## What is dynamic programming?

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Dynamic programming is when you use past knowledge to make solving a future problem easier.

A good example is solving the Fibonacci sequence for n=1,000,002.

This will be a very long process, but what if I give you the results for n=1,000,000 and n=1,000,001? Suddenly the problem just became more manageable.

Dynamic programming is used a lot in string problems, such as the string edit problem. You solve a subset(s) of the problem and then use that information to solve the more difficult original problem.

With dynamic programming, you store your results in some sort of table generally. When you need the answer to a problem, you reference the table and see if you already know what it is. If not, you use the data in your table to give yourself a stepping stone towards the answer.

The Cormen Algorithms book has a great chapter about dynamic programming. AND it's free on Google Books! Check it out here.

**Dynamic Programming - User Web Pages,** is a powerful technique that can be used to solve many problems in time O(n2) or O(n3) for which a naive approach would take exponential time. Dynamic programming is a technique for solving problems with overlapping sub problems. A dynamic programming algorithm solves every sub problem just once and then Saves its answer in a table (array). Avoiding the work of re-computing the answer every time the sub problem is encountered.

**Dynamic Programming,** algorithm will try to examine the results of the previously solved sub-problems. Dynamic Programming is a Bottom-up approach-we solve all possible small problems and then combine to obtain solutions for bigger problems. Dynamic Programming is a paradigm of algorithm design in which an optimization problem is solved by a combination of achieving sub-problem solutions and appearing to the " principle of optimality ".

**Data Structures - Dynamic Programming,** that helps to efficiently solve a class of problems that have overlapping subproblems and optimal substructure property. Such problems involve repeatedly calculating the value of the same subproblems to find the optimum solution. Dynamic Programming. Dynamic Programming is mainly an optimization over plain recursion. Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. The idea is to simply store the results of subproblems, so that we do not have to re-compute them when needed later.

It's an optimization of your algorithm that cuts running time.

While a Greedy Algorithm is usually called *naive*, because it may run multiple times over the same set of data, Dynamic Programming avoids this pitfall through a deeper understanding of the partial results that must be stored to help build the final solution.

A simple example is traversing a tree or a graph only through the nodes that would contribute with the solution, or putting into a table the solutions that you've found so far so you can avoid traversing the same nodes over and over.

Here's an example of a problem that's suited for dynamic programming, from UVA's online judge: Edit Steps Ladder.

I'm going to make quick briefing of the important part of this problem's analysis, taken from the book Programming Challenges, I suggest you check it out.

Take a good look at that problem, if we define a cost function telling us how far appart two strings are, we have two consider the three natural types of changes:

Substitution - change a single character from pattern "s" to a different character in text "t", such as changing "shot" to "spot".

Insertion - insert a single character into pattern "s" to help it match text "t", such as changing "ago" to "agog".

Deletion - delete a single character from pattern "s" to help it match text "t", such as changing "hour" to "our".

When we set each of this operations to cost one step we define the edit distance between two strings. So how do we compute it?

We can define a recursive algorithm using the observation that the last character in the string must be either matched, substituted, inserted or deleted. Chopping off the characters in the last edit operation leaves a pair operation leaves a pair of smaller strings. Let i and j be the last character of the relevant prefix of and t, respectively. there are three pairs of shorter strings after the last operation, corresponding to the string after a match/substitution, insertion or deletion. If we knew the cost of editing the three pairs of smaller strings, we could decide which option leads to the best solution and choose that option accordingly. We can learn this cost, through the awesome thing that's recursion:

#define MATCH 0 /* enumerated type symbol for match */ #define INSERT 1 /* enumerated type symbol for insert */ #define DELETE 2 /* enumerated type symbol for delete */ int string_compare(char *s, char *t, int i, int j) { int k; /* counter */ int opt[3]; /* cost of the three options */ int lowest_cost; /* lowest cost */ if (i == 0) return(j * indel(’ ’)); if (j == 0) return(i * indel(’ ’)); opt[MATCH] = string_compare(s,t,i-1,j-1) + match(s[i],t[j]); opt[INSERT] = string_compare(s,t,i,j-1) + indel(t[j]); opt[DELETE] = string_compare(s,t,i-1,j) + indel(s[i]); lowest_cost = opt[MATCH]; for (k=INSERT; k<=DELETE; k++) if (opt[k] < lowest_cost) lowest_cost = opt[k]; return( lowest_cost ); }This algorithm is correct, but is also

impossibly slow.Running on our computer, it takes several seconds to compare two 11-character strings, and the computation disappears into never-never land on anything longer.

Why is the algorithm so slow? It takes exponential time because it recomputes values again and again and again. At every position in the string, the recursion branches three ways, meaning it grows at a rate of at least 3^n – indeed, even faster since most of the calls reduce only one of the two indices, not both of them.

So how can we make the algorithm practical?

The important observation is that most of these recursive calls are computing things that have already been computed before.How do we know? Well, there can only be |s| · |t| possible unique recursive calls, since there are only that many distinct (i, j) pairs to serve as the parameters of recursive calls.

