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<div id="content">
<h1 class="title">Studying for CSI3105</h1>
<div id="table-of-contents">
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org864bc28">1. ./Introduction—Part-1.pdf</a></li>
<li><a href="#org69a0eb6">2. ./Introduction—Part-2.pdf</a></li>
<li><a href="#org0a5c7b4">3. ./Chapter-2—Part-1—MergeSort.pdf</a></li>
<li><a href="#orgde58d7a">4. ./Chapter-2—Part-2—StrassenAlgorithm.pdf</a></li>
<li><a href="#org7fb44a0">5. ./Chapter-2—Part-3—MasterTheorem.pdf</a></li>
<li><a href="#orgb9b50aa">6. ./Chapter-2—Part-4—SelectionAlgorithm.pdf</a></li>
<li><a href="#orgcd1cb7f">7. ./Chapter-3—GraphsAlgorithms—Part-1—Intro.pdf</a></li>
<li><a href="#orgf9dc88c">8. ./Chapter-3—GraphsAlgorithms—Part-2—ExploringUndirectedGraphs.pdf</a></li>
<li><a href="#org5982f64">9. ./Chapter-3—GraphsAlgorithms—Part-3—ExploringDirectedGraphs.pdf</a></li>
<li><a href="#org609690a">10. ./Chapter-3—GraphsAlgorithms—Part-4—Dijkstra.pdf</a></li>
<li><a href="#org8f47f6a">11. ./Chapter-3—GraphsAlgorithms—Part-5—DijkstraCorrectness.pdf</a></li>
</ul>
</div>
</div>
<div id="outline-container-org864bc28" class="outline-2">
<h2 id="org864bc28"><span class="section-number-2">1</span> <a href="./Introduction---Part-1.pdf">./Introduction---Part-1.pdf</a></h2>
<div class="outline-text-2" id="text-1">
<ul class="org-ul">
<li>What does "design & analsis of algos" mean?
<ul class="org-ul">
<li>Correctness</li>
<li>Does it terminate?</li>
<li>Efficiency</li>
<li>Limits thereof</li>
<li>Pseudocode, no programming</li>
</ul></li>
<li>Insertion sort
<ul class="org-ul">
<li>This is the one where we start at a position and then swap
backwards until it fits into place</li>
<li><code>~O(n^2)~</code> in the worst case</li>
</ul></li>
<li>Big O notation
<ul class="org-ul">
<li>Big Omega (which is the less than one)</li>
<li>Big Theta (Which is the greater than one)</li>
<li>What it means to be each one</li>
</ul></li>
<li>Barometer instruction
<ul class="org-ul">
<li>One which is executed at least as often as any other one</li>
</ul></li>
<li>Limit Criterion
<ul class="org-ul">
<li>lim(f(n)/g(n))
<ul class="org-ul">
<li>Goes to 0 if f = <code>O(g)</code></li>
<li>Goes to infty if f = Omega(g)</li>
<li>Goes to a real if f = Theta(g)</li>
<li>If not exist, dunno</li>
</ul></li>
</ul></li>
</ul>
</div>
</div>
<div id="outline-container-org69a0eb6" class="outline-2">
<h2 id="org69a0eb6"><span class="section-number-2">2</span> <a href="./Introduction---Part-2.pdf">./Introduction---Part-2.pdf</a></h2>
<div class="outline-text-2" id="text-2">
<ul class="org-ul">
<li>The fibonacci numbers
<ul class="org-ul">
<li>Calculating the running time of fib(n)</li>
<li>fib(n) is exponential, but fib2(n) is not, because two n-bit
numbers can be added together in a linear number of bit operations</li>
</ul></li>
</ul>
</div>
</div>
<div id="outline-container-org0a5c7b4" class="outline-2">
<h2 id="org0a5c7b4"><span class="section-number-2">3</span> <a href="./Chapter-2---Part-1---MergeSort.pdf">./Chapter-2---Part-1---MergeSort.pdf</a></h2>
<div class="outline-text-2" id="text-3">
<ul class="org-ul">
<li>Divide and conquer algorithms
<ul class="org-ul">
<li>Divide the problem</li>
<li>Conquer each subproblem recursively</li>
<li>Combine</li>
</ul></li>
<li>Merge sort
<ul class="org-ul">
<li>If size = 1: do nothing</li>
<li>if size >= 2: divide n numbers into two sequences, run merge
sort twice, and merge it back into a sorted sequence</li>
</ul></li>
<li>Computing runtime of merge sort
<ul class="org-ul">
<li>T(n) = c if n=1, or 2T(n/2) + cn if n >= 2</li>
<li>Solving by unfolding
<ul class="org-ul">
