Lecture 16
|
@ -298,7 +298,7 @@ schedule:
|
|||
section_b:
|
||||
slides: slide/12b-RecursionAndAverage.html
|
||||
- date: 09/30/22
|
||||
topic: Data-Bounded Analysis
|
||||
topic: Stacks, Queues
|
||||
dow: Fri
|
||||
section_a:
|
||||
slides: slide/lec14a.pdf
|
||||
|
@ -307,31 +307,33 @@ schedule:
|
|||
- week: 6
|
||||
lectures:
|
||||
- date: 10/03/22
|
||||
topic: "Composite Data Structures, Stacks, Shortest Path"
|
||||
topic: "Applications of Stacks, Queues (Mazes)"
|
||||
dow: Mon
|
||||
section_a:
|
||||
slides: slide/lec15a.pdf
|
||||
section_b:
|
||||
slides: slide/14b-QueueStackApps.html
|
||||
- date: 10/05/22
|
||||
topic: "Queues, Shortest Path revisited"
|
||||
topic: "Graphs; Graph ADTs"
|
||||
dow: Wed
|
||||
section_b:
|
||||
slides: slide/15b-Graphs.html
|
||||
- date: 10/07/22
|
||||
topic: Graphs Intro
|
||||
topic: "Graph Search; DFS"
|
||||
dow: Fri
|
||||
section_b:
|
||||
slides: slide/16b-GraphExploration.html
|
||||
- week: 7
|
||||
lectures:
|
||||
- date: 10/10/22
|
||||
topic: "Graph ADTs: Edge Lists"
|
||||
topic: "Graph Search; Pathfinding, BFS"
|
||||
due: PA2-Tests
|
||||
dow: Mon
|
||||
- date: 10/12/22
|
||||
topic: DFS, BFS, Analysis
|
||||
topic: BFS, Shortest Path, Priority Queues
|
||||
dow: Wed
|
||||
- date: 10/14/22
|
||||
topic: Priority Queues, Orderings
|
||||
topic: Order Relations, Priority Queues
|
||||
dow: Fri
|
||||
- week: 8
|
||||
lectures:
|
||||
|
@ -343,7 +345,7 @@ schedule:
|
|||
special: Midterm Exam
|
||||
dow: Wed
|
||||
- date: 10/21/22
|
||||
topic: Priority Queues, Orderings
|
||||
topic: Organizing Cat Pictures
|
||||
dow: Fri
|
||||
- week: 9
|
||||
lectures:
|
||||
|
|
|
@ -1,7 +1,7 @@
|
|||
---
|
||||
template: templates/cse250_2022_slides.erb
|
||||
title: Graphs
|
||||
date: Oct 4, 2022
|
||||
date: Oct 5, 2022
|
||||
textbook: Ch. 15.3
|
||||
cat: graphics/15b/cat.png
|
||||
attribution: Based on slides by Tamassa Goodrich
|
||||
|
@ -497,7 +497,7 @@ attribution: Based on slides by Tamassa Goodrich
|
|||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity? <b class="fragment">O(1) + O(T_{incidentEdges}(n, m))</b></p>
|
||||
<p class="fragment">What's the complexity? <b class="fragment">($O(1) + O(T_{incidentEdges}(n, m))$)</b></p>
|
||||
</section>
|
||||
|
||||
<section>
|
||||
|
@ -520,18 +520,18 @@ attribution: Based on slides by Tamassa Goodrich
|
|||
/* ... */
|
||||
}
|
||||
</code></pre>
|
||||
<p class="fragment">What's the complexity? <b class="fragment">$O(m) = O(n^2)$</b></p>
|
||||
<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>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>
|
||||
|
@ -677,11 +677,11 @@ attribution: Based on slides by Tamassa Goodrich
|
|||
<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>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>
|
||||
|
@ -692,13 +692,13 @@ attribution: Based on slides by Tamassa Goodrich
|
|||
<img src="graphics/15b/adjacency_matrix.png">
|
||||
</section>
|
||||
<section>
|
||||
<h4 class="slide_title">Adjacency List Summary</h4>
|
||||
<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>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>
|
||||
<tr>
|
||||
<td></td>
|
||||
<td class="fragment"><span style="border-top: solid 1px black; padding-top: 15px">$O(|V| + |E|)$</span></td>
|
||||
</tr>
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</table>
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style="fill:none;stroke:#a51d2d;stroke-width:1.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:1.5, 1.5;stroke-dashoffset:0;stroke-opacity:1"
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d="M -146.57492,103.91777 V 93.719796"
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id="path167981"
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sodipodi:nodetypes="cc" />
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<path
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style="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 -146.57492,90.688674 V 80.490669"
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id="path168078"
|
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sodipodi:nodetypes="cc" />
|
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<path
|
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style="fill:none;stroke:#52548a;stroke-width:0.365;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
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d="M -146.57492,67.261503 V 77.459521"
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id="path168335"
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sodipodi:nodetypes="cc" />
|
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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="-136.56303"
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y="46.050362"
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id="text170352"><tspan
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sodipodi:role="line"
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id="tspan170350"
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style="font-weight:bold;stroke-width:0.264583"
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x="-136.56303"
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y="46.050362">Unexplored</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="-136.56303"
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y="60.682026"
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id="text172656"><tspan
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sodipodi:role="line"
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id="tspan172654"
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style="font-weight:bold;stroke-width:0.264583"
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x="-136.56303"
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y="60.682026">Visited</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="-136.56303"
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y="73.917694"
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id="text173424"><tspan
