Lecture 11 Intro to Graphs - UMass Amherst 241 F19 Lecture 11.pdf · • Adjacency matrix is good fit when #edges is large • Completely filled matrix would require to connect all

Lecture11IntrotoGraphs

ECE241– AdvancedProgrammingIFall2019MikeZink

ECE 241 – Adv. Programming I Fall 2019 © 2019 Mike Zink

Overview

1

• Introtographs• Abstractdatatype(ADT)• Adjacencymatrix• WordLetterPuzzle• BreadthFirstSearch


Objective

• Learnwhatagraphisandhowitisused• Abletoimplementthe graph abstractdatatypeusingmultipleinternalrepresentations

• Toseehowgraphscanbeused tosolveawidevarietyofproblems


Graphs• Moregeneralstructurethantrees

• Treeisaspecialtypeofgraph• Graphs canrepresent:

• Roads• Airline flights from city to city• How the Internet is connected

• Oncegoodrepresentation forproblem,graphalgorithmscanbeapplied


Graphs• Computer canunderstand roadmapasagraph• Computer canusegraphrepresentation todetermine:

• Shortestpath• Easiestpath• Quickestpath


Key: Prerequisite = solid arrow (not vertical) Alternative prereq = dashed arrow Co-requisite = solid vertical arrow Alternative co-req = dashed vertical arrow Required for graduation as CSE = solid outline CSE Senior elective = dashed outline

ECE 211 Circuit

Analysis I

ECE 212 Circuit

Analysis II

CS 121 Intro Problem Solving (Java)

for students who entered CoE before Fall 2016:

for all other students:

ECE 323 Electronics I

ECE 324 Electronics II

ECE 415 Senior Design

Project I

ECE 416 Senior Design

Project II

MATH 131 Calculus I

MATH 132 Calculus II

PHYS 151 Physics I

PHYS 152 Physics II

MATH 235 Linear

Algebra

CS 250 Intro to

Computation

ECE 303 Junior

Seminar

ECE 242 Data

Structures

ECE 232/331 Hardware

Organization

ENGLWRIT 112 College Writing

ENGIN 351 Writing in

Engineering

ENGIN 112 Intro. to ECE

BS-CSE Prerequisite Flowchart, part 1 BS-CSE

ECE 568 Computer

Architecture

ECE 597PD Parallel/Distrib.

Programming

19-Apr-18 1pm

ECE 575 Intro. Analog

IC Design

ECE 373 Software

Engineering

ECE 353 Comp. Systems

Lab I

ECE 597ED Electricity

Infrastructure

(Fall 2018) (Spring 2019)

ECE 354 Comp. Systems

Lab II

ECE 544 Trustworthy Computing

ECE 221/124 Digital

Systems

(Fall 2014 or before)

NOTE: Always check SPIRE for availability, as these offerings can change.

ECE 547 Security

Engineering

ECE 374 Networks & the Internet

ECE 122 Intro

Programming

ECE 597RE Reverse

Engineering

MATH 331 Differential Equations

Graphs


Graphs - Definitions• Vertex (node):

• Fundamental part of a graph. • Can have a name =>“key.” • Additional information as “payload.”

• Edge:• Connects two vertices• One-way or two-way• One-way => directed graph (as in example on last slide)


Graphs - Definitions• Weight:

• Edges may be weighted• Cost to go from on vertex to another• E.g., cost on edges of graph of roads might represent

distance between two cities


Formally Define Graph

8

• Graphcanberepresented byG,whereG=(V,E)• V isasetofvertices, andE isasetofedges• Eachedgeisasetoftupleswherew,v∈ V• Weightcanbeaddedasthirdcomponent oftuple• Asubgraph isasetofedgeseandverticesvsuchthate⊂ Eandv⊂ V


Graph Example

9

𝑉 = 𝑉0, 𝑉1,𝑉2, 𝑉3, 𝑉4,𝑉5

𝐸 = 𝑣0, 𝑣1, 5 , 𝑣1, 𝑣2, 0 , 𝑣2,𝑣3,9 , 𝑣3, 𝑣4,7 , 𝑣4, 𝑣0, 1𝑣0,𝑣5, 2 , 𝑣5, 𝑣4, 8 , 𝑣3, 𝑣5,3 , 𝑣5,𝑣2,1


