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{\displaystyle n}
equally spaced discrete time intervals, and where
f
^
{\displaystyle {\hat {f}}}
and
g
^
{\displaystyle {\hat {\mathbf {g} }}}
denote discrete approximations to
f
{\displaystyle f}
and
g
{\displaystyle \mathbf {g} }
. 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.
==== Example from economics: Ramsey's problem of optimal saving ====
In economics, the objective is generally to maximize (rather than minimize) some dynamic social welfare function
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han 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
β
∈
(
0
,
1
)
{\displaystyle \beta \in (0,1)}
. A discrete approximation to the transition equation of capital is given by
k
t
+
1
=
g
^
(
k
t
,
c
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|
,
c
t
)
=
f
(
k
t
)
−
c
t
{\displaystyle k_{t+1}={\hat {g}}\left(k_{t},c_{t}\right)=f(k_{t})-c_{t}}
where
c
{\displaystyle c}
is consumption,
k
{\displaystyle k}
is capital, and
f
{\displaystyle f}
is a production function satisfying the Inada conditions. An initial capital stock
k
0
>
0
{\displaystyle k_{0}>0}
is assumed.
Let
c
t
{\displaystyle c_
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{\displaystyle c_{t}}
be consumption in period t, and assume consumption yields utility
u
(
c
t
)
=
ln
(
c
t
)
{\displaystyle u(c_{t})=\ln(c_{t})}
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
{\displaystyle 0<b<1}
. Let
k
t
{\displaystyle k_{t}}
be capital in period t. Assume initial capital is a given amount
k
0
>
0
{\displaystyle k_{0}>0}
, and sup
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|
{\displaystyle k_{0}>0}
, and suppose that this period's capital and consumption determine next period's capital as
k
t
+
1
=
A
k
t
a
−
c
t
{\displaystyle k_{t+1}=Ak_{t}^{a}-c_{t}}
, where A is a positive constant and
0
<
a
<
1
{\displaystyle 0<a<1}
. Assume capital cannot be negative. Then the consumer's decision problem can be written as follows:
max
∑
t
=
0
T
b
t
ln
(
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|
ln
(
c
t
)
{\displaystyle \max \sum _{t=0}^{T}b^{t}\ln(c_{t})}
subject to
k
t
+
1
=
A
k
t
a
−
c
t
≥
0
{\displaystyle k_{t+1}=Ak_{t}^{a}-c_{t}\geq 0}
for all
t
=
0
,
1
,
2
,
…
,
T
{\displaystyle t=0,1,2,\ldots ,T}
Written this way, the problem looks complicated, because it involves solving for all the choice variables
c
0
,
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|
0
,
c
1
,
c
2
,
…
,
c
T
{\displaystyle c_{0},c_{1},c_{2},\ldots ,c_{T}}
. (The capital
k
0
{\displaystyle k_{0}}
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
V
t
(
k
)
{\displaystyle V_{t}(k)}
, for
t
=
0
,
1
,
2
,
…
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|
1
,
2
,
…
,
T
,
T
+
1
{\displaystyle t=0,1,2,\ldots ,T,T+1}
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,
V
T
+
1
(
k
)
=
0
{\displaystyle V_{T+1}(k)=0}
.
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
{\displaystyle t=0,1,2,\ldots ,T}
, the Bellman equation is
V
t
(
k
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(
k
t
)
=
max
(
ln
(
c
t
)
+
b
V
t
+
1
(
k
t
+
1
)
)
{\displaystyle V_{t}(k_{t})\,=\,\max \left(\ln(c_{t})+bV_{t+1}(k_{t+1})\right)}
subject to
k
t
+
1
=
A
k
t
a
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|
a
−
c
t
≥
0
{\displaystyle k_{t+1}=Ak_{t}^{a}-c_{t}\geq 0}
This problem is much simpler than the one we wrote down before, because it involves only two decision variables,
c
t
{\displaystyle c_{t}}
and
k
t
+
1
{\displaystyle k_{t+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
k
t
{\displaystyle k_{t}}
is given, and he only needs to choose current consumption
c
t
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|
c
t
{\displaystyle c_{t}}
and saving
k
t
+
1
{\displaystyle k_{t+1}}
.
To actually solve this problem, we work backwards. For simplicity, the current level of capital is denoted as k.
V
T
+
1
(
k
)
{\displaystyle V_{T+1}(k)}
is already known, so using the Bellman equation once we can calculate
V
T
(
k
)
{\displaystyle V_{T}(k)}
, and so on until we get to
V
0
(
k
)
{\displaystyle V_{0}(k)}
, which is th
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|
{\displaystyle V_{0}(k)}
, which is the value of the initial decision problem for the whole lifetime. In other words, once we know
V
T
−
j
+
1
(
k
)
{\displaystyle V_{T-j+1}(k)}
, we can calculate
V
T
−
j
(
k
)
{\displaystyle V_{T-j}(k)}
, which is the maximum of
ln
(
c
T
−
j
)
+
b
V
T
−
j
+
1
(
A
k
a
−
c
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|
−
c
T
−
j
)
{\displaystyle \ln(c_{T-j})+bV_{T-j+1}(Ak^{a}-c_{T-j})}
, where
c
T
−
j
{\displaystyle c_{T-j}}
is the choice variable and
A
k
a
−
c
T
−
j
≥
0
{\displaystyle Ak^{a}-c_{T-j}\geq 0}
.
