The Graph Abstract Data Type

1
The Graph Abstract Data Type

• CS 5050
• Chapter 6

2
Terminology

• Vertex (vertices) position, holds an object
• Edge also a position, holds an object
• Directed, undirected ((a)symmetric)
• Graph is a set of vertices and a bag of edges
  (can be duplicated)
• Parallel and self-edges
• Directed, undirected, mixed (both directed and
  undirected edges) convert to digraph
• End vertices/endpoints (of an edge),
  origin/destination, adjacent vertices (endpoints
  of same edge)
• Incident edge, incoming, outgoing, in (out)-degree

3
Terminology continued

• Simple graphs have no parallel (multiple edges
  between same vertices) or self loop
• Path sequence of alternating vertices and edges
  which match up
• Cycle - path with same first and last vertex
• Simple and directed paths and cycles
• (Spanning contains all vertices) subgraph
• Connected graph, connected component
• Forest (acyclic), free (no root) trees
  (connected forest), spanning tree

4
Graph Facts

• Assume the graph G has n vertices (V) and m edges
  (E)
• The sum over V of degrees is 2m
• If G is directed, the sum over V of in-degrees
  the sum over V of out-degrees m
• If G is simple and undirected, m lt n(n-1)/2
• (worst case is every pair)
• If G is simple and directed, m lt n(n-1)
• So m is O( V2 )
• If G is connected, m gt n-1, if a tree m n-1
• If G is a forest, m lt n-1

5
Adjacency List -partitioned incidence container
(in/out)
6
Adjacency Matrix
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Traversals

• Visit each node e.g., web crawler
• Have to restart traversal in each connected
  component, but this allows us to identify
  components
• Reachability in a digraph is an important issue
  the transitive closure graph
• Book permits counter-direction motion in general
  traversals

8
Looking for a spanning Tree

• Many times algorithms depend on some ordering of
  the nodes of a graph
• Labeling the edges is some way is helpful in
  organizing thinking about a graph

9
Depth-First Search (undirected graph)

• Simple recursive backtracking algorithm calls
  recursive function at starting point
• For each incident edge to a vertex
• If opposite (other) vertex is unvisited
• Label edge as discovery
• Recur on the opposite vertex
• Else label edge as back
• Discovery edges form a component spanning tree,
  back edges go to an ancestor
• with m edges, O(m) using an adjacency list, but
  not using an adjacency matrix

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Depth-First Traversal
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Biconnectivity
SEA
PVD
ORD
FCO
SNA
MIA
12
Outline and Reading

• Definitions (6.3.2)
• Separation vertices and edges
• Biconnected graph
• Biconnected components
• Equivalence classes
• Linked edges and link components
• Algorithms (6.3.2)
• Auxiliary graph
• Proxy graph

13
Separation Edges and Vertices

• Definitions
• Let G be a connected graph
• A separation edge of G is an edge whose removal
  disconnects G
• A separation vertex of G is a vertex whose
  removal disconnects G
• Applications
• Separation edges and vertices represent single
  points of failure in a network and are critical
  to the operation of the network
• Example
• DFW, LGA and LAX are separation vertices
• (DFW,LAX) is a separation edge

ORD
PVD
SFO
LGA
HNL
LAX
DFW
MIA
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Biconnected Graph

• Equivalent definitions of a biconnected graph G
• Graph G has no separation edges and no separation
  vertices
• For any two vertices u and v of G, there are two
  disjoint simple paths between u and v (i.e., two
  simple paths between u and v that share no other
  vertices or edges)
• For any two vertices u and v of G, there is a
  simple cycle containing u and v
• Example

PVD
ORD
SFO
LGA
HNL
LAX
DFW
MIA
15
Biconnected Components

• Biconnected component of a graph G
• A maximal biconnected subgraph of G, or
• A subgraph consisting of a separation edge of G
  and its end vertices
• Interaction of biconnected components
• An edge belongs to exactly one biconnected
  component
• A nonseparation vertex belongs to exactly one
  biconnected component
• A separation vertex belongs to two or more
  biconnected components
• Example of a graph with four biconnected
  components