Graphs - Definitions• Path:

• w1, w2, …, wn, such that (wi, wi+1) ∈ E for all 1 ≤ i ≤ n-1• Path length = number of edges in path => n-1

• Cycle:• Path that starts and ends at same vertex• (V5, V2, V3, V5) • Acyclic graph => no cycles• Directed acyclic graph (DAG) => directed graph with no

cycles


Graphs – Abstract Data Type• Graph()createsnewemptygraph• addVertex(vert) addsaninstanceofVertex tothe

graph• addEdge(fromVert, toVert) addsanew,directed

edgetothegraphthatconnectstwovertices• addEdge(fromVert, toVert, weight) addsa

new,weighted,directededgetothegraphthatconnectstwovertices


Graphs – Abstract Data Type• getVertex(vertKey)finds thevertexinthegraph

named vertKey• getVertices() returnsthelistofallverticesinthegraph• in returns True forastatementofthe

form vertex in graph, ifthegivenvertexisinthegraph, False otherwise


Graphs – Adjacency Matrix

V0 V1 V2 V3 V4 V5V0 5 2V1 4V2 9V3 7 3V4 1V5 1 8


Graphs – Adjacency Matrix• Forsmallgraphs:

• Easytoseewhichnodesareconnected• Butmatrixissparse (manyemptycells),notefficient

• Adjacencymatrixisgoodfitwhen#edges islarge• Completely filledmatrixwouldrequiretoconnectalledgeswitheachother->doesn’toccuroften


Adjacency List

15

• Masterlistofallvertices

• Eachvertexmaintainslistofotherverticesitsconnectedto

• Vertexclassusesdictionaryinsteadoflist

• Keysareverticesandvaluesareweights

• Benefits:• Compactlyrepresent sparsematrix

• Easilyfindlinks


Vertex Class• Each Vertex usesdictionarytokeeptrackoftheverticestowhichitisconnected, andtheweightofeachedge

• dictionary iscalled connectedTo• addNeighbor method is used to add a connection from

this vertex to another • getConnections method returns all of the vertices in the

adjacency list• getWeight method returns the weight of the edge


Vertex Class

17

class Vertex:def __init__(self,key):

self.id = key self.connectedTo = {}

def addNeighbor(self,nbr,weight=0):self.connectedTo[nbr] = weight

def __str__(self):return str(self.id) + ' connectedTo: ' + str([x.id for x in

self.connectedTo])

def getConnections(self):return self.connectedTo.keys()

def getId(self):return self.id

def getWeight(self,nbr):return self.connectedTo[nbr]


Graphs Class• Containsdictionarythatmapsvertexnamestovertexobjects

• Classalsoprovidesmethods foraddingandconnectingvertices

• getVertices returnsnamesofallvertices ingraph• __iter__ allowsiterationoverparticulargraph


Graph Class

19

class Graph:def __init__(self):

self.vertList = {} self.numVertices = 0

def addVertex(self,key):self.numVertices = self.numVertices + 1newVertex = Vertex(key)self.vertList[key] = newVertexreturn newVertex

def getVertex(self,n):if n in self.vertList:

return self.vertList[n] else: return None

def __contains__(self,n):return n in self.vertList

def addEdge(self,f,t,cost=0):if f not in self.vertList:

nv = self.addVertex(f) if t not in self.vertList:

nv = self.addVertex(t) self.vertList[f].addNeighbor(self.vertList[t], cost)

def getVertices(self):return self.vertList.keys()

def __iter__(self):return iter(self.vertList.values())


Graph Class - Operations

20

g = Graph() for i in range(6):

g.addVertex(i)g.vertListg.addEdge(0,1,5) g.addEdge(0,5,2) g.addEdge(1,2,4)g.addEdge(2,3,9)g.addEdge(3,4,7)g.addEdge(3,5,3) g.addEdge(4,0,1) g.addEdge(5,4,8) g.addEdge(5,2,1) for v in g:

for w in v.getConnections(): print("( %s , %s )" % (v.getId(), w.getId()))


Word Letter Puzzle

21

• Goal:Transformword“FOOL”into”SAGE”• Changeoneletteratatime• Ateachstep:transform onewordintoanother

FOOLPOOLPOLLPOLEPALESALESAGE


Word Letter Puzzle

22

• Cansolvethisproblemusingagraphalgorithm• Represent relationshipsbetweenwordsasgraph

