class: middle, center # Lecture 2 ### 19 August 2024 Admin Matters<br> **Unit 3**: Functions<br> **Unit 4**: Types --- class: top,wide ### Tutorials and Labs - Starts next week. - Appeal via CourseReg if you still haven't gotten a slot. - Swap with your friends if your slot is not ideal. - We might ask you to switch rooms to balance out the groups (please say yes). --- class: top,wide ### Preparing for CS1010 By now you should: - have your Piazza, GitHub, and SoC accounts setup and ready to go - be able to `ssh` into the PE environments and use basic Unix CLI commands --- class: top,wide ### Unix Clinics (This week only) Venue: MR6 (AS6 #05-14) - Tuesday, 2 - 3 PM - Wednesday, 12 - 1 PM, 2 - 3 PM - Thursday, 12 - 1 PM --- class: middle,center background-image: url(figures/why-cli/terminal.png) ## <font color="white">Why Unix CLI?</font> --- class: middle,center .fit[] .tiny[ <a href="https://www.flaticon.com/free-icons/computer" title="computer icons">Computer icons created by Freepik - Flaticon</a> ] --- class: middle,center background-image: url(figures/why-vim/vim-example.png) ## <font color="white">Why Vim?</font> --- class: top - Terminal-based - Practical examination constraints - Installed by default in many OSes - Optimized for keyboard-only editing - Editing at the speed of thoughts --- background-image: url(figures/why-vim/stackoverflow-survey-2024.png) class: black ??? .tiny[[via stackoverflow](https://survey.stackoverflow.co/2024/technology#most-popular-technologies-new-collab-tools)] --- background-image: url(figures/reddit-vim-is-helpful.png) class: black ??? .tiny[[via reddit](https://www.reddit.com/r/nus/comments/nwpc54/nus_cs_fun/h1b2ikz/?context=3)] --- class: top ### Getting start with `vim` - Read CS1010 Handbook articles about [the philosophy of `vim`](https://nus-cs1010.github.io/2425-s1/guides/vim-philosophy.html) and [our quick lessons](https://nus-cs1010.github.io/2425-s1/guides/vim-quick-lessons.html) - [Set up `Vim` on your PE account](https://nus-cs1010.github.io/2425-s1/guides/vim-setup.html) - Run `vimtutor` - Many other resources available on the Web --- class: black, center background-image: url(figures/why-vim/vim-racer.png) ## <font color="white">Vim Racer</font> --- class: black, center background-image: url(figures/why-vim/vim-adventure.png) ## <font color="white">Vim Adventures</font> --- class: black, center background-image: url(figures/why-vim/vim-genius.png) ## <font color="white">Vim Genius</font> --- class: top,center .smaller[.fit[]<br>https://xkcd.com/518] --- class: top,wide ### Flowchart .fit[] --- class: middle,center ## Functions <!-- Start changes here --> --- class: top,center ### Functions in Math $$f : \mathbb{R} \rightarrow \mathbb{R}$$ $$f(x) = x^2 + 1$$ $f(2)$ evaluates to $5$ --- class: middle,center,wide $$m = max(L, k)$$ .fit[] --- class: middle,center,wide Represent the solution to a computational problem<br>as a function. .fit[] --- ### Find The Range - The _range_ of a given list of $k$ integers is the difference between the max and the min values in the list. - Give an algorithm to find the range given a list $L$ of $k$ integers. --- class: middle,center,wide We can try to change .fit[] --- class: middle,center or refer to existing solutions (as functions) ### $max(L, k) - min(L, k)$ --- class: middle,center ### Abstraction .fit[] --- class: top ### Functions - A powerful tool to help us think at a higher level. - We can focus on _what_ a function does, not _how_ it does it. - Free us from worrying about details (e.g., variables $i$ and $m$ are now internal to $max$) - Allows us to re-use the same idea/process. --- class: top ### Find the Mean - Give an algorithm to find the mean value given a list $L$ of $k$ integers. --- class: middle,center ## $\frac{sum(L,k)}{k}$ --- class: top ### A C program is just a collection of functions -- - written by you - provided by the standard C library - provided by other libraries --- class: middle,center Assume $len(L)$ returns the size of the list. ## $\frac{sum(L)}{len(L)}$ --- ### Find the Standard Deviation Give an algorithm to find the standard deviation of a list $L$ of $k$ integers. $$\sqrt{\frac{\sum\_{i=0}^{k-1}(l\_i - \mu)^2}{k}}$$ --- class: middle,center ### Let's break it down into sub-problems --- class: middle,center ### Colour Coded: .fit[] --- class: middle,center ### Pictorially: .fit[] --- class: middle,center .small[ $$\sum\_{i=0}^{k-1}(l\_i - \mu)^2$$ set $\mu$ to $mean(L)$ set $L'$ to $subtract(L, \mu)$ set $L''$ to $square(L')$ set $v$ to $mean(L'')$ set $\sigma$ to $sqrt(v)$] --- class: middle,center,wide ### So here's the flowchart: .fit[] --- class: middle,center,wide ### Decomposing the problem into smaller problems. .fit[] --- class: middle,center,wide ### The smaller problems should be abstracted away. .fit[] --- ### Functions we wish for - $square(L)$ - $subtract(L, \mu)$ - $sqrt(x)$ - $mean(L)$ --- class: middle,center .fit[] --- class: middle, center ### What about $sqrt(x)$? How do we implement this? --- class:center,top ### Hold on, an important detail: .fit[] What are the incoming arrows for? --- class:center,top ### Hold on, an important detail: .fit[] .small[Remember: Functions take inputs and produce outputs.] --- ### Approach to solve computational problems - _Decompose_ a problem into (simpler, smaller) sub-problems - Solve the sub-problems one-by-one with appropriate functions - Compose the functions to solve the original given task --- class: center ### Question: What's stopping a function from using itself? -- Nothing! We can totally do it. --- class: center, middle ## Recursion To understand recursion, you must first understand recursion. --- class: center, middle Let's go through a few problems that can be solved recursively --- class: center ### Let's try solving factorial: Recall: $$ n! = n \times (n - 1)!$$ and also $$0! = 1$$ --- class: top, center ### What do we think of this algorithm? .fit[] -- .small[The program will not terminate!] --- class: top, center ### Actual factorial flowchart: .fit[] --- class: top ### Wishful thinking: Key features: - The algorithm solved the problem by relying on solutions to _smaller_ or _simpler_ sub-problems. - The algorithm "does not care" how it's done. Only that the answer is correct. - The algorithms have "base cases" that they eventually solve. --- class: top ### Recursion: Top-down vs Bottom-up. Up until this point, we've shown you how to solve Factorial using a top-down approach: -- (Informally) "To solve problem $n$, we trust someone to solve the $n - 1$ case, and we make use of it to solve our problem." --- class: top ### Recursion: Top-down vs Bottom-up. There is another way (slightly different way) of seeing it: -- We slowly build up bigger and bigger solutions, using parts that we have built up. --- class: center ### Finding the max of a list. Again. Recall we did this last week iteratively. This week, let's try to solve it recursively. --- class: center ### Before we begin: .fit[] --- class: center ### Finding max of 3 elements: .fit[] --- class: center ### Finding max of 3 elements: .fit[] --- class: center ### Finding max of 3 elements: .fit[] --- class: center ### Finding max of 3 elements: .fit[] --- class: center ### Finding max of 3 elements: .fit[] --- class: center ### Finding max of 3 elements: .fit[] --- class: center ### Finding max of 3 elements: .fit[] --- class: center ### Here's how we represent it in a flowchart: .fit[] -- Gotcha! The base case is missing! --- class: center ### With the base case back in: .fit[] --- class: center ### Yet another way to do it recursively: .tiny[ ``` 12 20 11 20 14 2 24 36 43 16 27 6 11 7 10 15 38 10 22 7 16 26 32 30 11 9 30 6 20 6 27 19 26 10 27 6 19 34 5 9 3 22 10 4 13 1 10 22 43 36 39 29 41 12 13 25 17 8 30 31 29 38 2 42 7 45 24 33 9 40 34 29 37 2 26 17 19 34 6 7 34 22 21 41 38 5 15 13 9 1 42 39 5 29 38 4 22 29 41 10 26 32 30 26 16 16 18 22 32 34 14 10 5 17 25 16 19 6 31 16 3 13 8 42 41 0 13 30 44 1 41 14 5 39 40 38 6 37 38 9 . ``` ] -- Say we were given a list of numbers. --- class: middle,center ### Yet another way to do it recursively: .tiny[ ``` 12 20 11 20 14 2 24 10 15 38 10 22 7 16 20 6 27 19 26 10 27 10 4 13 1 10 22 43 17 8 30 31 29 38 2 34 29 37 2 26 17 19 38 5 15 13 9 1 42 41 10 26 32 30 26 16 5 17 25 16 19 6 31 13 30 44 1 41 14 5 . ``` ] What if we knew the maximum from this half... (Let's call it $m_l$) --- class: middle,center ### Yet another way to do it recursively: .tiny[ ``` 36 43 16 27 6 11 7 26 32 30 11 9 30 6 6 19 34 5 9 3 22 36 39 29 41 12 13 25 42 7 45 24 33 9 40 34 6 7 34 22 21 41 39 5 29 38 4 22 29 16 18 22 32 34 14 10 16 3 