Some algorithmic background

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Some algorithmic background

• Biology 162 Computational Genetics
• Todd VisionFall 2004
• 26 Aug 2004

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Some algorithmic background

• Algorithms
• Analysis of time and memory requirements
• NP completeness
• Graphs
• Travelling salesman problem
• DNA computers
• Strings and Sequences
• Recursion

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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

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Knuths 5 features of an algorithm

• Finiteness - guaranteed to terminate
• Definiteness - each step precisely defined
• Effectiveness - each step must be small
• Defined inputs
• Defined outputs

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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

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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

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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

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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

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Algorithms without optimality guarantees

• Approximation algorithm
• For many NP-hard problems, polynomial-time
  algorithms exist that can provably give answers
  within some small factor e 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 e
  of the optimal one

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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
810
Chicago
2050
1330
2790
Los Angeles
1090
1400
1610
2720
1540
Dallas
Miami
1190
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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

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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,vi1) 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

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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

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Eulerian graph

• Contains a cycle in which each edge appears
  exactly once
• A Eulerian path can be found with an algorithm
  that is O(nm) in the number of vertices n and
  edges m

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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

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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

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DNA computing

• Cities are encoded by oligonucleotides
• Los Angeles GCTACG
• Chicago CTAGTA
• Dallas TCGTAC
• Miami CTACGG
• New York ATGCCG
• The path (LA, Chicago, Dallas, Miami, New York)
  would be
• GCTACG CTAGTA TGCTAC CTACGG ATGCCG

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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

X
X
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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

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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

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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

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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

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String jargon

• W.r.t. some alphabet A
• For DNA, Aa,c,g,t
• For proteins, there are 20 symbols in the
  alphabet
• A DNA string Sacgtgc
• The length of a string is given by S6
• Index the ith position in S by Si
• An interval Si..j defines a substring of S
• S is a superstring of all its component
  substrings
• S1..j is a prefix and Sj..S is is a suffix
  of S

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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

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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

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Combinatorial explosion!
n of alignments
5 258
10 187,126
15 156,454,989
20 1.4 x 1011
25 1.3 x 1014
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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.

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Recursion

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

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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.