Functional genomics
Functional genomics is a field of molecular biology that attempts to make use of the vast wealth of data produced by genomic projects (such as genome sequencing projects) to describe gene (and protein) functions and interactions. Unlike genomics and proteomics, functional genomics focuses on the dynamic aspects such as gene transcription, translation, and protein-protein interactions, as opposed to the static aspects of the genomic information such as DNA sequenceor structures. Functional genomics attempts to answer questions about the function of DNA at the levels of genes, RNA transcripts, and protein products. A key characteristic of functional genomics studies is their genome-wide approach to these questions, generally involving high-throughput methods rather than a more traditional “gene-by-gene” approach.
Goals of functional genomics
The goal of functional genomics is to understand the relationship between an organism's genome and its phenotype. The term functional genomics is often used broadly to refer to the many possible approaches to understanding the properties and function of the entirety of an organism's genes and gene products. This definition is somewhat variable; Gibson and Muse define it as "approaches under development to ascertain the biochemical, cellular, and/or physiological properties of each and every gene product"[1], while Pevsner includes the study of nongenic elements in his definition: "the genome-wide study of the function of DNA (including genes and nongenic elements), as well as the nucleic acid and protein products encoded by DNA".[2]Because of its genome-wide approach, functional genomics requires the use of high-throughput technologies capable of assaying many functions or relationships simultaneously. Functional genomics involves studies of natural variation in genes, RNA, and proteins over time (such as an organism's development) or space (such as its body regions), as well as studies of natural or experimental functional disruptions affecting genes, chromosomes, RNAs, or proteins.
The promise of functional genomics is to expand and synthesize genomic and proteomic knowledge into an understanding of the dynamic properties of an organism at cellular and/or organismal levels. This would provide a more complete picture of how biological function arises from the information encoded in an organism's genome. The possibility of understanding how a particular mutation leads to a given phenotype has important implications for human genetic diseases, as answering these questions could point scientists in the direction of a treatment or cure.
Techniques and Applications
Functional genomics includes function-related aspects of the genome itself such as mutation and polymorphism (such as SNP) analysis, as well as measurement of molecular activities. The latter comprise a number of "-omics" such as transcriptomics (gene expression), proteomics (protein expression), phosphoproteomics (a subset of proteomics) and metabolomics. Functional genomics uses mostly multiplex techniques to measure the abundance of many or all gene products such as mRNAs or proteins within a biological sample. Together these measurement modalities endeavor to quantitate the various biological processes and improve our understanding of gene and protein functions and interactions.
At the DNA level
Genetic interaction mapping
Main article: biotecsynthesis
Systematic pairwise deletion of genes or inhibition of gene expression can be used to identify genes with related function, even if they do not interact physically. Epistasis refers to the fact that effects for two different gene knockouts may not be additive; that is, the phenotype that results when two genes are inhibited may be different from the sum of the effects of single knockouts.
The ENCODE project
The ENCODE (Encyclopedia of DNA elements) project is an in-depth analysis of the human genome whose goal is to identify all the functional elements of genomic DNA, in both coding and noncoding regions. To this point, only the pilot phase of the study has been completed, involving hundreds of assays performed on 44 regions of known or unknown function comprising 1% of the human genome. Important results include evidence from genomic tiling arrays that most nucleotides are transcribed as coding transcripts, noncoding RNAs, or random transcripts, the discovery of additional transcriptional regulatory sites, further elucidation of chromatin-modifying mechanisms.
At the RNA level: transcriptome profiling
Microarrays
Main article: DNA microarray
Microarrays measure the amount of mRNA in a sample that corresponds to a given gene or probe DNA sequence. Probe sequences are immobilized on a solid surface and allowed tohybridize with fluorescently-labeled “target” mRNA. The intensity of fluorescence of a spot is proportional to the amount of target sequence that has hybridized to that spot, and therefore to the abundance of that mRNA sequence in the sample. Microarrays allow for identification of candidate genes involved in a given process based on variation between transcript levels for different conditions and shared expression patterns with genes of known function.