Working backwards, it can be shown that the value function at time
t
=
T
−
j
{\displaystyle t=T-j}
is
V
T
−
j
(
k
)
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|
(
k
)
=
a
∑
i
=
0
j
a
i
b
i
ln
k
+
v
T
−
j
{\displaystyle V_{T-j}(k)\,=\,a\sum _{i=0}^{j}a^{i}b^{i}\ln k+v_{T-j}}
where each
v
T
−
j
{\displaystyle v_{T-j}}
is a constant, and the optimal amount to consume at time
t
=
T
−
j
{\displaystyle t=T-j}
is
c
T
−
j
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|
T
−
j
(
k
)
=
1
∑
i
=
0
j
a
i
b
i
A
k
a
{\displaystyle c_{T-j}(k)\,=\,{\frac {1}{\sum _{i=0}^{j}a^{i}b^{i}}}Ak^{a}}
which can be simplified to
c
T
(
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|
(
k
)
=
A
k
a
c
T
−
1
(
k
)
=
A
k
a
1
+
a
b
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|
a
b
c
T
−
2
(
k
)
=
A
k
a
1
+
a
b
+
a
2
|
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|
b
2
…
c
2
(
k
)
=
A
k
a
1
+
a
b
|
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|
a
b
+
a
2
b
2
+
…
+
a
T
−
2
b
T
−
2
c
|
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|
c
1
(
k
)
=
A
k
a
1
+
a
b
+
a
2
b
2
+
…
+
|
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|
…
+
a
T
−
2
b
T
−
2
+
a
T
−
1
b
T
−
1
|
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|
c
0
(
k
)
=
A
k
a
1
+
a
b
+
a
2
b
2
|
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|
+
…
+
a
T
−
2
b
T
−
2
+
a
T
−
1
b
T
−
1
+
|
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|
+
a
T
b
T
{\displaystyle {\begin{aligned}c_{T}(k)&=Ak^{a}\\c_{T-1}(k)&={\frac {Ak^{a}}{1+ab}}\\c_{T-2}(k)&={\frac {Ak^{a}}{1+ab+a^{2}b^{2}}}\\&\dots \\c_{2}(k)&={\frac {Ak^{a}}{1+ab+a^{2}b^{2}+\ldots +a^{T-2}b^{T-2}}}\\c_{1}(k)&={\frac {Ak^{a}}{1+ab+a^{2}b^{2}+\ldots +a^{T-2}b^{T-2}+a^{T-1}b^{T-1}}}\\c_{0}(k)&={\frac {Ak^{a}}{1+ab+a^{2}b^{2}+\ldots +a^{T-2}b^{T-2}+a^{T-1}b^{T-1}+a^{T}b^{T}}}\end{aligned}}}
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, t
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|
ally consuming all remaining wealth in period T, the last period of life.
=== Computer science ===
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. 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 t
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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: 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 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
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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:
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 sol
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lating 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. In any case, this is only possible for a referentially transparent function. Memoization is also encountered as an easily accessible design pattern withi
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tered as an easily accessible design pattern within term-rewrite based languages such as Wolfram Language.
=== Bioinformatics ===
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 and by Georgii Gurskii and Alexander Zasedatelev in the Soviet Union. Recently these algorithms have become very popular in bioinformatics and computational biology, particularly in the studies of nucleosome positioning and transcription factor binding.
== Examples: computer algorithms ==
=== Dijkstra's algorithm for the shortest path problem ===
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 th
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|
ional equation for the shortest path problem by the Reaching method.
In fact, Dijkstra's explanation of the logic behind the algorithm, namely
Problem 2. Find the path of minimum total length between two given nodes
P
{\displaystyle P}
and
Q
{\displaystyle Q}
.
We use the fact that, if
R
{\displaystyle R}
is a node on the minimal path from
P
{\displaystyle P}
to
Q
{\displaystyle Q}
, knowledge of the latter implies the knowledge of the minimal path from
P
{\displaystyle P}
to
R
{\displaystyle R}
.
is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem.
=== Fibonacci sequence ===
Using dynamic programming in the calculation of the nt
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|
g 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:
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 requir
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|
se 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
prev
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|
ewFib := 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.
=== A type of balanced 0–1 matrix ===
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
{\displaystyle n}
. For example, when n = 4, five possible solutions are
[
0
1
0
1
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|
1
1
0
1
0
0
1
0
1
1
0
1
0
]
and
[
0
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|
0
0
1
1
0
0
1
1
1
1
0
0
1
1
0
0
]
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|
]
and
[
1
1
0
0
0
0
1
1
1
1
0
0
0
0
|
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|
0
1
1
]
and
[
1
0
0
1
0
1
1
0
0
1
1
|
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|
1
0
1
0
0
1
]
and
[
1
1
0
0
1
1
0
0
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|
0
0
1
1
0
0
1
1
]
.
