# Dynamic Programming Algorithm

A dynamic programming algorithm is an iterative algorithm that follows strategy which is applicable to decomposable tasks whose optimal solution contains optimal solutions to its subtasks and solutions can be stored and reused in a bottom-up fashion.

**Context:**- It can be a commonly used Algorithm Strategy.
- It can expend Memory Resources.
- It can be an Algorithm Strategy used for Optimization Algorithms.

**Example(s):****Counter-Example(s):****See:**Recursive Algorithm, Linear Programming Algorithm, Markov Decision Processes.

## References

### 2012

- (Wikipedia, 2012) ⇒ http://en.wikipedia.org/wiki/Dynamic_programming
- QUOTE: In mathematics and computer science,
**dynamic programming**is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller^{[1]}and optimal substructure (described below). When applicable, the method takes far less time than naive methods.The key idea behind dynamic programming is quite simple. In general, to solve a given problem, we need to solve different parts of the problem (subproblems), then combine the solutions of the subproblems to reach an overall solution. Often, many of these subproblems are really the same. The dynamic programming approach seeks to solve each subproblem only once, thus reducing the number of computations: once the solution to a given subproblem has been computed, it is stored or “memo-ized": the next time the same solution is needed, it is simply looked up. This approach is especially useful when the number of repeating subproblems grows exponentially as a function of the size of the input.

Top-down dynamic programming simply means storing the results of certain calculations, which are later used again since the completed calculation is a sub-problem of a larger calculation. Bottom-up dynamic programming involves formulating a complex calculation as a recursive series of simpler calculations.

- QUOTE: In mathematics and computer science,

### 2011

- (Puterman & Patrick, 2011) ⇒ Martin L. Puterman, Jonathan Patrick. (2011). “Dynamic Programming.” In: (Sammut & Webb, 2011) p.298

### 2009

- http://www.nature.com/nrg/journal/v5/n4/glossary/nrg1318_glossary.html
- DYNAMIC PROGRAMMING A large class of programmimg algorithms that are based on breaking a large problem down (if possible) into incremental steps so that, at any given stage, optimal solutions are known sub-problems.

### 2008

- (Huang, 2008) ⇒ Liang Huang. (2008). “Advanced Dynamic Programming in Semiring and Hypergraph Frameworks.” In: COLING 2008 Tutorial Notes.
- … The basic idea of DP is to solve a bigger problem by divide-and-conquer, but also reuses the solutions of
*overlapping subproblems*to avoid recalculation. The simplest such example is a Fibonacci series, where each*F*(*n*) is used twice (if cached). The correctness of a DP algorithm is ensured by the*optimal substructure property*, which informally says that an optimal solution must contain optimal subsolutions for subproblems.

- … The basic idea of DP is to solve a bigger problem by divide-and-conquer, but also reuses the solutions of
- (Edmonds, 2008) ⇒ Jeff Edmonds. (2008). “Thinking about Algorithms Abstractly: Recursive Back Tracking & Dynamic Programming." Presentation, Lecture 7, COSC 3101, York University.

### 2005

- (Klinberg & Tardos, 2005) ⇒ Jon Kleinberg, and Éva Tardos. (2005). “Algorithm Design." Addison-Wesley. ISBN:0-321-29535-8
- Chapter 6: Dynamic Programming http://www.aw-bc.com/info/kleinberg/assets/downloads/ch6.pdf

### 1966

- (Bellman, 1966) ⇒ Richard Bellman. (1966). “Dynamic Programming.” In: Science, 153(3731).

### 1957

- (Bellman, 1957) ⇒ Richard Bellman. (1957). “Dynamic Programming.” Princeton University Press.

- ↑ S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani, '
**Algorithms'**, p 173, available at http://www.cs.berkeley.edu/~vazirani/algorithms.html