Some algorithmic background

Biology 162 Computational Genetics

Todd VisionFall 2004

26 Aug 2004

Some algorithmic background

• Algorithms– Analysis of time and memory requirements– NP completeness

• Graphs– Travelling salesman problem

• DNA computers• Strings and Sequences• Recursion

Algorithm• A finite set of rules that gives a sequence of

operations for solving a problem suitable for implementation by a computer

• A correct algorithm will solve all instances of a problem

• An algorithm can be implemented – Multiple ways– In different languages– On different hardware architectures

• The choice of algorithm is usually far more important to time/memory usage than implementation

Knuth’s 5 features of an algorithm

• Finiteness - guaranteed to terminate• Definiteness - each step precisely

defined• Effectiveness - each step must be

small• Defined inputs• Defined outputs

Analysis of algorithms

• Mathematical description of time and memory requirements– Algorithm efficiency

• Time and memory are a function of the size of the problem instance f(x)

• Efficiency generally expressed in Big O notation– Assuming the instance is a worst-case scenario– Describes how time/memory scale as problem

size grows asymptotically large

Big O notation

• O(n), or “order n”, where n is the highest order term in f(x)

• For small instances, an O(n2) algorithm may be faster than an O(n) algorithm

• The notation does not account for constant factors, which may affect comparisons

• The big O notation does not allow one to actually predict the running time or memory usage

• Average running time may be much better than worst-case

Algorithm efficiency

• An algorithm is efficient if the running time is bounded by a polynomial – O(n4) yes– O(4n) no– O(4log(n)) gray area

• Problems are considered to be of class– P if a deterministic efficient algorithm exists– NP if no such algorithm has yet been found– NP-complete if a nondeterministic

polynomial time algorithm exists

Are NP-complete problems in class P?

• If any NP-complete problem is provably in class P, then all NP-complete problems must be!

• Strictly, this applies only to decision problems

• Corresponding optimization problems must be at least as hard, and are referred to as NP-hard

• Many of the most interesting problems in computational biology are NP-complete or NP-hard

Algorithms without optimality guarantees

• Approximation algorithm– For many NP-hard problems, polynomial-

time algorithms exist that can provably give answers within some small factor of the optimal answer

• Heuristic algorithm– An algorithm that may be sensible, and may

work in practice, but is not necessarily efficient and has no guarantee of finding a solution within of the optimal one

Travelling salesman problem

• A salesman must visit each city on a list exactly once, covering the smallest number of miles in total

• Classic NP-hard problem• Excellent approximate algorithms exist• Many computational biology problems

are solved by casting them as instances of the TSP and then applying an existing algorithm

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Travelling salesman problem

New York

Los Angeles

Dallas

Chicago

Miami

2790

2720

1540

2050

1190

1610 1090 1400

810

1330

Graph jargon• A graph G(V, E) is composed of a set of

vertices (V) and edges (E)• Vertices are also known as nodes• The edges, and thus the graphs, may be

– Directed, if edges have a head at one vertex and a tail at the other

– Undirected otherwise

• The degree of a vertex is the number of adjacent vertices– For directed graphs, vertices have an indegree

and an outdegree

Graph jargon• Weighted graphs have a cost or distance w(Ei) on

each edge i (as in the TSP)• A path is a list of vertices (v1,v2..vk) where (vi,vi+1)

are adjacent– The weight of a path is the sum of the weights on each

edge

– A cycle is a path which returns to the same vertex

• Acyclic graphs have no paths that are cyclic• Acyclic undirected graphs are trees

– The phylogenetic trees that biologists know and love– Important data structures €

w(E i)i=1

k

∑

Graph jargon

• Connected components are sets of vertices for which– No adjacent vertices are excluded– Do not contain subsets of vertices

that are themselves connected components

Eulerian graph

• Contains a cycle in which each edge appears exactly once

• A Eulerian path can be found with an algorithm that is O(n+m) in the number of vertices n and edges m

1

27

3

4

5

68

Hamiltonian graph

• Contains a cycle in which each vertex appears exactly once

• The objective of the TSP is to find a Hamiltonian path with minimal weight

• Problems with Hamiltonian paths are NP-hard

DNA computing

• In 1994, Leonard Adleman implemented a DNA computer that could solve for a Hamiltonian cycle in a graph

