From 8015bcec221153d63e1a51dd6692e42aa88e8914 Mon Sep 17 00:00:00 2001 From: Yadunand Prem Date: Sat, 6 Apr 2024 21:11:06 +0800 Subject: [PATCH] feat: misc updates --- .gitignore | 4 ++ cs2109s/labs/requirements.txt | 8 ---- cs3230/pa3.pdf | Bin 0 -> 197927 bytes cs3230/pa3.typ | 82 ++++++++++++++++++++++++++++++++++ 4 files changed, 86 insertions(+), 8 deletions(-) delete mode 100644 cs2109s/labs/requirements.txt create mode 100644 cs3230/pa3.pdf create mode 100644 cs3230/pa3.typ diff --git a/.gitignore b/.gitignore index 2be9f09..ac5648f 100644 --- a/.gitignore +++ b/.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 diff --git a/cs2109s/labs/requirements.txt b/cs2109s/labs/requirements.txt deleted file mode 100644 index 51f0e8c..0000000 --- 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..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 + + +