By storing the values for each of these (i, j) pairs in a table, we can avoid recomputing them and just look them up as needed.The table is a two-dimensional matrix m where each of the |s|·|t| cells contains the cost of the optimal solution of this subproblem, as well as a parent pointer explaining how we got to this location:

typedef struct { int cost; /* cost of reaching this cell */ int parent; /* parent cell */ } cell; cell m[MAXLEN+1][MAXLEN+1]; /* dynamic programming table */The dynamic programming version has three differences from the recursive version.

First,it gets its intermediate values using table lookup instead of recursive calls.**Second,**it updates the parent field of each cell, which will enable us to reconstruct the edit sequence later.

**Third,**Third, it is instrumented using a more general goal

`cell()`

function instead of just returning m[|s|][|t|].cost. This will enable us to apply this routine to a wider class of problems.

Here, a very particular analysis of what it takes to gather the most optimal partial results, is what makes the solution a "dynamic" one.

Here's an alternate, full solution to the same problem. It's also a "dynamic" one even though its execution is different. I suggest you check out how efficient the solution is by submitting it to UVA's online judge. I find amazing how such a heavy problem was tackled so efficiently.

**Dynamic Programming,** In programming, Dynamic Programming is a powerful technique that allows one to solve different types of problems in time O(n2) or O(n3) for which a naive Dynamic programming (DP) is breaking down an optimisation problem into smaller sub-problems, and storing the solution to each sub-problems so that each sub-problem is only solved once.

**Dynamic programming,** Dynamic Programming Tutorial** This is a quick introduction to dynamic programming and how Duration: 14:28
Posted: Dec 13, 2017 Dynamic Programming is a powerful technique that allows one to solve many diﬀerent types of problems in time O(n 2) or O(n 3) for which a naive approach would take exponential time.

**Introduction to Dynamic Programming 1 Tutorials & Notes ,** Dynamic Programming for Coding Interviews: A Bottom-Up approach to problem solving Dynamic programming is used where we have problems, which can be divided into similar sub-problems, so that their results can be re-used. Mostly, these algorithms are used for optimization. Before solving the in-hand sub-problem, dynamic algorithm will try to examine the results of the previously solved

**What is Dynamic Programming?,** This is how we can start learning Dynamic Programming. It is an extensively used concept when Duration: 3:07
Posted: Apr 28, 2018 Dynamic Programming is also used in optimization problems. Like divide-and-conquer method, Dynamic Programming solves problems by combining the solutions of subproblems. Moreover, Dynamic Programming algorithm solves each sub-problem just once and then saves its answer in a table, thereby avoiding the work of re-computing the answer every time.

**What Is Dynamic Programming and How To Use It,** Dynamic Programming techniques are primarily based on the principle of Mathematical Induction unlike greedy algorithms which try to make an optimization 98% Success Rate. University of Oregon Researched. Just takes 15 min/day, 3 days/week. 17 years of in-class development. Over 400,000 children taught. Used in over 400 schools.

##### Comments

- Here is one tutorial by Michael A. Trick from CMU that I found particularly helpful: mat.gsia.cmu.edu/classes/dynamic/dynamic.html It is certainly in addition to all resources others have recommended (all other resources, specially CLR and Kleinberg,Tardos are very good!). The reason why I like this tutorial is because it introduces advanced concepts fairly gradually. It is bit oldish material but it is a good addition to the list of resources presented here. Also check out Steven Skiena's page and lectures on Dynamic Programming: cs.sunysb.edu/~algorith/video-lectures http:
- I have always found "Dynamic Programming" a confusing term - "Dynamic" suggests not-static, but what's "Static Programming"? And "... Programming" brings to mind "Object Oriented Programming" and "Functional Programming", suggesting DP is a programming paradigm. I don't really have a better name (perhaps "Dynamic Algorithms"?) but it's too bad we're stuck with this one.
- @dimo414 The "programming" here is more related to "linear programming" which falls under a class of mathematical optimization methods. See article Mathematical optimization for a list of other mathematical programming methods.
- @dimo414 "Programming" in this context refers to a tabular method, not to writing computer code. - Coreman
- The bus ticket cost minimization problem described in cs.stackexchange.com/questions/59797/… is best solved in dynamic programming.
- Didn't you just describe memoization though?
- I would say memoization is a form of dynamic programming, when the memoized function/method is a recursive one.
- Good answer, would only add a mention about optimal sub-structure (e.g., every subset of any path along the shortest path from A to B is itself the shortest path between the 2 endpoints assuming a distance metric that observes the triangle inequality).
- I wouldn't say "easier", but faster. A common misunderstanding is that dp solves problems that naive algorithms can't and that isn't the case. Is not a matter of functionality but of performance.
- Using memoization, dynamic programming problems can be solved in a top down manner. i.e. calling the function to calculate the final value, and that function in turn calls it self recursively to solve the subproblems. Without it, dynamic programming problems can only be solved in a bottom up way.
- This is a great answer and the problem collection on Github is also very useful. Thanks!