SAGE
Main article: Serial analysis of gene expression
SAGE (Serial analysis of gene expression) is an alternate method of gene expression analysis based on RNA sequencing rather than hybridization. SAGE relies on the sequencing of 10-17 base pair tags which are unique to each gene. These tags are produced from poly-A mRNA and ligated end-to-end before sequencing. SAGE gives an unbiased measurement of the number of transcripts per cell, since it does not depend on prior knowledge of what transcripts to study (as microarrays do).
At the protein level: protein-protein interactions
Yeast two-hybrid system
Main article: Yeast two hybrid
A yeast two-hybrid (Y2H) screen tests a "bait" protein against many potential interacting proteins ("prey") to identify physical protein-protein interactions. This system is based on a transcription factor, originally GAL4[3], whose separate DNA-binding and transcription activation domains are both required in order for the protein to cause transcription of a reporter gene. In a Y2H screen, the "bait" protein is fused to the binding domain of GAL4, and a library of potential "prey" (interacting) proteins is recombinantly expressed in a vector with the activation domain. In vivo interaction of bait and prey proteins in a yeast cell brings the activation and binding domains of GAL4 close enough together to result in expression of a reporter gene. It is also possible to systematically test a library of bait proteins against a library of prey proteins to identify all possible interactions in a cell
AP/MS
Affinity purification and mass spectrometry (AP/MS) is able to identify proteins that interact with one another in complexes. Complexes of proteins are allowed to form around a particular “bait” protein. The bait protein is identified using an antibody or a recombinant tag which allows it to be extracted along with any proteins that have formed a complex with it. The proteins are then digested into short peptide fragments and mass spectrometry is used to identify the proteins based on the mass-to-charge ratios of those fragments.
Comparative genomics
Comparative genomics is the study of the relationship of genome structure and function across different biological species or strains. Comparative genomics is an attempt to take advantage of the information provided by the signatures of selection to understand the function and evolutionary processes that act on genomes. While it is still a young field, it holds great promise to yield insights into many aspects of the evolution of modern species. The sheer amount of information contained in modern genomes (750 megabytes in the case of humans) necessitates that the methods of comparative genomics are automated. Gene finding is an important application of comparative genomics, as is discovery of new, non-coding functional elements of the genome.
One of the important goals of the field is the identification of the mechanisms of eukaryotic genome evolution. It is however often complicated by the multiplicity of events that have taken place throughout the history of individual lineages, leaving only distorted and superimposed traces in the genome of each living organism. For this reason comparative genomics studies of small model organisms (for example yeast) are of great importance to advance our understanding of general mechanisms of evolution.Comparative genomics exploits both similarities and differences in the proteins, RNA, and regulatory regions of different organisms to infer howselection has acted upon these elements. Those elements that are responsible for similarities between different species should be conserved through time (stabilizing selection), while those elements responsible for differences among species should be divergent (positive selection). Finally, those elements that are unimportant to the evolutionary success of the organism will be unconserved (selection is neutral).
Having come a long way from its initial use of finding functional proteins, comparative genomics is now concentrating on finding regulatory regionsand siRNA molecules. Recently, it has been discovered that distantly related species often share long conserved stretches of DNA that do not appear to code for any protein [1]. One such ultra-conserved region, that was stable from chicken to chimp has undergone a sudden burst of change in the human lineage, and is found to be active in the developing brain of the human embryo [2].
Computational approaches to genome comparison have recently become a common research topic in computer science. A public collection of case studies and demonstrations is growing, ranging from whole genome comparisons to gene expression analysis.[3] This has increased the introduction of different ideas, including concepts from systems and control, information theory, strings analysis and data mining. It is anticipated that computational approaches will become and remain a standard topic for research and teaching, while multiple courses will begin training students to be fluent in both topics.
Dynamic programming
In mathematics and computer science, dynamic programming is a method of solving complex problems by breaking them down into simpler steps. It is applicable to problems that exhibit the properties of overlapping subproblems which are only slightly smaller[1] and optimal substructure (described below). When applicable, the method takes much less time than naive methods.
