Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker
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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 .

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Chapter 8: Graph Algorithms July/23/2012 Name: Xuanyu Hu Professor: Elise de Doncker

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Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

Chapter 8: Graph AlgorithmsJuly/23/2012Name: Xuanyu HuProfessor: Elise de Doncker


Outline

Outline

  • Graphs

  • Graphs and Genetics

  • DNA Sequencing

  • Shortest Superstring Problem


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

1: Graphs

  • Diagrams with collections of points connected by lines are examples of graphs.

  • The points are called vertices and lines are called edges.


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

  • 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

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?


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

  • 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.


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

  • 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.


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

  • 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.


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

  • 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-white-black-black knight arrangement cannot be transformed into the alternating white-black-white-black arrangement.


How to use graph knights problem 2

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?


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

  • A rather complex graph with twelve vertices and sixteen edges revealing all possible knight moves.


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

  • Rearranging the vertices reveals the cycle that describes the correct sequence of moves.


Connected and disconnected

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

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(3-2-8-6-12-7-5-11-3) are cycles.


The bridge obsession problem

The Bridge Obsession Problem

Find a tour crossing every bridge just once

Leonhard Euler, 1735

Bridges of Königsberg


Eulerian cycle problem

Eulerian Cycle Problem

  • Find a cycle that visits every edgeexactly once.

  • Graph theory was born when Leonhard Euler solved the famous Königsberg Bridge problem.

More complicated Königsberg


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?

Hamiltonian Cycle Problem

Game invented by Sir

William Hamilton in 1857


Trees

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.


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

  • 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.


Chapter 8 graph algorithms july 23 2012 name xuanyu hu professor elise de doncker

  • 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

Seymour Benzer, 1950s

2: Graphs and Genetics

Benzer’s work

  • Developed deletion mapping

  • “Proved” linearity of the gene

  • Demonstrated internal structure of the gene


Viruses attack bacteria

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

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

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: Linear Genes


Interval graph branched genes

Interval Graph: Branched Genes


Interval graph comparison

Interval Graph: Comparison

Linear genome

Branched genome


3 dna sequencing history

3: DNA Sequencing: History

  • Gilbert method (1977):

  • chemical method to cleave DNA at specific points (G, G+A, T+C, C).

Sanger method (1977): labeled ddNTPs terminate DNA copying at random points.

  • Both methods generate labeled fragments of varying lengths that are further electrophoresed.


Sanger method generating read

Start at primer (restriction site)

Grow DNA chain

Include ddNTPs

Stops reaction at all possible points

Separate products by length, using gel electrophoresis

Sanger Method: Generating Read


Dna sequencing

DNA Sequencing

  • Shear DNA into millions of small fragments

  • Read 500 – 700 nucleotides at a time from the small fragments (Sanger method)


Fragment assembly

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

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

Shortest Superstring Problem: Example

  • Concatenating all eight strings results in a 24-letter superstring

  • the shortest superstring contains only 10 letters.


Conclusion and qustions

Conclusion and Qustions

  • Graphs

    graphs, vertex(vertices), edges, connected, disconnected, cycles, trees, degree, leaf

  • Graphs and Genetics

  • DNA Sequencing

  • Shortest Superstring Problem


References

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/dnapr/sequencing.html

5.http://www.wiley.com/college/pratt/0471393878/student/animations/dna_sequencing/index.html

6.http://math.mit.edu/~goemans/18434S06/superstring-lele.pdf


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