Cheatsheets for CM2035 Algorithms & Data Structures II

level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/README.md

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+# About
+
+Listed here is a collection of cheatsheet by topic. Those cheatsheets do not
+explain the topics in depth, but rather serve as quick lookup documents.
+Therefore, the course material provided by the lecturer should still be studied
+and understood. Not everything that is tested at the mid-terms or final exams is
+covered and the Author does not guarantee that the cheatsheets are free of
+errors.
+
+* [Time and Space Complexity](./cheatsheet_time_space_complexity.pdf)
+* [Asymptotic Analysis](./cheatsheet_asymptotic_analysis.pdf)
+* [Time Complexity of Recursive Algorithms](.cheatsheet_time_complexity_recursive_algorithms.pdf)
+* [Comparison and Non-Comparison Sorting Algorithms](./cheatsheet_sorting_algorithms.pdf)
+* [Hash Tables](./cheatsheet_hash_tables.pdf)
+
+**NOTE**: Those cheatsheets only cover the course material **up to the midterms**.
+The weeks after the midterms are not covered here.
+
+# Building
+
+_NOTE_: This step is only necessary if you chose to modify the base documents.
+
+The base documents are written in [AsciiDoc](https://asciidoc.org/) and can be
+found in the `src/` directory.
+
+The following dependencies must be installed (Ubuntu):
+
+```console
+$ apt install -y ruby-dev wkhtmltopdf
+$ gem install asciidoctor
+$ chmod +x build.sh
+```
+
+To build the documents (PDF version):
+
+```console
+$ ./build.sh pdf
+```
+
+Optionally, for the HTML version:
+
+```console
+$ ./build.sh html
+```
+
+and for the PNG version:
+
+```console
+$ ./build.sh png
+```
+
+The generated output can be deleted with `./build.sh clean`.
+
+# Disclaimer
+
+The Presented Documents ("cheatsheets") by the Author ("Fabio Lama") are
+summaries of specific topics. The term "cheatsheet" implies that the Presented
+Documents are intended to be used as learning aids or as references for
+practicing and does not imply that the Presented Documents should be used for
+inappropriate practices during exams such as cheating or other offenses.
+
+The Presented Documents are heavily based on the learning material provided by
+the University of London, respectively the VLeBooks Collection database in the
+Online Library and the material provided on the Coursera platform.
+
+The Presented Documents may incorporate direct or indirect definitions,
+examples, descriptions, graphs, sentences and/or other content used in those
+provided materials. **At no point does the Author present the work or ideas
+incorporated in the Presented Documents as their own.**

level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/build.sh

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+#!/bin/bash
+
+# Because `make` sucks.
+
+gen_html() {
+    # Remove suffix and prefix
+    FILE=$1
+    OUT=${FILE%.adoc}
+    HTML_OUT="cheatsheet_${OUT}.html"
+
+    asciidoctor $FILE -o ${HTML_OUT}
+}
+
+# Change directory to src/ in order to have images included correctly.
+cd "$(dirname "$0")/src/"
+
+case $1 in
+    html)
+        for FILE in *.adoc
+        do
+            # Generate HTML file.
+            gen_html ${FILE}
+        done
+
+        # Move up from src/
+        mv *.html ../
+        ;;
+    pdf)
+        for FILE in *.adoc
+        do
+            # Generate HTML file.
+            gen_html ${FILE}
+
+            # Convert HTML to PDF.
+            PDF_OUT="cheatsheet_${OUT}.pdf"
+            wkhtmltopdf \
+                --enable-local-file-access \
+                --javascript-delay 2000\
+                $HTML_OUT $PDF_OUT
+        done
+
+        # Move up from src/
+        mv *.pdf ../
+
+        # Cleanup temporarily generated HTML files.
+        rm *.html > /dev/null 2>&1
+        ;;
+    png | img)
+        for FILE in *.adoc
+        do
+            # Generate HTML file.
+            gen_html ${FILE}
+
+            # Convert HTML to PNG.
+            IMG_OUT="cheatsheet_${OUT}.png"
+            wkhtmltopdf \
+                --enable-local-file-access \
+                --javascript-delay 2000\
+                $HTML_OUT $IMG_OUT
+        done
+
+        # Move up from src/
+        mv *.png ../
+
+        # Cleanup temporarily generated HTML files.
+        rm *.html > /dev/null 2>&1
+        ;;
+    clean)
+        rm *.html > /dev/null 2>&1
+        rm *.png > /dev/null 2>&1
+        rm ../*.html > /dev/null 2>&1
+        rm ../*.png > /dev/null 2>&1
+        ;;
+    *)
+        echo "Unrecognized command"
+        ;;
+esac

