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Chapter 6 Graphs

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Chapter 6Graphs

- Graphs (§6.1)
- Definition
- Applications
- Terminology
- Properties
- ADT

- Data structures for graphs (§6.2)
- Edge list structure
- Adjacency list structure
- Adjacency matrix structure

“To find the way out of a labyrinth there is only one means. At every new junction, never seen before, the path we have taken will be marked with three signs. If … you see that the junction has already been visited, you will make only one mark on the path you have taken. If all the apertures have already been marked, then you must retrace your steps. But if one or two apertures of the junction are still without signs, you will choose any one, making two signs on it. Proceeding through an aperture that bears only one sign, you will make two more, so that now the aperture bears three. All the parts of the labyrinth must have been visited if, arriving at a junction, you never take a passage with three signs, unless none of the other passages is now without signs.”

--Ancient Arabic text

- How do you find your way out of a maze, given a large supply of pennies?

- Graph traversals
- Depth-first search
- Breadth-first search

- Other applications: Boggle™, tic-tac-toe, path finding, theorem proving, motion planning, AI, …

A graph is a pair (V, E), where

V is a set of nodes, called vertices

E is a collection of pairs of vertices, called edges

Vertices and edges are positions and store elements

Example:

A vertex represents an airport and stores the three-letter airport code

An edge represents a flight route between two airports and stores the mileage of the route

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- Directed edge
- ordered pair of vertices (u,v)
- first vertex u is the origin
- second vertex v is the destination
- e.g., a flight

- Undirected edge
- unordered pair of vertices (u,v)
- e.g., a flight route

- Directed graph
- all the edges are directed
- e.g., route network

- Undirected graph
- all the edges are undirected
- e.g., flight network

flight

AA 1206

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849

miles

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- Electronic circuits
- Printed circuit board
- Integrated circuit

- Transportation networks
- Highway network
- Flight network

- Computer networks
- Local area network
- Internet
- Web

- Databases
- Entity-relationship diagram

V

a

b

h

j

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- End vertices (or endpoints) of an edge
- U and V are the endpoints of a

- Edges incident on a vertex
- a, d, and b are incident on V

- Adjacent vertices
- U and V are adjacent

- Degree of a vertex
- X has degree 5

- Parallel edges
- h and i are parallel edges

- Self-loop
- j is a self-loop

- Path
- sequence of alternating vertices and edges
- begins with a vertex
- ends with a vertex
- each edge is preceded and followed by its endpoints

- Simple path
- path such that all its vertices and edges are distinct

- Examples
- P1=(V,b,X,h,Z) is a simple path
- P2=(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple

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- Cycle
- circular sequence of alternating vertices and edges
- each edge is preceded and followed by its endpoints

- Simple cycle
- cycle such that all its vertices and edges are distinct

- Examples
- C1=(V,b,X,g,Y,f,W,c,U,a,) is a simple cycle
- C2=(U,c,W,e,X,g,Y,f,W,d,V,a,) is a cycle that is not simple

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Notation

nnumber of vertices

mnumber of edges

deg(v)degree of vertex v

Property 1

Sv deg(v)= 2m

Proof: each edge is counted twice

Property 2

In an undirected graph with no self-loops and no multiple edges

m n (n -1)/2

Proof: each vertex has degree at most (n -1)

What is the bound for a directed graph?

Example

- n = 4
- m = 6
- deg(v)= 3

Two different drawings of the same graph are shown.

- What data structure to represent this graph?

A B C D E F G H I J K L M

A 1 1 0 0 1 1 0 0 0 0 0 0

B 0 0 0 0 0 0 0 0 0 0 0

C 0 0 0 0 0 0 0 0 0 0

D 1 1 0 0 0 0 0 0 0

E 1 1 0 0 0 0 0 0

F 0 0 0 0 0 0 0

G 0 0 0 0 0 0

H 1 0 0 0 0

I 0 0 0 0

J 1 1 1

K 0 0

L 1

M

Space required for a graph

with v vertices, e edges?

Q(v2)

Time to tell if there is an

edge from v1 to v2?

Q(1)

A: F -> B -> C -> G

B: A

C: A

D: E -> F

E: D -> F -> G

F: D -> E

G: A -> E

H: I

I: H

J: K -> L -> M

K: J

L: M -> J

M: J -> L

Space required for a graph with v vertices, e edges?

Q(v+e)

Time to tell if there is an edge from v1 to v2?

