1
Intro. to Graph Theory
BIO/CS 471
–
Algorithms for bioinformatics
Graph Theoretic
Concepts
and Algorithms
for Bioinformatics
2
Intro. to Graph Theory
What
is
a “graph”
•
Formally: A finite graph
G
(
V
,
E
) is a pair (
V
,
E
),
where
V
is a finite set and
E
is a
binary relation
on
V
.
–
Recall: A
relation
R
between two sets
X
and
Y
is a subset of
X
x
Y
.
–
For each selection of two distinct
V
’s, that pair of
V
’s is
either in set
E
or not in set
E
.
•
The elements of the set
V
are called
vertices
(or
nodes)
and those of set
E
are called
edges
.
•
Undirected graph
: The edges are unordered pairs of
V
(i.e. the binary relation is symmetric).
–
Ex: undirected G(V,E); V = {a,b,c}, E = {{a,b}, {b,c}}
•
Directed graph
(digraph):The edges are ordered
pairs of
V
(i.e. the binary relation is not necessarily
symmetric).
–
Ex: digraph G(V,E); V = {a,b,c}, E = {(a,b), (b,c)}
a
b
c
a
b
c
3
Intro. to Graph Theory
Why graphs?
•
Many problems can be stated in terms of a graph
•
The properties of graphs are well

studied
–
Many algorithms exists to solve problems posed as graphs
–
Many problems are already known to be intractable
•
By
reducing
an instance of a problem to a standard graph
problem, we may be able to use well

known graph algorithms
to provide an optimal solution
•
Graphs are excellent structures for storing, searching, and
retrieving large amounts of data
–
Graph theoretic techniques play an important role in increasing the
storage/search efficiency of computational techniques.
•
Graphs are covered in section 2.2 of Setubal & Meidanis
4
Intro. to Graph Theory
Graphs in bioinformatics
•
Sequences
–
DNA, proteins, etc.
Chemical compounds
Metabolic pathways
R
Y
L
I
5
Intro. to Graph Theory
Graphs in bioinformatics
Phylogenetic trees
6
Intro. to Graph Theory
Basic definitions
•
incidence
: an edge (directed or undirected) is incident to a vertex
that is one of its end points.
•
degree
of a vertex: number of edges incident to it
–
Nodes of a digraph can also be said to have an
indegree
and an
outdegree
•
adjacency
: two vertices connected by an edge are adjacent
Undirected graph
Directed graph
isolated vertex
loop
multiple
edges
G
=(
V
,
E
)
adjacent
loop
7
Intro. to Graph Theory
x
y
path
: no vertex can be repeated
example path: a

b

c

d

e
trail
: no edge can be repeated
example trail: a

b

c

d

e

b

d
walk
: no restriction
example walk: a

b

d

a

b

c
closed:
if starting vertex is also ending vertex
length
: number of edges in the path, trail, or walk
circuit:
a closed trail (ex: a

b

c

d

b

e

d

a)
cycle:
closed path (ex: a

b

c

d

a)
a
b
c
d
e
“Travel” in graphs
8
Intro. to Graph Theory
Types of graphs
•
simple graph:
an undirected graph with no loops or multiple edges between
the same two vertices
•
multi

graph:
any graph that is not simple
•
connected graph
: all vertex pairs are joined by a path
•
disconnected graph
: at least one vertex pairs is not joined by a path
•
complete graph
: all vertex pairs are adjacent
–
K
n
: the completely connected graph with
n
vertices
Simple graph
a
b
c
d
e
K
5
a
b
c
d
e
Disconnected graph
with two components
9
Intro. to Graph Theory
Types of graphs
•
acyclic graph
(forest): a graph with no cycles
•
tree:
a connected, acyclic graph
•
rooted tree
: a tree with a “root” or “distinguished” vertex
–
leaves:
the terminal nodes of a rooted tree
•
directed acyclic graph
(DAG): a digraph with no cycles
•
weighted graph:
any graph with weights associated with the edges (edge

