Merge branch 'master' of git.odin.cse.buffalo.edu:ODIn/Website
commit
2e06b822ce
GPG Key ID:
3E5F9B3ABD3FDB60
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@ -298,44 +298,54 @@ schedule:
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section_b:
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slides: slide/12b-RecursionAndAverage.html
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- date: 09/30/22
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topic: Data-Bounded Analysis
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||||
topic: Stacks, Queues
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dow: Fri
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section_a:
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slides: slide/lec14a.pdf
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section_b:
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slides: slide/13b-AverageQueueStack.html
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- week: 6
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||||
lectures:
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||||
- date: 10/03/22
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topic: "Composite Data Structures, Stacks, Shortest Path"
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topic: "Applications of Stacks, Queues (Mazes)"
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dow: Mon
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section_a:
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slides: slide/lec15a.pdf
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section_b:
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slides: slide/14b-QueueStackApps.html
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- date: 10/05/22
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topic: "Queues, Shortest Path revisited"
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topic: "Graphs; Graph ADTs"
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dow: Wed
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section_b:
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||||
slides: slide/15b-Graphs.html
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||||
- date: 10/07/22
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due: PA2-Tests
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topic: Graphs Intro
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topic: "Graph Search; DFS"
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dow: Fri
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section_b:
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slides: slide/16b-GraphExploration.html
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- week: 7
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lectures:
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||||
- date: 10/10/22
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topic: "Graph ADTs: Edge Lists"
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||||
topic: "Graph Search; Pathfinding, BFS"
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due: PA2-Tests
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dow: Mon
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- date: 10/12/22
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topic: DFS, BFS, Analysis
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topic: BFS, Shortest Path, Priority Queues
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dow: Wed
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- date: 10/14/22
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due: PA2
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topic: Priority Queues, Orderings
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topic: Order Relations, Priority Queues
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dow: Fri
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- week: 8
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lectures:
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||||
- date: 10/17/22
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topic: Midterm Review
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due: PA2
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dow: Mon
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- date: 10/19/22
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special: Midterm Exam
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dow: Wed
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- date: 10/21/22
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topic: Priority Queues, Orderings
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topic: Organizing Cat Pictures
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dow: Fri
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- week: 9
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lectures:
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|
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@ -2,7 +2,7 @@
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template: templates/cse250_2022_slides.erb
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title: Average Runtime, Stacks, Queues
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date: Sept 30, 2022
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textbook: Ch. 15
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textbook: Ch. 15, Ch. 7
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---
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<section>
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|
|
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@ -0,0 +1,617 @@
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---
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template: templates/cse250_2022_slides.erb
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title: Average Runtime, Stacks, Queues
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date: Oct 3, 2022
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textbook: Ch. 7
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cat: graphics/14b/cat.png
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---
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|
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<section>
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||||
<h4 class="slide_title">Stacks</h4>
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|
||||
<p>A stack of objects on top of one another.</p>
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|
||||
<dl>
|
||||
<dt>Push</dt>
|
||||
<dd>Put a new object on top of the stack</dd>
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||||
|
||||
<dt>Pop</dt>
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||||
<dd>Remove the object on top of the stack</dd>
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||||
|
||||
<dt>Top</dt>
|
||||
<dd>Peek at what's on top of the stack</dd>
|
||||
</dl>
|
||||
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Queue</h4>
|
||||
|
||||
<p>Outside of the US, "queueing" is lining up.</p>
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||||
|
||||
<dl>
|
||||
<dt>Enqueue</dt>
|
||||
<dd>Put a new object at the end of the queue</dd>
|
||||
|
||||
<dt>Dequeue</dt>
|
||||
<dd>Remove the next object in the queue</dd>
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||||
|
||||
<dt>Head</dt>
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||||
<dd>Peek at the next object in the queue</dd>
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</dl>
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</section>
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||||
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||||
<section>
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<p style="font-size: 90%"><b>Thought question:</b> How could you use an array to build a queue?</p>
|
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</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">ArrayBuffer Attempt 1</h4>
|
||||
|
||||
<dl>
|
||||
<dt>Enqueue</dt>
|
||||
<dd>Append</dd>
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||||
|
||||
<dt>Dequeue</dt>
|
||||
<dd>Remove(0)</dd>
|
||||
</dl>
|
||||
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">ArrayBuffer Attempt 2</h4>
|
||||
|
||||
<dl>
|
||||
<dt>Enqueue</dt>
|
||||
<dd>Insert(0)</dd>
|
||||
|
||||
<dt>Dequeue</dt>
|
||||
<dd>Remove(last)</dd>
|
||||
</dl>
|
||||
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>Can we avoid having to move all of the<br> elements forward or backward a spot?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
