Algorithm Design and Analysis (ADA)

242-535 ADA: 8. Intro. Graphs

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• Objectiveo introduce the main kinds of graphs, discuss

two implementation approaches, and remind you about trees

Algorithm Design and Analysis (ADA)

242-535, Semester 1 2016-2017

8. Introduction

to Graphs


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1. Graphs2. Graph Terminology3. Implementing Graphs

- adjency matrix- adjency list

4. Trees and Forests5. Tree Terminology

Overview

1. Graphs• A graph has two parts (V, E), where:

o V are the nodes, called verticeso E are the links between vertices, called edges

• Example:o airports and distance between them

ORD PVD

MIADFW

SFO

LAX

LGA

HNL

849

802

13871743

1843

109911201233

3372555

142


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• Directed grapho the edges are directedo e.g., bus cost network

• Undirected grapho the edges are undirectedo e.g., road network

Graph Types


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Example Nodes EdgesTransportation network: airline routes

airports nonstop flights

Communication networks

computers, hubs, routers

physical wires

Information network: web

pages hyperlinks

Information network: scientific papers

articles references

Social networks people “u is v’s friend”, “u sends email to v”, “u’s FaceBook links to v”

Computer programs functions (or modules)statement blocks

“u calls v”

“v can follow u”

Graphs are everywhere


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• A calling graph for a program:

A Calling Graph

main

printList

mergeSort

mergesplit

makeList

4 examples ofrecursion


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• Problem: minimise the moving time of the drill over a metal sheet.

Sheet Metal Hole Drilling

continued


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• Add edge numbers (weights) for the movement time between any two holes.

A Weighted Graph

129

4

4

35

6

86

a

d

e

c

b

2


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• End vertices (or endpoints) of an edgeo U and V are the endpoints

• Edges incident on a vertexo a, d, and b are incident

• Adjacent verticeso U and V are adjacent

• Degree of a vertexo X has degree 5

• Parallel edgeso h and i are parallel edges

• Self-loopo j is a self-loop

2. Graph Terminology

XU

V

W

Z

Y

a

c

b

ed

fg

h

i

j


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• Patho sequence of alternating

vertices and edges o begins with a vertexo ends with a vertexo each edge is preceded and

followed by its endpoints

• Simple patho path such that all its vertices

and edges are distinct

• Exampleso P1=(V,b,X,h,Z) is a simple

patho P2=(U,c,W,e,X,g,Y,f,W,d,V)

is a path that is not simple

P1

XU

V

W

Z

Y

a

c

b

e

d

fg

hP2


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• Cycleo circular sequence of

alternating vertices and edges

o each edge is preceded and followed by its endpoints

• Simple cycleo cycle such that all its vertices

and edges are distinct

• Exampleso C1=(V,b,X,g,Y,f,W,c,U,a) is

a simple cycleo C2=(U,c,W,e,X,g,Y,f,W,d,V,a

,) is a cycle that is not simpleGraphs 11

C1

XU

V

W

Z

Y

a

c

b

e

d

fg

hC2

Connectivity• A graph is connected if

there is a path between every pair of vertices

Connected graph

Non connected graph with two connected components

Strong Connectivity• Each vertex can reach all other

vertices

13

a

d

c

b

e

f

g


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• All pairs of vertices are connected by an edge.• No. of edges |E| = |V| (|V-1|)/2 = O(|V|2)

Complete Graphs


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• Directed Acyclic Grapho no cycle

• Eulerian Grapho must visit each edge once

• Bipartiteo 2 sets, no edges within set!

Some Common Graphs


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• A DAG has no cycles• Some algorithms become simpler when used

on DAGs instead of general graphs, based on the principle of topological orderingo find shortest paths and longest paths by

processing the vertices in a topological order

Directed Acyclic Graph (DAG)


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• An Euler graph has either an Euler path or Euler tour.

• An Euler path is defined as a path in a graph which visits each edge exactly once.

• An Euler tour/cycle is an Euler path which starts and ends on the same vertex.

