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CSE 326: Data Structures: Graphs

Lecture 19: Monday, Feb 24, 2003

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Outline

• Graphs:– Definitions– Properties– Examples

• Topological Sort

• Graph Traversals– Depth First Search– Breadth First Search

TO DO: READ WEISS CH 9

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GraphsGraphs = formalism for representing relationships

between objects• A graph G = (V, E)

– V is a set of vertices– E V V is a set of edges

Han

Leia

Luke

V = {Han, Leia, Luke}E = {(Luke, Leia), (Han, Leia), (Leia, Han)}

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Directed vs. Undirected Graphs

Han

Leia

Luke

Han

Leia

Luke

• In directed graphs, edges have a specific direction:

• In undirected graphs, they don’t (edges are two-way):

• Vertices u and v are adjacent if (u, v) E

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Weighted Graphs

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30

35

60

Mukilteo

Edmonds

Seattle

Bremerton

Bainbridge

Kingston

Clinton

There may be more information in the graph as well.

Each edge has an associated weight or cost.

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Paths and CyclesA path is a list of vertices {v1, v2, …, vn} such

that (vi, vi+1) E for all 0 i < n.A cycle is a path that begins and ends at the same

node.

Seattle

San FranciscoDallas

Chicago

Salt Lake City

p = {Seattle, Salt Lake City, Chicago, Dallas, San Francisco, Seattle}

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Path Length and Cost

Path length: the number of edges in the pathPath cost: the sum of the costs of each edge

Seattle

San FranciscoDallas

Chicago

Salt Lake City

3.5

2 2

2.5

3

22.5

2.5

length(p) = 5 cost(p) = 11.5

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ConnectivityUndirected graphs are connected if there is a

path between any two vertices

Directed graphs are strongly connected if there is a path from any one vertex to any other

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ConnectivityDirected graphs are weakly connected if there

is a path between any two vertices, ignoring direction

A complete graph has an edge between every pair of vertices

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Connectivity

A (strongly) connected component is a subgraph which is (strongly) connected

CC in an undirected graph:

SCC in a directed graph:

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Distance and Diameter

• The distance between two nodes, d(u,v), is the length of the shortest paths, or if there is no path

• The diameter of a graph is the largest distance between any two nodes

• Graph is strongly connected iff diameter <

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An Interesting Graph

• The Web !

• Each page is a vertice• Each hyper-link is an edge

• How many nodes ?• Is it connected ?• What is its diameter ?

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The Web Graph

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Other Graphs

• Geography:– Cities and roads– Airports and flights (diameter 20 !!)

• Publications:– The co-authorship graph

• E.g. the Erdos distance

– The reference graph

• Phone calls: who calls whom• Almost everything can be modeled as a graph !

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Graph Representation 1: Adjacency Matrix

A |V| x |V| array in which an element (u, v) is true if and only if there is an edge from u to vHan

Leia

Luke

Han Luke Leia

Han

Luke

LeiaRuntime:iterate over verticesiterate ever edgesiterate edges adj. to vertexedge exists?

Space requirements:

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Graph Representation 2: Adjacency List

A |V|-ary list (array) in which each entry stores a list (linked list) of all adjacent verticesHan

Leia

LukeHan

Luke

Leia

space requirements:

Runtime:iterate over verticesiterate ever edgesiterate edges adj. to vertexedge exists?

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Graph Density

A sparse graph has O(|V|) edges

A dense graph has (|V|2) edges

Anything in between is either sparsish or densy depending on the context.

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Trees as Graphs

• Every tree is a graph with some restrictions:– the tree is directed– there are no cycles

(directed or undirected)– there is a directed path

from the root to every node

A

B

D E

C

F

HG

JI

BAD!

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Directed Acyclic Graphs (DAGs)

DAGs are directed graphs with no cycles.

main()

add()

access()

mult()

read()

Trees DAGs Graphs

if program call graph is a DAG, then all procedure calls can be in-lined

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Application of DAGs: Representing Partial Orders

check inairport

calltaxi

taxi toairport

reserveflight

packbagstake

flight

locategate

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Topological Sort

Given a graph, G = (V, E), output all the vertices in V such that no vertex is output before any other vertex with an edge to it.

check inairport

calltaxi

taxi toairport

reserveflight

packbags

takeflight

locategate

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Topological Sort

check inairport

calltaxi

taxi toairport

reserveflight

packbags

takeflight

locategate

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Topo-Sort Take One

Label each vertex’s in-degree (# of inbound edges)While there are vertices remaining