|
||||
<style type="text/css">
|
||||
#dequeue_example {
|
||||
height: 500px;
|
||||
width: 500px;
|
||||
border: solid 1px black;
|
||||
margin-left: auto;
|
||||
margin-right: auto;
|
||||
overflow-y: scroll;
|
||||
}
|
||||
#dequeue_example .dequeue_node {
|
||||
width: 50px;
|
||||
height: 50px;
|
||||
display: inline-block;
|
||||
color: lightgray;
|
||||
background: black;
|
||||
margin-left: auto;
|
||||
margin-right: auto;
|
||||
margin-top: 5px;
|
||||
}
|
||||
#dequeue_example .dequeue_node.dequeue_end {
|
||||
border-bottom: solid 10px green;
|
||||
}
|
||||
#dequeue_example .dequeue_node.dequeue_start {
|
||||
border-bottom: solid 10px blue;
|
||||
}
|
||||
#dequeue_example .dequeue_node.dequeue_end.dequeue_start {
|
||||
border-bottom: solid 10px green;
|
||||
}
|
||||
</style>
|
||||
|
||||
<button onclick="dequeue_enqueue()">Enqueue</button>
|
||||
<button onclick="dequeue_dequeue()">Dequeue</button>
|
||||
<div id="dequeue_example">
|
||||
<div class="dequeue_node dequeue_start dequeue_end ">_</div>
|
||||
<div class="dequeue_node">_</div>
|
||||
<div class="dequeue_node">_</div>
|
||||
<div class="dequeue_node">_</div>
|
||||
</div>
|
||||
|
||||
<script type="text/javascript">
|
||||
var dequeue = document.getElementById("dequeue_example")
|
||||
|
||||
var dequeue_counter = 0
|
||||
|
||||
function dequeue_expand() {
|
||||
var values = []
|
||||
var x
|
||||
while(x = dequeue_dequeue()){
|
||||
values.push(x)
|
||||
}
|
||||
console.log("Copying values: "+values)
|
||||
var count = dequeue.children.length * 2
|
||||
dequeue.innerText = ""
|
||||
for(x = 0; x < count; x++){
|
||||
var elem = document.createElement('div')
|
||||
elem.classList.add("dequeue_node")
|
||||
console.log("Update "+x)
|
||||
if(x == 0){
|
||||
console.log("Start")
|
||||
elem.classList.add("dequeue_start")
|
||||
}
|
||||
if(x == values.length){
|
||||
console.log("End")
|
||||
elem.classList.add("dequeue_end")
|
||||
}
|
||||
if(x < values.length){
|
||||
console.log("Existing: "+values[x])
|
||||
elem.innerText = values[x]
|
||||
} else {
|
||||
console.log("Blank")
|
||||
elem.innerText = "_"
|
||||
}
|
||||
dequeue.append(elem)
|
||||
}
|
||||
|
||||
}
|
||||
function dequeue_enqueue() {
|
||||
var node = dequeue.querySelector(".dequeue_end")
|
||||
var next = node.nextElementSibling || dequeue.firstElementChild
|
||||
if(next.classList.contains("dequeue_start")){
|
||||
dequeue_expand()
|
||||
node = dequeue.querySelector(".dequeue_end")
|
||||
next = node.nextElementSibling || dequeue.firstElementChild
|
||||
}
|
||||
node.innerText = dequeue_counter++
|
||||
node.classList.add("active")
|
||||
node.classList.remove("dequeue_end")
|
||||
next.classList.add("dequeue_end")
|
||||
}
|
||||
function dequeue_dequeue() {
|
||||
var node = dequeue.querySelector(".dequeue_start")
|
||||
var next = node.nextElementSibling || dequeue.firstElementChild
|
||||
if(node.classList.contains("dequeue_end")){ return null }
|
||||
var ret = node.innerText
|
||||
node.innerText = "_"
|
||||
node.classList.remove("active")
|
||||
node.classList.remove("dequeue_start")
|
||||
next.classList.add("dequeue_start")
|
||||
return ret
|
||||
}
|
||||
</script>
|
||||
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">ArrayDequeue (Resizable Ring Buffer)</h4>
|
||||
|
||||
<p>Active Array = [start, end)</p>
|
||||
|
||||
<dl style="font-size: 80%">
|
||||
<dt>Enqueue</dt>
|
||||
<dd>Resize buffer if needed</dd>
|
||||
<dd>Add new element at buffer[end]</dd>
|
||||
<dd>Advance end pointer (wrap around to front)</dd>
|
||||
|
||||
<dt>Dequeue</dt>
|
||||
<dd>Remove element at buffer[start]</dd>
|
||||
<dd>Advance start pointer (wrap around to front)</dd>
|
||||
</dl>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Applications of Stacks and Queues</h4>
|
||||
|
||||
<ul>
|
||||
<li class="fragment highlight-blue">Stack: Checking for Balanced Parenthesis/Braces</li>
|
||||
<li>Queue: Scheduling Packets for Delivery</li>
|
||||
<li>Both: Searching Mazes</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Balanced Parenthesis/Braces</h4>
|
||||
|
||||
<p>What does it mean for parenthesis/braces to be balanced?</p>
|
||||
|
||||
<ul>
|
||||
<li>Every opening symbol is matched by a closing symbol</li>
|
||||
<li>No nesting overlaps (e.g., <tt style="font-weight: bold;">{(})</tt> is not ok).</li>
|
||||
</ul>
|
||||
|
||||
<table style="margin-top: 50px;">
|
||||
<tr>
|
||||
<td><tt style="font-weight: bold;">{()({})}</tt></td>
|
||||
<td><tt style="font-weight: bold;">{())</tt></td>
|
||||
<td><tt style="font-weight: bold;">())</tt></td>
|
||||
</tr>
|
||||
|
||||
<tr>
|
||||
<td style="font-size: 200%" class="fragment">✅</td>
|
||||
<td style="font-size: 200%" class="fragment">❌</td>
|
||||
<td style="font-size: 200%" class="fragment">❌</td>
|
||||
</tr>
|
||||
</table>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Balanced Parenthesis/Braces</h4>
|
||||
|
||||
<p><b>Idea: </b> Count the number of unmatched open parens, braces.</p>
|
||||
<p>Increment counter on <tt>(</tt>, decrement on <tt>)</tt></p>
|
||||
|
||||
<p class="fragment"><b>Problem: </b> allows <tt style="font-weight: bold;">{(})</tt></p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Balanced Parenthesis/Braces</h4>
|
||||
|
||||
<p><b>Idea: </b> Track nesting on a stack!</p>
|
||||
<p>On <tt>(</tt> or <tt>{</tt>, push the symbol on the stack.</p>
|
||||
<p>On <tt>)</tt> or <tt>}</tt>, pop the stack and check the popped symbol.</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
|
||||
|
||||
<style type="text/css">
|
||||
#nesting_stack {
|
||||
height: 400px;
|
||||
width: 200px;
|
||||
border: solid 1px black;
|
||||
margin-left: auto;
|
||||
margin-right: auto;
|
||||
overflow-y: scroll;
|
||||
transition-property: background;
|
||||
transition-duration: 0.1s;
|
||||
}
|
||||
#nesting_stack.nesting_alert {
|
||||
background: #ffbbbb;
|
||||
}
|
||||
#nesting_stack .nesting_symbol {
|
||||
width: 50px;
|
||||
height: 50px;
|
||||
display: block;
|
||||
color: lightgray;
|
||||
background: black;
|
||||
margin-left: auto;
|
||||
margin-right: auto;
|
||||
margin-top: 5px;
|
||||
}
|
||||
#nesting_string {
|
||||
height: 100px;
|
||||
font-family: fixed;
|
||||
}
|
||||
#nesting_string::before {
|
||||
content: '"';
|
||||
}
|
||||
#nesting_string::after {
|
||||
content: '"';
|
||||
}
|
||||
</style>
|
||||
|
||||
<button onclick="nesting_push('(')">(</button>
|
||||
<button onclick="nesting_push('{')">{</button>
|
||||
<button onclick="nesting_pop(')')">)</button>
|
||||
<button onclick="nesting_pop('}')">}</button>
|
||||
<div id="nesting_stack"></div>
|
||||
<div id="nesting_string"></div>
|
||||
|
||||
<script type="text/javascript">
|
||||
var nesting_stack = document.getElementById("nesting_stack")
|
||||
var nesting_string = document.getElementById("nesting_string")
|
||||
|
||||
function nesting_push(chr){
|
||||
nesting_string.innerText = nesting_string.innerText+chr
|
||||
var elem = document.createElement('div')
|
||||
elem.classList.add("nesting_symbol")
|
||||
elem.innerText = chr
|
||||
nesting_stack.prepend(elem)
|
||||
}
|
||||
|
||||
function nesting_alert(){
|
||||
nesting_stack.classList.add("nesting_alert")
|
||||
setTimeout(() => {
|
||||
nesting_stack.classList.remove("nesting_alert")
|
||||
}, 100)
|
||||
}
|
||||
|
||||
function nesting_pop(chr){
|
||||
var elem = nesting_stack.firstElementChild
|
||||
var match = (chr == ")" ? "(" : "{")
|
||||
|
||||
if(!elem){ nesting_alert() }
|
||||
else {
|
||||
if(elem.innerText != match){ nesting_alert() }
|
||||
else {
|
||||
nesting_string.innerText = nesting_string.innerText+chr
|
||||
elem.remove()
|
||||
}
|
||||
}
|
||||
}
|
||||
</script>
|
||||
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Network Packets</h4>
|
||||
|
||||
<dl style="font-size: 75%">
|
||||
<dt>Router: 1gb/s internal network, 100mb/s external</dt>
|
||||
<dd><ul>
|
||||
<li>1gb/s sent to router, but only 100mb/s can leave.</li>
|
||||
<li>How do packets get delivered?</li>
|
||||
</ul></dd>
|
||||
|
||||
<dt>Queues</dt>
|
||||
<dd><ul>
|
||||
<li>Enqueue data packets in the order they are received.</li>
|
||||
<li>When outgoing bandwidth available, dequeue and send.</li>
|
||||
</ul></dd>
|
||||
|
||||
<dt>Avoiding Queueing Delays</dt>
|
||||
<dd><ul>
|
||||
<li>Limit size of queue; Packets that don't fit are dropped</li>
|
||||
</ul></dd>
|
||||
|
||||
<p style="font-size: 70%">
|
||||
<span style="margin-right: auto">
|
||||
<b>TCP:</b> blocked packets are retried
|
||||
</span>
|
||||
|
||||
<span style="margin-left: 50px;">
|
||||
<b>UDP:</b> application deals with dropped packets.