<li>All steps are outlined in document</li>
</ul></li>
<li>We find T(n) <= 2cnlog<sub>2</sub>(n)</li>
<li>Therefore merge sort is Theta(n log<sub>2</sub>(n)) if n is a power of 2</li>
</ul></li>
</ul>
</div>
</div>
<div id="outline-container-orgde58d7a" class="outline-2">
<h2 id="orgde58d7a"><span class="section-number-2">4</span> <a href="./Chapter-2---Part-2---StrassenAlgorithm.pdf">./Chapter-2---Part-2---StrassenAlgorithm.pdf</a></h2>
<div class="outline-text-2" id="text-4">
<ul class="org-ul">
<li>What is matrix multiplication?</li>
<li>Trivial algorithm: triple for loop</li>
<li>Strassen algorithm</li>
<li>How can we divide and conquer?
<ul class="org-ul">
<li>Split A, B (inputs) into four separate matrices</li>
<li>Then compute the quadrants of C as products of these
<ul class="org-ul">
<li>We find no improvement.</li>
</ul></li>
<li>However the strassen algorithm derives various intermediate matrices
and then finds C as a prodcut of these matrices
<ul class="org-ul">
<li>(P<sub>1</sub> through P<sub>7</sub>). Their calculations are in the notes.</li>
<li>It then derives the quadrants of C to be products of P1 through P7</li>
</ul></li>
<li>After computing this through unfolding we then calculate that we
have <code>O(nlog_2(7) where log2(7)</code> is approx 2.81</li>
</ul></li>
<li>There is also another algorithm by coppersmith and winograd, but
that wasn't covered, it's very group theory heavy.</li>
</ul>
</div>
</div>
<div id="outline-container-org7fb44a0" class="outline-2">
<h2 id="org7fb44a0"><span class="section-number-2">5</span> <a href="./Chapter-2---Part-3---MasterTheorem.pdf">./Chapter-2---Part-3---MasterTheorem.pdf</a></h2>
<div class="outline-text-2" id="text-5">
<ul class="org-ul">
<li>Master theorem
<ul class="org-ul">
<li>Let T be a function defined by parts. If
<ul class="org-ul">
<li>T(1) = 1</li>
<li>T(n) = a . T(n / b) + n<sup>d</sup> where A >= 1, b > 1, d >= 0
<ul class="org-ul">
<li>If d > logb(a) then T(n) = <code>O(n^d)</code></li>
<li>If d = logb(a) then T(n) = <code>O(n^d log(n))</code></li>
<li>if d < logb(a) then T(n) = <code>O(n^{logb(a)})</code></li>
</ul></li>
</ul></li>
</ul></li>
<li>They then use the master theorem on merge sort and strassen and
binary search to find results.</li>
</ul>
</div>
</div>
<div id="outline-container-orgb9b50aa" class="outline-2">
<h2 id="orgb9b50aa"><span class="section-number-2">6</span> <a href="./Chapter-2---Part-4---SelectionAlgorithm.pdf">./Chapter-2---Part-4---SelectionAlgorithm.pdf</a></h2>
<div class="outline-text-2" id="text-6">
<ul class="org-ul">
<li>Selection algo
<ul class="org-ul">
<li>Find kth smallest number in array of n numbers</li>
</ul></li>
<li>Naive algo:
<ul class="org-ul">
<li>Sort S</li>
<li>return the element at position k</li>
<li>Runs in <code>O(n log n)</code></li>
<li>Without sorting:
<ul class="org-ul">
<li>find pivot, go through array like quicksort, but instead of
then sorting each sub-array, simply use the fact that it's
less than k to continually go inwards.</li>
<li>Quickselect
<ul class="org-ul">
<li>Worst case: <code>O(n^2)</code> if you pick bad pivots every time</li>
<li>Average case: <code>O(n)</code> because on average a random pivot will be
close to median</li>
<li>Has a thing where they find a pivot</li>
</ul></li>
</ul></li>
</ul></li>
</ul>
</div>
</div>
<div id="outline-container-orgcd1cb7f" class="outline-2">
<h2 id="orgcd1cb7f"><span class="section-number-2">7</span> <a href="./Chapter-3---GraphsAlgorithms---Part-1---Intro.pdf">./Chapter-3---GraphsAlgorithms---Part-1---Intro.pdf</a></h2>
<div class="outline-text-2" id="text-7">
<ul class="org-ul">
<li>Graphs!
