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In terms of mathematical optimization, dynamic programming usually refers to simplifying 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, …, V_n taking y as an argument representing the state of the system at times i from 1 to n.

The definition of V_n(y) is the value obtained in state y at the last time n.

The values V_i at earlier times i = n −1, n − 2, …, 2, 1 can be found by working backwards, using a recursive relationship called the Bellman equation.

For i = 2, …, n, V_i−1 at any state y is calculated from V_i by maximizing a simple function (usually the sum) of the gain from a decision at time i − 1 and the function V_i at the new state of the system if this decision is made.

Since V_i has already been calculated for the needed states, the above operation yields V_i−1 for those 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.

In control theory, a typical problem is to find an admissible control   which causes the system   to follow an admissible trajectory   on a continuous time interval   that minimizes a cost function

The solution to this problem is an optimal control law or policy  , which produces an optimal trajectory   and a cost-to-go function  . The latter obeys the fundamental equation of dynamic programming:

a partial differential equation known as the Hamilton–Jacobi–Bellman equation, in which   and  . One finds that minimizing   in terms of  ,  , and the unknown function   and then substitutes the result into the Hamilton–Jacobi–Bellman equation to get the partial differential equation to be solved with boundary condition  .^[2] In practice, this generally requires numerical techniques for some discrete approximation to the exact optimization relationship.

Alternatively, the continuous process can be approximated by a discrete system, which leads to a following recurrence relation analog to the Hamilton–Jacobi–Bellman equation:

at the  -th stage of   equally spaced discrete time intervals, and where   and   denote discrete approximations to   and  . This functional equation is known as the Bellman equation, which can be solved for an exact solution of the discrete approximation of the optimization equation.^[3]

In economics, the objective is generally to maximize (rather than minimize) some dynamic social welfare function. In Ramsey’s problem, this function relates amounts of consumption to levels of utility. Loosely speaking, the planner faces the trade-off between contemporaneous consumption and future consumption (via investment in capital stock that is used in production), known as intertemporal choice. Future consumption is discounted at a constant rate  . A discrete approximation to the transition equation of capital is given by

where   is consumption,   is capital, and   is a production function satisfying the Inada conditions. An initial capital stock

  subject to   for all  

Written this way, the problem looks complicated, because it involves solving for all the choice variables  . (The capital   is not a choice variable—the consumer’s initial capital is taken as given.)

The dynamic programming approach to solve this problem involves breaking it apart into a sequence of smaller decisions. To do so, we define a sequence of value functions  , for   which represent the value of having any amount of capital k at each time t. 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  , 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,   and  . 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   is given, and he only needs to choose current consumption   and saving  .

To actually solve this problem, we work backwards. For simplicity, the current level of capital is denoted as k.   is already known, so using the Bellman equation once we can calculate  , and so on until we get to  , which is the value of the initial decision problem for the whole lifetime. In other words, once we know  , we can calculate  , which is the maximum of  , where   is the choice variable and  .

Working backwards, it can be shown that the value function at time   is

where each   is a constant, and the optimal amount to consume at time   is

which can be simplified to

We see that it is optimal to consume a larger fraction of current wealth as one gets older, finally consuming all remaining wealth in period T, the last period of life.

There are two key attributes that a problem must have in order for dynamic programming to be applicable: optimal substructure and overlapping sub-problems. If a problem can be solved by combining optimal solutions to non-overlapping sub-problems, the strategy is called “divide and conquer” instead.^[1] This is why merge sort and quick sort 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 sub-problems. Such optimal substructures are usually described by means of recursion. 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 it can be split into sub-paths p₁ from u to w and p₂ from w to v such that these, in turn, are indeed the shortest paths between the corresponding vertices (by the simple cut-and-paste argument described in Introduction to Algorithms). Hence, one can easily formulate the solution for finding shortest paths in a recursive manner, which is what the Bellman–Ford algorithm or the Floyd–Warshall algorithm does.

Overlapping sub-problems means that the space of sub-problems must be small, that is, any recursive algorithm solving the problem should solve the same sub-problems over and over, rather than generating new sub-problems. For example, consider the recursive formulation for generating the Fibonacci sequence: F_i = F_i−1 + F_i−2, with base case F₁ = F₂ = 1. Then F₄₃ = F₄₂ + F₄₁, and F₄₂ = F₄₁ + F₄₀. Now F₄₁ is being solved in the recursive sub-trees of both F₄₃ as well as F₄₂. Even though the total number of sub-problems 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 sub-problem only once.

