# The complete beginners guide to dynamic programming

If you’ve been programming for long enough, you’ve probably heard the term dynamic programming. Often a key subject in technical interviews, the idea will also come up in design review meetings or regular interactions with fellow developers. This essay will examine what dynamic programming is and why you would use it. I’ll be illustrating this concept with specific code examples in Swift, but the concepts I introduce can be applied to your language of choice. Let’s begin!

## A way of thinking

Unlike specific coding syntax or design patterns, dynamic programming isn’t a particular algorithm but a **way of thinking**. Therefore, the technique takes many forms when it comes to implementation.

The main idea of dynamic programming is to consider a significant problem and break it into smaller, individualized components. When it comes to implementation, optimal techniques rely on data storage and reuse to increase algorithm efficiency. As we’ll see, many questions in software development are solved using various forms of dynamic programming. The trick is recognizing when optimal solutions can be devised using a simple variable or require a sophisticated data structure or algorithm.

For example, code variables can be considered an elementary form of dynamic programming. As we know, a variable’s purpose is to reserve a specific place in memory for a value to be recalled later.

```
//non-memoized function
func addNumbers(lhs: Int, rhs: Int) -> Int {
return lhs + rhs
}
//memoized function
func addNumbersMemo(lhs: Int, rhs: Int) -> Int {
let result: Int = lhs + rhs
return result
}
```

While `addNumbersMemo`

above does provide a simple introduction, the goal of a dynamic programming solution is to preserve previously seen values because the alternative could either be inefficient or could prevent you from answering the question. This design technique is known as **memoization**.

## Code challenge—Pair of numbers

Over the years, I’ve had the chance to conduct mock interviews with dozens of developers preparing for interviews at top companies like Apple, Facebook, and Amazon. Most of us would be happy to skip the realities of the dreaded whiteboard or take-home coding project. However, the fact is that many of these brain-teaser questions are designed to test for a basic understanding of computer science fundamentals. For example, consider the following:

```
/*
In a technical interview, you've been given an array of numbers and you need to find a pair of numbers that are equal to the given target value. Numbers can be either positive, negative, or both. Can you design an algorithm that works in O(n)—linear time or greater?
let sequence = [8, 10, 2, 9, 7, 5]
let results = pairValues(sum: 11) = //returns (9, 2)
*/
```

As developers, we know there are usually multiple ways to arrive at a solution. In this case, the goal is knowing which numbers should be paired to achieve the expected result. As people, it’s easy for us to quickly scan the sequence of numbers and promptly come up with the pair of 9 and 2. However, an algorithm will need to either check and compare each value in the sequence or develop a more streamlined solution to help us find the values we are seeking. Let’s review both techniques.

## A brute force approach

Our first approach involves looking at the first value, then reviewing each subsequent value to determine if it will provide the **difference** needed to solve the question. For example, once our algorithm checks the value of the first array item, 8, it will then scan the remaining values for 3 (e.g., 11 – 8 = 3). However, we can see the value of 3 doesn’t exist, so the algorithm will repeat the same process for the next value (in our case, 10) until it finds a successful matching pair. Without going into the details of big-O notation, we can assume this type of solution would have an average runtime of O(n ^ 2)time or greater, mainly because our algorithm works by comparing each value with every other value. In code, this can be represented as follows:

```
let sequence = [8, 10, 2, 9, 7, 5]
//non-memoized version - O(n ^ 2)
func pairNumbers(sum: Int) -> (Int, Int) {
for a in sequence {
let diff = sum - a
for b in sequence {
if (b != a) && (b == diff) {
return (a, b)
}
}
}
return (0, 0)
}
```

## A memoized approach

Next, let’s try a different approach using the idea of memoization. Before implementing our code, we can brainstorm how storing previously seen values will help streamline the process. While using a standard array is feasible, a set collection object (also referred to as a hash table or hash map) could provide an optimized solution.

```
//memoized version - O(n + d)
func pairNumbersMemoized(sum: Int) -> (Int, Int) {
var addends = Set<Int>()
for a in sequence {
let diff = sum - a
if addends.contains(diff) { //O(1) - constant time lookup
return (a, diff)
}
//store previously seen value
else {
addends.insert(a)
}
}
return (0, 0)
}
```

Using a memoized approach, we’ve improved the algorithm’s average run time efficiency to O(n + d) by adding previously seen values to a set collection object. Those familiar with hash-based structures will know that item insert and retrieval occurs in O(1) – constant time. This further streamlines the solution, as the set is designed to retrieve values in an optimized way regardless of size.

