1. Based on slides of Dan Suciu
Graphs
Based on slides of Dan Suciu

2. Read: textbook, Chapter 22
Outline
Graphs:
Definitions
Properties
Examples
Topological Sort
Graph Traversals
Depth First Search
Breadth First Search
Read: textbook, Chapter 22

3. Graphs
Graphs = formalism for representing relationships between objects
A graph G = (V, E)
V is a set of vertices, |V|= n
E  V  V is a set of edges, | E|= m
Han
Leia
Luke
V = {Han, Leia, Luke}
E = {(Luke, Leia),
(Han, Leia),
(Leia, Han)}

4. Directed vs. Undirected Graphs
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
Han
Luke
Leia
Han
Luke
Leia

5. Weighted Graphs Each edge has an associated weight or cost. Clinton20
Mukilteo
Kingston 30
Edmonds
How could we store weights in a
matrix?
List?
Bainbridge 35
Seattle 60
Bremerton
There may be more
information in the graph as well.

6. Paths and Cycles
A 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.
Chicago
Seattle
Salt Lake City
San Francisco
Dallas
p = {Seattle, Salt Lake City, Chicago, Dallas, San Francisco, Seattle}

7. Path Length and Cost Path length: the number of edges in the path
Path cost: the sum of the costs of each edge
3.5
Chicago
Seattle 2 2
Salt Lake City 2
2.5
2.5
2.5 3
San Francisco
Dallas
length(p) = 5
cost(p) = 11.5

8. Connectivity
Undirected 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
Can a directed graph which is strongly connected be acyclic?
NO. (Except the trivial one node case.) There must be a path from A to B and a path from B to A; so, there must be a cycle.
What about a weakly connected directed graph?
YES! See the one on the slide.
There are also further definitions of these; for example, the concept of biconnectivity showed up in my satisfiability research:
Does the graph have two distinct paths between any two vertices.

9. Connectivity
Directed 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
Can a directed graph which is strongly connected be acyclic?
NO. (Except the trivial one node case.) There must be a path from A to B and a path from B to A; so, there must be a cycle.
What about a weakly connected directed graph?
YES! See the one on the slide.
There are also further definitions of these; for example, the concept of biconnectivity showed up in my satisfiability research:
Does the graph have two distinct paths between any two vertices.

10. Connectivity
A (strongly) connected component is a subgraph which is (strongly) connected
CC in an undirected graph:
SCC in a directed graph:

11. 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 < 

12. 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 ?

13. The Web Graph

14. Other Graphs Geography: Publications: Phone calls: who calls whom
Cities and roads
Airports and flights (diameter  20 !!)
Publications:
The co-authorship graph
The reference graph
Phone calls: who calls whom
Almost everything can be modeled as a graph !

15. Implementing a Graph Implement a graph in three ways: Adjacency List
Adjacency-Matrix
Pointers/memory for each node (actually a form of adjacency list)

16. Adjacency List
List of pointers for each vertex

17. Undirected Adjacency List

18. Adjacency List
The sum of the lengths of the adjacency lists is 2|E| in an undirected graph, and |E| in a directed graph.
The amount of memory to store the array for the adjacency list is O(max(V,E))=O(V+E).

19. Adjacency Matrix

20. Undirected Graph - Adjacency Matrix

21. Adjacency Matrix vs. List?
The matrix always uses Θ(v2) memory. Usually easier to implement and perform lookup than an adjacency list.
The adjacency matrix is a good way to represent a weighted graph. In a weighted graph, the edges have weights associated with them. Update matrix entry to contain the weight. Weights could indicate distance, cost, etc.

22. 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.
Graph density is a measure of the number of edges.
Remember that we need to analyze graphs in terms of both |E| and |V|.
If we know the density of the graph, we can dispense with that.

