Transcript Graph Algorithms

Chapter 8: Graph
Algorithms
July/23/2012
Name: Xuanyu Hu
Professor: Elise de Doncker
Outline
•
•
•
•
Graphs
Graphs and Genetics
DNA Sequencing
Shortest Superstring Problem
1: Graphs
• Diagrams with
collections of points
connected by lines
are examples of
graphs.
• The points are called
vertices and lines are
called edges.
• We denote a
graph by G =
G(V, E) and
describe it by
its set of
vertices V and
set of edges E.
How to Use Graph: Knights
Problem 1
• This upper picture shows
two white and two black
knights on a 3*3
chessboard.
• Can they move, using
the usual chess knight's
moves, to occupy the
positions shown in the
below picture?
• This picture represents
the chessboard as a
set of nine points.
• Two points are
connected by a line if
moving from one point
to another is a valid
knight move.
• The upper picture
represents the
chessboard as a set of
nine points.
• Two points are
connected by a line if
moving from one point to
another is a valid knight
move.
• An equivalent
representation of
the resulting
diagram that
reveals that knights
move aroung a
"cycle" formed by
points 1,6,7,2,9,4,3,
and 8.
• Every knight's move on the
chessboard corresponds to
moving to a neighboring point
in the diagram, in either a
clockwise or
counterclockwise direction.
• Therefore, the white-whiteblack-black knight
arrangement cannot be
transformed into the
alternating white-black-whiteblack arrangement.
How to Use Graph: Knights
Problem 2
• This picture represents
anohter chessboard
obtained from a 4*4
chessboard by removing
the four corner squares.
• Can a knight travel
around this board, pass
through each square
exactly once, and return
to the same square it
started on?
• A rather complex
graph with
twelve vertices
and sixteen
edges revealing
all possible
knight moves.
• Rearranging the
vertices reveals
the cycle that
describes the
correct
sequence of
moves.
Connected and Disconnected
• A graph is called connected if all
pairs of vertices can be connected by
a path, which is a continuous
sequence of edges, where each
successive edge begins where the
previous one left off.
• Graphs that are not connected are
disconnected.
Cycles
• Paths that start and
end at the same
vertex are referred
to as cycles.
• For example, the
paths(3-2-10-11-3),
and paths(3-2-8-612-7-5-11-3) are
cycles.
The Bridge Obsession Problem
Find a tour crossing every bridge just once
Leonhard Euler, 1735
Bridges of Königsberg
Eulerian Cycle Problem
• Find a cycle that
visits every edge
exactly once.
• Graph theory was
born when
Leonhard Euler
solved the
famous
Königsberg
More complicated Königsberg
Bridge problem.
Hamiltonian Cycle Problem
• Can you travel
from any one
of the vertices
in this graph,
visit every
other vertex
exactly once,
and end up at
the original
vertex?
Game invented by Sir
William Hamilton in 1857
Trees
• Arthur Cayley studied
chemical structures of
hydrocarbons in the
mid-1800s
• Structures of this type
of hydrocarbon are
examples of trees,
which are simply
connected graphs
with no cycles.
• Every tree has at
least one vertex
with degree 1,
called leaf.
• Every tree on n
vertices has n-1
edges, regardless
of the structure of
the tree.
• Every tree on n
vertices has n-1
edges, regardless of
the structure of the
tree.
• Every tree has a leaf,
we can remove it and
its attached edge. We
keep this up until we
are left with a graph
with a single vertex
and no edges.
2: Graphs and Genetics
Benzer’s work
• Developed
deletion mapping
• “Proved”
linearity of the
gene
• Demonstrated
internal structure
of the gene
Seymour Benzer, 1950s
Viruses Attack Bacteria
• Normally bacteriophage T4 kills bacteria
• However if T4 is mutated (e.g., an important gene is
deleted) it gets disable and looses an ability to kill
bacteria
• Suppose the bacteria is infected with two different
mutants each of which is disabled – would the
bacteria still survive?
• Amazingly, a pair of disable viruses can kill a
bacteria even if each of them is disabled.
• How can it be explained?
Benzer’s Experiment
• Idea: infect bacteria with pairs of mutant
T4 bacteriophage (virus)
• Each T4 mutant has an unknown interval
deleted from its genome
• If the two intervals overlap: T4 pair is
missing part of its genome and is disabled
– bacteria survive
• If the two intervals do not overlap: T4 pair
has its entire genome and is enabled –
bacteria die
Benzer’s Experiment and
Graphs
• Construct an interval graph: each T4
mutant is a vertex, place an edge
between mutant pairs where bacteria
survived (i.e., the deleted intervals in
the pair of mutants overlap)
• Interval graph structure reveals
whether DNA is linear or branched
DNA
Interval Graph: Linear Genes
Interval Graph: Branched Genes
Interval Graph: Comparison
Linear genome
Branched genome
3: DNA Sequencing: History
Sanger method (1977):
labeled ddNTPs
terminate DNA
copying at random
points.
Gilbert method (1977):
chemical method to
cleave DNA at specific
points (G, G+A, T+C, C).
Both methods generate
labeled fragments of
varying lengths that are
further electrophoresed.
Sanger Method: Generating Read
1. Start at primer
(restriction site)
2. Grow DNA chain
3. Include ddNTPs
4. Stops reaction at all
possible points
5. Separate products
by length, using gel
electrophoresis
DNA Sequencing
• Shear DNA into
millions of small
fragments
• Read 500 – 700
nucleotides at a
time from the small
fragments (Sanger
method)
Fragment Assembly
• Computational Challenge: assemble
individual short fragments (reads) into a
single genomic sequence (“superstring”)
• Until late 1990s the shotgun fragment
assembly of human genome was viewed
as intractable problem
4: Shortest Superstring Problem
• Problem: Given a set of strings, find a
shortest string that contains all of them
• Input: Strings s1, s2,…., sn
• Output: A string s that contains all strings
s1, s2,…., sn as substrings, such that the
length of s is minimized
• Note: this formulation does not take into
account sequencing errors
Shortest Superstring Problem:
Example
• Concatenating all
eight strings
results in a 24letter superstring
• the shortest
superstring
contains only 10
letters.
Conclusion and Qustions
• Graphs
graphs, vertex(vertices), edges,
connected, disconnected,
cycles, trees, degree, leaf
• Graphs and Genetics
• DNA Sequencing
• Shortest Superstring Problem
References
1.http://bix.ucsd.edu/bioalgorithms/slides.php
2.http://en.wikipedia.org/wiki/Graph_theory
3.http://simple.wikipedia.org/wiki/Genetics
4.http://seqcore.brcf.med.umich.edu/doc/educ/dna
pr/sequencing.html
5.http://www.wiley.com/college/pratt/0471393878/s
tudent/animations/dna_sequencing/index.html
6.http://math.mit.edu/~goemans/18434S06/superst
ring-lele.pdf