13 8 42 41 0 39 40 38 6 37 38 9 . ``` ] ... and the maximum from this half? (Let's call it $m_r$) --- class: middle,center ### Yet another way to do it recursively: .tiny[ ``` 12 20 11 20 14 2 24 36 43 16 27 6 11 7 10 15 38 10 22 7 16 26 32 30 11 9 30 6 20 6 27 19 26 10 27 6 19 34 5 9 3 22 10 4 13 1 10 22 43 36 39 29 41 12 13 25 17 8 30 31 29 38 2 42 7 45 24 33 9 40 34 29 37 2 26 17 19 34 6 7 34 22 21 41 38 5 15 13 9 1 42 39 5 29 38 4 22 29 41 10 26 32 30 26 16 16 18 22 32 34 14 10 5 17 25 16 19 6 31 16 3 13 8 42 41 0 13 30 44 1 41 14 5 39 40 38 6 37 38 9 . ``` ] Then, the overall max of the entire list is just the larger of $m_l$ and $m_r$! --- class: center ### In general, this is the process: .fit[] .tiny[Note: $\lfloor x \rfloor$ means to round $x$ down to the nearest integer.] --- class: center ### Again, but with a flowchart: .fit[] --- class: top ### Some details to think about: - What is the base case? What do we do during the base case? - How are we decomposing the problem into sub-problems? - How do we use the answers of the sub-problems? --- class:middle,center # Types --- ### Recap - A variable is a named storage location in memory. - A variable holds a value. - A value is a sequence of bits. --- class:middle ### `1010010101010101001111010..` can mean - 1933091 - 34.95108 - `hello` - an image --- class:middle ### A variable needs a type to be interpreted correctly. --- class:middle ### A type is represented by a fixed, finite, number of bits. ### Different types can be represented by a different number of bits. --- class:middle, center ### 1 bit: `1` or `0` black or white<br> true or false<br> yes or no<br> S or U --- class:middle, center ### 2 bits: `11`, `10`, `01`, or `00` north, south, east, west<br> spring, summer, fall, winter --- class:middle, center ### In general, $k$ bits can represent $2^k$ different possible values --- class:middle ### more bits $\rightarrow$ more memory, but can represent bigger range ### fewer bits $\rightarrow$ less memory, but represent smaller range --- class:middle ### When writing programs for embedded systems, IoT devices, sensors, etc, you need to be frugal with memory usage. ### In CS1010, however, we will go with more bits for better precision. --- class:middle, center ## Integers --- class:middle, center ### 8 bits: 256 different integers 0 to 255 (for unsigned)<br> -128 to 127 (for signed) --- class:middle, center ### 64 bits signed -9,223,372,036,854,775,808 <br> to <br> 9,223,372,036,854,775,807 --- class:middle, center ## Characters --- class:middle, center ### 8 bits: 127 different symbols using ASCII standards ``` 0-9, A-Z, a-z <>?:”{}!@#$%^&* ()_+-=[]\';/., ``` return, tab, escape, etc --- class:middle, center ### up to 32 bits: Unicode (UTF-8) 11111011000000000 😀 --- class:middle, center ## Real numbers (also called floating point numbers) --- class:middle ### There are infinitely many real numbers but only a finite sequence of bits. ### Not all of real numbers can be represented. --- class:middle, center ### Not every real number can be presented precisely E.g., `0.1 + 0.2` is `0.30000000000000004` --- class:middle, center background-image: url(figures/foodpanda.png) --- class:middle, center ### Rule: Use integer type unless you absolutely need to use a real number type. --- class:middle, center ### We normally use 32, 64, or 128 bits to represent real numbers. --- class: middle - A variable needs a type to be interpreted correctly. - We need to decide on the type of a variable before we can use it. --- class:middle set $total$ to $sum(L, k)$<br> set $avg$ to $mean(L, k)$ $L$ is a list of integers.<br> $k$ is the number of elements in $L$. Should these variables be an integer or an real number?<br> - $k$ - $total$ - $avg$ --- class:top ### Compound type - A compound type is made out of other _primitive types_. Example, - A word (a sequence of characters) - A 2D coordinate (two real numbers $(x, y)$) --- class: wide .small[ ### Homework - Problem Set in [Unit 3](https://nus-cs1010.github.io/2425-s1/03-func.html) - `ssh` into CS1010 PE hosts and follow the Unix CLI and `vim` tutorial - Diagnostic Quiz for Lecture 2 (due this Wed 2359) ### From Last Week - Diagnostic Quiz for Lecture 1 (due this Wed 2359) - Diagnostic Quiz for Using CS1010 PE (due this Wed 2359) ] --- class: bottom .tiny[ ]