weighted) and/or the vertices (vertex

weighted)
b
a
c
d
e
f
10
5
8

3
2
6
10
Intro. to Graph Theory
Digraph definitions
•
for digraphs only…
•
Every edge has a
head
(starting point) and a
tail
(ending point)
•
Walks, trails, and paths can only use edges in
the appropriate direction
•
In a DAG, every path connects an
predecessor/ancestor
(the vertex at the head
of the path) to its
successor/descendents
(nodes at the tail of any path).
•
parent:
direct ancestor (one hop)
•
child:
direct descendent (one hop)
•
A descendent vertex is
reachable
from any of
its ancestors vertices
Directed graph
a
b
c
d
x
y
z
w
u
v
11
Intro. to Graph Theory
Computer representation
•
undirected graphs:
usually represented as digraphs with two
directed edges per “actual” undirected edge.
•
adjacency matrix:
a 
V
 x 
V
 array where each cell
i
,
j
contains
the weight of the edge between
v
i
and
v
j
(or 0 for no edge)
•
adjacency list:
a V array where each cell
i
contains a list of all
vertices adjacent to
v
i
•
incidence matrix:
a V by E array where each cell
i
,
j
contains
a weight (or a defined constant HEAD for unweighted graphs)
if the vertex i is the head of edge
j
or a constant TAIL if vertex I
is the tail of edge
j
c
b
a
d
4
2
6
10
8
adjacency
matrix
adjacency
list
incidence
matrix
12
Intro. to Graph Theory
Computer representation
•
Linked list of nodes:
Node is a defined data object with labels which
include a list of pointers to its children and/or parents
•
Graph = [] # list of nodes
Class Node:
label = NIL;
parents = []; # list of nodes coming into this node
children = []; # list of nodes coming out of this node
childEdgeWeights = []; # ordered list of edged weights
13
Intro. to Graph Theory
•
G’
(
V’
,
E’
) is a
subgraph
of
G
(
V
,
E
) if
V’
V
and
E’
E.
•
induced subgraph:
a subgraph that contains all possible edges
in E that have end points of the vertices of the selected V’
Subgraphs
a
b
c
d
e
b
c
d
e
a
c
d
G(V,E)
G’({a,c,d},{{c,d}})
Induced subgraph of
G with V’ = {b,c,d,e}
14
Intro. to Graph Theory
•
The
complement
of a graph G (
V
,
E
) is a graph with the same
vertex set, but with vertices adjacent only if they were not
adjacent in
G
(
V
,
E
)
Complement of a graph
a
b
c
d
e
G G
a
b
c
d
e
15
Intro. to Graph Theory
•
Consider a weighted connected directed graph with a distinguished vertex
source:
a distinguished vertex with zero in

degree
•
What is the path of total minimum weight from the source to any other
vertex?
•
Greedy strategy works for simple problems (no cycles, no negative weights)
•
Longest path is a similar problem (complement weights)
•
We will see this again soon for fragment assembly!
Famous problems: Shortest path
c
b
a
d
4
2
6
10
8
16
Intro. to Graph Theory
Dijkstra’s Algorithm
•
D(
x
) = distance from
s
to
x
(initially all
)
1.
Select the closest vertex to
s
, according to the current estimate
(call it
c
)
2.
Recompute the estimate for every other vertex,
x
, as the
MINIMUM of:
1.
The current distance, or
2.
The distance from
s
to
c
, plus the distance from
c
to
x
–
D(
c
) + W(
c,
x
)
17
Intro. to Graph Theory
Dijkstra’s Algorithm Example
A
B
C
D
E
Initial
0
Process A
0
10
3
20
Process C
0
5
3
20
18
Process B
0
5
3
10
18
Process D
0
5
3
10
18
Process E
0
5
3
10
18
A
B
C
E
D
10
5
20
2
3
15
11
18
Intro. to Graph Theory
•
Two graphs are
isomorphic
if a 1

to

1 correspondence between
their vertex sets exists that preserve adjacencies
•
Determining to two graphs are isomorphic is NP

complete
Famous problems: Isomorphism
a
b
c
d
e
1
2
3
4
5
19
Intro. to Graph Theory
Famous problems: Maximal clique
•
clique:
a complete subgraph
•
maximal clique:
a clique not contained in any other clique; the largest
complete subgraph in the graph
•
Vertex cover:
a subset of vertices such that each edge in E has at least one
end

point in the subset
•
clique cover:
vertex set divided into non

disjoint subsets, each of which
induces a clique
•
clique partition:
a disjoint clique cover
1
2
4
3
Maximal cliques: {1,2,3},{1,3,4}
Vertex cover: {1,3}
Clique cover: { {1,2,3}{1,3,4} }
Clique partition: { {1,2,3}{4} }
20
Intro. to Graph Theory
Famous problems: Coloring
•
vertex coloring:
labeling the vertices such that no edge in E has two end

points with the same label
•
chromatic number
: the smallest number of labels for a coloring of a graph
•
What is the chromatic number of this graph?
•
Would you believe that this problem (in general) is intractable?
1
2
4
3
21
Intro. to Graph Theory
Famous problems: Hamilton & TSP
•
Hamiltonian path:
a path through a graph which contains
every vertex exactly once
•
Finding a Hamiltonian path is another NP

complete problem…
•
Traveling Salesmen Problem (TSP):
find a Hamiltonian path
of minimum cost
a
b
c
d
e
f
g
h
i
a
b
c
d
e
3
4
1
3
5
4
3
2
2
22
Intro. to Graph Theory
Famous problems: Bipartite graphs
•
Bipartite:
any graph whose vertices can be partitioned into two
distinct sets so that every edge has one endpoint in each set.
•
How colorable is a bipartite graph?
•
Can you come up with an algorithm to determine if a graph is
bipartite or not?
•
Is this problem tractable or intractable?
K
4,4
23
Intro. to Graph Theory
Famous problems: Minimal cut set
•
cut set:
a subset of edges whose remove causes the number of
graph components to increase
•
vertex separation set:
a subset of vertices whose removal
causes the number of graph components to increase
•
How would you determine the
minimal
cut set or vertex
separation set?
a
b
c
d
e
f
g
h
1
2
4
3
cut

