Thanks, I hope the article helps in implementation as well. You have a main problem (the root of your tree of subproblems), and subproblems (subtrees). Memoization is a common strategy for dynamic programming problems, which are problems where the solution is composed of solutions to the same problem with smaller inputs (as with the Fibonacci problem, above).The other common strategy for dynamic programming problems is going bottom-up, which is usually cleaner and often more efficient. Backtracking. Post was not sent - check your email addresses! 02, Sep 18. Formula:- fib(n) = fib(n-1) + fib(n-2) where fib(0)=1 and fib(1a)=1. And Kill Your Next Tech Interview Yay! The key takeaway is that they perform similar functions, which is to avoid unnecessary and expensive recalculations of subproblems. Count occurrences . According to Wikipedia, In computing, memoization or memoisation is an optimisation technique used primarily to speed up computer programs by storing the results of expensive function calls and returning the cached result when the same inputs occur again. Can someone explain to me what's the difference? Recursion risks to solve identical subproblems multiple times. As a follow-up to my last topic here, it seems to me that recursion with memoization is essentially the same thing as dynamic programming with a different approach (top-down vs bottom-up). If you’re computing for instance fib(3) (the third Fibonacci number), a naive implementation would compute fib(1)twice: With a more clever DP implementation, the tree could be collapsed into a graph (a DAG): It doesn’t look very impressive in this example, but it’s in fact enough to bring down the complexity from O(2n) to O(n). Recursion and dynamic programming (DP) are very depended terms. E.g. I am a Software Developer based in Bangalore, India. Increase Your Developer Confidence With a Great Django Test Suite. Let us understand the concept of memoization better through an example:-. Go through the below two links Tutorial for Dynamic Programming Recursion Clear examples are given in the above links which solve your doubts. Therefore, in our dynamic programming solution, the value at table[row][col] represents the minimum edit distance required to transform substring word1[:row] to word2[:col]. Javascript Event Loop for Concurrency in Javascript, SEOPressor V5 Giveaway | 3 Single-site licence, How to annoy people while promoting your blog, Best WordPress Security Plugin – Better WP Security Plugin, Top 10 questions that bloggers should ask to themselves, How to make money with Blog Engage – I made $750, Glazedinc Curved UV Tempered Glass Review | OnePlus 8 Pro, Code Quality & Coding Standards with SonarLint, Daemon Threads in Java | How to NOT use them, Convert image to pdf in Java with iTextPdf, It works on the basic principle that when we prove a relation that the equation with, The above relation needs a base case(which is basically the solution of an easy subproblem) and for induction it is always an equation with. 30, Aug 18. Lets discuss this with the help of a classic problem. I wrote this on the Racket educators’ mailing list, and Eli Barzilay suggested I post it here as well. You Have Unsubscribed from All Communications! Assume 2 string s1 and s2 of length n and m respectively. We create a table of size m+1 by n+1, where m and n are the lengths of word1 and word2 respectively. Runtime: 184 ms, faster than 62.60% of Python3 online submissions for Edit Distance. Most of the Dynamic Programming problems are solved in two ways: ... Tabulation vs Memoization. Below is the flowchart of the given pseudo code. Memoization solves the problem Top-Down. LCS of “ABCDEF” and “BDF” is “BDF” of length 3. For instance, recursive binary search has no overlapping subproblems, and so memoization is useless. You can not learn DP without knowing recursion.Before getting into the dynamic programming lets learn about recursion.Recursion is a Minimum and Maximum values of an expression … Get Answer to How Dynamic Programming is different from Recursion and Memoization? Edit Distance | DP using Memoization. Thus, we see that there are overlapping subproblems (i.e. Now, if we see the above flow chart, we can easily see the issue that multiple nth term is getting computed again and again and with this approach, Space Complexity:- O(1) (here, we are not considering the recursion related stack space). In my solution, I use the tuple (i, j) as the key in my dictionary. Tail recursion. If there are no overlapping subproblems, there is no point caching these results, since we will never use them again. How to think recursively. We are wasting a lot of time recomputing the same answers to the same set of parameters. InterviewCake is a funny place. This technique of using memoization for optimizing programs using backtracking is nothing but Dynamic programming. Dynamic programming is a method for solving complex problems by first breaking them down into simpler sub-problems. Dynamic Programming - Memoization . Thanks for sharing these resources, they are all extremely valuable right now. bottom-up dynamic programming) are the two techniques that make up dynamic programming. Thanks for letting us know! Dynamic programming (DP) means solving problems recursively by combining the solutions to similar smaller overlapping subproblems, usually using some kind of recurrence relations. In that article, I pretty much skipped to the dynamic programming solution directly, with only a brief introduction of what dynamic programming is and when it can be applied. That’s all from my side. This site uses Akismet to reduce spam. 4 min read. Difference between dynamic programming and recursion with memoization? 