23. Trees as Graphs
Every tree is a graph(can be directed or undirected) with some restrictions:
there are no cycles (directed or undirected)
there is a directed path from the root to every node A B C D E F
Each of the red areas breaks one of these constraints.
We won’t always require the constraint that the tree be directed.
In fact, if it’s not directed, we can just pick a root and hang everything from that node (making the edges directed); so, it’s not a big deal. G H

24. Trees as Graphs
Every tree is a graph(can be directed or undirected) with some restrictions:
there are no cycles (directed or undirected)
there is a path from the root to every node A B C D E F
Each of the red areas breaks one of these constraints.
We won’t always require the constraint that the tree be directed.
In fact, if it’s not directed, we can just pick a root and hang everything from that node (making the edges directed); so, it’s not a big deal. G H
BAD! I J

25. Directed Acyclic Graphs (DAGs)
DAGs are directed graphs with no cycles.
main()
mult()
if program call graph is a DAG, then all procedure calls can be in-lined
add()
DAGs are a very common representation for dependence graphs.
The DAG here shows the non-recursive call-graph from a program.
Remember topological sort?
Any DAG can be topo-sorted.
Any graph that’s not a DAG cannot be topo-sorted.
read()
access()
Trees  DAGs  Graphs

26. Application of DAGs: Representing Partial Orders
reserve
flight
check in
airport
call
taxi
pack
bags
take
flight
Not all orderings are total orderings, however.
Now, everyone tell me where there should be an edge!
locate
gate
taxi to
airport

27. 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.
reserve
flight
check in
airport
call
taxi
take
flight
taxi to
airport
locate
gate
pack
bags

28. Topological Sort call taxi pack bags taxi to airport reserve flight
check in
airport
locate
gate
take
flight

29. Topo-Sort Take One runtime: O(|V|2)
Label each vertex’s in-degree (# of inbound edges)
While there are vertices remaining
Pick a vertex with in-degree of zero and output it
Reduce the in-degree of all vertices adjacent to it
Remove it from the list of vertices
That’s a bit less efficient than we can do.
Let’s try this.
How well does this run?
runtime: O(|V|2)

30. Can we use a stack instead of a queue ?
Topo-Sort Take Two
Label each vertex’s in-degree
Initialize a queue to contain all in-degree zero vertices
While there are vertices remaining in the queue
Remove a vertex v with in-degree of zero and output it
Reduce the in-degree of all vertices adjacent to v
Put any of these with new in-degree zero on the queue
Why use a queue?
runtime: O(|V| + |E|)
Can we use a stack instead of a queue ?

31. Topo-Sort
Topo-Sort takes linear time: O(|V|+|E|)

32. Recall: Tree Traversalsb c d e h i j f g k l
a b f g k c d h i l j e

33. 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!”
In DFS the nodes “being worked on” are kept on a stack

34. DFS(not quite right) } 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); }

35. DFS on Trees a b f g k a b c d e h i j f g k l

36. DFS on Graphs a b f g a b c d e h i j f g k l
Problem with cycles: may run into an infinite loop

37. DFS- for connected graphs
for v in Nodes do
v.visited = false;
DFS(startNode);
void DFS(Node v) {
Stack s = new Stack;
s.push(v);
while (!s.empty()) {
x = s.pop();
if (!x.visited) {
x.visited = true;
process( x ) }
for y in x.children() do
if (!y.visited) s.push(y);
Running time:
O(|V| + |E|)
Key observation: each node enters the stack at most its indegree times

38. DFS- for general graphs
for v in V do
v.visited = false;
if (!v.visited) DFS(v);
Running time:

39. Breadth-First Search
Traversing tree level by level from top to bottom (alphabetic order) a i d h j b f k l e c g

40. 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

41. Breadth-First Search- for connected graphs
for v in Nodes do
v.visited = false;
DFS(startNode);
void DFS(Node v) {
Stack s = new Stack;
s.push(v);
while (!s.empty()) {
x = s.pop();
if (!x.visited)
x.visited = true;
process (x)
for y in x.children() do
if (!y.visited) s.push(y); }
for v in Nodes do
v.visited = false;
BFS(startNode);
void BFS(Node v) {
Queue s = new Queue;
s.enqueue(startNode);
while (!s.empty()) {
x = s.dequeue();
if (!x.visited)
x.visited = true;
process (x)
for y in x.children() do
if (!y.visited) s.enqueue(y); }

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

43. Graph Traversals DFS 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!