Top-down dynamic programming simply means storing the results of certain calculations, which are then re-used later because the same calculation is a sub-problem in a larger calculation. Bottom-up dynamic programming involves formulating a complex calculation as a recursive series of simpler calculations.
History
The term was originally used in the 1940s by Richard Bellman to describe the process of solving problems where one needs to find the best decisions one after another. By 1953, he had refined this to the modern meaning, which refers specifically to nesting smaller decision problems inside larger decisions,[2] and the field was thereafter recognized by the IEEE as asystems analysis and engineering topic. Bellman's contribution is remembered in the name of the Bellman equation, a central result of dynamic programming which restates an optimization problem in recursive form.
Originally the word "programming" in "dynamic programming" had no connection to computer programming, and instead came from the term "mathematical programming"[3] - a synonym foroptimization. However, nowadays many optimization problems are best solved by writing a computer program that implements a dynamic programming algorithm, rather than carrying out hundreds of tedious calculations by hand. Some of the examples given below are illustrated using computer programs.
Overview
Dynamic programming is both a mathematical optimization method, and a computer programming method. In both contexts, it refers to simplifying a complicated problem by breaking it down into simpler subproblems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively; Bellman called this the "Principle of Optimality". Likewise, in computer science, a problem which can be broken down recursively is said to have optimal substructure.
If subproblems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the subproblems.[4] In the optimization literature this relationship is called the Bellman equation.
Figure 1. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (other nodes on these paths are not shown); the bold line is the overall shortest path from start to goal.
Dynamic programming in mathematical optimization
In terms of mathematical optimization, dynamic programming usually refers to a simplification of a decision by breaking it down into a sequence of decision steps over time. This is done by defining a sequence of value functions V1 , V2 , ... Vn , with an argument y representing the state of the system at times i from 1 to n. The definition of Vn(y) is the value obtained in state y at the last time n. The values Vi at earlier times i=n-1,n-2,...,2,1can be found by working backwards, using a recursive relationship called the Bellman equation. For i=2,...n, Vi -1 at any state y is calculated from Viby maximizing a simple function (usually the sum) of the gain from decision i-1 and the function Vi at the new state of the system if this decision is made. Since Vi has already been calculated, for the needed states, the above operation yields Vi -1 for all the needed states. Finally, V1 at the initial state of the system is the value of the optimal solution. The optimal values of the decision variables can be recovered, one by one, by tracking back the calculations already performed.
Dynamic programming in computer programming
There are two key attributes that a problem must have in order for dynamic programming to be applicable: optimal substructure and overlapping subproblems which are only slightly smaller. When the overlapping problems are, say, half the size of the original problem the strategy is called "divide and conquer" rather than "dynamic programming". This is why merge sort, and quick sort, and finding all matches of a regular expression are not classified as dynamic programming problems.
Optimal substructure means that the solution to a given optimization problem can be obtained by the combination of optimal solutions to its subproblems. Consequently, the first step towards devising a dynamic programming solution is to check whether the problem exhibits such optimal substructure. Such optimal substructures are usually described by means ofrecursion. For example, given a graph G=(V,E), the shortest path p from a vertex u to a vertex v exhibits optimal substructure: take any intermediate vertex w on this shortest path p. If p is truly the shortest path, then the path p1 from u to w and p2 from w to v are indeed the shortest paths between the corresponding vertices (by the simple cut-and-paste argument described inCLRS). Hence, one can easily formulate the solution for finding shortest paths in a recursive manner, which is what the Bellman-Ford algorithm does.
Overlapping subproblems means that the space of subproblems must be small, that is, any recursive algorithm solving the problem should solve the same subproblems over and over, rather than generating new subproblems. For example, consider the recursive formulation for generating the Fibonacci series: Fi = Fi-1 + Fi-2, with base case F1=F2=1. Then F43 = F42 + F41, and F42 = F41 + F40. Now F41 is being solved in the recursive subtrees of both F43 as well as F42. Even though the total number of subproblems is actually small (only 43 of them), we end up solving the same problems over and over if we adopt a naive recursive solution such as this. Dynamic programming takes account of this fact and solves each subproblem only once. Note that the subproblems must be only 'slightly' smaller (typically taken to mean a constant additive factor) than the larger problem; when they are a multiplicative factor smaller the problem is no longer classified as dynamic programming (otherwise mergesort and quicksort would be dynamic programming problems).