### The Fibonacci sequence

When learning various programming techniques, one topic that comes to mind is recursion. Recursive solutions work by having a model that refers to itself. As such, recursive techniques are implemented through algorithms or data structures. A well-known example of recursion can be seen with the Fibonacci sequence—a numerical sequence made by adding the two preceding numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, etc):

```
public func fibRec(_ n: Int) -> Int {
if n < 2 {
return n
} else {
return fibRec(n-1) + fibRec(n-2)
}
}
```

When examined, our code is error-free and works as expected. However, notice a few things about the performance of the algorithm:

Positions (n) | `fibRec()` – Number of times called |

2 | 1 |

4 | 5 |

10 | 109 |

15 | 1219 |

As shown, there’s a significant increase in the number of times our function is called. Similar to our previous example, the algorithm’s performance decreases exponentially based on the input size. This occurs because the operation does not store previously calculated values. Without access to stored variables, the only way we can obtain the required (preceding) values is through recursion. Assuming this code is used in a production setting, the function could introduce bugs or performance errors. Let’s refactor the code to support a memoized approach:

```
func fibMemoizedPosition(_ n: Int) -> Int {
var sequence: Array<Int> = [0, 1]
var results: Int = 0
var i: Int = sequence.count
//trivial case
guard n > i else {
return n
}
//all other cases..
while i <= n {
results = sequence[i - 1] + sequence[i - 2]
sequence.append(results)
i += 1
}
return results
}
```

This revised solution now supports memoization through the use of stored variables. Notice how the refactored code no longer requires a recursive technique. The two most previous values are added to a result, which is appended to the main array sequence. Even though the algorithm’s performance still depends on the sequence size, our revisions have increased algorithmic efficiency to O(n) – linear time. In addition, our iterative solution should be easier to revise, test and debug since a single function is added to the call stack, thus reducing complexities with memory management and object scope.

## Conclusion

We’ve learned that dynamic programming isn’t a specific design pattern as it is a way of thinking. Its goal is to create a solution to preserve previously seen values to increase time efficiency. While examples include basic algorithms, dynamic programming provides a foundation in almost all programs. This includes the use of simple variables and complex data structures.

Tags: dynamic programming, software development
## 12 Comments

For fibMemoizedPosition, at the trivial case it should be “guard n < i" in order to work properly

I think this article could lead someone to think that memoization is synonym for dynamic programming.

As a member of Code Review Stack Exchange, I do have to point out that the indentation in pairNumbersMemoized is not consistent. If only there was a site you could put up code for review before you publish it on your blog…

I think there is something wrong with your solution of pair of numbers. You are assuming that there is no repetition of numbers in your sequence and the query number is the sum of those. The correction for the brute force solution could be (python):

for i,a in enumerate(sequence):

diff =sum-a

for j,b in enumerate(sequence):

If(i != j and b==diff):

Return(a,b)

…

For the other solution my idea is using a dictionary instead of a set and for each diff save a list of values that give the diff number, in the end just look for the list with two elements. Of course this time the cost is worse since you need to build the dictionary O(n) and then search the list with length 2 O(n-1). Let me know what you think.

Memoization shouldn’t be used in the Fibonacci sequence example, there is no need to store the whole sequence as you only ever need to previous two values. Storing the whole lot in an array is a waste of memory and isn’t showing a novice programmer how this problem should be solved.

I think the example is in case someone wants random access to the Fibonacci sequence.

Okay thanks to this post I understood memoization (a word almost impossible to type with autocorrect on btw), and realized I was actually using it while not understanding the term, that might prove useful!

But I still don’t understand what is dynamic in that usage, why is that more dynamic than the non memoized version?

See here for the origin of the term: https://en.wikipedia.org/wiki/Dynamic_programming#History

As someone who doesn’t usually click on article-links from the Overflow, I have to say I was disappointed with this one. If this is the quality of articles I can expect from that newsletter, I may not be clicking in too often.

To call this guide “complete” even for beginners, would be a bit of a stretch. It doesn’t even begin to touch upon what I consider the basics of DP, which is the solving of problems by solving other (smaller) sub-problems. Sure, once you have that problem sub-structure defined, you might realize that there’s a lot of repetitive work being done and memoization can help. And then maybe you refine your implementation to work “bottom up” instead of recursion-with-memoization.

This article instead, makes it sound like memoization *is* DP, which isn’t entirely accurate or helpful. Readers are better off referring to other resources to get a handle on DP.

Dynamic programming is not the same as memo’ization. Dynamic programming is the notion of solving successively growing subproblems. It is a way to solve problems where, once solve a subproblem, the next larger one uses this and you never have to go back. Of course, recording these subproblem solutions is memo’ization, but there’s more to it.

A much better example is the Smith-Waterman algorithm for gene matching. There `value(i,j)=min( value(i-1,j),value(i,j-1) )` so you make a “wavefront” for values as you progress through `i,j` space.

This is very informative the article broadens my mind on what dynamic programming is.

Unfortunately, I have to agree with Miles Smarck’s comment with the lack of quality and clickbait title of this post. This is not the complete guide and DP is much more than just memoization. I would strongly recommend reading better material to learn DP, this post is definitely not it. This isn’t the first time I have read a poorly written article on this blog and will consider reducing my time invested here as it is not improving.