Q(v)

Vertices and edges

are positions

store elements

Accessor methods

aVertex()

incidentEdges(v)

endVertices(e)

isDirected(e)

origin(e)

destination(e)

opposite(v, e)

areAdjacent(v, w)

Update methods

insertVertex(o)

insertEdge(v, w, o)

insertDirectedEdge(v, w, o)

removeVertex(v)

removeEdge(e)

Generic methods

numVertices()

numEdges()

vertices()

edges()

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Depth-First Search

- Definitions (§6.1)
- Subgraph
- Connectivity
- Spanning trees and forests

- Depth-first search (§6.3.1)
- Algorithm
- Example
- Properties
- Analysis

- Applications of DFS (§6.5)
- Path finding
- Cycle finding

Subgraph

Spanning subgraph

- A subgraph S of a graph G is a graph such that
- The vertices of S are a subset of the vertices of G
- The edges of S are a subset of the edges of G

- A spanning subgraph of G is a subgraph that contains all the vertices of G

A graph is connected if there is a path between every pair of vertices

A connected component of a graph G is a maximal connected subgraph of G

Connected graph

Non connected graph with two connected components

- A (free) tree is an undirected graph T such that
- T is connected
- T has no cycles
This 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

Tree

Forest

- A spanning tree of a connected graph is a spanning subgraph that is a tree
- A spanning tree is not unique unless the graph is a tree
- Spanning trees have applications to the design of communication networks
- A spanning forest of a graph is a spanning subgraph that is a forest

Graph

Spanning tree

Main idea: keep traveling to a

new, unvisited node until you you get stuck.

Then backtrack as far as necessary and try a new path.

Depth-first search (DFS) is a general technique for traversing a graph

A DFS traversal of a graph G

Visits all the vertices and edges of G

Determines whether G is connected

Computes the connected components of G

Computes a spanning forest of G

DFS on a graph with n vertices and m edges takes O(n + m ) time

DFS can be further extended to solve other graph problems

Find and report a path between two given vertices

Find a cycle in the graph

Depth-first search is to graphs what Euler tour is to binary trees

- The algorithm uses a mechanism for setting and getting “labels” of vertices and edges

AlgorithmDFS(G, v)

Inputgraph G and a start vertex v of G

Outputlabeling of the edges of Gin the connected component of vas discovery edges and back edges

setLabel(v, VISITED)

for all e G.incidentEdges(v)

ifgetLabel(e) = UNEXPLORED

w opposite(v,e)

if getLabel(w) = UNEXPLORED

setLabel(e, DISCOVERY)

DFS(G, w)

else

setLabel(e, BACK)

AlgorithmDFS(G)

Inputgraph G

Outputlabeling of the edges of Gas discovery edges andback edges

for all u G.vertices()

setLabel(u, UNEXPLORED)

for all e G.edges()

setLabel(e, UNEXPLORED)

for all v G.vertices()

ifgetLabel(v) = UNEXPLORED

DFS(G, v)

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unexplored vertex

A

visited vertex

A

unexplored edge

discovery edge

backedge

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- How can you implement DFS non-recursively?
- Use a stack
- For unconnected graphs, restart at unvisited nodes
- Runtime?
- O(n+m)

AlgorithmDFS(G, v)

Inputgraph G and a start vertex v of G

Outputtraversal of the vertices of Gin the connected component of vStack S

S.push(v)

while S is not empty

v S.pop()

setLabel(v, VISITED)

for each edge (v,w) out of v

if w is UNEXPLORED,

S.push(w)

- The DFS algorithm is similar to a classic strategy for exploring a maze
- We mark each intersection, corner and dead end (vertex) visited
- We mark each corridor (edge) traversed
- We keep track of the path back to the entrance (start vertex) by means of a rope (recursion stack)

Property 1

DFS(G, v) visits all the vertices and edges in the connected component of v

Property 2

The discovery edges labeled by DFS(G, v) form a spanning tree of the connected component of v

A

B

D

E

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Setting/getting a vertex/edge label takes O(1) time

Each vertex is labeled twice

once as UNEXPLORED

once as VISITED

Each edge is labeled twice

once as UNEXPLORED

once as DISCOVERY or BACK

Method incidentEdges is called once for each vertex

DFS runs in O(n + m) time provided the graph is represented by the adjacency list structure

Recall that Sv deg(v)= 2m

- How can you find a set of plane flights to get to your destination from your airport?
- How can you tell if you can fly to all destinations from your airport?
- How can you find a maximal set of plane flights that can be cancelled while still making it possible to get from any city to any other?
- If computers have multiple network connections, how can you prevent routing storms?