|
||||
</span>
|
||||
</p>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Mazes</h4>
|
||||
|
||||
<ul style="font-size: 80%; max-width: 450px;">
|
||||
<li><b>O</b> is the start, <b>X</b> is the objective <ul>
|
||||
<li>There may be multiple paths</li>
|
||||
<li>Generally, we want the shortest</li>
|
||||
</ul></li>
|
||||
<li class="fragment"><b>Approach 1: </b> Take the first available route in one direction <ul>
|
||||
<li style="color: red">Right, Down, Left, or Up</li>
|
||||
<li style="color: blue">Down, Right, Up, Left</li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
<img src="graphics/14b/maze1.png" style="float: right" width="300px">
|
||||
</section>
|
||||
<section>
|
||||
<h4 class="slide_title">Mazes</h4>
|
||||
|
||||
<ul style="font-size: 80%; max-width: 450px;">
|
||||
<li><b>O</b> is the start, <b>X</b> is the objective <ul>
|
||||
<li>There may be multiple paths</li>
|
||||
<li>Generally, we want the shortest</li>
|
||||
</ul></li>
|
||||
<li><b>Approach 1: </b> Take the first available route in one direction <ul>
|
||||
<li style="color: red">Right, Down, Left, or Up</li>
|
||||
<li style="color: blue">Down, Right, Up, Left</li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
<img src="graphics/14b/maze1-rdlu.png" style="float: right" width="300px">
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Mazes</h4>
|
||||
<img src="graphics/14b/maze1-rdlu.png">
|
||||
|
||||
<p>How do you know which one is best?</p>
|
||||
<p class="fragment">Other problems with this algorithm?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Mazes</h4>
|
||||
<img src="graphics/14b/maze2.png" width="350px">
|
||||
<p> </p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Mazes</h4>
|
||||
<img src="graphics/14b/maze2-rdlu.png" width="350px">
|
||||
<p class="fragment">Priority order doesn't guarantee exploring the entire maze</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Formalizing Maze-Solving</h4>
|
||||
|
||||
<div style="font-size: 80%;">
|
||||
<p><b>Inputs:</b> <ul>
|
||||
<li>The map: An $n\times m$ grid of filled/empty squares.</li>
|
||||
<li>The <b>O</b> is at position $start$</li>
|
||||
<li>The <b>X</b> is at position $dest$</li>
|
||||
</ul></p>
|
||||
|
||||
<p><b>Goal: </b>
|
||||
<li>Compute $steps(\texttt{start}, \texttt{dest})$, the minimum steps from start to end.</li>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<p class="fragment">How do we define $steps$?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Formalizing Maze-Solving</h4>
|
||||
|
||||
<svg data-src="graphics/14b/maze1-distances.svg" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Formalizing Maze-Solving</h4>
|
||||
|
||||
<div style="font-size: 70%">
|
||||
$$steps(\texttt{pos},\texttt{dest}) = \begin{cases}
|
||||
0 & \textbf{if } \texttt{pos} = \texttt{dest}\\
|
||||
\end{cases}$$
|
||||
</div>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Formalizing Maze-Solving</h4>
|
||||
|
||||
<div style="font-size: 70%">
|
||||
$$steps(\texttt{pos},\texttt{dest}) = \begin{cases}
|
||||
0 & \textbf{if } \texttt{pos} = \texttt{dest}\\
|
||||
\infty & \textbf{if } is\_filled(\texttt{pos})\\
|
||||
\end{cases}$$
|
||||
</div>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Formalizing Maze-Solving</h4>
|
||||
|
||||
<div style="font-size: 70%">
|
||||
$$steps(\texttt{pos},\texttt{dest}) = \begin{cases}
|
||||
0 & \textbf{if } \texttt{pos} = \texttt{dest}\\
|
||||
\infty & \textbf{if } is\_filled(\texttt{pos})\\
|
||||
1 + min\_adjacent(\texttt{pos}, \texttt{dest}) & \textbf{otherwise}
|
||||
\end{cases}$$
|
||||
</div>
|
||||
|
||||
<p class="fragment" data-fragment-index="1">where...</p>
|
||||
|
||||
<p style="text-align: left;" class="fragment" data-fragment-index="1">
|
||||
$min\_adjacent(\texttt{pos}, \texttt{dest}) = $
|
||||
$$\min\begin{cases}
|
||||
steps(moveRight(\texttt{pos}), \texttt{dest})\\
|
||||
steps(moveDown(\texttt{pos}), \texttt{dest})\\
|
||||
steps(moveLeft(\texttt{pos}), \texttt{dest})\\
|
||||
steps(moveUp(\texttt{pos}), \texttt{dest})\\
|
||||
\end{cases}$$
|
||||
</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">$steps(\texttt{pos}, \texttt{dest})$</h4>
|
||||
|
||||
<ul style="font-size: 80%">
|
||||
<li><b>if</b> <tt>pos == dest</tt> <b>then</b> <tt>return 0</tt></li>
|
||||
<li><b>elif</b> <tt>is_filled(pos)</tt> <b>then</b> <tt>return ∞</tt></li>
|
||||
<li><b>else</b> <tt>return 1 +</tt> min of<ul style="font-size: 80%">
|
||||
<li>$steps(moveRight(\texttt{pos}, \texttt{dest}))$</li>
|
||||
<li>$steps(moveDown(\texttt{pos}, \texttt{dest}))$</li>
|
||||
<li>$steps(moveLeft(\texttt{pos}, \texttt{dest}))$</li>
|
||||
<li>$steps(moveUp(\texttt{pos}, \texttt{dest}))$</li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<img src="graphics/14b/maze2.png">
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p><b>Problem: </b> Infinite loop!</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p><b>Insight: </b> A path with a loop in it can't<br>be shorter than one without the loop.</p>
|
||||
|
||||
<p class="fragment">Mark nodes as visited</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">$steps(\texttt{pos}, \texttt{dest})$</h4>
|
||||
|
||||
<ul style="font-size: 70%">
|
||||
<li><b>if</b> <tt>pos == dest</tt> <b>then</b> <tt>return 0</tt></li>
|
||||
<li style="color: forestgreen;"><b>elif</b> <tt>pos</tt> marked visited <b>then</b> <tt>return ∞</tt></li>
|
||||
<li><b>elif</b> <tt>is_filled(pos)</tt> <b>then</b> <tt>return ∞</tt></li>
|
||||
<li><b>else</b> <ul>
|
||||
<li style="color: forestgreen;">Mark <tt>pos</tt> as visited</li>
|
||||
<li><tt>return 1 +</tt> min of all 4 steps</li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<img src="graphics/14b/maze2.png">
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p><b>Problem: </b> The first time you visit a node,<br>it may be via a longer path!</p>
|
||||
<p class="fragment">Unmark nodes as you leave them</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">$steps(\texttt{pos}, \texttt{dest})$</h4>
|
||||
|
||||
<ul style="font-size: 60%">
|
||||
<li><b>if</b> <tt>pos == dest</tt> <b>then</b> <tt>return 0</tt></li>
|
||||
<li><b>elif</b> <tt>pos</tt> marked visited <b>then</b> <tt>return ∞</tt></li>
|
||||
<li><b>elif</b> <tt>is_filled(pos)</tt> <b>then</b> <tt>return ∞</tt></li>
|
||||
<li><b>else</b> <ul>
|
||||
<li>Mark <tt>pos</tt> as visited</li>
|
||||
<li><tt>stepCount = 1 +</tt> min of all 4 steps</li>
|
||||
<li style="color: forestgreen;">Unmark <tt>pos</tt> as visited</li>
|
||||
<li style="color: forestgreen;"><tt>return stepCount</tt></li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Maze-Solving</h4>
|
||||
|
||||
<div style="font-size: 80%;">
|
||||
<p><b>Inputs:</b> <ul>
|
||||
<li>The map: An $n\times m$ grid of filled/empty squares.</li>
|
||||
<li>The <b>O</b> is at position $start$</li>
|
||||
<li>The <b>X</b> is at position $dest$</li>
|
||||
</ul></p>
|
||||
|
||||
<p><b>Goal: </b>
|
||||
<li>Compute $steps(\texttt{start}, \texttt{dest})$, the minimum steps from start to end.</li>
|
||||
</p>
|
||||
</div>
|
||||
|
||||
<p class="fragment">What path did we take?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p><p>Idea: </p> Follow the nodes marked visited</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">$steps(\texttt{pos}, \texttt{dest}, \texttt{visited})$</h4>
|
||||
|
||||
<ul style="font-size: 70%">
|
||||
<li><b>if</b> <tt>pos == dest</tt> <b>then</b> <tt>return <span style="color: forestgreen;">visited.copy()</span></tt></li>
|
||||
<li><b>elif</b> <tt>pos</tt> <span style="color: forestgreen;">∈ <tt>visited</tt></span> <tt>return <span style="color: forestgreen;">no_path</span></tt></tt></li>
|
||||
<li><b>elif</b> <tt>is_filled(pos)</tt> <b>then</b> <tt>return <span style="color: forestgreen;">no_path</span></tt></tt></li>
|
||||
<li><b>else</b> <ul>
|
||||
<li><tt style="color: forestgreen;">visited.append(pos)</tt></li>
|
||||
<li style="color: forestgreen;"><tt>bestPath = </tt> min of all 4 one-step paths</li>
|
||||
<li><tt style="color: forestgreen;">visited.removeLast()</tt></li>
|
||||
<li><tt>return bestPath</tt></li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">$steps(\texttt{pos}, \texttt{dest}, \texttt{visited})$</h4>