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sodipodi:role="line"
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id="tspan173422"
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style="font-weight:bold;stroke-width:0.264583"
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x="-136.56303"
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y="73.917694">Unexplored</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="-136.56303"
|
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y="87.124802"
|
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id="text173764"><tspan
|
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sodipodi:role="line"
|
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id="tspan173762"
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style="font-weight:bold;stroke-width:0.264583"
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x="-136.56303"
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y="87.124802">Spanning</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="-136.56303"
|
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y="100.92304"
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id="text174138"><tspan
|
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sodipodi:role="line"
|
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id="tspan174136"
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style="font-weight:bold;stroke-width:0.264583"
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x="-136.56303"
|
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y="100.92304">Back</tspan></text>
|
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</g>
|
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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';text-decoration:underline;text-decoration-line:underline;letter-spacing:0px;word-spacing:0px;writing-mode:lr-tb;stroke-width:0.264583"
|
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x="-99.084877"
|
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y="129.5529"
|
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id="text186186"><tspan
|
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sodipodi:role="line"
|
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id="tspan186184"
|
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style="font-weight:bold;stroke-width:0.264583"
|
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x="-99.084877"
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y="129.5529">Call Stack</tspan></text>
|
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</g>
|
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<g
|
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inkscape:groupmode="layer"
|
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id="layer2"
|
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inkscape:label="Animation"
|
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style="display:inline"
|
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transform="translate(1.374123e-6,-9.2643834)">
|
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<text
|
||||
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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id="text191333"
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class="fragment"><tspan
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sodipodi:role="line"
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id="tspan191331"
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style="stroke-width:0.264583"
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x="52.286896"
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y="89.785408">DFS(G)</tspan></text>
|
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<g
|
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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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sodipodi:role="line"
|
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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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<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
|
||||
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>
|
||||
<text
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xml:space="preserve"
|
||||
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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</g>
|
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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 -81.028599,68.238522 -64.500232,52.532826"
|
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id="path168078-0"
|
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class="fragment"
|
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sodipodi:nodetypes="cc"
|
||||
transform="translate(153.12027,-28.683445)" />
|
||||
<g
|
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id="g239546"
|
||||
class="fragment">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
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"
|
||||
x="64.306305"
|
||||
y="47.164284"
|
||||
id="text195411"><tspan
|
||||
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>
|
||||
<text
|
||||
xml:space="preserve"
|
||||
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"
|
||||
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>
|
||||
</g>
|
||||
<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"
|
||||
d="m -81.028599,77.40366 16.528367,15.705685"
|
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id="path229112"
|
||||
class="fragment"
|
||||
sodipodi:nodetypes="cc"
|
||||
transform="translate(153.12027,-28.683445)" />
|
||||
<g
|
||||
id="g239554"
|
||||
class="fragment">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
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"
|
||||
x="89.177132"
|
||||
y="72.312668"
|
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id="text229110"><tspan
|
||||
sodipodi:role="line"
|
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id="tspan229108"
|
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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="89.177132"
