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1
Graphs
ORD
DFW
SFO
LAX
802
1743
1843
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Many slides taken from Goodrich, Tamassia 2004
2© 2004 Goodrich, Tamassia
GraphsA graph is a pair (V, E), where
V is a set of nodes, called vertices (node=vertex) 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
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
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3© 2004 Goodrich, Tamassia
Edge TypesDirected 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
ORD PVDflight
AA 1206
ORD PVD849
miles
4© 2004 Goodrich, Tamassia
John
DavidPaul
brown.edu
cox.net
cs.brown.edu
att.netqwest.net
math.brown.edu
cslab1bcslab1a
ApplicationsElectronic circuits
Printed circuit board Integrated circuit
Transportation networks Highway network Flight network
Computer networks Local area network Internet Web
Databases Entity-relationship
diagram
5© 2004 Goodrich, Tamassia
TerminologyEnd 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
VAdjacent vertices
U and V are adjacentDegree of a vertex
X has degree 5 Parallel edges
h and i are parallel edgesSelf-loop
j is a self-loop
XU
V
W
Z
Y
a
c
b
e
d
f
g
h
i
j
6© 2004 Goodrich, Tamassia
P1
Terminology (cont.)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
XU
V
W
Z
Y
a
c
b
e
d
f
g
hP2
7© 2004 Goodrich, Tamassia
Terminology (cont.)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
C1
XU
V
W
Z
Y
a
c
b
e
d
f
g
hC2
8© 2004 Goodrich, Tamassia
PropertiesNotation
n number of vertices m number of edgesdeg(v) degree of vertex v
Property 1
v deg(v) 2m
Proof: each edge is counted twice
Property 2In an undirected graph
with no self-loops and no multiple edges
m n (n 1)2Proof: each vertex has
degree at most (n 1)
What is the bound for a directed graph?
Example n 4 m 6 deg(v) 3
9© 2004 Goodrich, Tamassia
Main Methods of the Graph ADT
Vertices and edges are positions store elements
Accessor methods endVertices(e): an array
of the two endvertices of e
opposite(v, e): the vertex opposite of v on e
areAdjacent(v, w): true iff v and w are adjacent
replace(v, x): replace element at vertex v with x
replace(e, x): replace element at edge e with x
Update methods insertVertex(o): insert a
vertex storing element o insertEdge(v, w, o):
insert an edge (v,w) storing element o
removeVertex(v): remove vertex v (and its incident edges)
removeEdge(e): remove edge e
Iterator methods incidentEdges(v): edges
incident to v vertices(): all vertices in
the graph edges(): all edges in the
graph
10© 2004 Goodrich, Tamassia
Implementation of Graphs
Vertex adjacency listsAdjacency matrix
11© 2004 Goodrich, Tamassia
Example problem
Store the airport flight graphGiven an airport, output all airports that have direct flightsGiven an airport, output all airports that have flights with at most 1 connectionGiven two airports, find minimum number of hops needed to connect them
12
Shortest Paths
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13© 2004 Goodrich, Tamassia
Weighted GraphsIn a weighted graph, each edge has an associated numerical value, called the weight of the edgeEdge weights may represent, distances, costs, etc.Example:
In a flight route graph, the weight of an edge represents the distance in miles between the endpoint airports
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
142
12
05
14© 2004 Goodrich, Tamassia
Shortest PathsGiven a weighted graph and two vertices u and v, we want to find a path of minimum total weight between u and v.
Length of a path is the sum of the weights of its edges.
Example: Shortest path between Providence and Honolulu
Applications Internet packet routing Flight reservations Driving directions
ORD PVD
MIADFW
SFO
LAX
LGA
HNL
849
802
13871743
1843
10991120
1233337
2555
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05
15© 2004 Goodrich, Tamassia
Shortest Path PropertiesProperty 1:
A subpath of a shortest path is itself a shortest pathProperty 2:
There is a tree of shortest paths from a start vertex to all the other vertices
Example:Tree of shortest paths from Providence
ORD PVD
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16© 2004 Goodrich, Tamassia
Dijkstra’s AlgorithmThe distance of a vertex v from a vertex s is the length of a shortest path between s and vDijkstra’s algorithm computes the distances of all the vertices from a given start vertex sAssumptions:
the graph is connected the edges are
undirected the edge weights are
nonnegative
We grow a “cloud” of vertices, beginning with s and eventually covering all the verticesWe store with each vertex v a label d(v) representing the distance of v from s in the subgraph consisting of the cloud and its adjacent verticesAt each step
We add to the cloud the vertex u outside the cloud with the smallest distance label, d(u)
We update the labels of the vertices adjacent to u
17© 2004 Goodrich, Tamassia
Edge RelaxationConsider an edge e (u,z) such that
u is the vertex most recently added to the cloud
z is not in the cloud
The relaxation of edge e updates distance d(z) as follows:d(z) min{d(z),d(u) weight(e)}
d(z) 75
d(u) 5010
zsu
d(z) 60
d(u) 5010
zsu
e
e
18© 2004 Goodrich, Tamassia
Example
CB
A
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D
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0
428
48
7 1
2 5
2
3 9
CB
A
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D
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328
5 11
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7 1
2 5
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3 9
CB
A
E
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328
5 8
48
7 1
2 5
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3 9
CB
A
E
D
F
0
327
5 8
48
7 1
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3 9
19© 2004 Goodrich, Tamassia
Example (cont.)
CB
A
E
D
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327
5 8
48
7 1
2 5
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3 9
CB
A
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0
327
5 8
48
7 1
2 5
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3 9
20© 2004 Goodrich, Tamassia
Dijkstra’s Algorithm
A priority queue stores the vertices outside the cloud: Key: distance, Element: vertex
nnn
nnlogn (priority heap queue)nnlogn (priority heap queue)Sumn(deg(n))=2m
2m2m2m2mlogn (priority heap queue)
Algorithm DijkstraDistances(G, s)Q new heap-based priority queuefor all v G.vertices()
if v ssetDistance(v, 0)
else setDistance(v, )
Q.insert(getDistance(v), v)while Q.isEmpty()
u Q.removeMin() for all e= (z,u)
G.incidentEdges(u)/* relax edge e */r getDistance(u) weight(e)if r getDistance(z)
setDistance(z,r) Recompute position of z in Q
O(mlogn)
21© 2004 Goodrich, Tamassia
Shortest Paths TreeUsing the template method pattern, we can extend Dijkstra’s algorithm to return a tree of shortest paths from the start vertex to all other verticesWe store with each vertex a third label:
parent edge in the shortest path tree
In the edge relaxation step, we update the parent label
Algorithm DijkstraShortestPathsTree(G, s)
…
for all v G.vertices()…
setParent(v, )…
for all e G.incidentEdges(u){ relax edge e }z G.opposite(u,e)r getDistance(u) weight(e)if r getDistance(z)
setDistance(z,r)setParent(z,e) Q.replaceKey(getLocator(z),r)
22© 2004 Goodrich, Tamassia
Why It Doesn’t Work for Negative-Weight Edges
If a node with a negative incident edge were to be added late to the cloud, it could mess up distances for vertices already in the cloud.
CB
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0
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5 9
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7 1
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0 -8
Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance.
C’s true distance is 1, but it is already in the
cloud with d(C)=5!
23© 2004 Goodrich, Tamassia
Other Graph issues
Traversal depth-first Breadth-first
Very hard (NP-complete) problems on graphs Coloring
Given a graph, can we assign one of three colors (say red, blue, or green) to each vertex, such that no adjacent vertices have the same color?
Traveling salesman (Hamiltonian cycle) path that travels through every vertex once, and
winds up where it started