|
||||
|
||||
<ul style="font-size: 70%">
|
||||
<li><b>if</b> <tt>pos == dest</tt> <b>then</b> <tt>return visited.copy()</tt></li>
|
||||
<li><b>elif</b> <tt>pos</tt> ∈ <tt>visited</tt> <tt>return no_path</tt></tt></li>
|
||||
<li><b>elif</b> <tt>is_filled(pos)</tt> <b>then</b> <tt>return no_path</tt></tt></li>
|
||||
<li><b>else</b> <ul>
|
||||
<li><tt style="color: forestgreen;">visited.push(pos)</tt></li>
|
||||
<li><tt>bestPath = </tt> min of all 4 one-step paths</li>
|
||||
<li><tt style="color: forestgreen;">visited.pop()</tt></li>
|
||||
<li><tt>return bestPath</tt></li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p><b>Thought question:</b> Can you solve a maze<br>with a <b>queue</b> instead?</p>
|
||||
</section>
|
||||
|
|
@ -0,0 +1,704 @@
|
|||
---
|
||||
template: templates/cse250_2022_slides.erb
|
||||
title: Graphs
|
||||
date: Oct 5, 2022
|
||||
textbook: Ch. 15.3
|
||||
cat: graphics/15b/cat.png
|
||||
attribution: Based on slides by Tamassa Goodrich
|
||||
---
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Graphs</h4>
|
||||
|
||||
<p>A <b>graph</b> is a pair $(V, E)$ where</p>
|
||||
<ul>
|
||||
<li>$V$ is a set of <b>vertices</b></li>
|
||||
<li>$E$ is a set of vertex pairs called <b>edges</b></li>
|
||||
<li>edges and vertices may also store data (<b>labels</b>)</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Graphs</h4>
|
||||
<p><b>Example:</b> A computer network<br>(edges store ping, nodes store addresses)</p>
|
||||
<img src="graphics/15b/network.png" width="600px">
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Edge Types</h4>
|
||||
|
||||
<img src="graphics/15b/edge_example.png" style="float: right; margin-top: 70px" width="200px">
|
||||
<dl width="500px" style="font-size: 80%">
|
||||
<dt>Directed Edge (e.g., transmit bandwidth)</dt>
|
||||
<dd><ul>
|
||||
<li>Ordered pair of vertices $(u, v)$</li>
|
||||
<li>origin ($u$) → destination ($v$)</li>
|
||||
</ul></dd>
|
||||
|
||||
<dt>Undirected edge (e.g., round-trip latency)</dt>
|
||||
<dd><ul>
|
||||
<li>Unordered pair of vertices $(u, v)$</li>
|
||||
</ul></dd>
|
||||
|
||||
<div class="fragment">
|
||||
<dt>Directed Graph</dt>
|
||||
<dd>All edges are directed</dd>
|
||||
|
||||
<dt>Undirected Graph</dt>
|
||||
<dd>All edges are undirected</dd>
|
||||
</div>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Other Applications</h4>
|
||||
|
||||
<ul>
|
||||
<li>Transporation (Flight/Road/Rail Routing)</li>
|
||||
<li>Protein/Protein Interactions</li>
|
||||
<li>Social Networks</li>
|
||||
<li>Dependency Tracking (e.g., <b>make</b>)</li>
|
||||
<li>Taxonomies</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Terminology</h4>
|
||||
|
||||
<img src="graphics/15b/graph_example.png" width="400px">
|
||||
<dl style="font-size: 50%">
|
||||
<dt>Endpoints (end-vertices) of an edge</dt>
|
||||
<dd>U, V are the endpoints of a</dd>
|
||||
|
||||
<dt>Edges incident on a vertex</dt>
|
||||
<dd>a, b, d are incident on V</dd>
|
||||
|
||||
<dt>Adjacent Vertices</dt>
|
||||
<dd>U, V are adjacent</dd>
|
||||
|
||||
<dt>Degree of a vertex (# of incident edges)</dt>
|
||||
<dd>X has degree 5</dd>
|
||||
|
||||
<dt>Parallel Edges</dt>
|
||||
<dd>h, i are parallel</dd>
|
||||
|
||||
<dt>Self-Loop</dt>
|
||||
<dd>j is a self-loop</dd>
|
||||
|
||||
<dt>Simple Graph</dt>
|
||||
<dd>A graph without parallel edges or self-loops</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Terminology</h4>
|
||||
|
||||
<img src="graphics/15b/graph_example.png" width="400px">
|
||||
<dl style="font-size: 60%; width: 400px;">
|
||||
<dt>Path</dt>
|
||||
<dd>Sequence of alternating vertices and edges</dd>
|
||||
<dd>begins with a vertex</dd>
|
||||
<dd>ends with a vertex</dd>
|
||||
<dd>each edge is preceded and followed by its endpoints</dd>
|
||||
|
||||
<div class="fragment">
|
||||
<dt>Simple Path</dt>
|
||||
<dd>path such that all of its vertices and edges are distinct</dd>
|
||||
</div>
|
||||
|
||||
<dt class="fragment">Examples</dt>
|
||||
<dd> </dd>
|
||||
<dd> <br> </dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Terminology</h4>
|
||||
|
||||
<img src="graphics/15b/graph_example_red.png" width="400px">
|
||||
<dl style="font-size: 60%; width: 400px;">
|
||||
<dt>Path</dt>
|
||||
<dd>Sequence of alternating vertices and edges</dd>
|
||||
<dd>begins with a vertex</dd>
|
||||
<dd>ends with a vertex</dd>
|
||||
<dd>each edge is preceded and followed by its endpoints</dd>
|
||||
|
||||
<dt>Simple Path</dt>
|
||||
<dd>path such that all of its vertices and edges are distinct</dd>
|
||||
|
||||
<dt>Examples</dt>
|
||||
<dd><span style="color: red">V, b, X, h, Z</span> is a simple path.</dd>
|
||||
<dd> <br> </dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Terminology</h4>
|
||||
|
||||
<img src="graphics/15b/graph_example_red_blue.png" width="400px">
|
||||
<dl style="font-size: 60%; width: 400px;">
|
||||
<dt>Path</dt>
|
||||
<dd>Sequence of alternating vertices and edges</dd>
|
||||
<dd>Begins with a vertex</dd>
|
||||
<dd>Ends with a vertex</dd>
|
||||
<dd>Each edge is preceded and followed by its endpoints</dd>
|
||||
|
||||
<dt>Simple Path</dt>
|
||||
<dd>Path such that all of its vertices and edges are distinct</dd>
|
||||
|
||||
<dt>Examples</dt>
|
||||
<dd><span style="color: red">V, b, X, h, Z</span> is a simple path.</dd>
|
||||
<dd><span style="color: blue">U, c, W, e, X, g, Y, f, W, d, V</span> is a path that is not simple.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Terminology</h4>
|
||||
|
||||
<img src="graphics/15b/graph_example.png" width="400px">
|
||||
<dl style="font-size: 60%; width: 500px;">
|
||||
<dt>Cycle</dt>
|
||||
<dd>Path that begins and ends with the same vertex</dd>
|
||||
<dd>Must contain at least one edge</dd>
|
||||
|
||||
<dt>Simple Cycle</dt>
|
||||
<dd>Cycle such that all of its vertices and edges are distinct</dd>
|
||||
|
||||
<dt class="fragment">Examples</dt>
|
||||
<dd> </dd>
|
||||
<dd> <br> </dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Terminology</h4>
|
||||
|
||||
<img src="graphics/15b/graph_example_cycle_red.png" width="400px">
|
||||
<dl style="font-size: 60%; width: 500px;">
|
||||
<dt>Cycle</dt>
|
||||
<dd>Path that begins and ends with the same vertex</dd>
|
||||
<dd>Must contain at least one edge</dd>
|
||||
|
||||
<dt>Simple Cycle</dt>
|
||||
<dd>Cycle such that all of its vertices and edges are distinct</dd>
|
||||
|
||||
<dt>Examples</dt>
|
||||
<dd><span style="color: red;">V, b, X, g, Y, f, W, c, U, a, V</span> is a simple cycle</dd>
|
||||
<dd> <br> </dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Terminology</h4>
|
||||
|
||||
<img src="graphics/15b/graph_example_cycle_red_blue.png" width="400px">
|
||||
<dl style="font-size: 60%; width: 500px;">
|
||||
<dt>Cycle</dt>
|
||||
<dd>Path that begins and ends with the same vertex</dd>
|
||||
<dd>Must contain at least one edge</dd>
|
||||
|
||||
<dt>Simple Cycle</dt>
|
||||
<dd>Cycle such that all of its vertices and edges are distinct</dd>
|
||||
|
||||
<dt>Examples</dt>
|
||||
<dd><span style="color: red;">V, b, X, g, Y, f, W, c, U, a, V</span> is a simple cycle</dd>
|
||||
<dd><span style="color: blue;">U, c, W, e, X, g, Y, f, W, d, V</span> is a cycle that is not simple</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Notation</h4>
|
||||
|
||||
<dl>
|
||||
<dt>$n$</dt>
|
||||
<dd>The number of vertices</dd>
|
||||
|
||||
<dt>$m$</dt>
|
||||
<dd>The number of edges</dd>
|
||||
|
||||
<dt>$deg(v)$</dt>
|
||||
<dd>The degree of vertex $v$</dd>
|
||||
</dl>
|
||||
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Graph Properties</h4>
|
||||
|
||||
$$\sum_{v} deg(v) = 2m$$
|
||||
|
||||
<p class="fragment"><b>Proof: </b> Each edge is counted twice</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Graph Properties</h4>
|
||||
|
||||
<p>In a directed graph with no self-loops and no parallel edges:</p>
|
||||
$$m \leq n(n-1)$$
|
||||
|
||||
<p class="fragment">
|
||||
<ul>
|
||||
<li>No parallel edges: each pair connected at most once</li>
|
||||
<li>No self loops: pick each vertex once</li>
|
||||
</ul>
|
||||
</p>
|
||||
|
||||
<p class="fragment">
|
||||
$n$ choices for the first vertex; $(n-1)$ choices for the second vertex.
