## Follow these steps to solve any Dynamic Programming interview problem

by Nikola Otasevic

Despite having significant experience building software products, many engineers feel jittery at the thought of going through a coding interview that focuses on algorithms. I’ve interviewed hundreds of engineers at Refdash , Google, and at startups I’ve been a part of, and some of the most common questions that make engineers uneasy are the ones that involve Dynamic Programming (DP).

Many tech companies like to ask DP questions in their interviews. While we can debate whether they’re effective in evaluating someone’s ability to perform in an engineering role, DP continues to be an area that trips engineers up on their way to finding a job that they love.

## Dynamic Programming — Predictable and Preparable

## 7 Steps to solve a Dynamic Programming problem

- How to recognize a DP problem
- Identify problem variables
- Clearly express the recurrence relation
- Identify the base cases
- Decide if you want to implement it iteratively or recursively
- Add memoization
- Determine time complexity

## Sample DP Problem

## Problem statement:

In this problem, we’re on a crazy jumping ball, trying to stop, while avoiding spikes along the way.

## Here are the rules:

3) Every time you land on a spot, you can adjust your speed by up to 1 unit before the next jump.

## Step 1: How to recognize a Dynamic Programming problem

Recognizing a Dynamic Programming problem is often the most difficult step in solving it. Can the problem solution be expressed as a function of solutions to similar smaller problems?

## Step 2: Identify problem variables

In our example, the two parameters that could change for every subproblem are:

Identify the changing parameters and determine the number of subproblems.

## Step 3: Clearly express the recurrence relation

Here is how we think about it in our sample problem:

More formally, if our speed is S, position P, we could go from (S, P) to:

- (S, P + S) ; # if we do not change the speed
- (S — 1, P + S — 1) ; # if we change the speed by -1
- (S + 1, P + S + 1) ; # if we change the speed by +1

canStop(S, P) = canStop(S, P + S) || canStop(S — 1, P + S — 1) || canStop(S + 1, P + S + 1)

Woohoo, it seems like we have our recurrence relation!

Recurrence relation: Assuming you have computed the subproblems, how would you compute the main problem?

## Step 4: Identify the base cases

- P should be within the bounds of the given runway
- P cannot be such that runway[P] is false because that would mean that we’re standing on a spike
- S cannot be negative, and a S==0 indicates that we’re done

- P < 0 || P >= length of runway seems like the right thing to do. An alternative could be to consider m aking P == end of runway a base case. However, it is possible that a problem splits into a subproblem which goes beyond the end of the runway, so we really need to check for inequality.
- This seems pretty obvious. We can simply check if runway[P] is false .
- Similar to #1, we could simply check for S < 0 and S == 0. However, here we can reason that it is impossible for S to be < 0 because S decreases by at most 1, so it would have to go through S == 0 case beforehand. Ther efore S == 0 is a sufficient base case for the S parameter.

## Step 5: Decide if you want to implement it iteratively or recursively

An iterative solution: (original code snippets can be found here )

## Step 6: Add memoization

- Store your function result into your memory before every return statement
- Look up the memory for the function result before you start doing any other computation

- I created a runway of length 1000 with spikes in random places (I chose to have a probability of a spike being in any given spot to be 20%)
- initSpeed = 30
- I ran all functions 10 times and measured the average time of execution

Here are the results (in seconds):

## Step 7: Determine Time complexity

- Count the number of states — this will depend on the number of changing parameters in your problem
- Think about the work done per each state. In other words, if everything else but one state has been computed, how much work do you have to do to compute that last state?

In our example problem, the number of states is |P| * |S|, where

For a runway of length L , the following has to hold:

=> (S-1) + (S-2) + (S-3) + ….+ 1 < L

If you find roots of the above function, they will be:

r1 = 1/2 + sqrt(1/4 + 2L) and r2 = 1/2 — sqrt(1/4 + 2L)

We can write our inequality as:

Considering that S — r2 > 0 for any S > 0 and L > 0, we need the following:

S — 1/2 — sqrt(1/4 + 2L) < ; 0

O(L * sqrt(L)) which is better than O(L²)

O(L * sqrt(L)) is the upper bound on the time complexity

Awesome, you made it through! :)

## Here are some next steps that you can take

- Extend the sample problem by trying to find a path to a stopping point. We solved a problem that tells you whether you can stop, but what if you wanted to also know the steps to take in order to stop eventually along the runway? How would you modify the existing implementation to do that?
- If you want to solidify your understanding of memoization, and understand that it is just a function result cache, you should read about decorators in Python or similar concepts in other languages. Think about how they would allow you to implement memoization in general for any function that you want to memoize.
- Work on more DP problems by following the steps we went through. You can always find a bunch of them online (ex. LeetCode or GeeksForGeeks ). As you practice, keep in mind one thing: learn ideas, don’t learn problems. The number of ideas is significantly smaller and it’s an easier space to conquer which will also serve you much better.