{\displaystyle {\begin{bmatrix}0&1&0&1\\1&0&1&0\\0&1&0&1\\1&0&1&0\end{bmatrix}}{\text{ and }}{\begin{bmatrix}0&0&1&1\\0&0&1&1\\1&1&0&0\\1&1&0&0\end{bmatrix}}{\text{ and }}{\begin{bmatrix}1&1&0&0\\0&0&1&1\\1&1&0&0\\0&0&1&1\end{bmatrix}}{\text{ and }}{\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&1&1&0\\1&0&0&1\end{bmatrix}}{\text{ and }}{\begin{bmatrix}1&1&0&0\\1&1&0&0\\0&0&1&1\\0&0&1&1\end{bmatrix}}.}
There are at least thr
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|
0&0&1&1\end{bmatrix}}.}
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
2
n
2
{\displaystyle 2^{n^{2}}}
possible assignments and
(
n
n
/
2
)
n
{\disp
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|
n
{\displaystyle {\tbinom {n}{n/2}}^{n}}
sensible assignments, this strategy is not practical except maybe up to
n
=
6
{\displaystyle 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 116,963,796,250 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 solution
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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
{\displaystyle k}
rows contain
n
/
2
{\displaystyle n/2}
zeros and
n
/
2
{\displaystyle 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 zeros and ones that have yet to be placed in that column. We seek the value of
f
(
(
n
/
2
,
n
/
2
)
,
(
n
/
2
,
n
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|
/
2
,
n
/
2
)
,
…
(
n
/
2
,
n
/
2
)
)
{\displaystyle f((n/2,n/2),(n/2,n/2),\ldots (n/2,n/2))}
(
n
{\displaystyle n}
arguments or one vector of
n
{\displaystyle n}
elements). The process of subproblem creation involves iterating over every one of
(
n
n
/
2
)
{\displaystyle {\tbinom {n}{n/2}}}
possible
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{\displaystyle {\tbinom {n}{n/2}}}
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
)
{\displaystyle
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|
1
)
{\displaystyle (0,1)}
and n / 2
(
1
,
0
)
{\displaystyle (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
,
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|
S) is
1
,
2
,
90
,
297200
,
116963796250
,
6736218287430460752
,
…
{\displaystyle 1,\,2,\,90,\,297200,\,116963796250,\,6736218287430460752,\ldots }
Links to the MAPLE implementation of the dynamic programming approach may be found among the external links.
=== 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),
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 forwar
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|
ward, 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:
q
(
A
)
=
min
(
q
(
B
)
,
q
(
C
)
,
q
(
D
)
)
+
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|
(
D
)
)
+
c
(
A
)
{\displaystyle q(A)=\min(q(B),q(C),q(D))+c(A)\,}
Now, let us define q(i, j) in somewhat more general terms:
q
(
i
,
j
)
=
{
∞
j
<
1
or
j
>
n
c
(
i
,
j
)
i
=
1
min
|
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|
min
(
q
(
i
−
1
,
j
−
1
)
,
q
(
i
−
1
,
j
)
,
q
(
i
−
1
,
j
+
1
)
)
+
c
(
i
,
j
)
otherwise.
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|
{\displaystyle q(i,j)={\begin{cases}\infty &j<1{\text{ or }}j>n\\c(i,j)&i=1\\\min(q(i-1,j-1),q(i-1,j),q(i-1,j+1))+c(i,j)&{\text{otherwise.}}\end{cases}}}
The first line of this equation deals with a board modeled as squares indexed on 1 at the lowest bound and n at the highest bound. The second line specifies what happens at the first rank; providing a base case. The third line, the recursion, is the important part. It represents the A,B,C,D terms in the example. From this definition we can derive straightforward recursive code for q(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 = 1
return c(i, j)
else
return min( minCost(i-1, j-1), minCost(i-1, j), minCost(i-1, j+1) ) + c(i, j)
This function only computes the path
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) + c(i, j)
This function only computes the path cost, not the actual path. We discuss the actual path below. This, like the Fibonacci-numbers example, is horribly slow because it too exhibits the overlapping sub-problems attribute. That is, it recomputes the same path costs 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; all the values needed for array q[i, j] are computed ahead of time only once. Precomputed values for (i,j) are simply looked up whenever needed.
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 records the path to any square s. The predecessor of s is modeled as an offset relative to the index (in q[i, j]) of the precomputed path cost of s. To reconstruct the complete path, we lookup the predecessor of s, then the predecessor of that square, then
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|
r of s, then the predecessor of that square, then the predecessor of that square, and so on recursively, until we reach the starting square. Consider the following pseudocode:
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
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|
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])
=== Sequence alignment ===
In genetics, sequence alignment is an important application where dynamic programming is essential. 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 tab
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ails 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 algorithm and Needleman–Wunsch algorithm.
=== Tower of Hanoi puzzle ===
The Tower of Hanoi or Towers of Hanoi is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.
The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:
Only one disk may be moved at a time.
Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disk
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ding it onto another rod, on top of the other disks that may already be present on that rod.
No disk may be placed on top of a smaller disk.
The dynamic programming solution consists of solving the functional equation
S(n,h,t) = S(n-1,h, not(h,t)) ; S(1,h,t) ; S(n-1,not(h,t),t)
where n denotes the number of disks to be moved, h denotes the home rod, t denotes the target rod, not(h,t) denotes the third rod (neither h nor t), ";" denotes concatenation, and
S(n, h, t) := solution to a problem consisting of n disks that are to be moved from rod h to rod t.
For n=1 the problem is trivial, namely S(1,h,t) = "move a disk from rod h to rod t" (there is only one disk left).
The number of moves required by this solution is 2n − 1. If the objective is to maximize the number of moves (without cycling) then the dynamic programming functional equation is slightly more complicated and 3n − 1 moves are required.
=== Egg dropping puzzle ===
The following is a description of the instance of this
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|
ollowing is a description of the instance of this famous puzzle involving N=2 eggs and a building with H=36 floors:
Suppose that we wish to know which stories in a 36-story building are safe to drop eggs from, and which will cause the eggs to break on landing (using U.S. English terminology, in which the first floor is at ground level). We make a few assumptions:
An egg that survives a fall can be used again.
A broken egg must be discarded.
The effect of a fall is the same for all eggs.
If an egg breaks when dropped, then it would break if dropped from a higher window.
If an egg survives a fall, then it would survive a shorter fall.
It is not ruled out that the first-floor windows break eggs, nor is it ruled out that eggs can survive the 36th-floor windows.
If only one egg is available and we wish to be sure of obtaining the right result, the experiment can be carried out in only one way. Drop the egg from the first-floor window; if it survives, drop it from the second-floor window.
