-sort.gif
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diff --git a/notes/algorithms/index.md b/notes/algorithms/index.md
index 6174075..ddb111b 100644
--- a/notes/algorithms/index.md
+++ b/notes/algorithms/index.md
@@ -1,5 +1,102 @@
 ---
 title: Algorithms
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algorithm
 tags:
   - algorithm
 ---
+
+## Overview
+
+An **incremental** approach to algorithm design involves acting on a single element at a time. In contrast, the **divide-and-conquer** approach breaks problems into subproblems that are easier to solve.
+
+%%ANKI
+Basic
+What does an incremental approach to algorithm design refer to?
+Back: An algorithm that acts on a single element at a time.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467144-->
+END%%
+
+%%ANKI
+Basic
+What does a divide-and-conquer approach to algorithm design refer to?
+Back: An algorithm that breaks a problem into similar but simpler subproblems.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467147-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a divide-and-conquer algorithm to "bottom out"?
+Back: An input that cannot (or should not) be divided any further is encountered.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467151-->
+END%%
+
+%%ANKI
+Basic
+In the context of algorithms, what does a "sentinel" refer to?
+Back: A special value used to simplify code.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467155-->
+END%%
+
+%%ANKI
+Cloze
+Insertion sort is to an {incremental} design approach whereas merge sort is to a {divide-and-conquer} design approach.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467159-->
+END%%
+
+%%ANKI
+Basic
+What ideas does the term "divide-and-conquer" invoke?
+Back: Breaking a problem into subproblems that are easier to solve.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467164-->
+END%%
+
+%%ANKI
+Basic
+According to Cormen et al., what three steps do divide-and-conquer algorithms take?
+Back: Divide, conquer, and combine.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467167-->
+END%%
+
+%%ANKI
+Basic
+What is the "divide" step of a divide-and-conquer algorithm?
+Back: Breaking the problem into smaller instances of the same problem.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467172-->
+END%%
+
+%%ANKI
+Basic
+What is the "conquer" step of a divide-and-conquer algorithm?
+Back: Solving subproblems recursively or, if small enough, in a straightforward manner.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467178-->
+END%%
+
+%%ANKI
+Basic
+What is the "combine" step of a divide-and-conquer algorithm?
+Back: Manipulating solutions to smaller problems into a solution for the original problem.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467182-->
+END%%
+
+%%ANKI
+Basic
+What is a running time recurrence?
+Back: A formula that describes overall running time in terms of running time on smaller inputs.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467187-->
+END%%
+
+## References
+
+* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
\ No newline at end of file
diff --git a/notes/algorithms/sorting/merge-sort.md b/notes/algorithms/sorting/merge-sort.md
new file mode 100644
index 0000000..01e1412
--- /dev/null
+++ b/notes/algorithms/sorting/merge-sort.md
@@ -0,0 +1,156 @@
+---
+title: Merge Sort
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algorithm::sorting
+tags:
+  - algorithm
+  - sorting
+---
+
+## Overview
+
+Property    | Value
+----------- | --------
+Best Case   | $\Omega(n\lg{n})$
+Worst Case  | $O(n\lg{n})$
+Avg. Case   | $O(n\lg{n})$
+Aux. Memory | -
+Stable      | -
+Adaptive    | -
+
+![[merge-sort.gif]]
+
+%%ANKI
+Basic
+What does the term "merge" in merge sort refer to?
+Back: The primary operation used to combine array halves.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467101-->
+END%%
+
+%%ANKI
+Basic
+What is merge sort's best case runtime?
+Back: $\Omega(n\lg{n})$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467109-->
+END%%
+
+%%ANKI
+Basic
+What is merge sort's worst case runtime?
+Back: $O(n\lg{n})$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467112-->
+END%%
+
+%%ANKI
+Basic
+What is merge sort's average case runtime?
+Back: $O(n\lg{n})$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467115-->
+END%%
+
+%%ANKI
+Basic
+What is the "divide" step of merge sort?
+Back: Divide the input array into two subarrays of half size.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467118-->
+END%%
+
+%%ANKI
+Basic
+What is the "conquer" step of merge sort?
+Back: Call merge sort on the "divide"-step's two subarrays.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467121-->
+END%%
+
+%%ANKI
+Basic
+What is the "combine" step of merge sort?
+Back: Merge the subarrays sorted after the "conquer" step.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467125-->
+END%%
+
+%%ANKI
+Basic
+When does merge sort "bottom out"?
+Back: When the sequence to be sorted has length 1 or less.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467133-->
+END%%
+
+```c
+static void merge(int i, int mid, int j, int *A) {
+  int si = mid - i + 1;
+  int sj = j - (mid + 1) + 1;
+
+  int *L = malloc(sizeof(int) * (si + 1));
+  int *R = malloc(sizeof(int) * (sj + 1));
+
+  L[si] = INT_MAX;
+  R[sj] = INT_MAX;
+
+  for (int k = 0; k < si; ++k) {
+    L[k] = A[i + k];
+  }
+  for (int k = 0; k < sj; ++k) {
+    R[k] = A[mid + 1 + k];
+  }
+
+  int topL = 0, topR = 0;
+  for (int k = 0; k < j - i + 1; ++k) {
+    if (L[topL] < R[topR]) {
+      A[i + k] = L[topL++];
+    } else {
+      A[i + k] = R[topR++];
+    }
+  }
+
+  free(L);
+  free(R);
+}
+
+void merge_sort(int i, int j, int *A) {
+  if (j <= i) {
+    return;
+  }
+  int mid = (i + j) / 2;
+  merge_sort(i, mid, A);
+  merge_sort(mid + 1, j, A);
+  merge(i, mid, j, A);
+}
+```
+
+%%ANKI
+Basic
+Where in merge sort's implementation are sentinels useful?
+Back: As the last elements of the two arrays to combine.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467137-->
+END%%
+
+%%ANKI
+Basic
+What sentinel values are typically used in merge sort's "merge" operation?
+Back: $\infty$ or the record type's equivalent.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742467141-->
+END%%
+
+%%ANKI
+Basic
+What sorting algorithm does the following demonstrate?
+![[merge-sort.gif]]
+Back: Merge sort.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708742590435-->
+END%%
+
+## References
+
+* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).