sets: {(a,b),(a,c)},
{(b,d),(c,d)},{(d,f)},...
24
Intro. to Graph Theory
Famous problem: Conflict graphs
•
Conflict graph:
a graph where each vertex represents a concept or resource
and an edge between two vertices represents a conflict between these two
concepts
•
When the vertices represents intervals on the real line (such as time) the
conflict graph is sometimes called an interval graph
•
A coloring of an interval graph produces a schedule that shows how to best
resolve the conflicts… a minimal coloring is the “best” schedule”
•
This concept is used to solve problems in the physical mapping of DNA
a
b
c
f
e
d
Colors?
25
Intro. to Graph Theory
Famous problems: Spanning tree
•
spanning tree:
A subset of edges that are sufficient to keep a
graph connected if all other edges are removed
•
minimum spanning tree:
A spanning tree where the sum of the
edge weights is minimum
a
b
c
d
e
f
g
h
2
2
2
4
8
1
4
6
2
4
a
b
c
d
e
f
g
h
2
2
2
8
1
4
6
2
4
26
Intro. to Graph Theory
•
G is said to have a
Euler circuit
if there is a circuit in G that traverses every
edge in the graph exactly once
•
The seven bridges of Konigsberg:
Find a way to walk about the city so as to
cross each bridge exactly once and then return to the starting point.
Famous problems: Euler circuit
area
b
area
d
area
c
b
c
d
a
This one is in P!
27
Intro. to Graph Theory
Famous problems: Dictionary
•
How can we organize a dictionary for fast lookup?
a
b
c
y
z
…
a
b
c
y
z
…
a
b
c
y
z
…
a
b
c
y
z
…
a
b
c
y
z
…
a
b
c
y
z
…
“CAB”
26

ary “trie”
28
Intro. to Graph Theory
Graph traversal
•
There are many strategies for solving graph problems… for
many problems, the efficiency and accuracy of the solution boil
down to how you “search” the graph.
•
We will consider a “travel” problem for example:
•
Given the graph below, find a path from vertex
a
to vertex
d
.
Shorter paths (in terms of edge weight sums) are desirable.
b
e
a
d
f
c
3
1
2
4
5
6
7
29
Intro. to Graph Theory
A greedy approach
•
greedy traversal
: Starting with the “root” node, take the edge
with smallest weight. Mark the edge so that you never attempt
to use it again. If you get to the end, great! If you get to a dead
end, back up one decision and try the next best edge.
•
Advantages: Fast! Drawbacks: Answer is usually non

optimal
•
For some problems, greedy approaches
are
optimal, for others
the answer may usually be
close
to the best answers, for yet
other problems, the greedy strategy is a poor choice.
b
e
a
d
f
c
3
1
2
4
5
6
7
Start node: a
End node: d
Traversal order: a, c, f, e, b, d
30
Intro. to Graph Theory
Exhaustive search: Breadth

first
•
For the
current
node, do any necessary work
–
In this case, calculate the cost to get to the node by the current path; if the cost is
better than any previous path, update the “best path” and “lowest cost”.
•
Place all adjacent unused edges in a queue (FIFO)
•
Take an edge from the queue, mark it as used, and follow it to the new
current node
b
e
a
d
f
c
3
1
2
4
5
6
7
Traversal order: a, b, c, d, e, f
31
Intro. to Graph Theory
Exhaustive search: Depth

first
•
For each current node
–
do any necessary work
–
Pick one unused edge out and
follow it to a new current
node
–
If no unused edges exist,
unmark all of your edges an
go back from whence you
came!
b
e
a
d
f
c
3
1
2
4
5
6
7
Traversal order: a, b, d, e, f, c
DFS (G, v)
V.state = “visited”
Process vertex v
Foreach edge (v,w) {
if w.state = “unseen” {
DFS (G, w)
process edge (v,w)
}
}
}
32
Intro. to Graph Theory
Branch and Bound
•
Begin a depth

first search (DFS)
•
Once you achieve a successful result, note the result as our initial “best
result”
•
Continue the DFS; if you find a better result, update the “best result”
•
At each step of the DFS compare your current “cost” to the cost of the
current “best result”; if we already exceed the cost of the best result, stop the
downward search! Mark all edges as used, and head back up.
b
e
a
d
f
c
3
1
2
4
5
6
7
Path Current Best
ACF
7
11
ACFE
15
11 < prune
AB
3
11
ABD
8
8
ABE
7
8
ABEF 14
8
Traversal order:
Path Current Best
A
0

AE
2

AEB
6

AEBD
11
11
AEF
9
11
AEFC 15
11
AC
1
11
33
Intro. to Graph Theory
Binary search trees
•
Binary trees have at two children per node (the child may be null)
•
Binary search trees are organized so that each node has a label.
•
When searching or inserting a value, compare the target value to each node;
one out

going edge corresponds to “less than” and one out

going edge
corresponds to “greater than”.
•
On the average, you eliminate 50% of the search space per node… if the tree
is
balanced
5
4
6
8
1
2
3
10
9
7
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