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

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

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

Outline

- Graphs
- Graphs and Genetics
- DNA Sequencing
- Shortest Superstring Problem

- 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 of

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

How to Use Graph: Knights Problem 2 to a neighboring point in the diagram, in either a clockwise or counterclockwise direction.

- 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 the correct sequence of moves.

- 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 the correct sequence of moves.

- 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 correct sequence of moves.

Find a tour crossing every bridge just once

Leonhard Euler, 1735

Bridges of Königsberg

Eulerian Cycle Problem the correct sequence of moves.

- Find a cycle that visits every edgeexactly once.
- Graph theory was born when Leonhard Euler solved the famous Königsberg Bridge problem.

More complicated Königsberg

Can you travel from any one of the the correct sequence of moves.vertices in this graph, visit every other vertex exactly once, and end up at the original vertex?

Hamiltonian Cycle ProblemGame invented by Sir

William Hamilton in 1857

Trees the correct sequence of moves.

- 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 the correct sequence of moves.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 the correct sequence of moves.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.

Seymour Benzer, 1950s the correct sequence of moves.

2: Graphs and GeneticsBenzer’s work

- Developed deletion mapping
- “Proved” linearity of the gene
- Demonstrated internal structure of the gene

Viruses Attack Bacteria the correct sequence of moves.

- 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 the correct sequence of moves.’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 the correct sequence of moves.’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 the correct sequence of moves.

Interval Graph: Branched Genes the correct sequence of moves.

3: the correct sequence of moves.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.

Start at primer (restriction site) the correct sequence of moves.

Grow DNA chain

Include ddNTPs

Stops reaction at all possible points

Separate products by length, using gel electrophoresis

Sanger Method: Generating ReadDNA Sequencing the correct sequence of moves.

- Shear DNA into millions of small fragments
- Read 500 – 700 nucleotides at a time from the small fragments (Sanger method)

Fragment Assembly the correct sequence of moves.

- 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: the correct sequence of moves.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 the correct sequence of moves.

- Concatenating all eight strings results in a 24-letter superstring
- the shortest superstring contains only 10 letters.

Conclusion and Qustions the correct sequence of moves.

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

- Graphs and Genetics
- DNA Sequencing
- Shortest Superstring Problem

References the correct sequence of moves.

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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