This can be achieved in either of two ways:[citation needed]
- Top-down approach: This is the direct fall-out of the recursive formulation of any problem. If the solution to any problem can be formulated recursively using the solution to its subproblems, and if its subproblems are overlapping, then one can easily memoize or store the solutions to the subproblems in a table. Whenever we attempt to solve a new subproblem, we first check the table to see if it is already solved. If a solution has been recorded, we can use it directly, otherwise we solve the subproblem and add its solution to the table.
- Bottom-up approach: This is the more interesting case. Once we formulate the solution to a problem recursively as in terms of its subproblems, we can try reformulating the problem in a bottom-up fashion: try solving the subproblems first and use their solutions to build-on and arrive at solutions to bigger subproblems. This is also usually done in a tabular form by iteratively generating solutions to bigger and bigger subproblems by using the solutions to small subproblems. For example, if we already know the values of F41 and F40, we can directly calculate the value of F42.
Some programming languages can automatically memoize the result of a function call with a particular set of arguments, in order to speed up call-by-nameevaluation (this mechanism is referred to as call-by-need). Some languages make it possible portably (e.g. Scheme, Common Lisp or Perl), some need special extensions (e.g. C++, see [5]). Some languages have automatic memoization built in. In any case, this is only possible for a referentially transparent function.
[edit]Example: mathematical optimization
[edit]Optimal consumption and saving
A mathematical optimization problem that is often used in teaching dynamic programming to economists (because it can be solved by hand[6]) concerns a consumer who lives over the periods t = 0,1,2,...,T and must decide how much to consume and how much to save in each period.
Let ct be consumption in period t, and assume consumption yields utility u(ct) = ln(ct) as long as the consumer lives. Assume the consumer is impatient, so that he discounts future utility by a factor b each period, where 0 < b < 1. Let kt be capital in period t. Assume initial capital is a given amount k0 > 0, and suppose that this period's capital and consumption determine next period's capital as 
, where A is a positive constant and 0 < a < 1. Assume capital cannot be negative. Then the consumer's decision problem can be written as follows:
, where A is a positive constant and 0 < a < 1. Assume capital cannot be negative. Then the consumer's decision problem can be written as follows:subject to
for all t = 0,1,2,...,T
Written this way, the problem looks complicated, because it involves solving for all the choice variables c0,c1,c2,...,cT and k1,k2,k3,...,kT + 1 simultaneously. (Note that k0 is not a choice variable—the consumer's initial capital is taken as given.)
The dynamic programming approach to solving this problem involves breaking it apart into a sequence of smaller decisions. To do so, we define a sequence of value functions Vt(k), fort = 0,1,2,...,T,T + 1 which represent the value of having any amount of capital k at each time t. Note that VT + 1(k) = 0, that is, there is (by assumption) no utility from having capital after death.
The value of any quantity of capital at any previous time can be calculated by backward induction using the Bellman equation. In this problem, for each t = 0,1,2,...,T, the Bellman equation is
subject to
This problem is much simpler than the one we wrote down before, because it involves only two decision variables, ct and kt + 1. Intuitively, instead of choosing his whole lifetime plan at birth, the consumer can take things one step at a time. At time t, his current capital kt is given, and he only needs to choose current consumption ct and saving kt + 1.
To actually solve this problem, we work backwards. For simplicity, the current level of capital is denoted as k. VT + 1(k) is already known, so using the Bellman equation once we can calculate VT(k), and so on until we get to V0(k), which is the value of the initial decision problem for the whole lifetime. In other words, once we know VT − j + 1(k), we can calculateVT − j(k), which is the maximum of ln(cT − j) + bVT − j + 1(Aka − cT − j), where cT − j is the variable and 
. It can be shown that the value function at time t = T − j is
. It can be shown that the value function at time t = T − j is
where each vT − j is a constant, and the optimal amount to consume at time t = T − j is
which can be simplified to
, and
, and
, etcetera.