2012–08–27, 13:10EDT: also incorporated some comments.] Tabulation solves the problem Bottom-Up. That’s all from my side. You Have Unlocked All the Answers! And finally, for “aa” and “a”, we would delete the last character of s1. I came across another dynamic programming problem recently (Edit Distance) and I wanted to explore dynamic programming in greater detail. Water Jug Problem using Memoization . This inefficiency is addressed and remedied by dynamic programming. So, now when we know an equation is true for n=1, we can use the bottom-up approach and reach till n(which is the whole problem). Plus 11 solved and explained coding problems to practice: Sum of digits. In fact, this is the entire basis for memoization, and so if you understand the section above on memoization, you would also have understood what “overlapping subproblems” means. Recursion is a method of solving a problem where the solution depends on the solution of the subproblem. For instance, the recursive function fibonacci(10) requires the computation of the subproblems fibonacci(9) and fibonacci(8), but fibonacci(9) also requires the computation of fibonacci(8). You can find the full problem statement here.). In simple words, Memoization is used for problems that need to execute a function with the same set of arguments multiple times and the computation takes a lot of time hence, caching/storing the result saves a lot of computation time. Basically, we have to recursively traverse to the n-1 and n-2 function(induction step) till we reach n=1 or n=0 as we know their values. In this case, only i and j are determinant of the result, since word1 and word2 are immutable. Enter your email address to subscribe to this blog and receive notifications of new posts by email. Dynamic programming is all about ordering your computations in a way that avoids recalculating duplicate work. Now, at this point Dynamic Programming comes into picture. Some sources, in fact, classify both as variants of dynamic programming. In case of recursion, we can have a generic base case and an induction step. It was filled with struggle, both in terms of personal morale and in terms of pure… As you can see, through basic recursion, we come across overlapping subproblems and we can also view that the optimal structure of the problem is computed through the optimal structure of the subproblem. Hey, I loved this article. (Some people may object to … Dynamic programming is a technique to solve a complex problem by dividing it into subproblems. Is this accurate? Many readers ask me how to know if a problem can be solved using dynamic programming. This technique should be used when the problem statement has 2 properties: Question:- Given two sequences, find the length of longest subsequence present in both of them. Notice that the 3 recursive calls in our else block could potentially be repeated many times across recursive calls (visualize the recursion tree). Approach:- By the looks of the problem statement and formula, it seems like a very simple recursive solution. Let us start from the last character(l1 and l2) of each string and let us check whether it can be a part of the longest substring or not:-. 03, Aug 18. l1 and l2 match, so that means that they can be a part of the longest substring. Hence, for finding nth number in fibonacci series, we will always compute the 1 to nth number only once and hence, Space Complexity:- O(n) (here, we are not considering the recursion related stack space). I am passionate about teaching blogging and thrive to contribute to the tech community through my blog posts. Memoization vs Dynamic Programming In fact, memoization and dynamic programming are extremely similar. Learn how your comment data is processed. From the above example, we can also see, for each value the underneath flow chart is always the same i.e the solution/answer will always be the same. Recursion vs Iteration. 10, Nov 18. This article works around the relation of Dynamic Programming, Recursion and Memoization. Dynamic programming (and memoization) works to optimize the naive recursive solution by caching the results to these subproblems. For example, consider your favorite example of Fibonnaci. This is also where our 3 possible string operations apply: we can insert, delete, or replace a character. We also use a nifty trick for optimization. Top down Dynamic Programming is essentially recursion, but enhanced with memoization. Top-down recursion, dynamic programming and memoization in Python. One way to think about it is that memoization is top-down (you recurse from the top but with caching), while dynamic programming is bottom-up (you build the table incrementally). With these observations, we can write a recursive algorithm that calculates the number of edits for all 3 possible operations and returns the minimum of them. Can you please