|
||||
$$m \leq n(n-1)$$
|
||||
</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>Hey, isn't this a <b>data structures</b> class?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">A (Directed) Graph ADT</h4>
|
||||
|
||||
<dl>
|
||||
<dt>Two type parameters (<tt>Graph[V, E]</tt>)</dt>
|
||||
<dd><tt>V</tt>: The vertex label type</dd>
|
||||
<dd><tt>E</tt>: The edge label type</dd>
|
||||
|
||||
<dt>Vertices</dt>
|
||||
<dd>... are elements (like Linked List Nodes)</dd>
|
||||
<dd>... store a value of type <tt>V</tt></dd>
|
||||
|
||||
<dt>Edges</dt>
|
||||
<dd>... are elements</dd>
|
||||
<dd>... store a value of type <tt>E</tt></dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">A (Directed) Graph ADT</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
trait Graph[V, E] {
|
||||
def vertices: Iterator[Vertex]
|
||||
def edges: Iterator[Edge]
|
||||
def addVertex(label: V): Vertex
|
||||
def addEdge(orig: Vertex, dest: Vertex, label: E): Edge
|
||||
def removeVertex(vertex: Vertex): Unit
|
||||
def removeEdge(edge: Edge): Unit
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">A (Directed) Graph ADT</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
trait Vertex[V, E] {
|
||||
def outEdges: Seq[Edge]
|
||||
def inEdges: Seq[Edge]
|
||||
def incidentEdges: Iterator[Edge] = outEdges ++ inEdges
|
||||
def edgeTo(v: Vertex): Boolean
|
||||
def label: V
|
||||
}
|
||||
|
||||
trait Edge[V, E] {
|
||||
def origin: Vertex
|
||||
def destination: Vertex
|
||||
def label: E
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 1: Edge List</h4>
|
||||
|
||||
<p style="text-decoration: underline;">Data Model</p>
|
||||
<dl>
|
||||
<div class="fragment">
|
||||
<dt>A List of Edges</dt>
|
||||
<dd>ArrayBuffer</dd>
|
||||
</div>
|
||||
|
||||
<div class="fragment">
|
||||
<dt>A List of Vertices</dt>
|
||||
<dd>ArrayBuffer</dd>
|
||||
</div>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 1: Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV1[V, E] extends Graph[V, E]
|
||||
{
|
||||
val vertices = mutable.Buffer[Vertex]()
|
||||
val edges = mutable.Buffer[Edge]()
|
||||
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 1: Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
def addVertex(label: V): Vertex =
|
||||
vertices.append(new Vertex(label))
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 1: Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
def addEdge(orig: Vertex, dest: Vertex, label: E): Edge =
|
||||
edges.append(new Edge(orig, dest, label))
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 1: Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
def removeEdge(edge: Edge): Unit =
|
||||
edges.subtractOne(edge)
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity? <b class="fragment">($O(n)$)</b></p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 2: Linked Edge List</h4>
|
||||
|
||||
<p style="text-decoration: underline;">Data Model</p>
|
||||
<dl>
|
||||
<div>
|
||||
<dt>A List of Edges</dt>
|
||||
<dd>DoublyLinkedList</dd>
|
||||
</div>
|
||||
|
||||
<div>
|
||||
<dt>A List of Vertices</dt>
|
||||
<dd>DoublyLinkedList</dd>
|
||||
</div>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">DoublyLinkedList</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DoublyLinkedList[T] extends Seq[T] {
|
||||
def append(element: T): Node =
|
||||
/* O(1) with tail pointer */
|
||||
|
||||
def remove(node: Node): Unit =
|
||||
/* O(1) */
|
||||
|
||||
def iterator: Iterator[T]: Unit =
|
||||
/* O(1) + O(1) per call to next */
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 2: Linked Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV2[V, E] extends Graph[V, E] {
|
||||
val vertices = DoublyLinkedList[Vertex]()
|
||||
|
||||
class Vertex(label: V) = {
|
||||
var node: DoublyLinkedList[Vertex].Node = null
|
||||
/* ... */
|
||||
}
|
||||
|
||||
def addVertex(label: V): Vertex = {
|
||||
val vertex = new Vertex(label)
|
||||
val node = vertices.append(vertex)
|
||||
vertex.node = node
|
||||
return vertex
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 2: Linked Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV2[V, E] extends Graph[V, E] {
|
||||
val edges = DoublyLinkedList[Edge]()
|
||||
|
||||
class Edge(orig: Vertex, dest: Vertex, label: E) = {
|
||||
var node: DoublyLinkedList[Edge].Node = null
|
||||
/* ... */
|
||||
}
|
||||
|
||||
def addEdge(orig: Vertex, dest: Vertex, label: E): Vertex = {
|
||||
val edge = new Edge(orig, dest, label)
|
||||
val node = edges.append(vertex)
|
||||
edge.node = node
|
||||
return edge
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 2: Linked Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV2[V, E] extends Graph[V, E] {
|
||||
val edges = DoublyLinkedList[Edge]()
|
||||
|
||||
def removeEdge(edge: Edge): Unit = {
|
||||
edges.remove(edge.node)
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 2: Linked Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV2[V, E] extends Graph[V, E] {
|
||||
val vertices = DoublyLinkedList[Vertex]()
|
||||
|
||||
def removeVertex(vertex: Vertex): Unit = {
|
||||
vertices.remove(vertex.node)
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What if there's an edge to/from vertex?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 2: Linked Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV2[V, E] extends Graph[V, E] {
|
||||
val vertices = DoublyLinkedList[Vertex]()
|
||||
|
||||
def removeVertex(vertex: Vertex): Unit = {
|
||||
vertices.remove(vertex.node)
|
||||
for(edge <- vertex.incidentEdges){
|
||||
removeEdge(edge)
|
||||
}
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity? <b class="fragment">($O(1) + O(T_{incidentEdges}(n, m))$)</b></p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 2: Linked Edge List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV2[V, E] extends Graph[V, E] {
|
||||
val vertices = DoublyLinkedList[Vertex]()
|
||||
val edges = DoublyLinkedList[Edge]()
|
||||
|
||||
class Vertex(label: V) = {
|
||||
/* ... */
|
||||
|
||||
def outEdges =
|
||||
vertices.filter { _.orig = this }
|
||||
|
||||
def inEdges =
|
||||
vertices.filter { _.dest = this }
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity? <b class="fragment">($O(m) = O(n^2)$)</b></p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Edge List Summary</h4>
|
||||
|
||||
<ul>
|
||||
<li>addEdge, addVertex: <b class="fragment">$O(1)$</b></li>
|
||||
<li>removeEdge: <b class="fragment">$O(1)$</b></li>
|
||||
<li>removeVertex: <b class="fragment">$O(m)$</b></li>
|
||||
<li>vertex.incidentEdges: <b class="fragment">$O(m)$</b></li>
|
||||
<li class="fragment">vertex.edgeTo: <b class="fragment">$O(m)$</b></li>
|
||||
<li class="fragment"><b>Space Used</b>: $O(n) + O(m)$</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p><b>Idea: </b> Store the in/out edges for each vertex.</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 3: Adjacency List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV3[V, E] extends Graph[V, E] {
|
||||
val vertices = DoublyLinkedList[Vertex]()
|
||||
|
||||
class Vertex(label: V) = {
|
||||
var node: DoublyLinkedList[Vertex].Node = null
|
||||
val inEdges = DoublyLinkedList[Edge]()
|
||||
val outEdges = DoublyLinkedList[Edge]()
|
||||
/* ... */
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 3: Adjacency List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV3[V, E] extends Graph[V, E] {
|
||||
/* ... */
|
||||
def addEdge(orig: Vertex, dest: Vertex, label: E): Vertex = {
|
||||
val edge = new Edge(orig, dest, label)
|
||||
val node = edges.append(vertex)
|
||||
edge.node = node
|
||||
orig.outEdges.append(edge)
|
||||
dest.inEdges.append(edge)
|
||||
return edge
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 3: Adjacency List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV3[V, E] extends Graph[V, E] {
|
||||
/* ... */