We see that it is optimal to consume a larger fraction of current wealth as one gets older, finally consuming all current wealth in period T, the last period of life.
[edit]Examples: Computer algorithms
[edit]Fibonacci sequence
Here is a naive implementation of a function finding the nth member of the Fibonacci sequence, based directly on the mathematical definition:
function fib(n)
if n = 0 return 0
if n = 1 return 1
return fib(n − 1) + fib(n − 2)
Notice that if we call, say,
fib(5), we produce a call tree that calls the function on the same value many different times:fib(5)fib(4) + fib(3)(fib(3) + fib(2)) + (fib(2) + fib(1))((fib(2) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))(((fib(1) + fib(0)) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))
In particular,
fib(2) was calculated three times from scratch. In larger examples, many more values of fib, or subproblems, are recalculated, leading to an exponential time algorithm.Now, suppose we have a simple map object, m, which maps each value of
fib that has already been calculated to its result, and we modify our function to use it and update it. The resulting function requires only O(n) time instead of exponential time:var m := map(0 → 0, 1 → 1)
function fib(n)
if map m does not contain key n
m[n] := fib(n − 1) + fib(n − 2)
return m[n]
This technique of saving values that have already been calculated is called memoization; this is the top-down approach, since we first break the problem into subproblems and then calculate and store values.
In the bottom-up approach we calculate the smaller values of
fib first, then build larger values from them. This method also uses O(n) time since it contains a loop that repeats n − 1 times, however it only takes constant (O(1)) space, in contrast to the top-down approach which requires O(n) space to store the map.function fib(n)
var previousFib := 0, currentFib := 1
if n = 0
return 0
else if n = 1
return 1
repeat n − 1 times
var newFib := previousFib + currentFib
previousFib := currentFib
currentFib := newFib
return currentFib
In both these examples, we only calculate
fib(2) one time, and then use it to calculate both fib(4) and fib(3), instead of computing it every time either of them is evaluated.[edit]A type of balanced 0-1 matrix
Consider the problem of assigning values, either zero or one, to the positions of an n x n matrix, n even, so that each row and each column contains exactly n / 2 zeros and n / 2 ones. For example, when n = 4, three possible solutions are:
+ - - - - + + - - - - + + - - - - + | 0 1 0 1 | | 0 0 1 1 | | 1 1 0 0 | | 1 0 1 0 | and | 0 0 1 1 | and | 0 0 1 1 | | 0 1 0 1 | | 1 1 0 0 | | 1 1 0 0 | | 1 0 1 0 | | 1 1 0 0 | | 0 0 1 1 | + - - - - + + - - - - + + - - - - +
We ask how many different assignments there are for a given n. There are at least three possible approaches: brute force, backtracking, and dynamic programming. Brute force consists of checking all assignments of zeros and ones and counting those that have balanced rows and columns (n / 2 zeros and n / 2 ones). As there are 
possible assignments, this strategy is not practical except maybe up to n = 6. Backtracking for this problem consists of choosing some order of the matrix elements and recursively placing ones or zeros, while checking that in every row and column the number of elements that have not been assigned plus the number of ones or zeros are both at least n / 2. While more sophisticated than brute force, this approach will visit every solution once, making it impractical for n larger than six, since the number of solutions is already 116963796250 for n = 8, as we shall see. Dynamic programming makes it possible to count the number of solutions without visiting them all.