- How to find a path between vertices u and z?
- Call DFS(G, u) with u as the start vertex
- We use a stack S to keep track of the path between the start vertex and the current vertex
- As soon as destination vertex z is encountered, we return the path as the contents of the stack

AlgorithmpathDFS(G, v, z)

setLabel(v, VISITED)

S.push(v)

if v= z

return S.elements()

for all e G.incidentEdges(v)

ifgetLabel(e) = UNEXPLORED

w opposite(v,e)

if getLabel(w) = UNEXPLORED

setLabel(e, DISCOVERY)

S.push(e)

pathDFS(G, w, z)

S.pop(e)

else

setLabel(e, BACK)

S.pop(v)

- How to find a cycle in a graph?
- We use a stack S to keep track of the path between the start vertex and the current vertex
- When a back edge (v, w) is encountered, return the cycle as the portion of the stack from the top to w

AlgorithmcycleDFS(G, v, z)

setLabel(v, VISITED)

S.push(v)

for all e G.incidentEdges(v)

ifgetLabel(e) = UNEXPLORED

w opposite(v,e)

S.push(e)

if getLabel(w) = UNEXPLORED

setLabel(e, DISCOVERY)

pathDFS(G, w, z)

S.pop(e)

else

T new empty stack

repeat

o S.pop()

T.push(o)

until o= w

return T.elements()

S.pop(v)

L0

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Breadth-First Search

- Breadth-first search (§6.3.3)
- Algorithm
- Example
- Properties
- Analysis
- Applications

- DFS vs. BFS (§6.3.3)
- Comparison of applications
- Comparison of edge labels

Breadth-first search (BFS) is a general technique for traversing a graph

A BFS traversal of a graph G

Visits all the vertices and edges of G

Determines whether G is connected

Computes the connected components of G

Computes a spanning forest of G

BFS on a graph with n vertices and m edges takes O(n + m ) time

BFS can be further extended to solve other graph problems

Find and report a path with the minimum number of edges between two given vertices

Find a simple cycle, if there is one

- The algorithm uses a mechanism for setting and getting “labels” of vertices and edges

AlgorithmBFS(G, s)

L0new empty sequence

L0.insertLast(s)

setLabel(s, VISITED)

i 0

while Li.isEmpty()

Li +1new empty sequence

for all v Li.elements() for all e G.incidentEdges(v)

ifgetLabel(e) = UNEXPLORED

w opposite(v,e)

if getLabel(w) = UNEXPLORED

setLabel(e, DISCOVERY)

setLabel(w, VISITED)

Li +1.insertLast(w)

else

setLabel(e, CROSS)

i i +1

AlgorithmBFS(G)

Inputgraph G

Outputlabeling of the edges and partition of the vertices of G

for all u G.vertices()

setLabel(u, UNEXPLORED)

for all e G.edges()

setLabel(e, UNEXPLORED)

for all v G.vertices()

ifgetLabel(v) = UNEXPLORED

BFS(G, v)

L0

A

L1

B

C

D

E

F

unexplored vertex

A

visited vertex

A

unexplored edge

discovery edge

crossedge

L0

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A

A

L1

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B

C

D

B

C

D

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F

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Notation

Gs: connected component of s

Property 1

BFS(G, s) visits all the vertices and edges of Gs

Property 2

The discovery edges labeled by BFS(G, s) form a spanning tree Ts of Gs

Property 3

For each vertex v in Li

The path of Ts from s to v has i edges

Every path from s to v in Gshas at least i edges

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Setting/getting a vertex/edge label takes O(1) time

Each vertex is labeled twice

once as UNEXPLORED

once as VISITED

Each edge is labeled twice

once as UNEXPLORED

once as DISCOVERY or CROSS

Each vertex is inserted once into a sequence Li

Method incidentEdges is called once for each vertex

BFS runs in O(n + m) time provided the graph is represented by the adjacency list structure

Recall that Sv deg(v)= 2m

- Using the template method pattern, we can specialize the BFS traversal of a graph Gto solve the following problems in O(n + m) time
- Compute the connected components of G
- Compute a spanning forest of G
- Find a simple cycle in G, or report that G is a forest
- Given two vertices of G, find a path in G between them with the minimum number of edges, or report that no such path exists

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DFS

BFS

Back edge(v,w)

w is an ancestor of v in the tree of discovery edges

Cross edge(v,w)

w is in the same level as v or in the next level in the tree of discovery edges

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DFS

BFS

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Directed Graphs

- Reachability (§6.4.1)
- Directed DFS
- Strong connectivity

- Transitive closure (§6.4.2)
- The Floyd-Warshall Algorithm

- Directed Acyclic Graphs (DAG’s) (§6.4.4)
- Topological Sorting

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- A digraph is a graph whose edges are all directed
- Short for “directed graph”

- Applications
- one-way streets
- flights
- task scheduling

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- A graph G=(V,E) such that
- Each edge goes in one direction:
- Edge (a,b) goes from a to b, but not b to a.

- Each edge goes in one direction:
- If G is simple, m < n*(n-1).
- If we keep in-edges and out-edges in separate adjacency lists, we can perform listing of in-edges and out-edges in time proportional to their size.

