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CS 575

Design and Analysis of

Computer Algorithms

Professor Michal Cutler

Graphs and traversals

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Graph Algorithms

Graphs and Theorems aboutGraphs

Graph implementations

Graph traversals

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What can graphs model?

Cost of wiring electronic components together.

Shortest route between two cities.

Finding the shortest distance between all pairs of cities in aroad atlas.

Flow of material (liquid flowing through pipes, current throughelectrical networks, information through communicationnetworks, parts through an assembly line, etc).

State of a machine.

Used in Operating systems to model resource handling(deadlock problems).

Used in compilers for parsing and optimizing the code.

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What is a Graph?

Informally a graphis a set of nodesjoined by a set of lines or arrows.

1 12 3

4 45 56 6

2 3

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A directed graph, also called a digraphG is a pair ( V, E),where the set Vis a finite set and Eis a binary relation on V.

The set Vis called the vertex setof G and the elements are called

vertices.The set Eis called the edge setof Gand the elements are edges(alsocalled arcs).

A edge from node ato node bis denoted by the ordered pair ( a, b).

1 2 3

4 5 6

V= { 1, 2, 3, 4, 5, 6, 7 }

| V| = 7

E= { (1,2), (2,2), (2,4), (4,5), (4,1), (5,4),(6,3) }| E| = 7

Self loop

7 Isolated node

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Undirected graphG= ( V, E) , the edge set Econsistof unordered pairs. Our text uses the notation

( a,b) to refer to a directed edge, and

{ a, b} for an undirected edge.

A

D E F

B C V= { A, B, C, D, E, F }

|V| = 6

E= { {A, B}, {A,E}, {B,E}, {C,F} }|E| = 4

Some texts use (a, b) also for undirected edges.

So ( a, b) and ( b, a) refers to the same edge.

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Degreeof a Vertex in an undirected graph is the number of edges incidenton it. In a directed graph , the out degreeof a vertex is the number ofedges leaving it and the in degreeis the number of edges entering it.

A

D E F

B CThe degree of B is 2.

12

4 5

The in degree of 2 is 2 andthe out degree of 2 is 3.

Self-loop

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Simple GraphsSimple graphsare graphs without multiple

(directed in same direction or undirected)edges connecting a single pair of vertices, orself-loops. We will consider only simple

graphs.

Proposition: If G is an undirected graph then

deg(v) = 2 |E|Proposition: If G is a digraph then indeg(v) = outdeg(v) = |E|

vVvV

vV

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A weighted graphis a graph for which each edge has an associatedweight, usually given by a weight functionw: ER.

12 3

4 5 6

.5

1.2

.2

.5

1.5.3

1

4 5 6

2 32

135

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Paths

A pathis a sequence of vertices such thatthere is an edge from each vertex to its

successor. A path is simpleif each vertex is distinct.

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Cyclic and Acyclic

A path from a vertex to itself is called acycle

(e.g., v1 v2v4v1)

If a graph contains a cycle, it iscyclic

Otherwise, it isacyclic

A path is simpleif it never passes through thesame vertex twice.

1 2

4 3

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s-t connectivity problem. Given two nodes s and t, isthere a path between s and t?

s-t shortest path problem. Given two nodes s and t,

what is the length of the shortest path between s andt?

Applications. Friendster.

Maze traversal.

Kevin Bacon number.

Fewest number of hops in a communication network.

s-t Connectivity

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Connectivity

An undirected graph is connectedif any twonodes are connected by a path.

A directed graph is strongly connectedifthere is a directed path from any node to anyother node.

A graph is sparseif | E| | V|

A graph is denseif | E| | V|2

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Connected Component

Connected component. Find all nodesreachable from s.

Connected component containing node1 = { 1, 2, 3, 4, 5, 6, 7, 8 }.

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Flood Fill Flood fill. Given lime green pixel in an image, change color of entire blob of neighboring

lime pixels to blue.

Node: pixel.

Edge: two neighboring lime pixels.

Blob: connected component of lime pixels.

recolor lime green blob to blue

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Flood Fill Flood fill. Given lime green pixel in an image, change color of entire

blob of neighboring lime pixels to blue.