possible assignments, this strategy is not practical except maybe up to n = 6. Backtracking for this problem consists of choosing some order of the matrix elements and recursively placing ones or zeros, while checking that in every row and column the number of elements that have not been assigned plus the number of ones or zeros are both at least n / 2. While more sophisticated than brute force, this approach will visit every solution once, making it impractical for n larger than six, since the number of solutions is already 116963796250 for n = 8, as we shall see. Dynamic programming makes it possible to count the number of solutions without visiting them all.We consider 
boards, where 
whose k rows contain n / 2 zeros and n / 2 ones. The function f to which memoization is applied maps vectors of n pairs of integers to the number of admissible boards (solutions). There is one pair for each column and its two components indicate respectively the number of ones and zeros that have yet to be placed in that column. We seek the value of 
(n arguments or one vector of n elements). The process of subproblem creation involves iterating over every one of 
possible assignments for the top row of the board, and going through every column, subtracting one from the appropriate element of the pair for that column, depending on whether the assignment for the top row contained a zero or a one at that position. If any one of the results is negative, then the assignment is invalid and does not contribute to the set of solutions (recursion stops). Otherwise, we have an assignment for the top row of the board and recursively compute the number of solutions to the remaining 
board, adding the numbers of solutions for every admissible assignment of the top row and returning the sum, which is being memoized. The base case is the trivial subproblem, which occurs for a 
board. The number of solutions for this board is either zero or one, depending on whether the vector is a permutation of n / 2 (0,1) and n / 2 (1,0)pairs or not.
boards, where whose k rows contain n / 2 zeros and n / 2 ones. The function f to which memoization is applied maps vectors of n pairs of integers to the number of admissible boards (solutions). There is one pair for each column and its two components indicate respectively the number of ones and zeros that have yet to be placed in that column. We seek the value of (n arguments or one vector of n elements). The process of subproblem creation involves iterating over every one of possible assignments for the top row of the board, and going through every column, subtracting one from the appropriate element of the pair for that column, depending on whether the assignment for the top row contained a zero or a one at that position. If any one of the results is negative, then the assignment is invalid and does not contribute to the set of solutions (recursion stops). Otherwise, we have an assignment for the top row of the board and recursively compute the number of solutions to the remaining board, adding the numbers of solutions for every admissible assignment of the top row and returning the sum, which is being memoized. The base case is the trivial subproblem, which occurs for a board. The number of solutions for this board is either zero or one, depending on whether the vector is a permutation of n / 2 (0,1) and n / 2 (1,0)pairs or not.For example, in the two boards shown above the sequences of vectors would be
((2, 2) (2, 2) (2, 2) (2, 2)) ((2, 2) (2, 2) (2, 2) (2, 2)) k = 4 0 1 0 1 0 0 1 1 ((1, 2) (2, 1) (1, 2) (2, 1)) ((1, 2) (1, 2) (2, 1) (2, 1)) k = 3 1 0 1 0 0 0 1 1 ((1, 1) (1, 1) (1, 1) (1, 1)) ((0, 2) (0, 2) (2, 0) (2, 0)) k = 2 0 1 0 1 1 1 0 0 ((0, 1) (1, 0) (0, 1) (1, 0)) ((0, 1) (0, 1) (1, 0) (1, 0)) k = 1 1 0 1 0 1 1 0 0 ((0, 0) (0, 0) (0, 0) (0, 0)) ((0, 0) (0, 0), (0, 0) (0, 0))
The number of solutions (sequence A058527 in OEIS) is
Links to the Perl source of the backtracking approach, as well as a MAPLE and a C implementation of the dynamic programming approach may be found among the external links.
[edit]Checkerboard
Consider a checkerboard with n × n squares and a cost-function c(i, j) which returns a cost associated with square i,j (i being the row, j being the column). For instance (on a 5 × 5 checkerboard),
| 5 | 6 | 7 | 4 | 7 | 8 |
|---|---|---|---|---|---|
| 4 | 7 | 6 | 1 | 1 | 4 |
| 3 | 3 | 5 | 7 | 8 | 2 |
| 2 | - | 6 | 7 | 0 | - |
| 1 | - | - | 5* | - | - |
| 1 | 2 | 3 | 4 | 5 |
Thus c(1, 3) = 5
Let us say you had a checker that could start at any square on the first rank (i.e., row) and you wanted to know the shortest path (sum of the costs of the visited squares are at a minimum) to get to the last rank, assuming the checker could move only diagonally left forward, diagonally right forward, or straight forward. That is, a checker on (1,3) can move to (2,2), (2,3) or (2,4).
| 5 | |||||
|---|---|---|---|---|---|
| 4 | |||||
| 3 | |||||
| 2 | x | x | x | ||
| 1 | o | ||||
| 1 | 2 | 3 | 4 | 5 |
This problem exhibits optimal substructure. That is, the solution to the entire problem relies on solutions to subproblems. Let us define a function q(i, j) as
- q(i, j) = the minimum cost to reach square (i, j)
If we can find the values of this function for all the squares at rank n, we pick the minimum and follow that path backwards to get the shortest path.