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DNA computing

• Outline of algorithm – Generate all possible routes– Select itineraries that start with the proper city

and end with the final city– Select itineraries with the correct number of

cities– Select itineraries that contain each city only

once

• Each step corresponds to the application of a standard molecular biology reaction

DNA computing

Cities are encoded by oligonucleotidesLos Angeles GCTACGChicago CTAGTADallas TCGTACMiami CTACGGNew York ATGCCG

The path (LA, Chicago, Dallas, Miami, New York) would be:

GCTACG CTAGTA TGCTAC CTACGG ATGCCG

DNA computing

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DNA computing• Random itineraries obtained by

– mixing oligonucleotides encoding both cities and routes in a test tube

– Allowing complementary DNA strands to hybridize

– Adding ligase to glue the pieces together

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DNA computing

• Select for paths that start in LA and end in NY – By performing the polymerase chain

reaction with LA and NY specific primers

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XX

DNA computing• Select paths of the appropriate

length (5 cities = 30 bases) by isolating the correct band from an electrophoretic gel

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DNA computing

• Select paths in which each city is represented by affinity purification with probes complementary to each city

• A path of length 5 containing each city once must be a Hamiltonian Path

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DNA computing

• Is this practical? – No. A 200 city HP problem would require

more DNA than the weight of the Earth

• Is this useful?– Yes.

• DNA operations are inherently massively parallel, making simultaneous evaluation of 1015 molecules feasible

• Silicon-chip computers perform only sequential operations and cannot deal with large combinatorial problems by exhaustive search

Stretching the analogy

• Many biological operations can be thought of in algorithmic terms

• Specific proteins act in defined sequences on a variable set of inputs to produce a definite output

• Cell division• Neuronal firing• Protein secretion

Segue to sequence analysis

• DNA and protein sequences will be the center of our attention for much of the course

• We need to be able to precisely describe algorithms that have these molecules as inputs and outputs

Sequences and strings• Biologists and computer scientists use the

words string and sequence differently• You will see “sequence” used in both ways in

this class• In CS jargon

– A string S is an contiguous ordered set of symbols– A sequence is an ordered set of letters that need not

be continuous• If ABCDEFGH is a string• ACEG is a sequence

• All strings are sequences, but not all sequences are strings

String jargon

• W.r.t. some alphabet A– For DNA, A={a,c,g,t}– For proteins, there are 20 symbols in the alphabet

• A DNA string: S=‘acgtgc’• The length of a string is given by |S|=6• Index the ith position in S by S[i]• An interval S[i..j] defines a substring of S• S is a superstring of all its component

substrings• S[1..j] is a prefix and S[j..|S|] is is a suffix of S

Alignment as a string edit

• We can define edit operations on S– Substitution– Insertion– Deletion

• Objective functions– One way to formulate the sequence

alignment problem is “transform S into S’ with a minimal edit distance” (ie fewest operations)

– Equivalently, we can seek an alignment with a maximal score

Pairwise alignment

• Scores reflects a ratio of– Probability of alignment under evolutionary

model– Probability of a chance alignment – Expressed as a Log Odds, or LOD, ratio

• Total score is simply the sum of scores for each edit operation

• A brute force algorithm– Enumerate all possible alignments and

choose the one(s) with highest score

Combinatorial explosion!

€

2n

n

⎛

⎝ ⎜

⎞

⎠ ⎟=

(2n)!

(n!)2≈

22n

πn

n # of alignments

5 258

10 187,126

15 156,454,989

20 1.4 x 1011

25 1.3 x 1014

Dynamic programming

• Efficient (ie polynomial-time) algorithm that guarantees finding an optimal pairwise alignment

• O(n2) where n is the the length of the sequences

• Comes in a few flavors– Global (Needleman-Wunsch)– Local (Smith-Waterman)– Multiple segments– Repeats, overlaps, etc.

Recursion

• Principle of dynamic programming is that the solution to a large instance can be recursively found from solutions to smaller instances

€

24 = 2 ⋅23 = 2 ⋅(2 ⋅22) = 2 ⋅(2 ⋅(2 ⋅2))

Reading assignments

• Gibson & Muse, Box 2.1 Pairwise sequence alignment, pgs 72-75.

• Durbin R, Eddy S, Krogh A, Mitchison G (1998) “Ch. 2: Pairwise alignment”, pgs, 12-31 in Biological sequence analysis, Cambridge Univ. Press.