This can be achieved in either of two ways:^[4]

• 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 sub-problems, and if its sub-problems are overlapping, then one can easily memoize or store the solutions to the sub-problems in a table (often an array or hashtable in practice). Whenever we attempt to solve a new sub-problem, 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 sub-problem and add its solution to the table.
• Bottom-up approach: Once we formulate the solution to a problem recursively as in terms of its sub-problems, we can try reformulating the problem in a bottom-up fashion: try solving the sub-problems first and use their solutions to build-on and arrive at solutions to bigger sub-problems. This is also usually done in a tabular form by iteratively generating solutions to bigger and bigger sub-problems by using the solutions to small sub-problems. 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-name evaluation (this mechanism is referred to as call-by-need). Some languages make it possible portably (e.g. Scheme, Common Lisp, Perl or D). Some languages have automatic memoization built in, such as tabled Prolog and J, which supports memoization with the M. adverb.^[5] In any case, this is only possible for a referentially transparent function. Memoization is also encountered as an easily accessible design pattern within term-rewrite based languages such as Wolfram Language.

Dynamic programming is widely used in bioinformatics for tasks such as sequence alignment, protein folding, RNA structure prediction and protein-DNA binding. The first dynamic programming algorithms for protein-DNA binding were developed in the 1970s independently by Charles DeLisi in the US^[6] and by Georgii Gurskii and Alexander Zasedatelev in the Soviet Union.^[7] Recently these algorithms have become very popular in bioinformatics and computational biology, particularly in the studies of nucleosome positioning and transcription factor binding.

From a dynamic programming point of view, Dijkstra’s algorithm for the shortest path problem is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.^[8][9][10]

In fact, Dijkstra’s explanation of the logic behind the algorithm,^[11] namely

Problem 2. Find the path of minimum total length between two given nodes   and  .

We use the fact that, if   is a node on the minimal path from   to  , knowledge of the latter implies the knowledge of the minimal path from   to  .

is a paraphrasing of Bellman’s famous Principle of Optimality in the context of the shortest path problem.

Using dynamic programming in the calculation of the nth member of the Fibonacci sequence improves its performance greatly. Here is a naïve implementation, based directly on the mathematical definition:

function fib(n)
    if n <= 1 return n
    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:

1. fib(5)
2. fib(4) + fib(3)
3. (fib(3) + fib(2)) + (fib(2) + fib(1))
4. ((fib(2) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))
5. (((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 (but requires O(n) space):

var m := map(0 → 0, 1 → 1)
function fib(n)
    if key n is not in map m 
        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, but 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)
    if n = 0
        return 0
    else
        var previousFib := 0, currentFib := 1
        repeat n − 1 times // loop is skipped if n = 1
            var newFib := previousFib + currentFib
            previousFib := currentFib
            currentFib  := newFib
        return currentFib

In both 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.

Consider the problem of assigning values, either zero or one, to the positions of an n × n matrix, with n even, so that each row and each column contains exactly n / 2 zeros and n / 2 ones. We ask how many different assignments there are for a given n. For example, when n = 4, five possible solutions are

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 and   sensible assignments, this strategy is not practical for arbitrarily large values of  .

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. Imagine backtracking values for the first row – what information would we require about the remaining rows, in order to be able to accurately count the solutions obtained for each first row value? We consider k × n boards, where 1 ≤ k ≤ n, whose k rows contain   zeros and   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 zeros and ones 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 k × n board and recursively compute the number of solutions to the remaining (k − 1) × n 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 1 × n 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 first 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 the OEIS) is

1, 2, 90, 297200, 116963796250, 6736218287430460752, …

Links to the MAPLE implementation of the dynamic programming approach may be found among the external links.

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),

Thus c(1, 3) = 5

Let us say there was a checker that could start at any square on the first rank (i.e., row) and you wanted to know the shortest path (the sum of the minimum costs at each visited rank) 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).

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).

Starting at rank n and descending to rank 1, we compute the value of this function for all the squares at each successive rank. Picking the square that holds the minimum value at each rank gives us the shortest path between rank n and rank 1.

The function 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:

Now, let us define q(i, j) in somewhat more general terms:

The term dynamic programming 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 refined this to the modern meaning, referring specifically to nesting smaller decision problems inside larger decisions,^[17] and the field was thereafter recognized by the IEEE as a systems 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.

