weds slides
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@ -309,29 +309,33 @@ schedule:
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- date: 10/03/22
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topic: "Composite Data Structures, Stacks, Shortest Path"
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dow: Mon
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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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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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dow: Fri
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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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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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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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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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@ -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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@ -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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<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>
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<dt>Push</dt>
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<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>
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<dd>Peek at what's on top of the stack</dd>
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</dl>
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</section>
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<section>
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<h4 class="slide_title">Queue</h4>
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<p>Outside of the US, "queueing" is lining up.</p>
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<dl>
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<dt>Enqueue</dt>
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<dd>Put a new object at the end of the queue</dd>
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<dt>Dequeue</dt>
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<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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<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>
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<section>
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<h4 class="slide_title">ArrayBuffer Attempt 1</h4>
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<dl>
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<dt>Enqueue</dt>
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<dd>Append</dd>
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<dt>Dequeue</dt>
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<dd>Remove(0)</dd>
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</dl>
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<p class="fragment">What's the complexity?</p>
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</section>
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<section>
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<h4 class="slide_title">ArrayBuffer Attempt 2</h4>
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<dl>
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<dt>Enqueue</dt>
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<dd>Insert(0)</dd>
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<dt>Dequeue</dt>
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<dd>Remove(last)</dd>
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</dl>
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<p class="fragment">What's the complexity?</p>
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</section>
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<section>
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<p>Can we avoid having to move all of the<br> elements forward or backward a spot?</p>
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</section>
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<section>
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<style type="text/css">
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#dequeue_example {
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height: 500px;
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width: 500px;
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border: solid 1px black;
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margin-left: auto;
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margin-right: auto;
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overflow-y: scroll;
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}
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#dequeue_example .dequeue_node {
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width: 50px;
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height: 50px;
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display: inline-block;
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color: lightgray;
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background: black;
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margin-left: auto;
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margin-right: auto;
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margin-top: 5px;
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}
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#dequeue_example .dequeue_node.dequeue_end {
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border-bottom: solid 10px green;
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}
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#dequeue_example .dequeue_node.dequeue_start {
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border-bottom: solid 10px blue;
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}
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#dequeue_example .dequeue_node.dequeue_end.dequeue_start {
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border-bottom: solid 10px green;
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}
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</style>
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<button onclick="dequeue_enqueue()">Enqueue</button>
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<button onclick="dequeue_dequeue()">Dequeue</button>
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<div id="dequeue_example">
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<div class="dequeue_node dequeue_start dequeue_end ">_</div>
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<div class="dequeue_node">_</div>
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<div class="dequeue_node">_</div>
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<div class="dequeue_node">_</div>
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</div>
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<script type="text/javascript">
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var dequeue = document.getElementById("dequeue_example")
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var dequeue_counter = 0
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function dequeue_expand() {
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var values = []
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var x
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while(x = dequeue_dequeue()){
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values.push(x)
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}
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console.log("Copying values: "+values)
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var count = dequeue.children.length * 2
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dequeue.innerText = ""
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for(x = 0; x < count; x++){
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var elem = document.createElement('div')
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elem.classList.add("dequeue_node")
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console.log("Update "+x)
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if(x == 0){
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console.log("Start")
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elem.classList.add("dequeue_start")
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}
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if(x == values.length){
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console.log("End")
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elem.classList.add("dequeue_end")
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}
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if(x < values.length){
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console.log("Existing: "+values[x])
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elem.innerText = values[x]
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} else {
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console.log("Blank")
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elem.innerText = "_"
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}
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dequeue.append(elem)
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}
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}
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function dequeue_enqueue() {
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var node = dequeue.querySelector(".dequeue_end")
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var next = node.nextElementSibling || dequeue.firstElementChild
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if(next.classList.contains("dequeue_start")){
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dequeue_expand()