Node: pixel.

Edge: two neighboring lime pixels.

Blob: connected component of lime pixels.recolor lime green blob to blue

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A Complete graphis an undirected/directedgraph in which every pair of vertices is adjacent.If (u, v) is an edge in a graph G, we say that

vertex vis adjacentto vertex u.

A

D

E

B

4 nodes and (4*3)/2 edges

V nodes and V*(V-1)/2 edges

Note: if self loops are allowed

V*V/2

D

A

B

3 nodes and 3*2 edges

V nodes and V*(V-1) edges

Note: if self loops are allowed V2

edges

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A bipartite graphis an undirected graph G= (V,E) inwhich Vcan be partitioned into 2 sets V1 and V2 such

that ( u,v) Eimplies either uV1 and vV2 OR vV1and uV2.

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Trees

A free treeis an acyclic, connected, undirectedgraph.

A forest is an acyclic undirected graph.

A rooted treeis a tree with one distinguished node,root.

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Tree Definitions

Let G= (V, E) be an undirected graph

G is a (free) tree if:.1. Gis acyclic and connected

2. Any two vertices in Gare connected by uniquesimple path.3. Gis connected, but if any edge is removed from E,the resulting graph is disconnected.4. Gis connected, and | E| = | V| -1

5. Gis acyclic, and | E| = | V| -1G is a tree or a forest if:

Gis undirected acyclic, but if any edge is added to E,the resulting graph contains a cycle.

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Implementation of a Graph. Adjacency-list representation of a graph G =( V, E)

consists of an array ADJ of |V| lists, one for eachvertex in V. For each uV, ADJ[ u] points to all itsadjacent vertices.

1

5

1

22

5

4 4

3 3

2 5

1 5 3 4

2 4

2

4

5

1

3

2

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Adjacency-list representation

for a directed graph.

1

5

1

22

5

4 4

3 3

2 5

5 3 4

4

5

5

Variation: Can keep a second list of edges coming into a vertex.

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Adjacency lists

Advantage: Saves space for sparse graphs. Most graphs are

sparse.

Visit edges that start at v Must traverse linked list of v

Size of linked list of v is degree(v)

(degree(v)) in the worst case

Disadvantage: Check for existence of an edge (v, u)

Must traverse linked list of v

Size of linked list of v is degree(v)

(degree(v)) in the worst case

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Adjacency List

Storage We need V pointers to linked lists

For a directed graph the number of nodes (oredges) contained (referenced) in all the linkedlists is

(out-degree (v)) = | E |.

So we need ( V + E )

For an undirected graph the number of nodes is(degree (v)) = 2 | E |

Also ( V + E )

vV

vV

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Adjacency-matrix-representationof a graph G= (V, E) is a |V| x |V |matrix A such that aij = 1 if (i, j )Eand 0otherwise.

0

4

1

3

2

0 1 2 3 4

012

34

0 1 0 0 1

1 0 1 1 1

0 1 0 1 0

0 1 1 0 1

1 1 0 1 0

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Adjacency Matrix Representation for

a Directed Graph

0 1 0 0 1

0 0 1 1 1

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

0 1 2 3 4

01234

0

4

1

3

2

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Adjacency Matrix

Representation Advantage: Saves space on pointers for dense graphs

Check for existence of an edge (v, u)

(adjacency [i] [j]) == 1?) So (1)

Disadvantage:

visit all the edges that start at v Row v of the matrix must be traversed. So (|V|).

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Graph traversals

Breadth first search

Depth first search

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Some applications

Is G connected?

Does G contain a cycle?

Is G a tree?

Is G bipartite?

Find connected components

Topological sorting

Is directed G strongly connected?

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Given a graph G=(V,E) and a sourcevertexs, BFS explores the edges of G

to discover (visit) each node of Greachable from s.

Idea - expand a frontierone step at a

time. Frontieris a FIFO queue (O(1) time to

update)

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Computes the shortestdistance(dist) from sto any reachable node.

Computes a breadthfirsttree(of parents)with rootsthat contains all the reachablevertices from s.