<ul class="org-ul">
<li>Set of vertices and set of edges <code>(V, E)</code></li>
<li>Graph is <b>Undirected</b> if each edge is a 2-set</li>
<li>Graph is <b>Directed</b> if each edge is an ordered 2-tuple
<ul class="org-ul">
<li>Facebook friends are undirected</li>
<li>Web page links are directed</li>
</ul></li>
</ul></li>
<li><b>Adjacent</b> edges <code>(u, v)</code>
<ul class="org-ul">
<li>if there is an edge between <code>u</code> and <code>v</code>.</li>
</ul></li>
<li><b>Incident</b> vertex v to an edge
<ul class="org-ul">
<li>If one of the vertexes of the edge is v</li>
</ul></li>
<li><b>degree</b> of <code>u \in V</code>
<ul class="org-ul">
<li>Equal to the num edges incident to u</li>
</ul></li>
<li><b>Outdegree</b> of <code>u \in V</code>
<ul class="org-ul">
<li>Equal to num of edges <code>e</code> such that <code>u</code> is the starting point of
<code>e</code>.</li>
</ul></li>
<li><b>Indegree</b> of <code>u \in V</code>
<ul class="org-ul">
<li>Num of edges such that <code>u</code> is endpoint of edge</li>
</ul></li>
<li>Handshaking Lemma
<ul class="org-ul">
<li>Let <code>G = (V, E)</code>. Then
<ul class="org-ul">
<li>Sum(deg(u)) forall u = 2|E|
<ul class="org-ul">
<li>proof: each edge counted twice lel</li>
</ul></li>
</ul></li>
</ul></li>
<li>How to store a graph
<ul class="org-ul">
<li>Adjacency matrix
<ul class="org-ul">
<li>In <code>O(1)</code> time we get whether two vertices are adjacent</li>
<li>Uses n<sup>2</sup> space</li>
<li>Finding all neighbours takes <code>O(n)</code> time</li>
</ul></li>
<li>Adjacency list
<ul class="org-ul">
<li>Uses |V| + |E| space</li>
<li>Finding all neighbours of a vertex takes <code>O(1 + deg(u))</code> time
(depeneding on implementation of list)</li>
<li>Testing if (u, v) is an edge takes the same amount of time as
finding all neighbours</li>
</ul></li>
</ul></li>
</ul>
</div>
</div>
<div id="outline-container-orgf9dc88c" class="outline-2">
<h2 id="orgf9dc88c"><span class="section-number-2">8</span> <a href="./Chapter-3---GraphsAlgorithms---Part-2---ExploringUndirectedGraphs.pdf">./Chapter-3---GraphsAlgorithms---Part-2---ExploringUndirectedGraphs.pdf</a></h2>
<div class="outline-text-2" id="text-8">
<ul class="org-ul">
<li>Depth first search
<ul class="org-ul">
<li>Find all vertices that can be reached from <code>v \in V</code></li>
<li>Tracing of function via lines on graph?</li>
<li>It'll generate a trees, and all the other edges are back cycles and stuff</li>
<li>Proves that DFS terminates</li>
<li>Proves that it visits all vertices that are reachable from <code>v</code>
<ul class="org-ul">
<li>This proof is worth reading</li>
</ul></li>
</ul></li>
<li>Finding connected components of G = (V, E)
<ul class="org-ul">
<li>We have some algo DFS</li>
<li>Finding connected components is done by doing DFS on all
unvisited nodes (which get filled out as one DFS call is done)</li>
<li>And then a number gets incremented</li>
</ul></li>
<li>Runtime:
<ul class="org-ul">
<li>First loop: <code>O(|V|)</code> time</li>