|
||||
def removeEdge(edge: Edge): Unit = {
|
||||
edges.remove(edge.node)
|
||||
edge.orig.outEdges.subtractOne(edge)
|
||||
edge.dest.inEdges.subtractOne(edge)
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 4: Adjacency List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV4[V, E] extends Graph[V, E] {
|
||||
/* ... */
|
||||
class Edge(orig: Vertex, dest: Vertex, label: E) = {
|
||||
var node: DoublyLinkedList[Edge].Node = null
|
||||
var origNode: DoublyLinkedList[Edge].Node = null
|
||||
var destNode: DoublyLinkedList[Edge].Node = null
|
||||
/* ... */
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 4: Adjacency List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV4[V, E] extends Graph[V, E] {
|
||||
/* ... */
|
||||
def addEdge(orig: Vertex, dest: Vertex, label: E): Vertex = {
|
||||
val edge = new Edge(orig, dest, label)
|
||||
val node = edges.append(vertex)
|
||||
edge.node = node
|
||||
edge.origNode = orig.outEdges.append(edge)
|
||||
edge.destNode = dest.inEdges.append(edge)
|
||||
return edge
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 4: Adjacency List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV4[V, E] extends Graph[V, E] {
|
||||
/* ... */
|
||||
def removeEdge(edge: Edge): Unit = {
|
||||
edges.remove(edge.node)
|
||||
edge.orig.outEdges.remove(edge.origNode)
|
||||
edge.dest.inEdges.remove(edge.destNode)
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Attempt 4: Adjacency List</h4>
|
||||
|
||||
<pre><code class="scala">
|
||||
class DirectedGraphV4[V, E] extends Graph[V, E] {
|
||||
/* ... */
|
||||
def removeVertex(vertex: Vertex): Unit = {
|
||||
vertices.remove(vertex.node)
|
||||
for(edge <- vertex.incidentEdges){
|
||||
removeEdge(edge)
|
||||
}
|
||||
}
|
||||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Adjacency List</h4>
|
||||
|
||||
<svg data-src="graphics/15b/adjacency_list.svg" width="700px"/>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Adjacency List Summary</h4>
|
||||
|
||||
<ul>
|
||||
<li>addEdge, addVertex: <b class="fragment">$O(1)$</b></li>
|
||||
<li>removeEdge: <b class="fragment">$O(1)$</b></li>
|
||||
<li>vertex.incidentEdges: <b class="fragment">$O(deg(vertex))$</b></li>
|
||||
<li>removeVertex: <b class="fragment">$O(deg(vertex))$</b></li>
|
||||
<li class="fragment">vertex.edgeTo: <b class="fragment">$O(deg(vertex))$</b></li>
|
||||
<li class="fragment"><b>Space Used</b>: $O(n) + O(m)$</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Adjacency Matrix</h4>
|
||||
|
||||
<img src="graphics/15b/adjacency_matrix.png">
|
||||
</section>
|
||||
<section>
|
||||
<h4 class="slide_title">Adjacency Matrix Summary</h4>
|
||||
|
||||
<ul>
|
||||
<li>addEdge, removeEdge: <b class="fragment">$O(1)$</b></li>
|
||||
<li>addVertex, removeVertex: <b class="fragment">$O(n^2)$</b></li>
|
||||
<li>vertex.incidentEdges: <b class="fragment">$O(n)$</b></li>
|
||||
<li>vertex.edgeTo: <b class="fragment">$O(1)$</b></li>
|
||||
<li class="fragment"><b>Space Used</b>: $O(n^2)$</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
|
@ -0,0 +1,510 @@
|
|||
---
|
||||
template: templates/cse250_2022_slides.erb
|
||||
title: Graph Exploration
|
||||
date: Oct 7, 2022
|
||||
textbook: Ch. 15.3
|
||||
cat: graphics/16b/cat.png
|
||||
attribution: Based on slides by Tamassa Goodrich
|
||||
---
|
||||
<section>
|
||||
<h4 class="slide_title">Edge List Summary</h4>
|
||||
|
||||
<ul>
|
||||
<li>addEdge, addVertex: <b>$O(1)$</b></li>
|
||||
<li>removeEdge: <b>$O(1)$</b></li>
|
||||
<li>removeVertex: <b>$O(m)$</b></li>
|
||||
<li>vertex.incidentEdges: <b>$O(m)$</b></li>
|
||||
<li>vertex.edgeTo: <b>$O(m)$</b></li>
|
||||
<li><b>Space Used</b>: $O(n) + O(m)$</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Adjacency List Summary</h4>
|
||||
|
||||
<ul>
|
||||
<li>addEdge, addVertex: <b>$O(1)$</b></li>
|
||||
<li>removeEdge: <b>$O(1)$</b></li>
|
||||
<li>vertex.incidentEdges: <b>$O(deg(vertex))$</b></li>
|
||||
<li>removeVertex: <b>$O(deg(vertex))$</b></li>
|
||||
<li>vertex.edgeTo: <b>$O(deg(vertex))$</b></li>
|
||||
<li><b>Space Used</b>: $O(n) + O(m)$</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Adjacency Matrix Summary</h4>
|
||||
|
||||
<ul>
|
||||
<li>addEdge, removeEdge: <b>$O(1)$</b></li>
|
||||
<li>addVertex, removeVertex: <b>$O(n^2)$</b></li>
|
||||
<li>vertex.incidentEdges: <b>$O(n)$</b></li>
|
||||
<li>vertex.edgeTo: <b>$O(1)$</b></li>
|
||||
<li><b>Space Used</b>: $O(n^2)$</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>Ok, we have a graph, now what do we do with it?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Connectivity Problems</h4>
|
||||
<p>Given graph $G$:</p>
|
||||
<ul>
|
||||
<li class="fragment">Is vertex $u$ <b>adjacent</b> to $v$?</li>
|
||||
<li class="fragment">Is vertex $u$ <b>connected</b> to $v$ via some path?</li>
|
||||
<li class="fragment">Which vertices are connected to vertex $v$?</li>
|
||||
<li class="fragment">What is the shortest path from vertex $u$ to $v$?</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">A few more definitions...</h4>
|
||||
|
||||
<svg data-src="graphics/16b/subgraph-and-spanning.svg" width="200px" style="float: right"/>
|
||||
|
||||
<dl style="width: 700px">
|
||||
<dt>A <u>subgraph</u> $S$ of a graph $G$ is a graph where...</dt>
|
||||
<dd>$S$'s vertices are a subset of $G$'s vertices</dd>
|
||||
<dd>$S$'s edges are a subset of $G$'s edges</dd>
|
||||
|
||||
<div class="fragment">
|
||||
<dt>A <u>spanning subgraph</u> of $G$...</dt>
|
||||
<dd>Is a subgraph of $G$</dd>
|
||||
<dd>Contains all of $G$'s vertices.</dd>
|
||||
</div>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">A few more definitions...</h4>
|
||||
|
||||
<svg data-src="graphics/16b/dis-connected-graph.svg" width="200px" style="float: right"/>
|
||||
<dl style="width: 700px">
|
||||
<dt>A graph is <u>connected</u>...</dt>
|
||||
<dd>If there is a path between every pair of vertices</dd>
|
||||
|
||||
<div class="fragment">
|
||||
<dt>A <u>connected component</u> of $G$...</dt>
|
||||
<dd>Is a maximal <i>connected</i> subgraph of $G$ <ul>
|
||||
<li style="font-size: 80%">"maximal" means you can't add any new vertex without breaking the property</li>
|
||||
<li style="font-size: 80%">Any subset of $G$'s edges that connects the subgraph is fine.</li>
|
||||
</ul></dd>
|
||||
</div>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">A few more definitions...</h4>
|
||||
|
||||
<dl style="font-size: 90%">
|
||||
<dt>A <u>free tree</u> is an <i>un</i>directed graph $T$ such that:</dt>
|
||||
<dd>There is exactly one simple path between any two nodes <ul>
|
||||
<li>T is connected</li>
|
||||
<li>T has no cycles</li>
|
||||
</ul></dd>
|
||||
|
||||
<div class="fragment">
|
||||
<dt>A <u>rooted tree</u> is a directed graph $T$ such that:</dt>
|
||||
<dd>One vertex of $T$ is the <u>root</u></dd>
|
||||
<dd>There is exactly one simple path from the root to every other vertex in the graph.</dd>
|
||||
</div>
|
||||
|
||||
<div class="fragment">
|
||||
<dt>A (free/rooted) <u>forest</u> is a graph $F$ such that</dt>
|
||||
<dd>Every connected component is a tree.</dd>
|
||||
</div>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">A few more definitions...</h4>
|
||||
|
||||
<dl>
|
||||
<dt>A <u>spanning tree</u> of a connected graph...</dt>
|
||||
<dd>Is a spanning subgraph that is a tree.</dd>
|
||||
<dd>Is <i>not</i> unique unless the graph is a tree.</dd>
|
||||
</dl>
|
||||
|
||||
<svg data-src="graphics/16b/spanning-tree.svg" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>Ok... back to the question: Connectivity!</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>The maze is a graph!</p>
|
||||
|
||||
<svg data-src="graphics/16b/maze-is-graph.svg" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Recall</h4>
|
||||
|
||||
<dl>
|
||||