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|
t survives, drop it from the second-floor window. Continue upward until it breaks. In the worst case, this method may require 36 droppings. Suppose 2 eggs are available. What is the lowest number of egg-droppings that is guaranteed to work in all cases?
To derive a dynamic programming functional equation for this puzzle, let the state of the dynamic programming model be a pair s = (n,k), where
n = number of test eggs available, n = 0, 1, 2, 3, ..., N − 1.
k = number of (consecutive) floors yet to be tested, k = 0, 1, 2, ..., H − 1.
For instance, s = (2,6) indicates that two test eggs are available and 6 (consecutive) floors are yet to be tested. The initial state of the process is s = (N,H) where N denotes the number of test eggs available at the commencement of the experiment. The process terminates either when there are no more test eggs (n = 0) or when k = 0, whichever occurs first. If termination occurs at state s = (0,k) and k > 0, then the test failed.
Now, let
W(n,k) = minimum
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|
, then the test failed.
Now, let
W(n,k) = minimum number of trials required to identify the value of the critical floor under the worst-case scenario given that the process is in state s = (n,k).
Then it can be shown that
W(n,k) = 1 + min{max(W(n − 1, x − 1), W(n,k − x)): x = 1, 2, ..., k }
with W(n,0) = 0 for all n > 0 and W(1,k) = k for all k. It is easy to solve this equation iteratively by systematically increasing the values of n and k.
==== Faster DP solution using a different parametrization ====
Notice that the above solution takes
O
(
n
k
2
)
{\displaystyle O(nk^{2})}
time with a DP solution. This can be improved to
O
(
n
k
log
k
)
{\displaystyle O(nk\log k)}
time by binary searching on the optimal
x
{\dis
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|
al
x
{\displaystyle x}
in the above recurrence, since
W
(
n
−
1
,
x
−
1
)
{\displaystyle W(n-1,x-1)}
is increasing in
x
{\displaystyle x}
while
W
(
n
,
k
−
x
)
{\displaystyle W(n,k-x)}
is decreasing in
x
{\displaystyle x}
, thus a local minimum of
max
(
W
(
n
−
1
,
x
−
1
)
,
W
(
n
,
k
−
x
)
)
{\displaystyle \max(W(n-1,x-1),W(n,k-x))}
is a global minimum. Also, by storing the optimal
x
{\displaystyl
|
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|
x
{\displaystyle x}
for each cell in the DP table and referring to its value for the previous cell, the optimal
x
{\displaystyle x}
for each cell can be found in constant time, improving it to
O
(
n
k
)
{\displaystyle O(nk)}
time. However, there is an even faster solution that involves a different parametrization of the problem:
Let
k
{\displaystyle k}
be the total number of floors such that the eggs break when dropped from the
k
{\displaystyle k}
th floor (The example above is equivalent to taking
k
=
37
{\displaystyle k=37}
).
Let
m
{\displaystyle m}
be the minimum floor from which the egg must be dropped to be broken.
Let
|
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|
st be dropped to be broken.
Let
f
(
t
,
n
)
{\displaystyle f(t,n)}
be the maximum number of values of
m
{\displaystyle m}
that are distinguishable using
t
{\displaystyle t}
tries and
n
{\displaystyle n}
eggs.
Then
f
(
t
,
0
)
=
f
(
0
,
n
)
=
1
{\displaystyle f(t,0)=f(0,n)=1}
for all
t
,
n
≥
0
{\displaystyle t,n\geq 0}
.
Let
a
{\displaystyle a}
be the floor from which the first egg is dropped in the optimal strategy.
If the first egg broke,
m
{\displaystyle m}
i
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|
m
{\displaystyle m}
is from
1
{\displaystyle 1}
to
a
{\displaystyle a}
and distinguishable using at most
t
−
1
{\displaystyle t-1}
tries and
n
−
1
{\displaystyle n-1}
eggs.
If the first egg did not break,
m
{\displaystyle m}
is from
a
+
1
{\displaystyle a+1}
to
k
{\displaystyle k}
and distinguishable using
t
−
1
{\displaystyle t-1}
tries and
n
{\displaystyle n}
eggs.
Therefore,
f
(
t
,
n
)
=
f
(
t
−
|
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|
=
f
(
t
−
1
,
n
−
1
)
+
f
(
t
−
1
,
n
)
{\displaystyle f(t,n)=f(t-1,n-1)+f(t-1,n)}
.
Then the problem is equivalent to finding the minimum
x
{\displaystyle x}
such that
f
(
x
,
n
)
≥
k
{\displaystyle f(x,n)\geq k}
.
To do so, we could compute
{
f
(
t
,
i
)
:
0
≤
i
≤
n
}
{\displaystyle \{f(t,i):0\leq i\leq n\}}
in order of increasing
t
{\displaystyle t}
, which would take
O
(
n
x
)
{\displaystyle O(nx)}
time.
Thu
|
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|
{\displaystyle O(nx)}
time.
Thus, if we separately handle the case of
n
=
1
{\displaystyle n=1}
, the algorithm would take
O
(
n
k
)
{\displaystyle O(n{\sqrt {k}})}
time.
But the recurrence relation can in fact be solved, giving
f
(
t
,
n
)
=
∑
i
=
0
n
(
t
i
)
{\displaystyle f(t,n)=\sum _{i=0}^{n}{\binom {t}{i}}}
, which can be computed in
O
(
n
)
{\d
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|
(
n
)
{\displaystyle O(n)}
time using the identity
(
t
i
+
1
)
=
(
t
i
)
t
−
i
i
+
1
{\displaystyle {\binom {t}{i+1}}={\binom {t}{i}}{\frac {t-i}{i+1}}}
for all
i
≥
0
{\displaystyle i\geq 0}
.