<li>Second loop:
<ul class="org-ul">
<li>explore(u) is called for each vertex u</li>
<li>Time spent in exlore(u) is <code>O(1 + deg(u))</code></li>
</ul></li>
<li><code>O(|V| + sum(1 + deg(u)) forall u) = O(|V| + |V| + 2|E|)</code> =
<code>O(|V| + |E|)</code></li>
</ul></li>
</ul>
</div>
</div>
<div id="outline-container-org5982f64" class="outline-2">
<h2 id="org5982f64"><span class="section-number-2">9</span> <a href="./Chapter-3---GraphsAlgorithms---Part-3---ExploringDirectedGraphs.pdf">./Chapter-3---GraphsAlgorithms---Part-3---ExploringDirectedGraphs.pdf</a></h2>
<div class="outline-text-2" id="text-9">
<ul class="org-ul">
<li>Topological ordering
<ul class="org-ul">
<li>Exists on a graph that is <b>directed</b> and <b>acyclic</b></li>
<li>such that for edge (u, v) #(u) < #(v)</li>
<li>Algo is:
<ul class="org-ul">
<li>find vertex with indegree 0</li>
<li>give u the num k</li>
<li>remove u from G</li>
<li>and so on and so forth</li>
<li>there is an example in the notes</li>
</ul></li>
</ul></li>
<li><b>Prenumbers</b> and <b>Postnumbers</b>
<ul class="org-ul">
<li><code>G = (V, E)</code> is directed.</li>
<li><code>forall v \in V</code>, the following two numbers with respect to DFS:</li>
<li><code>pre(v)</code> is the first time we visit <code>v</code></li>
<li><code>post(v)</code> is the time when <code>explore(v)</code> is done
<ul class="org-ul">
<li>This uses a variable clock. Each time pre or postvisit number
is assigned to a node, it gets incremented for next node</li>
<li>There is an example in the notes</li>
<li>Note: edges found in DFS are <b>Tree edges</b></li>
<li>There is then a concept of <b>forward</b> and <b>back</b> edges
<ul class="org-ul">
<li>Edges going down the DFS tree are forward</li>
<li>Going up -> backwards</li>
<li><b>Cross</b> edges go from one branch of the tree to another
<ul class="org-ul">
<li>Cross edges are neither forwards, backwards, nor tree
edges. They are the remainder.</li>
</ul></li>
</ul></li>
</ul></li>
</ul></li>
<li>To find the types of an edge:
<ul class="org-ul">
<li><b>Forwards</b>:
<ul class="org-ul">
<li><code>(v, u)</code> is not a tree edge</li>
<li><code>pre(v) < pre(u) < post(u) < post(v)</code></li>
</ul></li>
<li><b>Back edge</b>:
<ul class="org-ul">
<li><code>(v, u)</code> not tree edge</li>
<li><code>pre(u) < pre(v) < post(v) < post(u)</code></li>
</ul></li>
<li><b>Cross edge</b>:
<ul class="org-ul">
<li><code>(v, u)</code> not tree edge</li>
<li><code>pre(u) < post(u) < pre(v) < post(v)</code></li>
</ul></li>
</ul></li>
<li>To determine whether a directed graph has a directed cycle:
<ul class="org-ul">
<li>Graph has cycle iff DFS-forest has back-edge
<ul class="org-ul">
<li>Proof in notes</li>
</ul></li>
<li>If directed graph is cyclic:
<ol class="org-ol">
<li>run dfs including pre/post nums</li>
<li>for each edge test if edge is back edge (via inequality up top)
2.1 if yes then yes
2.2 if no then no</li>
<li>Runtime: <code>O(|V| + |E|)</code></li>
</ol></li>
<li>How to compute a topolotical ordering?
