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Dynamic Programming is one of those techniques that every programmer should have in their toolbox. But, it is also confusing for a lot of people. For a long time, I struggled to get a grip on how to apply Dynamic Programming to problems. Most articles that I could find on the internet, gave the final dynamic programming solution without actually showing the approach taken to arrive at the final solution.

In this article, I will show the benefits of using a Dynamic Programming approach to solving problems with an example. In the end, I will show some steps you can use to find a Dynamic Programming solution. Hopefully, after reading this article, you will find Dynamic Programming simple and intuitive.

What is Dynamic Programming?

Dynamic programming is an efficient method for solving specific types of complicated computational problems. These problems are generally those that can be broken down into smaller overlapping sub-problems. It can be characterised basically as recursion along with memoization. Memoization is the ability to save the results of specific states to reuse later.

Profiling

To prove that we are improving our solution, we need statistics that we can compare. I will be usinggoogle benchmark to help profile our solutions. The benchmark will look like this

1. Benchmark Name: The name of the benchmark. It will be the format FunctionName/value passed in.
2. Running Time: The time it took for our function to return a result.
3. Iterations/sec: The number of times the function could be invoked in 1 second.
4. Items/sec:Â The number of items that were processed in 1 second.

While 2 and 3 will give us an indication of the time complexity of the function, 4 will give us the space complexity.

Example: Fibonacci Series

The classic example to explain dynamic programming is theFibonacci computation which can be formalised as follows

Fibonacci(n)=0;ifn=0Fibonacci(n)=1;ifn=1Fibonacci(n)=Fibonacci(n-1)+Fibonacci(n-2);ifn>=2

Naive Recursive Approach

The Fibonacci sequence can easily be solved by the following recursive method:

longFibonacci(longn){if(n==0){return0;}if(n==1){return1;}returnFibonacci(n-2)+Fibonacci(n-1);}

On running the above code and profiling it on my machine, I get:

Although this method returns almost instantly forn <= 30, it takes a little less than a second forn = 40 and approximately 2 minutes forn = 50. Why is the amount of running time increasing so rapidly? This can be explained easily by following the execution stack. Let's do this forn = 6 to keep it simple. The following image shows the sequence of calls that are made. Fibonacci Series : From Wikimedia Commons

Looking at the image, we can see that to calculatefibonacci(6), we calculateÂfibonacci(5) 1 time,Âfibonacci(4) 2 times_, _fibonacci(3) 3 times,Âfibonacci(2) 5 times_, fibonacci(1) 8 times_ and fibonacci(0) 5 times. Throughout the call-stack, we are repeatedly computing values that we have already computed. This amount of duplicated work being done keeps on increasing asn becomes larger. You must have realised that this solution is not at all scalable. If you are thinking that there has to be a better way, you are correct.

Top-Down Recursive approach with Memoization

The 1st step to improving the above solution is to add memoization i.e to store the previously computed values in some kind of a data structure. Although you can use any data structure that you like, for the purposes of this example, I will use a map.

#define MOD 1000000007longFibonacciMemonized(longn){std::map<long,long>computedValues;computedValues.insert(make_pair(0,1));computedValues.insert(make_pair(1,1));returnFibonacciMemonized(n,computedValues);}longFibonacciMemonized(longn,std::map<long,long>&computedValues){if(computedValues.find(n)!=computedValues.end()){returncomputedValues[n];}longnewValue=(FibonacciMemonized(n-1,computedValues)+FibonacciMemonized(n-2,computedValues))%MOD;computedValues.insert(make_pair(n,newValue));returnnewValue;}

The top method is our main method. All it does is add the 2 base cases to a map and then calls the bottom method with the map as one of the arguments. This bottom method is our recursive method. In this method, we check if the map contains the computed value. If it does. then we just return that value, otherwise, we compute the value forn-1 andÂn-2. We mod the result usingÂ1000000007 to avoid overflows. Before, returning their sum, we store the value in our map. How better is this version? Let's look at the benchmark results

We can see that we have reduced the amount of time drastically. Even forn = 20000, the result is instantaneous. However, there is a problem with this approach. Can you spot it? If you said memory usage, you are absolutely correct.  Although the new version is much faster, it is still a recursive algorithm. And, the problem with recursive algorithms is that each recursive call takes some space on the stack. A high enoughn, and we run the risk of running out of memory. Let's see why this happens with an example whereÂn = 100. Because, we don't have the result when we start, we call the method recursively for 999, 998, 997 ... 1. At that point, we have all the computed results in our map. Now, as we return from our recursive functions, we just lookup the value in the table and return it. So, even though we have reduced the number of recursive calls, we still maken recursive calls before getting our initial result. This can easily be seen by comparing the iteration/seconds between this and the previous algorithm. Let's try something better.

Bottom Up Approach with Dynamic Programming

In the previous approach, our main problem was the recursive nature of our algorithm. Let's see if we can get rid of it by using an iterative approach. How do we do this? Instead of starting from the final value, we will start with the smallest value ofÂn and build up the results.

#define MOD 1000000007longFibonacciDP(longn){if(n==0){return0;}if(n==1){return1;}long*results=newlong[n+1];results[0]=0;results[1]=1;for(inti=2;i<=n;i++){results[i]=(results[i-1]+results[i-2])%MOD;}longvalue=results[n];delete[]results;returnvalue;}

In the above function, we have an array ofÂn+1 to store the results. We initialise the array for our base cases ofn=0Â andn=1Â **and then start iterating from **2 ton. At each step, we can use the 2 previously computed values and finally return the result. Lets again look at the benchmark results to see how does this approach do?

Even whenÂn=210,000, this approach returns almost instantly. At the same time, since, this algorithm is not recursive in nature, we have drastically reduced the amount of space that is required. We can see this by comparing the items/sec which is decreasing much slower even thoughÂn is increasing more rapidly than before. Now you may be thinking, that since we have got a linear time complexity and linear space complexity, this is the end. In most cases, that would be true. But, in this case, we can optimise our solution further.

Bottom Up Approach with Dynamic Programming (Optimised)

In the last algorithm, the amount of space required is proportional toÂn. This is because we are storing all the results. But, we don't need to store all of them. Let's get rid of the array by using just 3 variables - 2 to store the previous results and 1 to store the current result.

#define MOD 1000000007longFibonacciDPOptimized(longn){if(n==0){return0;}if(n==1){return1;}longn1=0;longn2=1;longcurrent=0;for(inti=2;i<=n;i++){current=(n1+n2)%MOD;n1=n2;n2=current;}returncurrent;}

Although this algorithm is doing exactly the same thing as the previous one, we have reduced our space complexity to be constant since the amount of space needed is no longer dependent onÂn. Again the benchmark results for comparison

Here, we can see that although **n **is increasing, the items/sec is more or less the same. This is the best we can do and no further optimisations are possible.

Conclusion

From the above example, we can see that we only need to identify overlapping subproblems and then avoid duplicated work by caching the computed results. To recap, we can use these steps to find a dynamic programming approach to our problem

1. Find the overlapping subproblem.
2. Start with a recursive solution
3. Modify the recursive solution to use a top-down memoized version.
4. Remove the recursion by making it an iterative solution.
5. If you don't need to keep all the previous results, keep only the ones that are required.

If you would like to run the code yourself, you can find the codehereÂ

Hopefully, this article has removed the mystery around Dynamic Programming. In the next few articles in the series, we will look at some of the more common problems that can be solved by Dynamic Programming.and use the above steps to come up with a solution. Have you tried Dynamic Programming before? How was your experience? Let me know in the comments.