Pick a vertex with in-degree of zero and output itReduce the in-degree of all vertices adjacent to itRemove it from the list of vertices

Label each vertex’s in-degree (# of inbound edges)While there are vertices remaining

Pick a vertex with in-degree of zero and output itReduce the in-degree of all vertices adjacent to itRemove it from the list of vertices

runtime:

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Topo-Sort Take Two

Label each vertex’s in-degreeInitialize a queue to contain all in-degree zero vertices

While there are vertices remaining in the queueRemove a vertex v with in-degree of zero and output itReduce the in-degree of all vertices adjacent to vPut any of these with new in-degree zero on the queue

Label each vertex’s in-degreeInitialize a queue to contain all in-degree zero vertices

While there are vertices remaining in the queueRemove a vertex v with in-degree of zero and output itReduce the in-degree of all vertices adjacent to vPut any of these with new in-degree zero on the queue

runtime:Can we use a stack instead of a queue ?

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Topo-Sort

• So we sort again in linear time: O(|V|+|E|)

• Can we apply this to normal sorting ?If we sort an array A[1], A[2], ..., A[n] using topo-sort, what is the running time ?

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Recall: Tree Traversals

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k l

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a b f g k c d h i l j e

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Depth-First Search• Pre/Post/In – order traversals are examples of

depth-first search– Nodes are visited deeply on the left-most branches

before any nodes are visited on the right-most branches• Visiting the right branches deeply before the left would still be

depth-first! Crucial idea is “go deep first!”

• Difference in pre/post/in-order is how some computation (e.g. printing) is done at current node relative to the recursive calls

• In DFS the nodes “being worked on” are kept on a stack

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DFS

void DFS(Node startNode) {Stack s = new Stack;

s.push(startNode);while (!s.empty()) {

x = s.pop();/* process x */for y in x.children() do

s.push(y);}

void DFS(Node startNode) {Stack s = new Stack;

s.push(startNode);while (!s.empty()) {

x = s.pop();/* process x */for y in x.children() do

s.push(y);}

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DFS on Trees

a

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k l

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k

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DFS on Graphs

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h j

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k l

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a

b

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k

Problem with cycles: may run into an infinite loop

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DFS

Running time:

void DFS(Node startNode) {for v in Nodes do

v.visited = false;Stack s = new Stack;

s.push(startNode);while (!s.empty()) {

x = s.pop();x.visited = true;/* process x */for y in x.children() do

if (!y.visited) s.push(y);}

void DFS(Node startNode) {for v in Nodes do

v.visited = false;Stack s = new Stack;

s.push(startNode);while (!s.empty()) {

x = s.pop();x.visited = true;/* process x */for y in x.children() do

if (!y.visited) s.push(y);}

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Breadth-First Search• Traversing tree level by level from top to bottom

(alphabetic order)a

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k l

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Breadth-First Search

BFS characteristics

• Nodes being worked on maintained in a FIFO Queue, not a stack

• Iterative style procedures often easier to design than recursive procedures

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Breadth-First Searchvoid DFS(Node startNode) {

for v in Nodes dov.visited = false;

Stack s = new Stack;

s.push(startNode);while (!s.empty()) {

x = s.pop();x.visited = true;/* process x */for y in x.children() do

if (!y.visited) s.push(y);}

void DFS(Node startNode) {for v in Nodes do

v.visited = false;Stack s = new Stack;

s.push(startNode);while (!s.empty()) {

x = s.pop();x.visited = true;/* process x */for y in x.children() do

if (!y.visited) s.push(y);}

void BFS(Node startNode) {for v in Nodes do

v.visited = false;Queue s = new Queue;

s.enqueue(startNode);while (!s.empty()) {

x = s.dequeue();x.visited = true;/* process x */for y in x.children() do

if (!y.visited) s.enqueue(y);}

void BFS(Node startNode) {for v in Nodes do

v.visited = false;Queue s = new Queue;

s.enqueue(startNode);while (!s.empty()) {

x = s.dequeue();x.visited = true;/* process x */for y in x.children() do

if (!y.visited) s.enqueue(y);}

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QUEUE

ab c d ec d e f gd e f ge f g h i jf g h i jg h i jh i j ki j kj k lk ll

a

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h j

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k l

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Graph TraversalsDFS and BFS

• Either can be used to determine connectivity:– Is there a path between two given vertices?– Is the graph (weakly) connected?

• Important difference: Breadth-first search always finds a shortest path from the start vertex to any other (for unweighted graphs)– Depth first search may not!