To get O(|V|+|E|) we use an adjacency list

representation. If we used an adjacencymatrix it would be O(|V|2)

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Coloring the nodes

We use colors (white, grayand black)to denote the state of the node during

the search. A node is whiteif it has not been

reached (discovered).

Discoverednodes are grayor black.Graynodes are at the frontier of thesearch. Blacknodes are fully explorednodes.

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BFS - initialize

procedureBFS(G, s, color, dist, parent);

foreach vertex u do

color[u]=white; dist[u]=(V) parent[u]=-1

color[s]=gray; dist[s]=0;

init(Q); enqueue(Q, s);

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BFS - mainwhilenot (empty(Q)) do

u:=head(Q);

foreach v in adj[u] do

if (color[v]= =white) then O(E)color[v]=gray; dist[v]=dist[u]+1;

parent[v]=u; enqueue(Q, v);

dequeue(Q); color[u]=black;

endBFS

)(][deg|][|1

][

EOureeuADJ

VuVu

reachableu

Vu uADJv

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Analysis of BFS

Initialization is (|V|).

Each node can be added to the queue atmost once (it needs to be white), and its

adjacency list is searched only once. At mostall adjacency lists are searched.

If graph is undirected each edge is reachedtwice, so loop repeated at most 2|E| times.

If graph is directed each edge is reachedexactly once. So the loop repeated at most|E| times.

Worst case time O(|V|+|E|)

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BFS example

0

1

1 0

1

r s t u r s t u

v w x y v w x yr s t u r s t u

v w x y v w x y

0s w r

1 0

1

1 2

2

r t x t x v

2

22

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BFS example

2

1

1 0 2

1 2

1 2

1 2

1

2

0 2 3

1 2

r s t u r s t u

v w x y v w x y

r s t u r s t u

v w x y v w x y

1 3

2 2

0.x v u v u y

0 3

2 3

3

32

u y y

3

Now y is removed from the Q and colored red

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Depth First Search

Goal - explore every vertex and edge ofG

We go deeper whenever possible.

Directedor undirectedgraph G= (V, E).

To get (|V|+|E|) we use an adjacency

list representation. If we used anadjacency matrix it would be (|V|2)

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Depth First Search

Until there are no more undiscovered nodes.

Picks an undiscovered node and starts a depthfirst search from it.

The search proceeds from the mostrecentlydiscoverednode to discover new nodes.

When the last discovered node vis fully explored,backtracks to the node used to discover v.Eventually, the start node is fully explored.

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Depth First Search

In this version allnodes are discovered evenif the graph is directed, or undirected and notconnected

The algorithm saves: A depth first forestof the edges used to

discover new nodes.

Timestamps for the first time a node uisdiscovered d[u] and the time when thenode is fully explored f[u]

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DFS

DFS(G, color, d, f, parent);

foreach vertex u do

color[u]=white; parent[u]=-1;

(V) time=0;foreach vertex u do

if (color[u]==white) then

DFS-Visit(u)endDFS

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DFS-Visit(u)color[u]=gray; time=time+1; d[u]=time

foreach v in adj[u] do

if (color[v]==white) thenparent[v]=u; DFS-Visit(v);

color[u]=black; time=time+1;

f[u]=time;endDFS-Visit

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Analysis

DFS is (|V|) (excluding the time takenby the DFS-Visits).

DFS-Visit is called once for each nodev. Its forloop is executed |adj(v)| times.The DFS-Visit calls for all the adjacent

nodes take (|E|). Worst case time (|V|+|E|)

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DFS example (1)

x y z

u v w

1/

u v w1/

x y z

2/

u v w1/

x y z

2/

3/

u v w1/

x y z

2/

3/4/

B

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DFS example (2)

u v w

x y z

x y z

u v w

x y z4/5

1/ 2/

3/

B

u v w

4/5 3/6

1/ 2/

B

4/5 3/6

1/ 2/7

B

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DFS example (3)

u v w

x y z

u v w

x y z x y z

u v w

x y z

u v w

F

4/5 3/6

1/8 2/7

B F

4/5

9

3/6

1/8 2/7

B

F

4/5

9

3/6 10

1/8 2/7

BF

4/5

9

3/6 10/11

1/8 2/7

B

C

CC

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DFS example (4)

x y z

F

4/5

9/12

3/6 10/11

1/8 2/7

B C

u v w

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Some applications

Is undirected G connected? ChangeDFS to call dfsVisit(v) only once, and

then to check if there are still whitenodes.