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node = dequeue.querySelector(".dequeue_end")
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next = node.nextElementSibling || dequeue.firstElementChild
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}
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node.innerText = dequeue_counter++
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node.classList.add("active")
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node.classList.remove("dequeue_end")
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next.classList.add("dequeue_end")
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}
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function dequeue_dequeue() {
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var node = dequeue.querySelector(".dequeue_start")
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var next = node.nextElementSibling || dequeue.firstElementChild
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if(node.classList.contains("dequeue_end")){ return null }
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var ret = node.innerText
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node.innerText = "_"
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node.classList.remove("active")
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node.classList.remove("dequeue_start")
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next.classList.add("dequeue_start")
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return ret
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}
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</script>
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</section>
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<section>
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<h4 class="slide_title">ArrayDequeue (Resizable Ring Buffer)</h4>
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<p>Active Array = [start, end)</p>
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<dl style="font-size: 80%">
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<dt>Enqueue</dt>
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<dd>Resize buffer if needed</dd>
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<dd>Add new element at buffer[end]</dd>
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<dd>Advance end pointer (wrap around to front)</dd>
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<dt>Dequeue</dt>
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<dd>Remove element at buffer[start]</dd>
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<dd>Advance start pointer (wrap around to front)</dd>
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</dl>
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<p class="fragment">What's the complexity?</p>
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</section>
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<section>
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<h4 class="slide_title">Applications of Stacks and Queues</h4>
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<ul>
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<li class="fragment highlight-blue">Stack: Checking for Balanced Parenthesis/Braces</li>
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<li>Queue: Scheduling Packets for Delivery</li>
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<li>Both: Searching Mazes</li>
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</ul>
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</section>
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<section>
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<h4 class="slide_title">Balanced Parenthesis/Braces</h4>
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<p>What does it mean for parenthesis/braces to be balanced?</p>
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<ul>
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<li>Every opening symbol is matched by a closing symbol</li>
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<li>No nesting overlaps (e.g., <tt style="font-weight: bold;">{(})</tt> is not ok).</li>
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</ul>
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<table style="margin-top: 50px;">
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<tr>
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<td><tt style="font-weight: bold;">{()({})}</tt></td>
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<td><tt style="font-weight: bold;">{())</tt></td>
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<td><tt style="font-weight: bold;">())</tt></td>
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</tr>
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<tr>
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<td style="font-size: 200%" class="fragment">✅</td>
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<td style="font-size: 200%" class="fragment">❌</td>
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<td style="font-size: 200%" class="fragment">❌</td>
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</tr>
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</table>
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</section>
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<section>
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<h4 class="slide_title">Balanced Parenthesis/Braces</h4>
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<p><b>Idea: </b> Count the number of unmatched open parens, braces.</p>
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<p>Increment counter on <tt>(</tt>, decrement on <tt>)</tt></p>
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<p class="fragment"><b>Problem: </b> allows <tt style="font-weight: bold;">{(})</tt></p>
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</section>
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<section>
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<h4 class="slide_title">Balanced Parenthesis/Braces</h4>
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<p><b>Idea: </b> Track nesting on a stack!</p>
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<p>On <tt>(</tt> or <tt>{</tt>, push the symbol on the stack.</p>
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<p>On <tt>)</tt> or <tt>}</tt>, pop the stack and check the popped symbol.</p>
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</section>
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<section>
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<style type="text/css">
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#nesting_stack {
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height: 400px;
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width: 200px;
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border: solid 1px black;
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margin-left: auto;
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margin-right: auto;
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overflow-y: scroll;
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transition-property: background;
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transition-duration: 0.1s;
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}
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#nesting_stack.nesting_alert {
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background: #ffbbbb;
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}
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#nesting_stack .nesting_symbol {
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width: 50px;
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height: 50px;
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display: block;
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color: lightgray;
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background: black;
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margin-left: auto;
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margin-right: auto;
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margin-top: 5px;
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}
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#nesting_string {
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height: 100px;
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font-family: fixed;
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}
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#nesting_string::before {
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content: '"';
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}
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#nesting_string::after {
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content: '"';
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}
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</style>
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<button onclick="nesting_push('(')">(</button>
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<button onclick="nesting_push('{')">{</button>
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<button onclick="nesting_pop(')')">)</button>
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<button onclick="nesting_pop('}')">}</button>
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<div id="nesting_stack"></div>
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<div id="nesting_string"></div>
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<script type="text/javascript">
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var nesting_stack = document.getElementById("nesting_stack")
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var nesting_string = document.getElementById("nesting_string")
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function nesting_push(chr){
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nesting_string.innerText = nesting_string.innerText+chr
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var elem = document.createElement('div')