Euler Graph


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• A graph whose vertices can be divided into two disjoint sets U and Vo every edge connects a vertex in U to one in V

Bipartite graph (bigraph)


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• Search/traversalo visit every node once (Euler)o visit every edge once (Hamiltonian)

• Shortest paths• Minimal Spanning Trees (MSTs)• Network flow

• Matching• Graph coloring• Travelling salesman problem

Graph-related Problems


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• We will typically express running times in terms of |E| and |V| (often dropping the |’s)o If |E| |V|2 the graph is dense• can also write this as |E| is O(|v2|)

o If |E| |V| the graph is sparse• or |E| is O(|V|)

• Dense and sparse graphs are best implemented using two different data structures:o Adjacency matricies: for dense graphso Adjacency lists: for sparse graphs

3. Implementing Graphs


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• In the most dense graph, a graph of v verticies will have |V|(|V|-1)/2 edges.

• In that case, for large n, |E| is O(|V|2)

Dense Big-Oh

|V| = 5|E| = (5*4)/2 = 10


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3.1. Adjacency Matrix

a b

c

ed

Graph

0 1 0 0 1a b c d e

1 0 1 0 10 1 1 0 10 0 0 0 11 1 1 1 0

abc

de

Adjacency Matrix


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• An adjacency matrix represents the graph as a V * V matrix A:o A[i, j] = 1 if edge (i, j) E

= 0 if edge (i, j) E

• The degree of a vertex v (of a simple graph) = sum of row v or sum of column vo e.g. vertex a has degree 2 since it is connected to b

and e

• An adjacency matrix can represent loopso e.g. vertex c on the previous slide

Properties

continued


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• An adjacency matrix can represent parallel edges if non-negative integers are allowed as matrix entrieso ijth entry = no. of edges between vertex i and j

• The matrix duplicates information around the main diagonalo the size can be easily reduced with some coding

tricks

• Properties of graphs can be obtained using matrix operationso e.g. the no. of paths of a given length, and vertex

degree


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• If an adjacency matrix A is multiplied by itself repeatedly:o A, A2, A3, ..., An

Then the ijth entry in matrix An is equal to the number of paths from i to j of length n.

The No. of Paths of Length n


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Example

a b

c

ed

0 1 0 1 0a b c d e

1 0 1 0 10 1 0 1 11 0 1 0 00 1 1 0 0

abc

de

A =


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0 1 0 1 01 0 1 0 10 1 0 1 11 0 1 0 00 1 1 0 0

A2 =

0 1 0 1 01 0 1 0 10 1 0 1 11 0 1 0 00 1 1 0 0

=

2 0 2 0 1

a b c d e

0 3 1 2 12 1 3 0 10 2 0 2 11 1 1 1 2

a

bc

d

e


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• Consider row a, column c in A2:

Why it Works...

( 0 1 0 1 0 )a

b d

c

0

1

01

1

b

d= 0*0 + 1*1 + 0*0 + 1*1 + 0*1= 2

continued

a-b-c a-d-c


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• A non-zero product means there is at least one vertex connecting verticies a and c.

• The sum is 2 because of:o (a, b, c) and (a, d, c)o 2 paths of length two


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• The entries on the main diagonal of A2 give the degrees of the verticies (when A is a simple graph).

• Consider vertex c:o degree of c == 3 since it is connected to the edges

(c,b), (c,d), and (c,e).

The Degree of Verticies

continued


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• In A2 these become paths of length 2:o (c,b,c), (c,d,c), and (c,e,c)

• So the number of paths of length 2 for c = the degree of co this is true for all verticies


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• #define NUMNODES nint arcs[NUMNODES][NUMNODES];

• arcs[u][v] == 1 if there is an edge (u,v); 0 otherwise

• Storage used: O(|V|2)

• The implementation may also need a way to map node names (strings) to array indicies.

Coding Adjacency Matricies

continued


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• If n is large then the array will be very large, with almost half of it being unnecessary.