level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/asymptotic_analysis.adoc

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+= Cheatsheet - Asymptotic Analysis
+Fabio Lama <[email protected]>
+:description: Module: CM2035 Algorithms and Data Structures II, started April 2024
+:doctype: article
+:sectnums: 4
+:toclevels: 4
+:stem:
+
+== About
+
+Asymptotic analysis is an alternative way of describing the time or memory
+requirements of an algorithm.
+
+== Big O Notation
+
+Big O notation stem:[O(x)] defines a set of functions that act as an **upper bound**
+stem:[g(N)] for stem:[T(N)]. Formally defined as:
+
+stem:[T(N)] is stem:[O(g(N))] if there exist positive
+constants stem:[c] and stem:[n_0] such that:
+
+[stem]
+++++
+T(N) <= c xx g(N) " for all " N > n_0
+++++
+
+Note that there can be **multiple functions** stem:[g_x(N)] that act as **an upper bound**
+for stem:[T(N)]. Additionally, do notice that it's **not necessary** that
+stem:[c xx g(N)] is equal to or greater than stem:[T(N)] for all values of
+stem:[N].
+
+image::assets/big_o_notation.png[align=center, width=300]
+
+For example, consider:
+
+[stem]
+++++
+T(N) = 10 N^2 + 15N + 5\
+g(N) = N^2\
+c = 1
+++++
+
+Here, stem:[c xx g(N)] is never greater than stem:[T(N)], because there is no
+solution for:
+
+[stem]
+++++
+10 N^2 + 15N + 5 <= 1 xx N^2
+++++
+
+However, consider:
+
+[stem]
+++++
+c = 25
+++++
+
+In case of stem:[N = 1] we get:
+
+[stem]
+++++
+10 xx 1^2 + 15 xx 1 + 5 <= 25 xx 1^2\
+= 10 + 15 + 5 <= 25\
+= 30 <= 25
+++++
+
+Which is false. However, for stem:[N = 2] we get:
+
+[stem]
+++++
+10 xx 2^2 + 15 xx 2 + 5 <= 25 xx 2^2\
+= 40 + 30 + 5 <= 100\
+= 75 <= 100
+++++
+
+Which is true. Therefore:
+
+[stem]
+++++
+T(N) " is " O(N^2) " because"\
+T(N) <= 25 xx g(N) " for all " N >= 2
+++++
+
+There choice for stem:[c] **is arbitrary**, as long as it satisfies the conditions.
+
+== Omega Notation
+
+The Omega notation stem:[Omega(x)] defines a set of functions that act as a
+**lower bound** stem:[g(N)] for (stem:[T(N)]). Formally defined as:
+
+stem:[T(N)] is stem:[Omega(g(N))] if there exist positive constants stem:[c] and
+stem:[n_0] such that:
+
+[stem]
+++++
+T(N) >= c xx g(N) " for all " N > n_0
+++++
+
+Similarly to the Big O notation, there can be **multiple functions** stem:[g_x(N)]
+that act as **a lower bound** for stem:[T(N)] and it's **not necessary** that
+stem:[c xx g(N)] is equal to or less than stem:[T(N)] for all values of
+stem:[N], but only for the larger values.
+
+image::assets/omega_notation.png[align=center, width=300]
+
+== Theta Notation
+
+The Theta notation stem:[Theta(x)] defines a **single function** that acts as
+both an **upper and lower bound** for (stem:[T(N)]). Formally defined as:
+
+stem:[T(N)] is stem:[Theta(g(N))] if there exist positive constants stem:[c_1],
+stem:[c_2] and stem:[n_o] such that both those conditions hold true:
+
+[stem]
+++++
+T(N) >= c_1 xx g(N) " for all " N > n_0\
+T(N) <= c_2 xx g(N) " for all " N > n_0
+++++
+
+Alternatively:
+
+[stem]
+++++
+c_1 xx g(N) <= T(N) <= c_2 xx g(N) " for all " N > n_0
+++++
+
+image::assets/theta_notation.png[align=center, width=300]
+
+As already noted, Theta notation has **only one function**.