ORD
PVD
SFO
LGA
RDU
HNL
LAX
DFW
MIA
16
Equivalence Classes

• Given a set S, a relation R on S is a set of
  ordered pairs of elements of S, i.e., R is a
  subset of S?S
• An equivalence relation R on S satisfies the
  following properties
• Reflexive (x,x) ? R
• Symmetric (x,y) ? R ? (y,x) ? R
• Transitive (x,y) ? R ? (y,z) ? R ? (x,z) ? R
• An equivalence relation R on S induces a
  partition of the elements of S into equivalence
  classes
• Example (connectivity relation among the vertices
  of a graph)
• Let V be the set of vertices of a graph G
• Define the relation C (v,w) ? V?V such that G
  has a path from v to w
• Relation C is an equivalence relation
• The equivalence classes of relation C are the
  vertices in each connected component of graph G

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Link Relation

• Edges e and f of connected graph G are linked if
• e f, or
• G has a simple cycle containing e and f
• Theorem
• The link relation on the edges of a graph is an
  equivalence relation
• Proof Sketch
• The reflexive and symmetric properties follow
  from the definition
• For the transitive property, consider two simple
  cycles sharing an edge

Equivalence classes of linked edges a b, c,
d, e, f g, i, j
i
g
e
b
a
d
j
f
c
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Link Components

• The link components of a connected graph G are
  the equivalence classes of edges with respect to
  the link relation
• A biconnected component of G is the subgraph of G
  induced by an equivalence class of linked edges
• A separation edge is a single-element equivalence
  class of linked edges
• A separation vertex has incident edges in at
  least two distinct equivalence classes of linked
  edge

ORD
PVD
SFO
LGA
RDU
HNL
LAX
DFW
MIA
19
Auxiliary Graph
h
g

• Auxiliary graph B for a connected graph G
• Associated with a DFS traversal of G
• The vertices of B are the edges of G
• For each back edge e of G, B has edges (e,f1),
  (e,f2) , , (e,fk),where f1, f2, , fk are the
  discovery edges of G that form a simple cycle
  with e
• Its connected components correspond to the the
  link components of G

i
e
b
i
j
d
c
f
a
DFS on graph G
g
e
i
h
b
f
j
d
c
a
Auxiliary graph B
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Auxiliary Graph (cont.)

• In the worst case, the number of edges of the
  auxiliary graph is proportional to nm

Auxiliary graph B
DFS on graph G
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Proxy Graph
h
g
Algorithm proxyGraph(G) Input connected graph
G Output proxy graph F for G F ? empty
graph DFS(G, s) s is any vertex of G for all
discovery edges dEdge of G F.insertVertex(dEdge)
setLabel(dEdge, UNLINKED) for all vertices v of
G in DFS visit order for all back edges bEdge
(u,v) F.insertVertex(bEdge) repeat dEdge
? discovery edge with dest. u F.insertEdge(bEdg
e,dEdge,?) if getLabel(dEdge)
UNLINKED setLabel(dEdge, LINKED) u ?
origin of dEdge else u ? v ends the
loop until u v return F
i
e
b
i
j
d
c
f
a
DFS on graph G
g
e
i
h
b
f
j
d
c
a
Proxy graph F
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Proxy Graph (cont.)
h
g

• Proxy graph F for a connected graph G
• Spanning forest of the auxiliary graph B
• Has m vertices and O(m) edges
• Can be constructed in O(n m) time
• Its connected components (trees) correspond to
  the the link components of G
• Given a graph G with n vertices and m edges, we
  can compute the following in O(n m) time
• The biconnected components of G
• The separation vertices of G
• The separation edges of G

i
e
b
i
j
d
c
f
a
DFS on graph G
g
e
i
h
b
f
j
d
c
a
Proxy graph F
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Breadth-First Search

• By levels, typically using queues

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BFS Facts

• There are discovery and cross edges
• (why no back edges?)
• Once marked, dont follow again.
• Discovery edges form spanning tree
• Tree edges are paths, minimal in length
• Cross edges differ by at most one in level
• Try writing the code to do a BFS

25
Thm 6.19 Algorithms based on BFS

• Test for connectivity
• compute spanning forest
• compute connected components
• find shortest path between two points (in number
  of links)
• compute a cycle in graph, or report none
• (have cross edges)
• Good for shortest path information, while DFS
  better for complex connectivity questions

26
Digraphs

• 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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Digraph Properties

• 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.
• If G is simple, m lt 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.