• Usebreadthfirstsearchalgorithm• Findsefficientpathfromstartingworktoendingword


Word Letter Puzzle

23

• Firstproblem:howtoturnlargecollectionofwordsintoagraph

• Onlyconnectwordsthatdifferbysingleletter• Ifsuchgraphcanbecreated,anypathfromonewordtoanother isasolution


Word Letter Puzzle

24


Tackling the Problem

25

• Letsassume listofwords,allsamelength1. Startingpoint:Createavertexforeverywordinlist2. Compareallwordswitheachother3. Ifdifferentbyoneletter=>createedgebetween

them


Tackling the Problem

26

Analysis:• Assume listof5,110words• Comparingonewordtoeachotheris≈O(n2)• For5,110wordsthatismorethan26million

comparisons


Improved Approach

27

• Huge#ofbucket• Eachwith4-letterwordontop• Oneletteriswildcard“_”• Example:“POPE”and”POPS”match“POP_”

• Whenmatchingbucket isfound, addword• Onceallwords inrightbucket=>mustbeconnected ingraph


Building the Graph

28

from pythonds.graphs import Graph

def buildGraph(wordFile): d = {} g = Graph()

wfile = open(wordFile,'r')# create buckets of words that differ by one letterfor line in wfile:

word = line[:-1] for i in range(len(word)):

bucket = word[:i] + '_' + word[i+1:] if bucket in d: d[bucket].append(word) else: d[bucket] = [word]

# add vertices and edges for words in the same bucketfor bucket in d.keys():

for word1 in d[bucket]: for word2 in d[bucket]: if word1 != word2: g.addEdge(word1,word2)

return g


Sparsity of Matrix

29

Analysis:• 5,110four-letterwords• Adjacencymatrixwouldhave5,1102=26,112,100

cells• GraphcreatedbybuildGraph()has 53,286edges• =>Only.2%ofmatrixcellswouldbefilled!


Implementing Breadth First Search

30

• BreadthFirstSearch(BFS)isoneoftheeasiestalgorithmstosearchagraph

• GivenagraphGandstartingvertexsBFSexploresedges inthegraphtofindallverticesforwhichthereisapathfroms.

• Note:BFSfindsallverticesatdistancekfroms,beforeanyverticesatdistancek+1


Implementing Breadth First Search

31

• TovisualizeBFS,imaginethatitisbuildingonelevelatatime

• BFSaddsallchildrenofthestartingvertex• Thenitbeginstodiscoveranyofthegrandchildren• Tokeeptrackofprogressedgesarecoloredwhitegrayorblack:

• White:undiscoveredvertex• Gray:initiallydiscovered

• Black:vertexiscoloredblackwhencompletelyexpored


BFS Example

32


BFS Example

33

• Startingwithfool,addallnodesadjacenttoit

• Addedasnewnodestoexpand


BFS Example

34

• Removes ”pool” fromfrontofthequeue

• Repeatsprocess for“pool”• When“cool”isexaminedalg.detectsthatitisalreadygrey=>shorterpathtocoolalreadyexists

• “poll”isonlynewnodeadded


BFS Example

35

• Nextwordinqueue is”foil”• Onlynewnodethat“foil”canaddis“fail”

• Neitherofnexttwonodesaddanythingnewtoqueueortree

• Figureshowstreeafterexpandingallverticeson2ndlevel


BFS Example

36

• FinalBFStreeshown• WithBFStree,canstartatanyvertexandfollowpredecessorarrowsbackto

• Findshortestwordladderfromanywordinthetreebacktothestartingvertex


Building the Graph

37

def traverse(y): x = y

while (x.getPred()): print(x.getId()) x = x.getPred()

print(x.getId())

traverse(g.getVertex('sage'))

• Functiontraverse()shows how to follow the predecessor links to print out the word ladder


Next Steps

38

• NextlectureonTuesday: BreadthFirstSearch,DepthFirstSearch

• HW1dueontonight10/17at11PM

ECE 241 – Data Structures Fall 2018 © 2018 Mike Zink39

Documents

Lecture 11 Intro to Graphs - UMass Amherst 241 F19 Lecture 11.pdf · • Adjacency matrix is good fit when #edges is large • Completely filled matrix would require to connect all