Graph Theory for Bioinformatics

abalonestrawBiotechnology

Oct 2, 2013 (4 years and 9 days ago)

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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

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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
)

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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

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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

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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

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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”

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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

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7