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Digraph Application

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

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ics22
ics21
ics53
ics51
ics52
ics161
ics131
ics141
ics121
ics171
The good life
ics151
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Digraph Facts

• Directed DFS gives directed paths from root to
  each reachable vertex
• Used for O(n(nm)) algorithm dfs is O(nm),
  these algorithms use n dfs searches
• Find all induced subgraphs (from each vertex, v,
  find subgraph reachable from v)
• Test for strong connectivity
• Compute the transitive closure
• Directed BFS has discovery, back, cross edges

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Directed DFS

• 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 a a vertex s determines
  the vertices reachable from s

E
D
C
B
A
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Reachability

• DFS tree rooted at v vertices reachable from v
  via directed paths

E
D
E
D
C
A
C
F
E
D
A
B
C
F
A
B
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Strong Connectivity

• Each vertex can reach all other vertices

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Strong Connectivity Algorithm

• Pick a vertex v in G.
• Perform a DFS from v in G.
• If theres a w not visited, print no.
• Let G be G with edges reversed.
• Perform a DFS from v in G.
• If theres a w not visited, print no.
• Else, print yes.
• Running time O(nm).

a
G
g
c
d
e
b
f
a
g
G
c
d
e
b
f
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Strongly Connected Components

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

a , c , g
f , d , e , b
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Transitive Closure
D
E

• 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

B
G
C
A
D
E
B
C
A
G
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Computing the Transitive Closure

• We can perform DFS starting at each vertex
• O(n(nm))

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

37
Floyd-Warshall Transitive Closure

• 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 its not already in)
i
j
Uses only vertices numbered 1,,k-1
Uses only vertices numbered 1,,k-1
k
38
Floyd-Warshalls Algorithm

• Floyd-Warshalls algorithm numbers the vertices
  of G as v1 , , vn and computes a series of
  digraphs G0, , Gn
• G0G
• Gk has directed edge (vi, vj) if G has directed
  path from vi to vj with 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)

Algorithm FloydWarshall(G) Input digraph G
(vertices numbered) Output transitive closure G
of G G0 ? G for k ? 1 to n do Gk ? Gk - 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) return Gn
39
Floyd-Warshall Example
BOS
v
ORD
4
JFK
v
v
2
6
SFO
DFW
LAX
v
3
v
1
MIA
v
5
40
(No Transcript)
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Floyd-Warshall, Conclusion
v
ORD
4
JFK
v
v
2
6
SFO
DFW
LAX
v
3
v
1
MIA
v
5
42
DAGs and Topological Ordering
D
E

• 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 lt 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

B
C
A
DAG G
v4
v5
D
E
v2
B
v3
C
v1
Topological ordering of G
A
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Topological Sorting

• Number vertices, so that (u,v) in E implies u lt v

1
A typical student day
wake up
3
2
eat
study computer sci.
5
4
nap
more c.s.
7
play
8
write c.s. program
6
9
work out
make cookies for professors

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sleep
dream about graphs
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Algorithm for Topological Sorting

• Note This algorithm is different than the one in
  Goodrich-Tamassia
• Running time O(n m). How?

Method TopologicalSort(G) H ? G //
Temporary copy of G n ? G.numVertices()
while H is not empty do Let v be a vertex with
no outgoing edges Label v ? n n ? n - 1 Remove
v from H
45
Topological Sorting Algorithm using DFS

• Simulate the algorithm by using depth-first
  search
• O(nm) time.

Algorithm topologicalDFS(G, v) Input graph G and
a start vertex v of G Output labeling of the
vertices of G in the connected component of v
setLabel(v, VISITED) for all edgeE ?
G.incidentEdges(v) if getLabel(edgeE)
UNEXPLORED w ? opposite(v,edgeE) if
getLabel(w) UNEXPLORED setLabel(edgeE,
DISCOVERY) topologicalDFS(G, w) else e
is a forward or cross edge Label v with
topological number n n ? n - 1
Algorithm topologicalDFS(G) Input dag G Output
topological ordering of G n ?
G.numVertices() for all u ? G.vertices()
setLabel(u, UNEXPLORED) for all e ? G.edges()
setLabel(e, UNEXPLORED) for all v ?
G.vertices() if getLabel(v)
UNEXPLORED topologicalDFS(G, v)
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Topological Sorting Example
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Topological Sorting Example
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Topological Sorting Example
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