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elem.classList.add("nesting_symbol")
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elem.innerText = chr
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nesting_stack.prepend(elem)
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}
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function nesting_alert(){
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nesting_stack.classList.add("nesting_alert")
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setTimeout(() => {
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nesting_stack.classList.remove("nesting_alert")
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}, 100)
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}
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function nesting_pop(chr){
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var elem = nesting_stack.firstElementChild
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var match = (chr == ")" ? "(" : "{")
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if(!elem){ nesting_alert() }
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else {
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||||
if(elem.innerText != match){ nesting_alert() }
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else {
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nesting_string.innerText = nesting_string.innerText+chr
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elem.remove()
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||||
}
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}
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}
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</script>
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</section>
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<section>
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<h4 class="slide_title">Network Packets</h4>
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<dl style="font-size: 75%">
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<dt>Router: 1gb/s internal network, 100mb/s external</dt>
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<dd><ul>
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<li>1gb/s sent to router, but only 100mb/s can leave.</li>
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<li>How do packets get delivered?</li>
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</ul></dd>
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<dt>Queues</dt>
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<dd><ul>
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<li>Enqueue data packets in the order they are received.</li>
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<li>When outgoing bandwidth available, dequeue and send.</li>
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</ul></dd>
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<dt>Avoiding Queueing Delays</dt>
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<dd><ul>
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<li>Limit size of queue; Packets that don't fit are dropped</li>
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</ul></dd>
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<p style="font-size: 70%">
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<span style="margin-right: auto">
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<b>TCP:</b> blocked packets are retried
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</span>
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<span style="margin-left: 50px;">
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<b>UDP:</b> application deals with dropped packets.
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</span>
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</p>
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</dl>
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</section>
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<section>
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<h4 class="slide_title">Mazes</h4>
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<ul style="font-size: 80%; max-width: 450px;">
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<li><b>O</b> is the start, <b>X</b> is the objective <ul>
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<li>There may be multiple paths</li>
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<li>Generally, we want the shortest</li>
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</ul></li>
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<li class="fragment"><b>Approach 1: </b> Take the first available route in one direction <ul>
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<li style="color: red">Right, Down, Left, or Up</li>
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<li style="color: blue">Down, Right, Up, Left</li>
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</ul></li>
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</ul>
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<img src="graphics/14b/maze1.png" style="float: right" width="300px">
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</section>
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<section>
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<h4 class="slide_title">Mazes</h4>
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<ul style="font-size: 80%; max-width: 450px;">
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<li><b>O</b> is the start, <b>X</b> is the objective <ul>
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<li>There may be multiple paths</li>
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<li>Generally, we want the shortest</li>
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</ul></li>
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<li><b>Approach 1: </b> Take the first available route in one direction <ul>
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<li style="color: red">Right, Down, Left, or Up</li>
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<li style="color: blue">Down, Right, Up, Left</li>
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</ul></li>
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</ul>
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<img src="graphics/14b/maze1-rdlu.png" style="float: right" width="300px">
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</section>
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<section>
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<h4 class="slide_title">Mazes</h4>
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<img src="graphics/14b/maze1-rdlu.png">
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<p>How do you know which one is best?</p>
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<p class="fragment">Other problems with this algorithm?</p>
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</section>
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<section>
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<h4 class="slide_title">Mazes</h4>
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<img src="graphics/14b/maze2.png" width="350px">
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<p> </p>
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</section>
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<section>
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<h4 class="slide_title">Mazes</h4>
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<img src="graphics/14b/maze2-rdlu.png" width="350px">
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<p class="fragment">Priority order doesn't guarantee exploring the entire maze</p>
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</section>
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<section>
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<h4 class="slide_title">Formalizing Maze-Solving</h4>
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<div style="font-size: 80%;">
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<p><b>Inputs:</b> <ul>
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<li>The map: An $n\times m$ grid of filled/empty squares.</li>
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<li>The <b>O</b> is at position $start$</li>
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<li>The <b>X</b> is at position $dest$</li>
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</ul></p>
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<p><b>Goal: </b>
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<li>Compute $steps(\texttt{start}, \texttt{dest})$, the minimum steps from start to end.</li>
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</p>
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</div>
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<p class="fragment">How do we define $steps$?</p>
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</section>
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<section>
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<h4 class="slide_title">Formalizing Maze-Solving</h4>
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<svg data-src="graphics/14b/maze1-distances.svg" />
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</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 4, 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 List 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>
|
||||
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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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<% if defined? cat %>
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<div style="width: 300px;">
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<% end %>
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<h3><%= title %></h3>
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<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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|
|
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|||
Loading…
Reference in New Issue