If this article was helpful, tweet it .

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- Optimal Substructure Property in Dynamic Programming | DP-2
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## How to solve a Dynamic Programming Problem ?

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- Difficulty Level : Medium
- Last Updated : 10 Jan, 2023
- Discuss(50+)

Dynamic Programming (DP) is a technique that solves some particular type of problems in Polynomial Time . Dynamic Programming solutions are faster than the exponential brute method and can be easily proved their correctness.

## To dynamically solve a problem, we need to check two necessary conditions:

- Overlapping Subproblems : W hen the solutions to the same subproblems are needed repetitively for solving the actual problem. The problem is said to have overlapping subproblems property.
- Optimal Substructure Property : I f the optimal solution of the given problem can be obtained by using optimal solutions of its subproblems then the problem is said to have Optimal Substructure Property.

## Steps to solve a Dynamic programming problem:

- Identify if it is a Dynamic programming problem.
- Decide a state expression with the Least parameters.
- Formulate state and transition relationship.
- Do tabulation (or memorization).

## Step 1: How to classify a problem as a Dynamic Programming Problem?

- Typically, all the problems that require maximizing or minimizing certain quantities or counting problems that say to count the arrangements under certain conditions or certain probability problems can be solved by using Dynamic Programming.
- All dynamic programming problems satisfy the overlapping subproblems property and most of the classic Dynamic programming problems also satisfy the optimal substructure property. Once we observe these properties in a given problem be sure that it can be solved using Dynamic Programming.

## Step 2: Deciding the state

A state can be defined as the set of parameters that can uniquely identify a certain position or standing in the given problem. This set of parameters should be as small as possible to reduce state space.

## Step 3: Formulating a relation among the states

The total number of ways to form 6 is: 8 1+1+1+1+1+1 1+1+1+3 1+1+3+1 1+3+1+1 3+1+1+1 3+3 1+5 5+1

The steps to solve the given problem will be:

- We decide a state for the given problem.
- We will take a parameter N to decide the state as it uniquely identifies any subproblem.
- DP state will look like state(N), state(N) means the total number of arrangements to form N by using {1, 3, 5} as elements. Derive a transition relation between any two states.
- Now, we need to compute state(N).

1) Adding 1 to all possible combinations of state (n = 6) Eg : [ (1+1+1+1+1+1) + 1] [ (1+1+1+3) + 1] [ (1+1+3+1) + 1] [ (1+3+1+1) + 1] [ (3+1+1+1) + 1] [ (3+3) + 1] [ (1+5) + 1] [ (5+1) + 1] 2) Adding 3 to all possible combinations of state (n = 4); [(1+1+1+1) + 3] [(1+3) + 3] [(3+1) + 3] 3) Adding 5 to all possible combinations of state(n = 2) [ (1+1) + 5] (Note how it sufficient to add only on the right-side – all the add-from-left-side cases are covered, either in the same state, or another, e.g. [ 1+(1+1+1+3)] is not needed in state (n=6) because it’s covered by state (n = 4) [(1+1+1+1) + 3]) Now, think carefully and satisfy yourself that the above three cases are covering all possible ways to form a sum total of 7; Therefore, we can say that result for state(7) = state (6) + state (4) + state (2) OR state(7) = state (7-1) + state (7-3) + state (7-5) In general, state(n) = state(n-1) + state(n-3) + state(n-5)

Below is the implementation of the above approach:

## Step 4: Adding memoization or tabulation for the state

Adding memoization to the above code

You may check the below problems first and try solving them using the above-described steps:-

Solve DSA problems on GfG Practice.

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## Improve your Coding Skills with Practice

Start your coding journey now.

## Steps to solve any Dynamic Programming Problem

Step 1: recognize a dp problem.

## Step 2: Identify problem variables

Identify the changing parameters and determine the number of subproblems.

## Step 3: Clearly express the recurrence relation

Recurrence relation: Assuming you have computed the subproblems, how would you compute the main problem?