Bellman explains the reasoning behind the term dynamic programming in his autobiography, Eye of the Hurricane: An Autobiography:

I spent the Fall quarter (of 1950) at RAND. My first task was to find a name for multistage decision processes. An interesting question is, “Where did the name, dynamic programming, come from?” The 1950s were not good years for mathematical research. We had a very interesting gentleman in Washington named Wilson. He was Secretary of Defense, and he actually had a pathological fear and hatred of the word “research”. I’m not using the term lightly; I’m using it precisely. His face would suffuse, he would turn red, and he would get violent if people used the term research in his presence. You can imagine how he felt, then, about the term mathematical. The RAND Corporation was employed by the Air Force, and the Air Force had Wilson as its boss, essentially. Hence, I felt I had to do something to shield Wilson and the Air Force from the fact that I was really doing mathematics inside the RAND Corporation. What title, what name, could I choose? In the first place I was interested in planning, in decision making, in thinking. But planning, is not a good word for various reasons. I decided therefore to use the word “programming”. I wanted to get across the idea that this was dynamic, this was multistage, this was time-varying. I thought, let’s kill two birds with one stone. Let’s take a word that has an absolutely precise meaning, namely dynamic, in the classical physical sense. It also has a very interesting property as an adjective, and that is it’s impossible to use the word dynamic in a pejorative sense. Try thinking of some combination that will possibly give it a pejorative meaning. It’s impossible. Thus, I thought dynamic programming was a good name. It was something not even a Congressman could object to. So I used it as an umbrella for my activities.

The word dynamic was chosen by Bellman to capture the time-varying aspect of the problems, and because it sounded impressive.^[12] The word programming referred to the use of the method to find an optimal program, in the sense of a military schedule for training or logistics. This usage is the same as that in the phrases linear programming and mathematical programming, a synonym for mathematical optimization.^[18]

The above explanation of the origin of the term may be inaccurate: According to Russell and Norvig, the above story “cannot be strictly true, because his first paper using the term (Bellman, 1952) appeared before Wilson became Secretary of Defense in 1953.”^[19] Also, Harold J. Kushner stated in a speech that, “On the other hand, when I asked [Bellman] the same question, he replied that he was trying to upstage Dantzig’s linear programming by adding dynamic. Perhaps both motivations were true.”^[20]

• Adda, Jerome; Cooper, Russell (2003), Dynamic Economics, MIT Press, ISBN 9780262012010. An accessible introduction to dynamic programming in economics. MATLAB code for the book Archived 2020-10-09 at the Wayback Machine.
• Bellman, Richard (1954), “The theory of dynamic programming”, Bulletin of the American Mathematical Society, 60 (6): 503–516, doi:10.1090/S0002-9904-1954-09848-8, MR 0067459. Includes an extensive bibliography of the literature in the area, up to the year 1954.
• Bellman, Richard (1957), Dynamic Programming, Princeton University Press. Dover paperback edition (2003), ISBN 0-486-42809-5.
• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), Introduction to Algorithms (2nd ed.), MIT Press & McGraw–Hill, ISBN 978-0-262-03293-3. Especially pp. 323–69.
• Dreyfus, Stuart E.; Law, Averill M. (1977), The Art and Theory of Dynamic Programming, Academic Press, ISBN 978-0-12-221860-6.
• Giegerich, R.; Meyer, C.; Steffen, P. (2004), “A Discipline of Dynamic Programming over Sequence Data” (PDF), Science of Computer Programming, 51 (3): 215–263, doi:10.1016/j.scico.2003.12.005.
• Meyn, Sean (2007), Control Techniques for Complex Networks, Cambridge University Press, ISBN 978-0-521-88441-9, archived from the original on 2010-06-19.
• Sritharan, S. S. (1991). “Dynamic Programming of the Navier-Stokes Equations”. Systems and Control Letters. 16 (4): 299–307. doi:10.1016/0167-6911(91)90020-f.
• Stokey, Nancy; Lucas, Robert E.; Prescott, Edward (1989), Recursive Methods in Economic Dynamics, Harvard Univ. Press, ISBN 978-0-674-75096-8.

• King, Ian, 2002 (1987), “A Simple Introduction to Dynamic Programming in Macroeconomic Models.” An introduction to dynamic programming as an important tool in economic theory.