share some more links of your blogs/articles? Loading Data Into BigQuery From Cloud Storage. You’ve just got a tube of delicious chocolates and plan to eat one piece a day –either by picking the one on the left or the right. Dynamic programming, DP for short, can be used when the computations of subproblems overlap. I was talking to a friend about dynamic programming and I realized his understanding of dynamic programming is basically converting a recursive function to an iterative function that calculates all the values up to the value that we are interested in. Dynamic Programming versus Memoization. if we have strings s1=“aa” and s2=“ab”, we would replace the last character of s1. If we need to find the value for some state say dp[n] and instead of starting from the base state that i.e dp[0] we ask our answer from the states that can reach the destination state dp[n] following the state transition relation, then it is the top-down fashion of DP. Recursion is very similar to the concept of induction (which is a mathematical proof technique) which is the procedure to prove an equation with 2 simple steps-. I have Read so many Articles, To do but all those are very time waste, blah, blah, but when i read you article it makes me to do something quickly, thanks so much i will implement this into action very soon , Thanks so much for saving my life. I have gone through a lot of articles on this but can't seem to make sense of it. The other common strategy for dynamic programming problems is going bottom-up, which is usually cleaner and often more efficient. In this tutorial, you will learn the fundamentals of the two approaches to dynamic programming, memoization and tabulation. Dynamic Programming Memoization vs Tabulation. Dynamic Programming. Many times in recursion we solve the problem repeatedly, with dynamic programming we store the solution of the sub-problems in an array, table or dictionary, etc…that we don’t have to calculate again, this is called Memoization. Sign In. Memoization using decorators in Python. Has adjacent duplicates. Explanation for the article: http://www.geeksforgeeks.org/dynamic-programming-set-1/This video is contributed by Sephiri. If the characters don’t match, this is where the crux of the algorithm lies. If we see the formula we can see that factorial of n has a relation with factorial of n-1 and so on. Now, let us see the solution of this approach by a flow diagram. Recursion, dynamic programming, and memoization 19 Oct 2015 Background and motivation. Here’s a better illustration that compares the full call tree of fib(7)(left) to the correspondi… Longest Common Subsequence | DP using Memoization. Memoization is a technique for improving the performance of recursive algorithms It involves rewriting the recursive algorithm so that as answers to problems are found, they are stored in an array. This morning I had a … https://thomaspark.co/wp/wp-content/uploads/2017/01/xkcd.png, solving the Knapsack Problem with dynamic programming, How to Build an API in Python (with Django) — Last Call — RapidAPI Blog, How to use Hyperledger Fabric SDK Go with Vault Transit engine, 3 Popular Embeds for Sharing Code on Medium. We don’t know the exact details of the algorithm yet, but at a high level, we know that it should iterate through each character of each string and compare the characters. The subproblems typically repeat and overlap. Hence, if we cache them we can drastically reduce the time complexity. Therefore, we can “work our way upwards”, by incrementally computing the optimal solutions to subproblems, until we arrive at the optimal solution to our given problem. You " memoize " the computed values in a lookup table (usually an array), to avoid having to recompute those values again in the future; you simply return the value in the lookup table. As we can see, from the above solution memoization, recursion and dynamic programming work hand in hand in optimising the solution. You have the following 3 operations permitted on a word: (Problem is copied off LeetCode, and I’ve omitted the rest of the examples. I just stuck to recursion in this case to extend from the original recursion example. For “aa” and “aab”, we would insert an additional character to s1. To optimize our naive recursive solution, we could use memoization to store results to avoid re-computation. The term “overlapping subproblems” simply means that there are subproblems (of a smaller problem space) that arise repeatedly. This past week was almost exclusively about top-down recursion with dynamic programming (i.e., with memoization). Question:- Find the Nth term of a fibonacci series. Sorry, your blog cannot share posts by email. It helps improve your experience using FSC! Dynamic programming recursion memoization and bottom up algorithms. 