<ol class="org-ol">
<li>Run dfs</li>
<li>Run bucket sort to sort vertices by postnumber</li>
<li>obtain topological ordering from reverse sorted order of
postnumbers</li>
</ol></li>
<li>Bucket sort takes <code>O(n) time, so runtime = O(|V| + |E|)</code>
<ul class="org-ul">
<li>Proof of correctness in notes</li>
</ul></li>
</ul></li>
</ul>
</div>
</div>
<div id="outline-container-org609690a" class="outline-2">
<h2 id="org609690a"><span class="section-number-2">10</span> <a href="./Chapter-3---GraphsAlgorithms---Part-4---Dijkstra.pdf">./Chapter-3---GraphsAlgorithms---Part-4---Dijkstra.pdf</a></h2>
<div class="outline-text-2" id="text-10">
<ul class="org-ul">
<li>Dijstra's algo
<ul class="org-ul">
<li>input:
<ul class="org-ul">
<li>A directed graph <code>G = (V, E)</code> where each edge has a weight <code>wt(u,
v) > 0</code></li>
<li>a vertex <code>s</code> which is the source</li>
</ul></li>
<li>Output:
<ul class="org-ul">
<li>For each vertex: <code>\del(s, v) =</code> len of shortest path from s to v</li>
</ul></li>
<li>if all weights equal, this is easy, use BFS.</li>
<li>algo:
<ol class="org-ol">
<li>for each vertex, maintain <code>d(v)</code>, the current shortest known path</li>
<li>start out with <code>d(s)</code> = 0, and <code>d(!s) = \infty</code></li>
<li>Loop:
<ol class="org-ol">
<li>Pick a vertex u for which <code>d(u) = \del(s, u)</code>.</li>
<li>For each edge, <code>(u, v)</code>, <code>d(v) = min{d(v), d(u) + wt(u, v)}</code>
<ul class="org-ul">
<li>But how to find u?
<ul class="org-ul">
<li>Maintain <code>S \subset V</code> such that for all <code>v \in S</code>:</li>
<li><code>d(v) = \del(s, v),</code> (i.e. we know <code>\del(s, v)</code>)</li>
</ul></li>
</ul></li>
</ol></li>
</ol></li>
<li>Algo runthrough done in notes</li>
<li>Store the set of nodes to visit in a min-heap
<ul class="org-ul">
<li>Initialization <code>O(n)</code></li>
<li>One iteration:
<ol class="org-ol">
<li>find <code>u</code> and delete it from <code>Q</code>.
<ul class="org-ul">
<li>O(log(n)) time</li>
</ul></li>
<li>For each edge, update d(v):
<ul class="org-ul">
<li>decrease<sub>key</sub> : O(log(n)) time</li>
</ul></li>
</ol></li>
<li>Time total for iteration
<ul class="org-ul">
<li><code>O(log(n)) + O(outdegree(u) * log(n))</code></li>
</ul></li>
<li>After some calculations, total runtime is <code>O((m + n) log(n))</code>
<ul class="org-ul">
<li>Note using fib heap to store Q we could do O(n log n + m time)</li>
</ul></li>
</ul></li>
</ul></li>
</ul>
</div>
</div>
<div id="outline-container-org8f47f6a" class="outline-2">
<h2 id="org8f47f6a"><span class="section-number-2">11</span> <a href="./Chapter-3---GraphsAlgorithms---Part-5---DijkstraCorrectness.pdf">./Chapter-3---GraphsAlgorithms---Part-5---DijkstraCorrectness.pdf</a></h2>
<div class="outline-text-2" id="text-11">
<ul class="org-ul">
<li>You know the drill, correctness proof for dijkstra's
<ul class="org-ul">
<li><p>
We say that a vertex <code>v</code> is special if all vertices on that path
are in <code>S</code> (except <code>v</code>).
</p>
<ul class="org-ul">
<li>We prove by induction that if <code>u \in S</code> then <code>d(u)</code> is shortest</li>
</ul>
<p>
path from <code>s</code> to <code>u</code>.
</p>
<ul class="org-ul">
<li>If it is not, then it is the shortest "special path" from <code>s</code>
to <code>u</code></li>
</ul></li>
</ul></li>
</ul>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Sean</p>
<p class="date">Created: 2020-10-14 Wed 23:43</p>
</div>
</body>
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