## Step 4: Identify the base cases

## Step 5: Decide if you want to implement it iteratively or recursively

## Step 6: Add memoization

## Step 7: Determine Time complexity

- Count the number of states — this will depend on the number of changing parameters in your problem
- Think about the work done per each state. In other words, if everything else but one state has been computed, how much work do you have to do to compute that last state?

## More from Manvi Tyagi

Software Developer working to become a better Software Developer everyday :)

## Get the Medium app

## Manvi Tyagi

LTCWM > Blog > Resources > Technical guides > How to Solve Any Dynamic Programming Problem

## How to Solve Any Dynamic Programming Problem

Updated on April 12th, 2020 | Sign up for learn to code tips

When it comes to coding interviews , not all topics are created equal.

## What is Dynamic Programming?

## Dynamic Programming Doesn’t Have to Be Hard

- Want to find the optimal solution? You actually need to start with the brute force approach.
- Want to find an iterative solution? You have to start with recursion.
- Want to solve the problem as quickly as possible? You need to slow it down and go step by step.

To start, let’s look at how most people prepare for their coding interviews:

They focus on memorizing solutions.

Memorizing gives you a quick and easy win. But the problem is that it ultimately handicaps you.

Not to mention that this approach prevents you from actually being able to connect the dots.

So you start memorizing. Here’s one sample set of words:

“suis”, “es”, “est”, “sommes”, “êtez”, “sont”

If we look at the translations, we see:

- “Je suis” – “I am”
- “Tu es” – “You are”
- “Elle est” – “She is”
- “Nous sommes” – “We are”
- “Vous êtez” – “You all are”
- “Ils sont” – “They are”

So what if we could do the same thing with dynamic programming?

The similarity between all dynamic programming problems is in the process.

## How to Solve Dynamic Programming Problems Using the Fast Method

What is the most important characteristic of any successful interviewee?

They have a repeatable strategy.

The FAST Method is an acronym for the 4 steps you need to solve any dynamic programming problem:

- F ind the First Solution
- A nalyze the First Solution
- Identify the S ubproblems
- T urn around the solution

Stop waiting and start learning! Get my 10 tips on teaching yourself how to code.

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## 1. Find the First Solution

There are several keys to think about while doing this step that you’ll want to keep in mind:

- The solution should be recursive. If not, the problem probably isn’t a good candidate for dynamic programming. For a lot more info on effectively coming up with a recursive solution, look here .
- Each recursive call must be self-contained. This means that you cannot store your results to a global variable or pass a results variable into your function. I often refer to the required approach as “building up as you return” and you can learn more about that here .
- It is important that we minimize the number of variables that we are passing into our recursive function. Dynamic programming solutions rely on there being multiple recursive calls with the same input, and the more variables there are, the less the inputs will overlap.

## 2. Analyze the First Solution

These are the criteria that we need to look for:

## Optimal Substructure

## Overlapping Subproblems

## 3. Identify the Subproblems

For full code and example problems, download Dynamic Programming for Interviews .

## 4. Turn Around the Solution

When doing dynamic programming, we really have two different options:

See examples of exactly how to do this in my free ebook, Dynamic Programming for Interviews .

## Next Steps : Where to Learn More About Dynamic Programming

Want to learn more about dynamic programming? Check out these online courses:

- Intro To Dynamic Programming – Coding Interview Preparation (Udemy): Specifically designed to help you gain foundational knowledge for your software engineering coding interview
- Dynamic Programming – I (Udemy): Also intro-level and geared toward coding interview basics
- Master the Art of Dynamic Programming (Udemy): In-depth theory and practice of dynamic programming
- Learn Dynamic Programming Using Python (Udemy): Hones in specifically on dynamic programming in the Python programming language
- Greedy Algorithms, Minimum Spanning Trees, and Dynamic Programming (Coursera): A course offered by Stanford about dynamic programming and other algorithm-related topics
- Dynamic Programming: Applications in Machine Learning and Genomics (edX): Learn about two interesting applications of dynamic programming that meld science and technology, from UC San Diego

## Dynamic Programming: An Approach to Solving Computing Problems

## Dynamic programming

More From Artturi Jalli: Python Cheat Sheet: A Handy Guide to Python

## What Types of Problems Can Dynamic Programming Solve?

There are two conditions that need to be satisfied to use dynamic programming:

## What Are Overlapping Subproblems?

There are two different ways of storing the solutions to the overlapping subproblems:

## What Is Memoization?

## What Is Tabulation?

Again, the examples below should make this easier to understand.

## What Is Optimal Substructure Property?

The shortest path from the Start node to the Goal node is through nodes three and four.