<dt>Searching the maze with a stack <span class="fragment">(Depth-First Search)</span></dt>
|
||||
<dd>Try out every path, one at a time to the end...</dd>
|
||||
<dd>... then backtrack and try another</dd>
|
||||
|
||||
<dt>Searching the maze with a queue <span class="fragment">(Breadth-First Search)</span></dt>
|
||||
<dd>Try out every path in parallel...</dd>
|
||||
<dd>... keep picking a path to extend by one step.</dd>
|
||||
</dl>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Depth-First Search</h4>
|
||||
<p><u>Primary Goals</u></p>
|
||||
<ul>
|
||||
<li>Visit every vertex in graph $G = (V, E)$</li>
|
||||
<li>Construct a spanning tree for every connected component <ul style="font-size: 80%">
|
||||
<li class="fragment"><b>Side Effect:</b> Compute connected components</li>
|
||||
<li class="fragment"><b>Side Effect:</b> Compute a paths between all connected vertices</li>
|
||||
<li class="fragment"><b>Side Effect:</b> Determine if the graph is connected</li>
|
||||
<li class="fragment"><b>Side Effect:</b> Identify Cycles</li>
|
||||
</ul></li>
|
||||
<li class="fragment">Complete in time $O(|V| + |E|)$</li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Depth-First Search</h4>
|
||||
|
||||
<dl>
|
||||
<dt>DFS</dt>
|
||||
<dd><b>Input:</b> Graph $G$</dd>
|
||||
<dd><b>Output:</b> Label every edge as: <ul>
|
||||
<li><u>Spanning Edge</u>: Part of the spanning tree</li>
|
||||
<li><u>Back Edge</u>: Part of a cycle</li>
|
||||
</ul></dd>
|
||||
|
||||
<div class="fragment">
|
||||
<dt>DFSOne</dt>
|
||||
<dd><b>Input:</b> Graph $G = (V, E)$, Start Vertex $v \in V$</dd>
|
||||
<dd><b>Output:</b> Label every edge in $v$'s connected component</dd>
|
||||
</div>
|
||||
</dl>
|
||||
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Depth-First Search</h4>
|
||||
<svg data-src="graphics/16b/dfs-one-intuition.svg" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Depth-First Search</h4>
|
||||
|
||||
<pre><code class="scala">object VertexLabel extends Enumeration
|
||||
{ val UNEXPLORED, VISITED = Value }
|
||||
|
||||
object EdgeLabel extends Enumeration
|
||||
{ val UNEXPLORED, SPANNING, BACK = Value }
|
||||
|
||||
def DFS(graph: Graph[VertexLabel.Value, EdgeLabel.Value])
|
||||
{
|
||||
for(v <- graph.vertices) { v.setLabel(VertexLabel.UNEXPLORED) }
|
||||
for(e <- graph.edges) { e.setLabel(EdgeLabel.UNEXPLORED) }
|
||||
for(v <- graph.vertices) {
|
||||
if(v.label == VertexLabel.UNEXPLORED){
|
||||
DFSOne(graph, v)
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Depth-First Search</h4>
|
||||
|
||||
<pre><code class="scala">def DFSOne(graph: Graph[…], v: Graph[…]#Vertex)
|
||||
{
|
||||
v.setLabel(VertexLabel.VISITED)
|
||||
|
||||
for(e <- v.incident) {
|
||||
if(e.label == EdgeLabel.UNEXPLORED){
|
||||
val w = e.getOpposite(v)
|
||||
|
||||
if(w.label == VertexLabel.UNEXPLORED){
|
||||
e.setLabel(EdgeLabel.SPANNING)
|
||||
DFSOne(graph, w)
|
||||
} else {
|
||||
e.setLabel(EdgeLabel.BACK)
|
||||
}
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Depth-First Search</h4>
|
||||
<svg data-src="graphics/16b/dfs-one-example.svg" />
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">DFS vs Mazes</h4>
|
||||
|
||||
<p>The DFS algorithm is like our stack-based maze solver</p>
|
||||
|
||||
<ul>
|
||||
<li>Mark each grid square with <b>VISITED</b>.</li>
|
||||
<li>Mark each path with <b>SPANNING</b>.</li>
|
||||
<li>Only visit each vertex once.<ul>
|
||||
<li class="fragment">DFS won't necessarily find the shortest paths.</li>
|
||||
</ul></li>
|
||||
</ul>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<p>What's the complexity?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFS(graph: Graph[VertexLabel.Value, EdgeLabel.Value])
|
||||
{
|
||||
for(v <- graph.vertices) { v.setLabel(VertexLabel.UNEXPLORED) }
|
||||
for(e <- graph.edges) { e.setLabel(EdgeLabel.UNEXPLORED) }
|
||||
for(v <- graph.vertices) {
|
||||
if(v.label == VertexLabel.UNEXPLORED){
|
||||
DFSOne(graph, v)
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFS(graph: Graph[VertexLabel.Value, EdgeLabel.Value])
|
||||
{
|
||||
/* O(|V|) */
|
||||
for(e <- graph.edges) { e.setLabel(EdgeLabel.UNEXPLORED) }
|
||||
for(v <- graph.vertices) {
|
||||
if(v.label == VertexLabel.UNEXPLORED){
|
||||
DFSOne(graph, v)
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFS(graph: Graph[VertexLabel.Value, EdgeLabel.Value])
|
||||
{
|
||||
/* O(|V|) */
|
||||
/* O(|E|) */
|
||||
for(v <- graph.vertices) {
|
||||
if(v.label == VertexLabel.UNEXPLORED){
|
||||
DFSOne(graph, v)
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFS(graph: Graph[VertexLabel.Value, EdgeLabel.Value])
|
||||
{
|
||||
/* O(|V|) */
|
||||
/* O(|E|) */
|
||||
/* O(|V|) Times */ {
|
||||
if(v.label == VertexLabel.UNEXPLORED){
|
||||
DFSOne(graph, v)
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFS(graph: Graph[VertexLabel.Value, EdgeLabel.Value])
|
||||
{
|
||||
/* O(|V|) */
|
||||
/* O(|E|) */
|
||||
/* O(|V|) Times */ {
|
||||
if(v.label == VertexLabel.UNEXPLORED){
|
||||
/* ??? */
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFSOne(graph: Graph[…], v: Graph[…]#Vertex)
|
||||
{
|
||||
v.setLabel(VertexLabel.VISITED)
|
||||
|
||||
for(e <- v.incident) {
|
||||
if(e.label == EdgeLabel.UNEXPLORED){
|
||||
val w = e.getOpposite(v)
|
||||
|
||||
if(w.label == VertexLabel.UNEXPLORED){
|
||||
e.setLabel(EdgeLabel.SPANNING)
|
||||
DFSOne(graph, w)
|
||||
} else {
|
||||
e.setLabel(EdgeLabel.BACK)
|
||||
}
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFSOne(graph: Graph[…], v: Graph[…]#Vertex)
|
||||
{
|
||||
/* O(1) */
|
||||
|
||||
for(e <- v.incident) {
|
||||
if(e.label == EdgeLabel.UNEXPLORED){
|
||||
val w = e.getOpposite(v)
|
||||
|
||||
if(w.label == VertexLabel.UNEXPLORED){
|
||||
e.setLabel(EdgeLabel.SPANNING)
|
||||
DFSOne(graph, w)
|
||||
} else {
|
||||
e.setLabel(EdgeLabel.BACK)
|
||||
}
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFSOne(graph: Graph[…], v: Graph[…]#Vertex)
|
||||
{
|
||||
/* O(1) */
|
||||
|
||||
/* O(deg(v)) times */ {
|
||||
if(e.label == EdgeLabel.UNEXPLORED){
|
||||
val w = e.getOpposite(v)
|
||||
|
||||
if(w.label == VertexLabel.UNEXPLORED){
|
||||
e.setLabel(EdgeLabel.SPANNING)
|
||||
DFSOne(graph, w)
|
||||
} else {
|
||||
e.setLabel(EdgeLabel.BACK)
|
||||
}
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFSOne(graph: Graph[…], v: Graph[…]#Vertex)
|
||||
{
|
||||
/* O(1) */
|
||||
|
||||
/* O(deg(v)) times */ {
|
||||
/* O(1) */ {
|
||||
/* O(1) */
|
||||
|
||||
/* O(1) */ {
|
||||
/* O(1) */
|
||||
DFSOne(graph, w)
|
||||
} else {
|
||||
/* O(1) */
|
||||
}
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">def DFSOne(graph: Graph[…], v: Graph[…]#Vertex)
|
||||
{
|
||||
/* O(1) */
|
||||
|
||||
/* O(deg(v)) times */ {
|
||||
/* O(1) */ {
|
||||
/* O(1) */
|
||||
|
||||
/* O(1) */ {
|
||||
/* O(1) */
|
||||
/* ??? */
|
||||
} else {
|
||||
/* O(1) */
|
||||
}
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Depth-First Search</h4>
|
||||
<p><b>Observation: </b> <tt>DFSOne</tt> is called on each vertex <i>at most</i> once.</p>
|
||||
<ul>
|
||||
<li>If <tt>v.label == VISITED</tt>, both <tt>DFS</tt>, <tt>DFSOne</tt> skip it.</li>
|
||||
</ul>
|
||||
|
||||
<p class="fragment">$O(|V|)$ calls to <tt>DFSOne</tt></p>
|
||||
|
||||
<p class="fragment">What's the runtime of <tt>DFSOne</tt> <b>excluding recursive calls</b>?</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<pre><code class="scala">
|
||||
def DFSOne(graph: Graph[…], v: Graph[…]#Vertex)
|
||||
{
|
||||
/* O(1) */
|
||||
|
||||
/* O(deg(v)) times */ {
|
||||
/* O(1) */ {
|
||||
/* O(1) */
|
||||
|
||||
/* O(1) */ {
|
||||
/* O(1) */
|
||||
DFSOne(graph, w)
|
||||
} else {
|
||||
/* O(1) */
|
||||
}
|
||||
}
|
||||
}
|
||||
} </code></pre>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Depth-First Search</h4>
|
||||
|
||||
<p>
|
||||
What's the sum over all calls to <tt>DFSOne</tt>?
