Algorithms for Bioinformatics - Applied Graph Theory - Lecture Slides, Slides of Computer Science

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BIO/CS 471 – Algorithms for bioinformatics

Graph Theoretic

Concepts and Algorithms

for Bioinformatics

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

Graphs in bioinformatics

• Sequences

• DNA, proteins, etc. Chemical compounds

Metabolic pathways

R Y L I

Docsity.com

Graphs in bioinformatics

Phylogenetic trees Docsity.com

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

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
  • Kn : 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 Docsity.com

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 b^ a

c

d x y

z

w

u v

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 4 d

2

6

8 10

a a^ b^ c 8^ d 4 bc 6 d 10 2

a b c (8), d (4) dc^ b (6)c (2), b (10)

a 1 2 8^^3 4 5 4 b c (^) 6 t (^) t t t d 2 10 t adjacency matrix

adjacency list

incidence matrix

Subgraphs

• 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’

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}

Complement of a graph

• 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 ) a

b

c

d

e G G

a

b

c

d

e

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 )

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 3 2 15

11

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

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} }

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?