## Dynamic Programming Example: Calculating Fibonacci Numbers

Due to the recursion, this function runs in exponential time.

This makes this function actually run in linear time instead of exponential time.

More in Software Engineering: Glob Module in Python: Explained

## How to Solve Problems Using Dynamic Programming

## Great Companies Need Great People. That's Where We Come In.

## PROGRAMMING TUTORIAL

Beginners guide to dynamic programming, optimize your code using some simple techniques.

Dynamic programming is an art, the more problems you solve easier it gets.

## What is Dynamic Programming?

Two ways in which dynamic programming can be applied:

Memoize != memorize

## Understanding Where to Use This Technique

Dynamic Programming can be applied to any such problem that requires the re-calculation of certain values to reach the final solution.

## Recursion and Dynamic Programming

Remember, dynamic programming should not be confused with recursion.

## Understanding Dynamic Programming With Examples

Let’s start with a basic example of the Fibonacci series.

F(n) = F(n-1) + F(n-2)

F(4) = F(3) + F(2) = ((F(2) + F(1)) + F(2) = ((F(1) + F(0)) + F(1)) + (F(1) + F(0))

Time complexity: O(n) <<< O(2^N)

Now, let’s see another example (this is an intermediate level problem):

An intuitive approach to this problem:

- My first intuitive approach was to create a list l of integers till n .
- Then append all the possible combinations of integers of list l into a new list sol .
- And, at the final step, I used a for loop to check the sum of every element of the list sol that if it is equal to the required value. If the condition is true that element is appended to another new list final . And the length of final is returned as a final solution to the problem.

Bottom-up approach for the same problem:

- At the first step, an empty list ‘a’ is initiated to store all the values from the further loops.
- After each iteration of the outer loop, a[j] is the number of staircases you can make with height at most i where j is the number of bricks used.
- List a is initiated to [1,0,0,...] because there can be only one stair with 0 blocks and 0 height.
- In each iteration of the inner loop, list a is transformed from representing max-height i-1 to representing max-height i , by incorporating the possibility of adding a step of height i to any shorter staircase that leaves you with at least i blocks.
- In the final step, the number of different staircases that can be built from exactly n bricks is returned by the function (1 is subtracted at the end to exclude the case of single stair of height n ).

“Those who cannot remember the past are condemned to repeat it.” -George Santayana

Check this out: https://www.fiverr.com/share/LDDp34

## More from Towards Data Science

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## Get the Medium app

## Shantnu & Kartik

## Introduction to Dynamic Programming

There are two ways of doing this.

l"> =((Fib(1) + Fib(0)) + Fib(1) ) + Fib(2)

=((Fib(1) + Fib(0)) + Fib(1) ) + ( Fib(1) + Fib(0) )

For ex. In combinatorics, C(n.m) = C(n-1,m) + C(n-1,m-1).

## Problem Statement:

## Approach / Idea:

## Memoization

Pseudo-code for finding the length of the longest increasing subsequence:

Similar concept could be applied in finding longest path in Directed acyclic graph.

---------------------------------------------------------------------------

Longest Common Subsequence - Dynamic Programming - Tutorial and C Program Source code

Can we break the problem of finding the LCS of S1[1...N] and S2[1...M] in to smaller subproblems ?

[to do , fibonacci , LCS etc., ]

You can refer to some of these in the Algorithmist site

2. The lucky draw(June 09 Contest). https://www.codechef.com/problems/D2/

3. Find the number of increasing subsequences in the given subsequence of length 1 or more.

To see problems on DP visit this link

6.TopCoder - AvoidRoads - A simple and nice problem to practice

Matrix findNthPower( Matrix M , power n )

Matrix R = findNthPower ( M , n/2 );

R = RxR; // matrix multiplication

if( n%2 == 1 ) R = RxM; // matrix multiplication

You can read more about it here

Tutorials and C Program Source Codes for Common Dynamic Programming problems

## The Ultimate Guide to Dynamic Programming

Imagine it again with those spooky Goosebumps letters.

## What Is Dynamic Programming?

A problem can be optimized using dynamic programming if it:

## Optimal Substructure

## Overlapping Subproblems

## When Should I Use Dynamic Programming?

Solve any dp problem using the fast method.

It was this mission that gave rise to The FAST Method .

## IMAGES

## VIDEO

## COMMENTS

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Steps to Solve Dynamic Programming Problems · 1. Identify the sub-problem and write it down in words · 2. Sub-problem expressed as Mathematical

The first step to solving any dynamic programming problem using The FAST Method is to find the initial brute force recursive solution.