• If the nodes are lightly connected then most of the array will contain 0’s, which is a further waste of memory.


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• A directed graph:

Representing Directed Graphs

0 1

32

4


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• Not symmetric; all the array may be necessary.

• Still a waste of space if nodes are lightly connected.

Its Adjacency Matrix

1 1 1 0 00 0 0 1 01 1 0 0 10 0 1 0 10 1 0 0 0

00 1 2 3 4

1234

finish

star

t


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• The adjacency matrix is an efficient way to store dense graphs.

• But most large interesting graphs are sparseo e.g., planar graphs, in which no edges cross, have

|e| = O(|v|) by Euler’s formula

o For this reason the adjacency list is often a better respresentation than the adjacency matrix

When to use an Adjacency Matrix


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• Adjacency list: for each vertex v V, store a list of vertices adjacent to v

• Example:o adj[0] = {0, 1, 2}o adj[1] = {3}o adj[2] = {0, 1, 4}o adj[3] = {2, 4}o adj[4] = {1}

• Can be used for directed and undirected graphs.

3.2. Adjacency List

0 1

32

4


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• An implementation diagram:

0 1 2

3

0 1 4

2 4

1

0

1

2

3

4

adj[]

meansNULL

size of array= no. ofvertices (|V|) no. of cells

== no. of edges (|E|)


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• struct cell { /* for a linked list */ Node nodeName; struct cell *next;};

struct cell *adj[NUMNODES];

• adj[u] points to a linked list of cells which give the names of the nodes connected to u.

Data Structures


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• How much storage is required?o The degree of a vertex v == number of incident edges

• directed graphs have in-degree, out-degree values

• For directed graphs, the number of items in an adjacency lists is out-degree(v) = |E|

•This uses (V + E) storage

Storage Needs


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• For undirected graphs, the number of items in the adjency list is

degree(v) = 2*|E| (the handshaking lemma)

o Why? If we mark every edge connected to every vertex, then by the end, every edge will be marked twice

• This also uses (V + E) storage

• In summary, adjacency lists use (V+E) storage


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• Which representation is better for graphs?• The simple answer:

• dense graph – use a matrix• sparse graph – use an adjcency list

• But a more accurate answer depends on the operations that will be applied to the graph.

• We will consider three operations:o is there an edge between u and v?o find the successors of u (in a directed graph)o find the predecessors of u (in a directed graph)

3.3. Running Time: Matrix or List?

continued


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• Adjacency matrix: O(1) to read arcs[u][v]

• Adjacency list: O(1 + E/V) // forget the |...|o O(1) to get to adj[u]o length of linked list is on average E/V

o if a sparse graph (E<<V): O(1+ E/V) => O(1)o if a dense graph (E ≈ V2): O(1+ E/V) => O(V)

Is there an edge (u,v)?


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• Adjacency matrix: O(V) since must examine the entire row for vertex u

• Adjacency list: O(1 + (E/V)) since must look at entire list pointed to by adj[u]o if a sparse graph (E<<V): O(1+ E/V) => O(1)o if a dense graph (E ≈ V2): O(1+ E/V) => O(V)

Find u’s successors (u->v)


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• Adjacency matrix: O(V) since must examine the entire column for vertex uo a 1 in the row for ‘t’ means that ‘t’ is a predecessor

• Adjacency list: O(E) since must examine every list pointed to by adj[]o if a sparse graph (E<<V): O(E) is fasto if a dense graph (E ≈ V2): O(E) is slow

Find u’s predecessors (t->u)


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• Operation Dense Graph Sparse GraphFind edge Adj. Matrix EitherFind succ. Either Adj. listFind pred. Adj. Matrix Either

• As a graph gets denser, an adjacency matrix has better execution time than an adjacency list.

Summary: which is faster?


