What is a graph ?

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What is a graph ?
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What is a graph ?
G(V,E)
V a set of vertices E a set of edges
edge unordered pair of vertices
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What is a graph ?
G(V,E)
V 1,2,3,4,5 E 1,5, 3,5, 2,3, 2,4,
3,4
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What is a graph ?
G(V,E)
V 1,2,3,4,5 E 1,5, 2,3, 2,4, 3,4
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Connectedness
connected
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not connected
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How can we check if a graph is connected?
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Representing a graph
adjacency matrix
V V symmetric matrix A with Ai,j 1 if
i,j? E Ai,j 0 otherwise
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Representing a graph
adjacency matrix
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0 0 1 0 1
0 0 1 1 0
1 1 0 1 0
0 1 1 0 0
1 0 0 0 0
space ?(V2)
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Representing a graph
adjacency matrix
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0 0 1 0 1
0 0 1 1 0
1 1 0 1 0
0 1 1 0 0
1 0 0 0 0
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space ?(V2)
is u,v an edge ? ?(?) list neighbors
of v ?(?)
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Representing a graph
adjacency matrix
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1
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0 0 1 0 1
0 0 1 1 0
1 1 0 1 0
0 1 1 0 0
1 0 0 0 0
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space ?(V2)
is u,v an edge ? ?(1) list neighbors
of v ?(n)
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Representing a graph
adjacency lists
for each vertex v? V linked list of neighbors of v
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Representing a graph
adjacency lists
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1 3,5 2 3,4 3 1,2,4 4 2,3 5 1
space ?(E)
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Representing a graph
adjacency lists
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1 3,5 2 3,4 3 1,2,4 4 2,3 5 1
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space ?(E)
is u,v an edge ? ?(?) list neighbors
of v ?(?)
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Representing a graph
adjacency lists
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1 3,5 2 3,4 3 1,2,4 4 2,3 5 1
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space ?(E)
is u,v an edge ? ?(min(dv,du)) list
neighbors of v ?(dv)
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Representing a graph
adjacency lists
space ?(E)
1 3,5 2 3,4 3 1,2,4 4 2,3 5 1
is u,v in E ? ?(mindu,dv) neigbors
of v ? ?(dv)
adjacency matrix
space ?(V2)
0 0 1 0 1
0 0 1 1 0
1 1 0 1 0
0 1 1 0 0
1 0 0 0 0
is u,v in E ? ?(1) neigbors of v ?
?(n)
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Counting connected components
How can we check if a graph is connected?
INPUT graph G given by adjacency list
OUTPUT number of components of G
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BFS (G,v)
G undirected graph, V1,...,n seenv false
for all v ? V Qqueue (FIFO)
seenv ? true enqueue(Q,v) while Q
not empty do w ? dequeue(Q) for
each neighbor u of w if not seenu
then seenu ? true
enqueue(Q,u)
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Counting connected components
C ? 0 for all v ? V do seenv ? false
for all v ? V do if not seenv then
C BFS(G,v) output G has C
connected components
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DFS
G undirected graph, V1,...,n visitedv
false for all v ? V
explore(G,v) visitedv ? true for each
neighbor u of v if not visited(u) then
explore(G,u)
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DFS
G undirected graph, V1,...,n visitedv
false for all v ? V
explore(G,v) visitedv ? true prev ?
clock clock for each neighbor u of v
if not visited(u) then explore(G,u) postv
? clock clock
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DFS
explore(G,v) visitedv ? true prev ?
clock clock for each neighbor u of v
if not visited(u) then explore(G,u) postv
? clock clock
vertex ? Iv prev,postv
interval property for u,v? V either
Iv and Iu are disjoint, or one is
contained in the other
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DFS
explore(G,v) visitedv ? true prev ?
clock clock for each neighbor u of v
if not visited(u) then explore(G,u) postv
? clock clock
A
B
D
C
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DFS
explore(G,v) visitedv ? true prev ?
clock clock for each neighbor u of v
if not visited(u) then explore(G,u) postv
? clock clock
A
tree edges
B
D
C
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Digraphs (directed graphs)
G(V,E)
V a set of vertices E a set of edges
edge ordered pair of vertices
(u,v)
v
u
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Digraphs (directed graphs)
adjacency lists
for each vertex v? V linked list of out-neighbors
of v
adjacency matrix
V V matrix A with Ai,j 1 if (i,j)? E
Ai,j 0 otherwise
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Digraphs (directed graphs)
a path sequence of vertices v1,v2,...,vk
such that (v1,v2)? E, ... , (vk-1,vk)? E
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DAGs (acyclic digraphs)
a cycle sequence of vertices v1,v2,...,vk
such that (v1,v2)? E, ... ,
(vk-1,vk),(vk,v1)? E
DAG digraph with no cycle
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Topological sort (linearization)
INPUT DAG G given by adjacency list
OUTPUT ordering of vertices such that
edges go forward
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DFS on digraphs
G digraph, V1,...,n visitedv false for
all v ? V
explore(G,v) visitedv ? true prev ?
clock clock for each out-neighbor u of v
if not visited(u) then explore(G,u)
postv ? clock clock
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DFS on digraphs
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B
D
C
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DFS on digraphs
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root
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D
C
descendant, ancestor child, parent
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DFS on digraphs
A
tree edge
B
D
C
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DFS on digraphs
A
tree edge
B
D
C
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DFS on digraphs
back edge
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tree edge
B
D
C
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DFS on digraphs
back edge
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tree edge
B
D
C
cross edge
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DFS on digraphs
back edge
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tree edge
B
forward edge
D
C
cross edge
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Relationships between the intervals?
back edge
A
tree edge
B
forward edge
D
C
cross edge
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Topological sort using DFS
Lemma digraph is a DAG if and only
if DFS has a back edge.
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Topological sort using DFS
Lemma digraph is a DAG if and only
if DFS has a back edge.
Lemma in a DAG every edge goes to a
vertex with lower post
explore(G,v) visitedv ? true prev ?
clock clock for each neighbor u of v
if not visited(u) then explore(G,u) postv
? clock clock
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(strong) connectedness
a digraph G is strongly connected if for every
u,v? V there exists a path from u to v in G
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(strong) connectedness
How to check if a digraph is strongly connected?
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(strong) connectedness
How to check if a digraph is strongly connected?
for every u?V do DFS(G,u) check if every
v?V was visited
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(strong) connectedness
How to check if a digraph is strongly connected?
pick some u?V DFS(G,u) check if every v?V was
visited DFS(reverse(G),u) check if every v?V was
visited
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Strongly connected components
DAG of strongly connected components
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Strongly connected components
Lemma G and reverse(G) have the same strongly
connected components.
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Strongly connected components
DAG of strongly connected components
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Strongly connected components
for all v? V do colorv? white for all v? V do
if colorvwhite then
DFS(reverse(G),v) DFS(G,u)
(vertices in order post)