Since
f
(
t
,
n
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|
f
(
t
,
n
)
≤
f
(
t
+
1
,
n
)
{\displaystyle f(t,n)\leq f(t+1,n)}
for all
t
≥
0
{\displaystyle t\geq 0}
, we can binary search on
t
{\displaystyle t}
to find
x
{\displaystyle x}
, giving an
O
(
n
log
k
)
{\displaystyle O(n\log k)}
algorithm.
=== Matrix chain multiplication ===
Matrix chain multiplication is a well-known example that demonstrates utility of dynamic programming. For example, engineering applications often have to multiply a chain of matrices. It is not surprising to find matrices of large dimensions, for example 100×100. Therefore, our task is to multiply matrices
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|
to multiply matrices
A
1
,
A
2
,
.
.
.
.
A
n
{\displaystyle A_{1},A_{2},....A_{n}}
. Matrix multiplication is not commutative, but is associative; and we can multiply only two matrices at a time. So, we can multiply this chain of matrices in many different ways, for example:
((A1 × A2) × A3) × ... An
A1×(((A2×A3)× ... ) × An)
(A1 × A2) × (A3 × ... An)
and so on. There are numerous ways to multiply this chain of matrices. They will all produce the same final result, however they will take more or less time to compute, based on which particular matrices are multiplied. If matrix A has dimensions m×n and matrix B has dimensions n×q, then matrix C=A×B will have dimensions m×q, and will require m*n*q scalar multiplications
|
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×q, and will require m*n*q scalar multiplications (using a simplistic matrix multiplication algorithm for purposes of illustration).
For example, let us multiply matrices A, B and C. Let us assume that their dimensions are m×n, n×p, and p×s, respectively. Matrix A×B×C will be of size m×s and can be calculated in two ways shown below:
Ax(B×C) This order of matrix multiplication will require nps + mns scalar multiplications.
(A×B)×C This order of matrix multiplication will require mnp + mps scalar calculations.
Let us assume that m = 10, n = 100, p = 10 and s = 1000. So, the first way to multiply the chain will require 1,000,000 + 1,000,000 calculations. The second way will require only 10,000+100,000 calculations. Obviously, the second way is faster, and we should multiply the matrices using that arrangement of parenthesis.
Therefore, our conclusion is that the order of parenthesis matters, and that our task is to find the optimal order of parenthesis.
At this point, we have seve
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|
order of parenthesis.
At this point, we have several choices, one of which is to design a dynamic programming algorithm that will split the problem into overlapping problems and calculate the optimal arrangement of parenthesis. The dynamic programming solution is presented below.
Let's call m[i,j] the minimum number of scalar multiplications needed to multiply a chain of matrices from matrix i to matrix j (i.e. Ai × .... × Aj, i.e. i<=j). We split the chain at some matrix k, such that i <= k < j, and try to find out which combination produces minimum m[i,j].
The formula is:
if i = j, m[i,j]= 0
if i < j, m[i,j]= min over all possible values of k (m[i,k]+m[k+1,j] +
p
i
−
1
∗
p
k
∗
p
j
{\displaystyle p_{i-1}*p_{k}*p_{j}
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|
{\displaystyle p_{i-1}*p_{k}*p_{j}}
)
where k ranges from i to j − 1.
p
i
−
1
{\displaystyle p_{i-1}}
is the row dimension of matrix i,
p
k
{\displaystyle p_{k}}
is the column dimension of matrix k,
p
j
{\displaystyle p_{j}}
is the column dimension of matrix j.
This formula can be coded as shown below, where input parameter "chain" is the chain of matrices, i.e.
A
1
,
A
2
,
.
.
.
A
n
{\displa
|
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|
n
{\displaystyle A_{1},A_{2},...A_{n}}
:
function OptimalMatrixChainParenthesis(chain)
n = length(chain)
for i = 1, n
m[i,i] = 0 // Since it takes no calculations to multiply one matrix
for len = 2, n
for i = 1, n - len + 1
j = i + len -1
m[i,j] = infinity // So that the first calculation updates
for k = i, j-1
q = m[i, k] + m[k+1, j] +
p
i
−
1
∗
p
k
∗
p
j
{\displaystyle p_{i-1}*p_{k}*p_{j}}
if q < m[i, j] // The new order of parentheses is better than what we had
m[i, j] = q // Update
s[i, j] = k // Record which k to split on, i.e.
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[i, j] = k // Record which k to split on, i.e. where to place the parenthesis
So far, we have calculated values for all possible m[i, j], the minimum number of calculations to multiply a chain from matrix i to matrix j, and we have recorded the corresponding "split point"s[i, j]. For example, if we are multiplying chain A1×A2×A3×A4, and it turns out that m[1, 3] = 100 and s[1, 3] = 2, that means that the optimal placement of parenthesis for matrices 1 to 3 is
(
A
1
×
A
2
)
×
A
3
{\displaystyle (A_{1}\times A_{2})\times A_{3}}
and to multiply those matrices will require 100 scalar calculations.
This algorithm will produce "tables" m[, ] and s[, ] that will have entries for all possible values of i and j. The final solution for the
|
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|
ble values of i and j. The final solution for the entire chain is m[1, n], with corresponding split at s[1, n]. Unraveling the solution will be recursive, starting from the top and continuing until we reach the base case, i.e. multiplication of single matrices.