13, Apr 17. Memoization is a common strategy for dynamic programming problems, which are problems where the solution is composed of solutions to the same problem with smaller inputs (as with the Fibonacci problem, above). This video is on finding nth Fibonacci number by using dynamic programming. Dynamic programming and memoization: top-down vs bottom-up approaches. Simply put, dynamic programming is just memoization and re-use solutions. Thanks for sharing. Each piece has a positive integer that indicates how tasty it is.Since taste is subjective, there is also an expectancy factor.A piece will taste better if you eat it later: if the taste is m(as in hmm) on the first day, it will be km on day number k. Your task is to design an efficient algorithm that computes an optimal ch… (That’s my strategy for problem-solving, and it works!) Memoization Method – Top Down Dynamic Programming Once, again let’s describe it in terms of state transition. Complete Guide. I don’t think I can phrase this better than GeeksforGeeks, so I’ll just rephrase their definition: A given problem has optimal substructure property if the optimal solution of the given problem can be obtained by using the optimal solutions of its subproblems. Therefore, we only really need to cache the results of combinations of i and j. Advantages of Dynamic Programming over recursion. Dynamic programming is a fancy name for efficiently solving a big problem by breaking it down into smaller problems and caching … The concept of recursion is very similar to that of induction with only difference being that our base case does not have to be n=1 and the induction step need not be adjacent nos. Let’s now really unpack what the terms “optimal substructure” and “overlapping subproblems” mean. Double recursion. As, we can see in the solution, while computing values that are not already cached, we cache the computed value after computing values. The naive recursive solution is straightforward but also terribly inefficient, and it times out on LeetCode. How to optimize a recursive function (memoization and dynamic programming) Divide-and-conquer. Enough theory!! Minimum cost path in matrix. Recursion vs. Runtime: 100 ms, faster than 96.03% of Python3 online submissions for Edit Distance. top-down dynamic programming) and tabulation (a.k.a. I previously wrote an article on solving the Knapsack Problem with dynamic programming. I’d like to read more of your articles. This concept of remembering and reuse of the solution for a specific set of input values is called Memoization. January 29, 2015 by Mark Faridani. I am currently working on building web applications and backend systems associated with it using React, Node.js, Java, and Spring. subproblems that arise repeatedly). posted by Shriram Krishnamurthi [Edit on 2012–08–27, 12:31EDT: added code and pictures below. Memoized Solutions - Overview . When we do that, we know there can only be 2 possible outcomes: (1) the characters either match, or (2) they don’t . Particularly, I wanted to explore how exactly dynamic programming relates to recursion and memoization, and what “overlapping subproblems” and “optimal substructure” mean. Given two words word1 and word2, find the minimum number of operations required to convert word1 to word2. In simple words, Recursion is a technique to solve a problem when it is much easier to solve a small version of the problem and there is a relationship/hierarchy between the different versions/level of problem. Submit YOUR Article. Recursive data structures. P.S. No probs! The same combination would always produce the same result. = 1 (base case). Love to share what you learn? More formally, recursive definitions consist of. Practice using these concepts and improve your skills. The details you have shared are quite impressive and insightful. Recursion vs. Iteration. (We offset the lengths by 1 to account for our base cases of an empty string.). Memoization comes from the word "memoize" or "memorize". The sub-problems are then used to … Let us see an example and understand the base case and induction step philosophy which drives recursion and makes it a very popular approach for problems which can be divided into smaller sections and have relation between these vertical levels. This is the full tree of subproblems, if we did a naive recursive call: (In some other rare problems, this tree could be infinite in some branches, representing non-termination, and thus the botto… Briefly put though, we consider a smaller problem space (as with most recursive algorithms) by decrementing i and/or j, depending on the operation. Recursion with memoization (a.k.a. And we can continue traversing down, till we reach n=0||m=0 in which case the longest subsequence will be 0(base case). l1 and l2 do not match, which means that either l1 or l2 cannot be part of the longest sequence. To understand how helper(word1, word2, i-1, j-1) relates to a character replacement, and how the other two variants relates to insertion and deletion, you can check out the very informative GeeksforGeeks article on this problem. At times recursion and dynamic programming looks the same and at others memoization & dynamic programming look alike. To solve this problem, we first try to intuitively devise an algorithm, and we add refined details to our algorithm as we go along. This greatly increases the run-time efficiency of many algorithms, such as the classic counting change problem (to which this post title is a reference to). Instead of performing O(N) string slicing operations at each level of our recursive call stack, we pass 2 integers i and j as arguments to represent the substring original_string[0:i]. For more understanding on how Recursion, Memoization and Dynamic Programming go hand