cse 326: data structures: graphs
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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
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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
a
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h j
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k l
ec
g
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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h j
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k l
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a
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k
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DFS on Graphs
a
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h j
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k l
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a
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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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h j
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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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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!
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