|
||||
</p>
|
||||
|
||||
<p class="fragment">
|
||||
$\sum_{v \in V} O(deg(v))$
|
||||
</p>
|
||||
|
||||
<p class="fragment">
|
||||
$ = O(\sum_{v \in V} deg(v))$
|
||||
</p>
|
||||
<p class="fragment">
|
||||
$ = O(2 |E|)$ (by rule)
|
||||
</p>
|
||||
<p class="fragment">
|
||||
$ = O(|E|)$
|
||||
</p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
<h4 class="slide_title">Depth-First Search</h4>
|
||||
|
||||
<p>Summing up...</p>
|
||||
<table>
|
||||
<tr class="fragment">
|
||||
<td>Mark Vertices <b>UNVISITED</b></td>
|
||||
<td class="fragment">$O(|V|)$</td>
|
||||
</tr>
|
||||
<tr class="fragment">
|
||||
<td>Mark Edges <b>UNVISITED</b></td>
|
||||
<td class="fragment">$O(|E|)$</td>
|
||||
</tr>
|
||||
<tr class="fragment">
|
||||
<td><tt>DFS</tt> Vertex Loop</td>
|
||||
<td class="fragment">$O(|V|)$</td>
|
||||
</tr>
|
||||
<tr class="fragment">
|
||||
<td>All Calls to <tt>DFSOne</tt></td>
|
||||
<td class="fragment">$O(|E|)$</td>
|
||||
</tr>
|
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<tr>
|
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<td></td>
|
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<td class="fragment"><span style="border-top: solid 1px black; padding-top: 15px">$O(|V| + |E|)$</span></td>
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</tr>
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<path
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x="-136.56303"
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style="font-size:5.64444px;line-height:1.25;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;stroke-width:0.264583"
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x="-136.56303"
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id="text174138"><tspan
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id="tspan174136"
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x="-136.56303"
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y="129.5529"
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id="text186186"><tspan
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id="tspan186184"
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x="-99.084877"
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id="layer2"
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style="display:inline"
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transform="translate(1.374123e-6,-9.2643834)">
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<text
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xml:space="preserve"
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style="font-size:5.64444px;line-height:1.25;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;stroke-width:0.264583"
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x="52.286896"
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y="89.785408"
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class="fragment"><tspan
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x="52.286896"
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y="89.785408">DFS(G)</tspan></text>
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id="g239536"
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class="fragment">
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<text
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xml:space="preserve"
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style="font-size:5.64444px;line-height:1.25;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;stroke-width:0.264583"
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x="52.286896"
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y="96.135406"
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id="text191417"><tspan
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id="tspan191415"
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style="stroke-width:0.264583"
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x="52.286896"
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y="96.135406">DFSOne(G, A)</tspan></text>
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xml:space="preserve"
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:8.46667px;line-height:1.25;font-family:'Noto Color Emoji';-inkscape-font-specification:'Noto Color Emoji';letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;fill:#26a269;stroke-width:0.264583"
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x="-60.631718"
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y="69.725365"
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id="text137893-9"
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transform="translate(149.80885,-47.43192)"
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class="fragment"><tspan
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sodipodi:role="line"
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id="tspan137891-5"
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:8.46667px;font-family:'Noto Color Emoji';-inkscape-font-specification:'Noto Color Emoji';fill:#26a269;stroke-width:0.264583"
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x="-60.631718"
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y="69.725365">☑</tspan></text>
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<text
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xml:space="preserve"
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style="font-size:5.64444px;line-height:1.25;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;stroke-width:0.264583"
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x="92.863266"
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y="96.130882"
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id="text211585"
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class="fragment"><tspan
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sodipodi:role="line"
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id="tspan211583"
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style="fill:#77767b;stroke-width:0.264583"
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x="92.863266"
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y="96.130882">(→ B, C, D)</tspan></text>
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d="M -81.028599,68.238522 -64.500232,52.532826"
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id="path168078-0"
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class="fragment"
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sodipodi:nodetypes="cc"
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transform="translate(153.12027,-28.683445)" />
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<g
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id="g239546"
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class="fragment">
|
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<text
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xml:space="preserve"
|
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:8.46667px;line-height:1.25;font-family:'Noto Color Emoji';-inkscape-font-specification:'Noto Color Emoji';letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;fill:#26a269;stroke-width:0.264583"
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x="64.306305"
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y="47.164284"
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id="text195411"><tspan
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sodipodi:role="line"
|
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id="tspan195409"
|
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:8.46667px;font-family:'Noto Color Emoji';-inkscape-font-specification:'Noto Color Emoji';fill:#26a269;stroke-width:0.264583"
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x="64.306305"
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y="47.164284">☑</tspan></text>
|
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<text
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xml:space="preserve"
|
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style="font-size:5.64444px;line-height:1.25;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;stroke-width:0.264583"
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x="52.286896"
|
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y="102.4854"
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id="text195415"><tspan
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sodipodi:role="line"
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id="tspan195413"
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style="stroke-width:0.264583"
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x="52.286896"
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y="102.4854">DFSOne(G, B)</tspan></text>
|
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<text
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xml:space="preserve"
|
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style="font-size:5.64444px;line-height:1.25;font-family:sans-serif;-inkscape-font-specification:'sans-serif, Normal';letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;stroke-width:0.264583"
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x="92.876953"
|
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y="102.46486"
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id="text218207"><tspan
|
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sodipodi:role="line"
|
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id="tspan218205"
|
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style="fill:#77767b;stroke-width:0.264583"
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x="92.876953"
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y="102.46486">(→ A, C)</tspan></text>
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<path
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style="display:inline;fill:none;stroke:#26a269;stroke-width:1.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
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d="m -81.028599,77.40366 16.528367,15.705685"
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@ -51,18 +51,28 @@ class_name = "CSE-250 Fall 2022 - Section B"
|
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<!-- Any Talk-Specific Footer Content Goes Here -->
|
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<span style="margin-right: 150px;"><%= date %></span>
|
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<span style="margin-right: 150px; background-color: lightgray; border-radius: 3px; padding: 3px;"><a href="<%= out_path %>?print-pdf">🖶</a></span>
|
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<span>© Oliver Kennedy, Andrew Hughes; The University at Buffalo, SUNY</span>
|
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<span style="display: inline-block; text-align: right">© Oliver Kennedy, Andrew Hughes; The University at Buffalo, SUNY
|
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<% if defined? attribution %><br><span style="font-size: 70%"><%=attribution%></span><% end %></span>
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</div>
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<div class="slides">
|
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<section>
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<% if defined? cat %>
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<div style="display: flex; flex-direction: columns">
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<div style="width: 300px;">
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<img src="<%=cat%>" width="300px">
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</div><div style="width: 500px; font-size: 50%">
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<% end %>
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<h3><%= title %></h3>
|
||||
<h4><%= class_name %></h4>
|
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<p><%= date %></p>
|
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<% if defined? textbook %>
|
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<h5><b>Textbook</b>: <%= textbook %></h5>
|
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<% end %>
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<% if defined? cat %>
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</div></div>
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<% end %>
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</section>
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||||
|
|
|
|||
Loading…
Reference in New Issue