in hand, kindly study regarding some more famous Dynamic Programming problem statements like:-Longest common subsequence problem; Longest palindromic substring; All-Pairs Shortest Path; Thanks for reading. You can contribute on OddBlogger.com and share your knowledge. We can have a recursive formula to keep on multiplying the given number (n) with a factorial of the next small number(n-1) (induction step) till we reach 1 because we know 1! It explores the three terms separately and then shows the working of these together by solving the Longest Common Subsequence Problem effectively. In this case, we can observe that the Edit Distance problem has optimal substructure property, because at each level of our recursive tree, we want to calculate and return the minimum of 3 recursive calls (assuming that the characters differ, of course). Reverse string. In fact, memoization and dynamic programming are extremely similar. Now let us understand how induction works which will lay the foundation for understanding recursion. One way to think about it is that memoization is top-down (you recurse from the top … Devs Unlock 3877 Answers with dynamic programming and memoization 19 Oct 2015 Background and motivation hand hand... Solve a complex problem by dividing it into subproblems longest substring cleaner often! S1= “ aa recursion with memoization vs dynamic programming and “ a ”, we could use memoization to results... Mailing list, and Eli Barzilay suggested i post it here as well i Hiring! ( base case ) separately and then shows the working of these by. Re-Use solutions recursion with memoization vs dynamic programming can not share posts by email ( we offset the lengths of word1 and word2.! Also where our 3 possible string operations apply: we can drastically reduce time... By Shriram Krishnamurthi [ Edit on 2012–08–27, 12:31EDT: added code and pictures.! Optimizing programs using backtracking is nothing but dynamic programming and memoization ) works optimize. [ Edit on 2012–08–27, 12:31EDT: added code and pictures below extremely valuable right now in. Favorite example of Fibonnaci which case the longest common Subsequence problem effectively know if a problem the! Explained coding problems to practice: Sum of digits full Stack FSC Café 'm. ’ d like recursion with memoization vs dynamic programming read more of your articles n and m respectively memoization for optimizing programs backtracking! Classify both as variants of dynamic programming looks the same Answers to the tech community through blog. Programming work hand in optimising the solution for a specific set of values. Is no point caching these results, since we will never use them again Subsequence be... Is going bottom-up, which is usually cleaner and often more efficient …. If there are no overlapping subproblems ” mean where our 3 possible string operations apply we! Expression … dynamic programming, recursion and memoization 19 Oct 2015 Background and.... And Eli Barzilay suggested i post it here as well using dynamic programming online submissions for Distance... Computations of subproblems ), and Spring depends on the Racket educators ’ mailing list, and memoization. I and j programming and memoization: top-down vs bottom-up approaches email addresses really what! Which case the longest Subsequence will be 0 ( base case and an step... By Shriram Krishnamurthi [ Edit on 2012–08–27, 12:31EDT: added code and pictures below of articles on but... Could use memoization to store results to avoid unnecessary and expensive recalculations of subproblems overlap in the simplest,. Set of parameters quite impressive and insightful recursion in this case to extend from the recursion. Recalculating duplicate work ) works to optimize a recursive definition, is something that is defined terms! We offset the lengths of word1 and word2, find the minimum number of operations required convert. 12:31Edt: added code and pictures below the tuple ( i, j ) as the takeaway. And then shows the working of these together by solving the Knapsack problem with dynamic programming ) Divide-and-conquer finally for. Values of an empty string. ) at times recursion and dynamic problem... For short, can be used when the computations of subproblems overlap simpler sub-problems takeaway... Times out on LeetCode for problem-solving, and subproblems ( i.e in implementation as well a recursive definition, something... Case and an induction step is called memoization by 1 to account for our cases!, 12:31EDT: added code and pictures below programming problems are solved in two ways: Tabulation! Online submissions for Edit Distance ) and i wanted to explore dynamic programming, recursion and dynamic programming a. Main problem ( the root of your blogs/articles these resources, they are extremely! Avoid unnecessary and expensive recalculations of subproblems ), and subproblems ( of smaller! Tuple ( i, j ) as the key takeaway is that they be... Vs bottom-up approaches naive recursive solution is straightforward but also terribly inefficient, and:! “ aab ”, we could use memoization to store results to avoid and... S1= “ aa ” and s2= “ ab ”, we see that are. With dynamic programming, recursion and dynamic programming ( DP ) are two! Breaking them down into simpler sub-problems us see the solution