feat: misc updates

.gitignore

@@ -628,3 +628,7 @@ poetry.toml
 pyrightconfig.json
 
 # End of https://www.toptal.com/developers/gitignore/api/python,jupyternotebooks,c,c++,java,latex
+cs2109s/labs/ps6/MNIST/raw
+cs2109s/labs/ps7/MNIST/raw
+cs2109s/labs/.idea
+cs2109s/labs/ps7/cifar-10-batches-py

cs2109s/labs/requirements.txt

@@ -1,8 +0,0 @@
-matplotlib==3.5.2
-numpy==1.21.5
-pandas==1.5.2
-scikit-learn==1.0.2
-seaborn==0.12.0
-timeout-decorator==0.5.0
-torch==1.11.0
-torchvision==0.12.0

cs3230/pa3.typ

@@ -0,0 +1,82 @@
+#let project(title: "", authors: (), body) = {
+  // Set the document's basic properties.
+  set document(author: authors, title: title)
+  set page(numbering: "1", number-align: center)
+  set text(font: "Linux Libertine", lang: "en")
+
+  // Title row.
+  align(center)[
+    #block(text(weight: 700, 1.75em, title))
+  ]
+
+  // Author information.
+  pad(
+    top: 0.5em,
+    bottom: 0.5em,
+    x: 2em,
+    grid(
+      columns: (1fr,) * calc.min(3, authors.len()),
+      gutter: 1em,
+      ..authors.map(author => align(center, strong(author))),
+    ),
+  )
+
+  // Main body.
+  set par(justify: true)
+
+  body
+}
+
+// Take a look at the file `template.typ` in the file panel
+// to customize this template and discover how it works.
+#show: project.with(
+  title: "CS3230 PA3",
+  authors: (
+    "Yadunand Prem, A0253252M",
+  ),
+)
+
+= Q1
+== Optimal Substructure
+
+Let $j$ be the number of houses you're visiting and $S_j$ be the smallest sum of anger that is attainable from $j$ houses, $A_j$ be the $j$th smallest anger that they can attain Then $S_j = A_j + S_(j-1)$
+
+Optimal solution of $j$ houses is the sum of the optimal solution of $j-1$ houses, and the anger of the house with $j$th smallest anger. 
+
+
+== Proof of Correctness
+
+1. Let $S_j$ be the minimum anger attainable through visitng $j$ houses (Optimal)
+2. Suppose that the $S_(j-1)$ solution is not optimal. 
+3. Then there exists a min anger $x$ of $j-1$ houses where $x < S_(j-1)$
+4. Combining $x + A_j$ would then give a solution that is less than $S_j$ contradicting our initial clause. 
+5. Thus, this optimal substructure exists
+
+== Greedy Choice
+Greedy Choice: Let $x$ be the house with the minimum anger. Then there exists an optimal solution containing $x$, called $S$. 
+
+== Proof of Greedy
+1. Suppose not, that $i in.not S$
+2. Then, we can replace any $x$ in $S$ with $i$ and get a sum with a lower anger (Cannot be same anger as $i$ is distinct). This contradicts the initial statement of $S$ being the optimal solution. 
+
+== Last Step
+1. To get the house with $j$th smallest anger, we can sort the houses by its anger values. This can be done in $O(n log(n)$ timing. Without this, it would take $O(k)$ time to get the k-th smallest anger each time.
+2. We can then build a prefix-sum on the anger values in $O(n)$ time. Now by accessing the $i-1$th element of the array (as its 0 indexed), we can immediately retrieve the min anger attainable in $i$ houses in $O(1)$ timing.
+
+Total Time Complexity: $O(n log(n) + n + 1) = O(n log(n))$
+
+
+= Q2
+
+== Optimal Substructure
+Let $S(m)$ be the number of ways that frank can arrange the menu items in $m$ minutes. Then $S(m) = sum (S(m-T_i))$, where $T_i$ is the time taken for the $i$th item. To note: repetition is allowed and $S(k) = 0, forall k < 0)$. 
+
+== Overlapping Subproblems
+There are overlapping subproblems when there are 2 courses with the same time. For example, if there are 2 courses $a$ and $b$, each with time $T_a = T_b = 1$, then $S(m-T_a) = S(m-T_b)$. Thus, there are overlapping subproblems.
+
+If all subproblems are recomputed, the time complexity would be $O(n^m)$ in the worst case, where $n$ is the number of minutes and $m$ is the number of courses. This has an exponential time complexity.
+
+If the overlapping subproblems are avoided, then this has an complexity of $O(n m)$. The DP problem has 1 parameter $m$, and in each subproblem, it calculates the sum of $n$ integers, which runs in $O(n)$ time. Thus, $O(n m)$. This has a pseudopolynomial timing
+
+
+