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• The size of an adjacency matrix for a graph of V nodes is:o V2 bits (assuming 0 and 1 are stored as bits)

3.4. Storage Space: Matrix or List?

continued


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• An adjacency list cell uses:o 32 bits for the integer, 32 bits for the pointero so, cell size = 64 bits

• Total no. of cells = total no. of edges, eo so, total size of lists = 64*E bits

• successors[] has V entries (for V verticies)o so, array size is 32*V bits

• Total size of an adjacency list data struct:64*E + 32*V


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• An adjacency list will use less storage than an adjacency matrix when:

64*E + 32*V < V2

which is: E < V2/64 – V/2

When V is large, ignore the V/2 term:E < V2/64

Size Comparison

continued


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• V2 is (roughly) the maximum number of edges.

• So if the actual number of edges in a graph is 1/64 of the maximum number of edges, then an adj. list representation will be smaller than an adj. matrix codingo but the graph must be quite sparse


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• A (free) tree is an undirected graph T such thato T is connectedo T has no cyclesThis definition of tree is different

from the one of a rooted tree

• A forest is an undirected graph without cycles

• The connected components of a forest are trees

4. Trees and Forests

Graphs 51

Tree

Forest


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Uses of TreesPresident

Vice-Presidentfor Academics

Vice-Presidentfor Admin.

Dean ofEngineering

Dean ofBusiness

Head of CoE Head of EE Head of AC.

PlanningOfficer

PurchasesOfficer

. . . .. . . .

. . . .


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• Non-rooted (free) treeso a free tree is a graph with no cycles

Saturated HydrocarbonsH

HH

H

C

HH C

HH C

HH CH C

H

H HH C

H

HC

H

H

C

H

Butane

Isobutane


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A Computer File System/

usr bin tmp

bin ad spool ls mail who junk

ed vi exs opr uucp

printer


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• e.g. Part of the ancient Greek god family:

5. (Rooted) Tree Terminology

Uranus

Aphrodite Kronos Atlas Prometheus

Eros Zeus Poseidon Hades Ares

Apollo Athena Hermes Heracles

levels0

1

2

3::


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• Let T be a tree with root v0.• Suppose that x, y, z are verticies in T.• (v0, v1,..., vn) is a simple path in T (no loops).

• a) vn-1 is the parent of vn.• b) v0, ..., vn-1 are ancestors of vn

• c) vn is a child of vn-1

Some Definitions

continued


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• d) If x is an ancestor of y, then y is a descendant of x.

• e) If x and y are children of z, then x and y are siblings.

• f) If x has no children, then x is a terminal vertex (or a leaf).

• g) If x is not a terminal vertex, then x is an internal (or branch) vertex.

continued


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• h) The subtree of T rooted at x is the graph with vertex set V and edge set Eo V contains x and all the descendents of xo E = {e | e is an edge on a simple path from x to some

vertex in V}

• i) The length of a path is the number of edges it uses, not verticies.

continued


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• j) The level of a vertex x is the length of the simple path from the root to x.

• k) The height of a vertex x is the length of the simple path from x to the farthest leafo the height of a tree is the height of its root

• l) A tree where every internal vertex has exactly m children is called a full m-ary tree.


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• The root is Uranus.• A simple path is {Uranus, Aphrodite, Eros}

• The parent of Eros is Aphrodite.• The ancestors of Hermes are Zeus, Kronos, and

Uranus.• The children of Zeus are Apollo, Athena, Hermes,

and Heracles.

Applied to the Example

continued


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• The descendants of Kronos are Zeus, Poseidon, Hades, Ares, Apollo, Athena, Hermes, and Heracles.

• The leaves (terminal verticies) are Eros, Apollo, Athena, Hermes, Heracles, Poseidon, Hades, Ares, Atlas, and Prometheus.

• The branches (internal verticies) are Uranus, Aphrodite, Kronos, and Zeus.

continued


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• The subtree rooted at Kronos:

Kronos

Zeus Poseidon Hades Ares

Apollo Athena Hermes Heracles

continued


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• The length of the path {Uranus, Aphrodite, Eros} is 2 (not 3).

• The level of Ares is 2.

• The height of the tree is 3.

Algorithm Design and Analysis (ADA)

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