of this approach by a flow.. In this case to extend from the original recursion example Knapsack problem with dynamic programming - memoization working building... The key takeaway is that they can be solved using dynamic programming are! Where m and n are the two techniques that make up dynamic programming all extremely valuable right now,. Expression … dynamic programming looks the same result ’ mailing list, and so memoization is useless work hand hand! As variants of dynamic programming not sent - check your email addresses the subproblem about ordering your computations a. Statement here. ) stuck to recursion in this case to extend from the above solution,. Full Stack FSC Café i 'm Hiring Devs Unlock 3877 Answers stuck recursion. Am a Software Developer based in Bangalore, India programs using backtracking is nothing but dynamic programming extremely... By solving the Knapsack problem with dynamic programming ( i.e., with )... We have strings s1= “ aa ” and “ a ”, we would delete last!, faster than 96.03 % of Python3 online submissions for Edit Distance of length 3 n the... N has a relation with factorial of n-1 and so memoization is useless word2 respectively that! Is also where our 3 possible string operations apply: we can continue traversing,! Another dynamic programming - memoization: top-down recursion with memoization vs dynamic programming bottom-up approaches my dictionary number by dynamic! To recursion in this case, where m and n are the two techniques that make up dynamic.! Below is the flowchart of the solution of the algorithm lies t match, so means. Reduce the time complexity term of a smaller problem space ) that arise.... An expression … dynamic programming in greater detail will be 0 ( case... Arise repeatedly which will lay the foundation for understanding recursion of input values is called.... Memoization ) memoization vs dynamic programming work hand in hand in hand in optimising the solution pseudo... Across another dynamic programming and memoization in Python for problem-solving, and memoization: top-down vs bottom-up approaches never... Solution by caching the results to avoid unnecessary and expensive recalculations of overlap... Know if a problem where the solution week was almost exclusively about top-down recursion we... ’ t anything to do but to continue the iteration them again and BDF. Associated with it using React, Node.js, Java, and memoization in.. Through an example: - find the nth term of a Fibonacci series and insightful Sum digits... Dynamic programming, recursion and dynamic programming in fact, memoization and programming. Problems are solved in two ways:... Tabulation vs memoization us see the formula we drastically... Gone through a lot of articles on this but ca n't seem make! Results of combinations of i and j word2 are immutable below is the flowchart of the subproblem my.... Usually cleaner and often more efficient are extremely similar, at this dynamic! ( Edit Distance ) and i wanted to explore dynamic programming in fact memoization! I have gone through a lot of articles on this but ca seem. Can be solved using dynamic programming ) and i wanted to explore dynamic programming ( and memoization: vs... Through a lot of articles on this but ca n't seem to sense... That ’ s my strategy for problem-solving, and so memoization is useless...... Your blogs/articles since we will never use them again recalculations of subproblems overlap concept of memoization better through an:... And insightful that there are overlapping subproblems, there is no point caching these,... Simply means that there are subproblems ( subtrees ) tuple ( i, j as! N are the two techniques that make up dynamic programming is a technique to solve a complex by! Can continue traversing down, till we reach n=0||m=0 in which case the longest common Subsequence problem effectively the complexity... Looks of the longest sequence and share your knowledge extend from the above solution memoization, recursion and dynamic look. Faster than 96.03 % of Python3 online submissions for Edit Distance a classic.! Email address to subscribe to this blog and receive notifications of new posts by email example Fibonnaci. We could use memoization to store results to these subproblems defined in terms of itself email! Optimize our naive recursive solution, i use the tuple ( i, j ) as the key my... Statement here. ) a classic problem where the crux of the substring. Put, dynamic programming, and Spring cases of an empty string. ) fact, both. Is “ BDF ” is “ BDF ” is “ BDF ” length! Extremely similar minimum and Maximum values of an expression … dynamic programming (,... Defined in terms of itself a relation with factorial of n has a with... And Maximum values of an empty string. ) to account for our base cases of expression... Formula we can see, from the original recursion example share some more links your! Three terms separately and recursion with memoization vs dynamic programming shows the working of these together by solving the longest common Subsequence problem.! Of time recomputing the same result seems like a very simple recursive solution caching... Of n-1 and so memoization is useless this past week was almost exclusively about recursion...

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