O(V + E)

Find connected components. CallDFS. The nodes discovered in each callto dfsVisit(v) belong to a singlecomponent. (V+E)

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Labeling the edges

(digraph) Tree edges - those belonging to the forest

Back edges - edges from a node to anancestor in the tree.

Forward edges - a non tree edge from anode to a descendant in the tree.

Cross edges - the rest of the edges,

between trees and subtrees When a graph is undirected its edges are

tree or back edges for DFS, treeorcross

for BFS

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Classifying edges of a

digraph (u, v) is:

Tree edge if v is white

Back edge if v is gray Forward or cross - if v is black

(u, v) is:

Forward edge if v is black and d[u] < d[v] (v was

discovered after u) Cross edge if v is black and d[u] > d[v] (u

discovered after v)

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More applications

Does directed G contain a directed cycle? Do DFS ifback edges yes. Time O(V+E).

Does undirected G contain a cycle? Same as

directed but be careful not to consider (u,v) and (v, u)a cycle.

If graph is a tree or a forest it can have at most |V|-1edges. Time O(V) since encounter at most |V| edges(if (u, v) and (v, u) are counted as one edge), before

cycle is found. Is undirected G a tree? DFS with one call to

dfsVisit(v). If all vertices are reached and no backedges G is a tree. O(V)

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Directed Acyclic Graphs

Def. A topological order of a directed graph G = (V,E) is an ordering of its nodes as v1, v2, , vn so thatfor every edge (vi, vj) we have i < j.

a DAG a topological ordering

v2 v3

v6 v5 v4

v7 v1

v1 v2 v3 v4 v5 v6 v7

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Topological sort

Applications.

Course prerequisite graph: course vi

must be taken before vj. Compilation: module vimust be

compiled before vj.

Pipeline of computing jobs: output ofjob vineeded to determine input of jobvj.

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Topological sort algorithm

Given a DAG G

Topological sort is a linear ordering of all

the vertices of G such that if G contains thedirected edge (u, v) u appears before v inthe ordering

TOPOLOGICAL-SORT(G)

1. Apply DFS(G)2. As each vertex is finished insert it at the front of a

list

3. return the list

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Second algorithm

Lemma. If G is a DAG, then G has a source node.

Pf. (by contradiction)

Suppose that G is a DAG and every node has at least one incoming edge.

Pick any node v, and begin following edges backward from v. Since v has

at least one incoming edge (u, v) we can walk backward to u. Then, since u has at least one incoming edge (x, u), we can walk backward

to x.

Repeat until we visit a node, say w, twice.

Let C denote the sequence of nodes encountered between successive visitsto w. C is a cycle.

w x u v

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Second algorithm

Lemma. If G is a DAG, then G has a topological ordering.

Pf. (by induction on n)

Base case: true if n = 1.

Given DAG on n > 1 nodes, find a node v with no incoming edges.

G - { v } is a DAG, since deleting v cannot create cycles. By inductive hypothesis, G - { v } has a topological ordering.

Place v first in topological ordering; then append nodes of G - { v }in topological order.

This is valid since v has no incoming edges.

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Topological Sorting Algorithm:

Running Time Theorem. Algorithm finds a topological order in O(m

+ n) time.

Pf.

Maintain the following information: count[w] = remaining number of incoming edges

S = set of remaining nodes with no incoming edges

Initialization: O(m + n) via single scan through graph.

Update: to delete v

remove v from S decrement count[w] for all edges from v to w, and add w to S if

c count[w] hits 0

this is O(1) per edge

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Strong Connectivity:

Algorithm Theorem. Can determine if G is strongly connected in O(m + n) time.

Pf.

Pick any node s.

Run BFS from s in G.

Run BFS from s in Grev.

Return true iff all nodes reached in both BFS executions.

reverse orientation of every edge in G

strongly connected not strongly connected

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