|
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y="72.312668">☑</tspan></text>
|
||||
<text
|
||||
xml:space="preserve"
|
||||
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"
|
||||
x="52.286896"
|
||||
y="108.83542"
|
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id="text229493"><tspan
|
||||
sodipodi:role="line"
|
||||
id="tspan229491"
|
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style="stroke-width:0.264583"
|
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x="52.286896"
|
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y="108.83542">DFSOne(G, C)</tspan></text>
|
||||
<text
|
||||
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="108.81487"
|
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id="text230975"><tspan
|
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sodipodi:role="line"
|
||||
id="tspan230973"
|
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style="fill:#77767b;stroke-width:0.264583"
|
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x="92.876953"
|
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y="108.81487">(→ B, A, D, E)</tspan></text>
|
||||
</g>
|
||||
<path
|
||||
style="display:inline;fill:none;stroke:#a51d2d;stroke-width:1.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:1.5, 1.5;stroke-dashoffset:0;stroke-opacity:1"
|
||||
d="M 92.853371,62.83839 V 25.436881"
|
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id="path167981-4"
|
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sodipodi:nodetypes="cc"
|
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class="fragment" />
|
||||
<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"
|
||||
d="M -39.419644,77.543496 -55.948011,93.249195"
|
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id="path231575"
|
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class="fragment"
|
||||
sodipodi:nodetypes="cc"
|
||||
transform="translate(153.12027,-28.683445)" />
|
||||
<g
|
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id="g239562"
|
||||
class="fragment">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
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"
|
||||
x="52.286896"
|
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y="115.18542"
|
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id="text231868"><tspan
|
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sodipodi:role="line"
|
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id="tspan231866"
|
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style="stroke-width:0.264583"
|
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x="52.286896"
|
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y="115.18542">DFSOne(G, D)</tspan></text>
|
||||
<text
|
||||
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="115.16486"
|
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id="text231872"><tspan
|
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sodipodi:role="line"
|
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id="tspan231870"
|
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style="fill:#77767b;stroke-width:0.264583"
|
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x="92.876953"
|
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y="115.16486">(→ C, D)</tspan></text>
|
||||
<text
|
||||
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"
|
||||
x="114.41547"
|
||||
y="47.164284"
|
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id="text234421"><tspan
|
||||
sodipodi:role="line"
|
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id="tspan234419"
|
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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"
|
||||
x="114.41547"
|
||||
y="47.164284">☑</tspan></text>
|
||||
</g>
|
||||
<path
|
||||
style="display:inline;fill:none;stroke:#a51d2d;stroke-width:1.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:1.5, 1.5;stroke-dashoffset:0;stroke-opacity:1"
|
||||
d="M 113.56079,39.485139 97.032423,23.77946"
|
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id="path234204"
|
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sodipodi:nodetypes="cc"
|
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class="fragment" />
|
||||
<rect
|
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style="fill:#ffffff;fill-opacity:1;stroke:#ffffff;stroke-width:0.365001;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0"
|
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id="rect239586"
|
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width="69.220787"
|
||||
height="8.3245096"
|
||||
x="49.503799"
|
||||
y="110.01897"
|
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class="fragment" />
|
||||
<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 3.4126818,77.50376 -53.721683,97.902045"
|
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id="path234503"
|
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class="fragment"
|
||||
sodipodi:nodetypes="cc"
|
||||
transform="translate(153.12027,-28.683445)" />
|
||||
<g
|
||||
id="g239727"
|
||||
class="fragment">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
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"
|
||||
x="157.31093"
|
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y="47.164284"
|
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id="text234682"><tspan
|
||||
sodipodi:role="line"
|
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id="tspan234680"
|
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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="157.31093"
|
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y="47.164284">☑</tspan></text>
|
||||
<text
|
||||
xml:space="preserve"
|
||||
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"
|
||||
x="52.286896"
|
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y="115.18543"
|
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id="text234901"><tspan
|
||||
sodipodi:role="line"
|
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id="tspan234899"
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