Therefore, the next step is to actually split the chain, i.e. to place the parenthesis where they (optimally) belong. For this purpose we could use the following algorithm:
function PrintOptimalParenthesis(s, i, j)
if i = j
print "A"i
else
print "("
PrintOptimalParenthesis(s, i, s[i, j])
PrintOptimalParenthesis(s, s[i, j] + 1, j)
print ")"
Of course, this algorithm is not useful for actual multiplication. This algorithm is just a user-friendly way to see what the result looks like.
To actually multiply the matrices using the proper splits, we need the following algorithm:
== History of the name ==
The term dynamic programming was originally used in the 1940s by Richard Bellm
|
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|
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, 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
|
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|
d 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
|
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|
ltistage, 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. 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.
The above expla
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|
nym for mathematical optimization.
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." 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."
== See also ==
== References ==
== Further reading ==
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 ex
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0002-9904-1954-09848-8, 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". Sys
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c 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.
== External links ==
A Tutorial on Dynamic programming
MIT course on algorithms - Includes 4 video lectures on DP, lectures 15–18
Applied Mathematical Programming by Bradley, Hax, and Magnanti, Chapter 11
More DP Notes
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.
Dynamic Programming: from novice to advanced A TopCoder.com article by Dumitru on Dynamic Programming
Algebraic Dynamic Programming – a formalized framework for dynamic programming, including an entry-level course to DP, University of Bielefeld
Dreyfus, Stuart, "Richard Bellman on the birth of Dynamic Programming. Archived 2020-10
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the birth of Dynamic Programming. Archived 2020-10-13 at the Wayback Machine"
Dynamic programming tutorial
A Gentle Introduction to Dynamic Programming and the Viterbi Algorithm
Tabled Prolog BProlog, XSB, SWI-Prolog
IFORS online interactive dynamic programming modules including, shortest path, traveling salesman, knapsack, false coin, egg dropping, bridge and torch, replacement, chained matrix products, and critical path problem.
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Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation.
Python is dynamically type-checked and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library.
Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language, and he first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000. Python 3.0, released in 2008, was a major revision not completely backward-compatible with earlier versions. Python 2.7.18, released in 2020, was the last release of Python 2.
Python consistently ranks as one of the most popular programming languages, and it has gained widespread use in the machine learning community.
== History ==
Python was conceived in th
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unity.
== History ==
Python was conceived in the late 1980s by Guido van Rossum at Centrum Wiskunde & Informatica (CWI) in the Netherlands; it was conceived as a successor to the ABC programming language, which was inspired by SETL, capable of exception handling and interfacing with the Amoeba operating system. Python implementation began in December 1989. Van Rossum assumed sole responsibility for the project, as the lead developer, until 12 July 2018, when he announced his "permanent vacation" from responsibilities as Python's "benevolent dictator for life" (BDFL); this title was bestowed on him by the Python community to reflect his long-term commitment as the project's chief decision-maker. (He has since come out of retirement and is self-titled "BDFL-emeritus".) In January 2019, active Python core developers elected a five-member Steering Council to lead the project.
The name Python is said to derive from the British comedy series Monty Python's Flying Circus.
Python 2.0 was re
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es Monty Python's Flying Circus.
Python 2.0 was released on 16 October 2000, with many major new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 2.7's end-of-life was initially set for 2015, and then postponed to 2020 out of concern that a large body of existing code could not easily be forward-ported to Python 3. It no longer receives security patches or updates. While Python 2.7 and older versions are officially unsupported, a different unofficial Python implementation, PyPy, continues to support Python 2, i.e., "2.7.18+" (plus 3.10), with the plus signifying (at least some) "backported security updates".
Python 3.0 was released on 3 December 2008, with some new semantics and changed syntax. At least every Python release since (the now unsupported) 3.5 has added some syntax to the language; a few later releases have removed outdated modules and have changed semantics, at least in a minor way.
As of 8 April 2025
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ntics, at least in a minor way.
As of 8 April 2025, Python 3.13.3 is the latest stable release (it's highly recommended to upgrade to it, or upgrade any other older 3.x release). This version currently receives full bug-fix and security updates, while Python 3.12—released in October 2023—had active bug-fix support only until April 2025, and since then only security fixes. Python 3.9 is the oldest supported version of Python (albeit in the 'security support' phase), because Python 3.8 has become an end-of-life product. Starting with Python 3.13, it and later versions receive two years of full support (which has increased from one and a half years), followed by three years of security support; this is the same total duration of support as previously.
Security updates were expedited in 2021 and again twice in 2022. More issues were fixed in 2023 and in September 2024 (for Python versions 3.8.20 through 3.12.6)—all versions (including 2.7) had been insecure because of issues leading to pos
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had been insecure because of issues leading to possible remote code execution and web-cache poisoning.
Python 3.10 added the | union type operator and added structural pattern matching capability to the language, with the new match and case keywords. Python 3.11 expanded exception handling functionality. Python 3.12 added the new keyword type. Notable changes from version 3.10 to 3.11 include increased program execution speed and improved error reporting. Python 3.11 is claimed to be 10–60% faster than Python 3.10, and Python 3.12 increases by an additional 5%. Python 3.12 also includes improved error messages (again improved in 3.14) and many other changes.
Python 3.13 introduced more syntax for types; a new and improved interactive interpreter (REPL), featuring multi-line editing and color support; an incremental garbage collector, which results in shorter pauses for collection in programs that have many objects, as well as increasing the improved speed in 3.11 and 3.12); an expe
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ing the improved speed in 3.11 and 3.12); an experimental just-in-time (JIT) compiler (such features need to be enabled specifically for the increase in speed); and an experimental free-threaded build mode, which disables the global interpreter lock (GIL), allowing threads to run more concurrently, as enabled inpython3.13t or python3.13t.exe.