Note that q(i, j) is equal to the minimum cost to get to any of the three squares below it (since those are the only squares that can reach it) plus c(i, j). For instance:
| 5 | |||||
|---|---|---|---|---|---|
| 4 | A | ||||
| 3 | B | C | D | ||
| 2 | |||||
| 1 | |||||
| 1 | 2 | 3 | 4 | 5 |
q(A) = min(q(B),q(C),q(D)) + c(A)
Now, let us define q(i, j) in somewhat more general terms:
The first line of this equation is there to make the recursive property simpler (when dealing with the edges, so we need only one recursion). The second line says what happens in the last rank, to provide a base case. The third line, the recursion, is the important part. It is similar to the A,B,C,D example. From this definition we can make a straightforward recursive code forq(i, j). In the following pseudocode, n is the size of the board,
c(i, j) is the cost-function, and min() returns the minimum of a number of values:function minCost(i, j)
if j < 1 or j > n
return infinity
else if i = 5
return c(i, j)
else
return min( minCost(i+1, j-1), minCost(i+1, j), minCost(i+1, j+1) ) + c(i, j)
It should be noted that this function only computes the path-cost, not the actual path. We will get to the path soon. This, like the Fibonacci-numbers example, is horribly slow since it spends mountains of time recomputing the same shortest paths over and over. However, we can compute it much faster in a bottom-up fashion if we store path-costs in a two-dimensional array
q[i, j] rather than using a function. This avoids recomputation; before computing the cost of a path, we check the array q[i, j] to see if the path cost is already there.We also need to know what the actual shortest path is. To do this, we use another array
p[i, j], a predecessor array. This array implicitly stores the path to any square s by storing the previous node on the shortest path to s, i.e. the predecessor. To reconstruct the path, we lookup the predecessor of s, then the predecessor of that square, then the predecessor of that square, and so on, until we reach the starting square. Consider the following code:function computeShortestPathArrays()
for x from 1 to n
q[1, x] := c(1, x)
for y from 1 to n
q[y, 0] := infinity
q[y, n + 1] := infinity
for y from 2 to n
for x from 1 to n
m := min(q[y-1, x-1], q[y-1, x], q[y-1, x+1])
q[y, x] := m + c(y, x)
if m = q[y-1, x-1]
p[y, x] := -1
else if m = q[y-1, x]
p[y, x] := 0
else
p[y, x] := 1
Now the rest is a simple matter of finding the minimum and printing it.
function computeShortestPath()
computeShortestPathArrays()
minIndex := 1
min := q[n, 1]
for i from 2 to n
if q[n, i] < min
minIndex := i
min := q[n, i]
printPath(n, minIndex)
function printPath(y, x)
print(x)
print("<-")
if y = 2
print(x + p[y, x])
else
printPath(y-1, x + p[y, x])
[edit]Sequence alignment
In genetics, sequence alignment is an important application where dynamic programming is essential.[7] Typically, the problem consists of transforming one sequence into another using edit operations that replace, insert, or remove an element. Each operation has an associated cost, and the goal is to find the sequence of edits with the lowest total cost.
The problem can be stated naturally as a recursion, a sequence A is optimally edited into a sequence B by either:
- inserting the first character of B, and performing an optimal alignment of A and the tail of B
- deleting the first character of A, and performing the optimal alignment of the tail of A and B
- replacing the first character of A with the first character of B, and performing optimal alignments of the tails of A and B.
The partial alignments can be tabulated in a matrix, where cell (i,j) contains the cost of the optimal alignment of A[1..i] to B[1..j]. The cost in cell (i,j) can be calculated by adding the cost of the relevant operations to the cost of its neighboring cells, and selecting the optimum.
Different variants exist, see Smith-Waterman and Needleman-Wunsch.