- Scheduling: edge (a,b) means task a must be completed before b can be started

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The good life

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- We can specialize the traversal algorithms (DFS and BFS) to digraphs by traversing edges only along their direction
- In the directed DFS algorithm, we have four types of edges
- discovery edges
- back edges
- forward edges
- cross edges

- A directed DFS starting at a vertex s determines the vertices reachable from s

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- DFS tree rooted at v: vertices reachable from v via directed paths

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- Each vertex can reach all other vertices

- Pick a vertex v in G.
- Perform a DFS from v in G.
- If there’s a w not visited, print “no”.

- Let G’ be G with edges reversed.
- Perform a DFS from v in G’.
- If there’s a w not visited, print “no”.
- Else, print “yes”.

- Running time: O(n+m).

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- Maximal subgraphs such that each vertex can reach all other vertices in the subgraph
- Can also be done in O(n+m) time using DFS, but is more complicated (similar to biconnectivity).

{ a , c , g }

{ f , d , e , b }

- Given a digraph G, the transitive closure of G is the digraph G* such that
- G* has the same vertices as G
- if G has a directed path from u to v (u v), G* has a directed edge from u to v

- The transitive closure provides reachability information about a digraph

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G*

If there's a way to get from A to B and from B to C, then there's a way to get from A to C.

- We can perform DFS starting at each vertex
- O(n(n+m))

Alternatively ... Use dynamic programming: The Floyd-Warshall Algorithm

- Idea #1: Number the vertices 1, 2, …, n.
- Idea #2: Consider paths that use only vertices numbered 1, 2, …, k, as intermediate vertices:

Uses only vertices numbered 1,…,k

(add this edge if it’s not already in)

i

j

Uses only vertices

numbered 1,…,k-1

Uses only vertices

numbered 1,…,k-1

k

- Floyd-Warshall’s algorithm numbers the vertices of G as v1 , …, vn and computes a series of digraphs G0, …, Gn
- G0=G
- Gkhas a directed edge (vi, vj) if G has a directed path from vi to vjwith intermediate vertices in the set {v1 , …, vk}

- We have that Gn = G*
- In phase k, digraph Gk is computed from Gk -1
- Running time: O(n3), assuming areAdjacent is O(1) (e.g., adjacency matrix)

AlgorithmFloydWarshall(G)

Inputdigraph G

Outputtransitive closure G* of G

i 1

for all v G.vertices()

denote v as vi

i i+1

G0G

for k 1 to n do

GkGk -1

for i 1 to n (i k)do

for j 1 to n (j i, k)do

if Gk -1.areAdjacent(vi, vk)

Gk -1.areAdjacent(vk, vj)

if Gk.areAdjacent(vi, vj)

Gk.insertDirectedEdge(vi, vj , k)

return Gn

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- A directed acyclic graph (DAG) is a digraph that has no directed cycles
- A topological ordering of a digraph is a numbering
v1 , …, vn

of the vertices such that for every edge (vi , vj), we have i < j

- Example: in a task scheduling digraph, a topological ordering a task sequence that satisfies the precedence constraints
Theorem

A digraph admits a topological ordering if and only if it is a DAG

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DAG G

v4

v5

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Topological ordering of G

A

- Number vertices, so that (u,v) in E implies u < v

1

A typical student day

wake up

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eat

study computer sci.

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nap

more c.s.

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play

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write c.s. program

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work out

make brownies for professors

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sleep

dream about graphs

- Note: This algorithm is different from the one in Goodrich-Tamassia
- Running time: O(n + m). How…?

MethodTopologicalSort(G)

H G// Temporary copy of G

n G.numVertices()

whileH is not emptydo

Let v be a vertex with no outgoing edges

Label v n

n n - 1

Remove v from H

- Simulate the algorithm by using depth-first search
- O(n+m) time.

AlgorithmtopologicalDFS(G, v)

Inputgraph G and a start vertex v of G

Outputlabeling of the vertices of Gin the connected component of v

setLabel(v, VISITED)

for all e G.incidentEdges(v)

ifgetLabel(e) = UNEXPLORED

w opposite(v,e)

if getLabel(w) = UNEXPLORED

setLabel(e, DISCOVERY)

topologicalDFS(G, w)

else

{e is a forward or cross edge}

Label v with topological number n

n n - 1

AlgorithmtopologicalDFS(G)

Inputdag G

Outputtopological ordering of Gn G.numVertices()

for all u G.vertices()

setLabel(u, UNEXPLORED)

for all e G.edges()

setLabel(e, UNEXPLORED)

for all v G.vertices()

ifgetLabel(v) = UNEXPLORED

topologicalDFS(G, v)

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