Python Enhancement Proposal (PEP) 711 proposes PyBI—a standard format for distributing Python binaries.
Python 3.14.0 is now in the beta 1 phase (introduces e.g. a new opt-in interpreter, up to 30% faster).
Python 3.15 will "Make UTF-8 mode default"; This mode is supported in all current Python versions, but it currently must be opted into. UTF-8 is already used by default on Windows (and other operating systems) for most purposes; an exception is opening files. Enabling UTF-8 also makes code fully cross-platform.
Potentially breaking changes
Python 3.0 introduced very breaking changes, but all breaking changes in 3.x discussed below, are design
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reaking changes in 3.x discussed below, are designed to affect few users.
Python 3.12 dropped some outdated modules, and more will be dropped in the future, deprecated as of 3.13; already deprecated array 'u' format code will emit DeprecationWarning since 3.13 and will be removed in Python 3.16. The 'w' format code should be used instead. Part of ctypes is also deprecated and http.server.CGIHTTPRequestHandler will emit a DeprecationWarning, and will be removed in 3.15. Using that code already has a high potential for both security and functionality bugs. Parts of the typing module are deprecated, e.g. creating a typing.NamedTuple class using keyword arguments to denote the fields and such (and more) will be disallowed in Python 3.15. Python 3.12 removed wstr meaning Python extensions need to be modified.
Python 3.13 introduces some changes in behavior, i.e., new "well-defined semantics", fixing bugs, and removing many deprecated classes, functions and methods (as well as some of the Py
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, functions and methods (as well as some of the Python/C API and outdated modules). "The old implementation of locals() and frame.f_locals was slow, inconsistent and buggy, and it had many corner cases and oddities. Code that works around those may need revising; code that uses locals() for simple templating or print debugging should continue to work correctly."
Python 3.13 introduces the experimental free-threaded build mode, which disables the Global Interpreter Lock (GIL); the GIL is a feature of CPython that previously prevented multiple threads from executing Python bytecode simultaneously. This optional build, introduced through PEP 703, enables better exploitation of multi-core CPUs. By allowing multiple threads to run Python code in parallel, the free-threaded mode addresses long-standing performance bottlenecks associated with the GIL. This change offers a new path for parallelism in Python, without resorting to multiprocessing or external concurrency frameworks.
Regarding a
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ng or external concurrency frameworks.
Regarding annotations in upcoming Python version: "In Python 3.14, from __future__ import annotations will continue to work as it did before, converting annotations into strings."
Python 3.14 drops the PGP digital verification signatures, it had deprecated in version 3.11, when its replacement Sigstore was added for all CPython artifacts; the use of PGP has been criticized by security practitioners.
Some additional standard-library modules will be removed in Python 3.15 or 3.16, as will be many deprecated classes, functions and methods.
== Design philosophy and features ==
Python is a multi-paradigm programming language. Object-oriented programming and structured programming are fully supported, and many of their features support functional programming and aspect-oriented programming (including metaprogramming and metaobjects). Many other paradigms are supported via extensions, including design by contract and logic programming. Python is often
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y contract and logic programming. Python is often referred to as a 'glue language' because it can seamlessly integrate components written in other languages.
Python uses dynamic typing and a combination of reference counting and a cycle-detecting garbage collector for memory management. It uses dynamic name resolution (late binding), which binds method and variable names during program execution.
Python's design offers some support for functional programming in the Lisp tradition. It has filter,mapandreduce functions; list comprehensions, dictionaries, sets, and generator expressions. The standard library has two modules (itertools and functools) that implement functional tools borrowed from Haskell and Standard ML.
Python's core philosophy is summarized in the Zen of Python (PEP 20), which includes aphorisms such as these:
Beautiful is better than ugly.
Explicit is better than implicit.
Simple is better than complex.
Complex is better than complicated.
Readability counts.
However, Py
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than complicated.
Readability counts.
However, Python features regularly violate these principles and have received criticism for adding unnecessary language bloat. Responses to these criticisms note that the Zen of Python is a guideline rather than a rule. The addition of some new features had been controversial: Guido van Rossum resigned as Benevolent Dictator for Life after conflict about adding the assignment expression operator in Python 3.8.
Nevertheless, rather than building all functionality into its core, Python was designed to be highly extensible via modules. This compact modularity has made it particularly popular as a means of adding programmable interfaces to existing applications. Van Rossum's vision of a small core language with a large standard library and easily extensible interpreter stemmed from his frustrations with ABC, which represented the opposite approach.
Python claims to strive for a simpler, less-cluttered syntax and grammar, while giving developers a choi
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syntax and grammar, while giving developers a choice in their coding methodology. In contrast to Perl's motto "there is more than one way to do it", Python advocates an approach where "there should be one—and preferably only one—obvious way to do it.". In practice, however, Python provides many ways to achieve a given goal. There are, for example, at least three ways to format a string literal, with no certainty as to which one a programmer should use. Alex Martelli is a Fellow at the Python Software Foundation and Python book author; he wrote that "To describe something as 'clever' is not considered a compliment in the Python culture."
Python's developers usually try to avoid premature optimization; they also reject patches to non-critical parts of the CPython reference implementation that would offer marginal increases in speed at the cost of clarity. Execution speed can be improved by moving speed-critical functions to extension modules written in languages such as C, or by using a
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les written in languages such as C, or by using a just-in-time compiler like PyPy. It is also possible to cross-compile to other languages; but this approach either fails to achieve the expected speed-up, since Python is a very dynamic language, or only a restricted subset of Python is compiled (with potential minor semantic changes).