level-5/algorithms-and-data-structures-ii/student-notes/fabio-lama/src/hash_tables.adoc

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+= Cheatsheet - Hash Tables
+Fabio Lama <[email protected]>
+:description: Module: CM2035 Algorithms and Data Structures II, started April 2024
+:doctype: article
+:sectnums: 4
+:toclevels: 4
+:stem:
+
+== About
+
+A hash table is a data structure that maps keys to values using a hash function. This function transforms the input (key) into a fixed-size integer, which serves as an index in an array, enabling fast data retrieval. Hash tables are widely used due to their average stem:[Theta(1)] time complexity for both insertion and lookup operations.
+
+== Search Algorithms Complexity
+
+WARNING: This table assumes no collisions in the hash table.
+
+|===
+|Name |Worst case (time) |Best case (time)| Worst case (space) |Best case (space)
+
+|Linear Search
+|stem:[Theta(N)]
+|stem:[Theta(1)]
+|stem:[Theta(1)]
+|stem:[Theta(1)]
+
+|Binary Search (iterative)
+|stem:[Theta(log N)]
+|stem:[Theta(1)]
+|stem:[Theta(1)]
+|stem:[Theta(1)]
+
+|Binary Search (recursive)
+|stem:[Theta(log N)]
+|stem:[Theta(1)]
+|stem:[Theta(log N)]
+|stem:[Theta(1)]
+
+|Direct Addressing
+|stem:[Theta(1)]
+|stem:[Theta(1)]
+|stem:[Theta(k)]
+|stem:[Theta(k)]
+
+|Hash Table
+|stem:[Theta(1)]
+|stem:[Theta(1)]
+|stem:[Theta(M)]
+|stem:[Theta(M)]
+|===
+
+Where stem:[k] is the maximum possible key value and stem:[M] is the size of the hash table.
+
+== Hash Tables
+
+Hash tables use an index of an array to represent a number. Consider an array of
+size 7 and the follwing, simple hash function:
+
+[stem]
+++++
+h(x) = x mod 7
+++++
+
+For example, lets say stem:[x = 11] and we compute:
+
+[stem]
+++++
+h(11) = 11 mod 7 = 4
+++++
+
+This means the number 11 is stored in the array at index 4. Knowing the index,
+the search for that number is very fast. However, given that the size of this 
+array is very small, **the number of collisions can be very high**, depending on
+number of inputs.
+
+For example, both 11 and 18 would be stored at index 4:
+
+[stem]
+++++
+h(11) = 11 mod 7 = 4\
+h(18) = 18 mod 7 = 4
+++++
+
+If necessary, the hash table can be **extended** with a larger array, but this
+requires rehashing all the existing elements. Alternatively, the method of
+**linear probing** can be applied to use the next available index in case of a
+collision (note that this information must be stored somewhere). Or, each index
+can be a pointer to a separate, nested table, which is a **separate chaining** method.
+
+The best case and worst case complexity for hash tables must consider those
+collision handling methods as well, as in how it behaves with no collisions at
+all and with all elements colliding, respectively.