Python's developers aim for the language to be fun to use. This goal is reflected in the name—a tribute to the British comedy group Monty Python—and in playful approaches to some tutorials and reference materials. For instance, some code examples use the terms "spam" and "eggs" (in reference to a Monty Python sketch), rather than the typical terms "foo" and "bar". A common neologism in the Python community is pythonic, which has a wide range of meanings related to program style. Pythonic code may use Python idioms well; be natural or show fluency in the language; or conform with Python's minimalist philosophy and emphasis on readability.
== Syntax and s
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phy and emphasis on readability.
== Syntax and semantics ==
Python is meant to be an easily readable language. Its formatting is visually uncluttered and often uses English keywords where other languages use punctuation. Unlike many other languages, it does not use curly brackets to delimit blocks, and semicolons after statements are allowed but rarely used. It has fewer syntactic exceptions and special cases than C or Pascal.
=== Indentation ===
Python uses whitespace indentation, rather than curly brackets or keywords, to delimit blocks. An increase in indentation comes after certain statements; a decrease in indentation signifies the end of the current block. Thus, the program's visual structure accurately represents its semantic structure. This feature is sometimes termed the off-side rule. Some other languages use indentation this way; but in most, indentation has no semantic meaning. The recommended indent size is four spaces.
=== Statements and control flow ===
Python's
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s.
=== Statements and control flow ===
Python's statements include the following:
The assignment statement, using a single equals sign =
The if statement, which conditionally executes a block of code, along with else and elif (a contraction of else if)
The for statement, which iterates over an iterable object, capturing each element to a local variable for use by the attached block
The while statement, which executes a block of code as long as its condition is true
The try statement, which allows exceptions raised in its attached code block to be caught and handled by except clauses (or new syntax except* in Python 3.11 for exception groups); the try statement also ensures that clean-up code in a finally block is always run regardless of how the block exits
The raise statement, used to raise a specified exception or re-raise a caught exception
The class statement, which executes a block of code and attaches its local namespace to a class, for use in object-oriented programming
The d
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lass, for use in object-oriented programming
The def statement, which defines a function or method
The with statement, which encloses a code block within a context manager, allowing resource-acquisition-is-initialization (RAII)-like behavior and replacing a common try/finally idiom Examples of a context include acquiring a lock before some code is run, and then releasing the lock; or opening and then closing a file
The break statement, which exits a loop
The continue statement, which skips the rest of the current iteration and continues with the next
The del statement, which removes a variable—deleting the reference from the name to the value, and producing an error if the variable is referred to before it is redefined
The pass statement, serving as a NOP (i.e., no operation), which is syntactically needed to create an empty code block
The assert statement, used in debugging to check for conditions that should apply
The yield statement, which returns a value from a generator function
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, which returns a value from a generator function (and also an operator); used to implement coroutines
The return statement, used to return a value from a function
The import and from statements, used to import modules whose functions or variables can be used in the current program
The match and case statements, analogous to a switch statement construct, which compares an expression against one or more cases as a control-flow measure
The assignment statement (=) binds a name as a reference to a separate, dynamically allocated object. Variables may subsequently be rebound at any time to any object. In Python, a variable name is a generic reference holder without a fixed data type; however, it always refers to some object with a type. This is called dynamic typing—in contrast to statically-typed languages, where each variable may contain only a value of a certain type.
Python does not support tail call optimization or first-class continuations; according to Van Rossum, the language never
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tions; according to Van Rossum, the language never will. However, better support for coroutine-like functionality is provided by extending Python's generators. Before 2.5, generators were lazy iterators; data was passed unidirectionally out of the generator. From Python 2.5 on, it is possible to pass data back into a generator function; and from version 3.3, data can be passed through multiple stack levels.
=== Expressions ===
Python's expressions include the following:
The +, -, and * operators for mathematical addition, subtraction, and multiplication are similar to other languages, but the behavior of division differs. There are two types of division in Python: floor division (or integer division) //, and floating-point division /. Python uses the ** operator for exponentiation.
Python uses the + operator for string concatenation. The language uses the * operator for duplicating a string a specified number of times.
The @ infix operator is intended to be used by libraries such as
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erator is intended to be used by libraries such as NumPy for matrix multiplication.
The syntax :=, called the "walrus operator", was introduced in Python 3.8. This operator assigns values to variables as part of a larger expression.
In Python, == compares two objects by value. Python's is operator may be used to compare object identities (i.e., comparison by reference), and comparisons may be chained—for example, a <= b <= c.
Python uses and, or, and not as Boolean operators.
Python has a type of expression called a list comprehension, and a more general expression called a generator expression.
Anonymous functions are implemented using lambda expressions; however, there may be only one expression in each body.
Conditional expressions are written as x if c else y. (This is different in operand order from the c ? x : y operator common to many other languages.)
Python makes a distinction between lists and tuples. Lists are written as [1, 2, 3], are mutable, and cannot be used as the keys
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2, 3], are mutable, and cannot be used as the keys of dictionaries (since dictionary keys must be immutable in Python). Tuples, written as (1, 2, 3), are immutable and thus can be used as the keys of dictionaries, provided that all of the tuple's elements are immutable. The + operator can be used to concatenate two tuples, which does not directly modify their contents, but produces a new tuple containing the elements of both. For example, given the variable t initially equal to (1, 2, 3), executing t = t + (4, 5) first evaluates t + (4, 5), which yields (1, 2, 3, 4, 5); this result is then assigned back to t—thereby effectively "modifying the contents" of t while conforming to the immutable nature of tuple objects. Parentheses are optional for tuples in unambiguous contexts.
Python features sequence unpacking where multiple expressions, each evaluating to something assignable (e.g., a variable or a writable property) are associated just as in forming tuple literal; as a whole, the resu
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