Dynamic programming table calculator Topics in this lecture include: Dynamic programming is a very powerful algorithmic paradigm in which a problem is solved by identifying a collection of subproblems and tackling them one by one, smallest rst, using the answers to small problems to help gure out larger ones, until the whole lot of them is solved. ” The key to dynamic programming is to identify the subproblem that gives the main problem “optimal substructure property. 3b- if possible, optimize for space. I have explained the concept using 01 knapsack Sep 13, 2015 · According to wikipedia, the definition of the recursive formula which calculates the Levenshtein distance between two strings a and b is the following: I don't understand why we don't take into table[0] = 0 table[1] = 1 for i in range(2,n+1): table[i] = table[i-2] + table[i-1] return table[n]or alternately, in a list instead of a dictionary We will see this pattern many more times in this course: 1. Feb 6, 2018 · After filling out the dynamic programming table, we can trace a path back from the bottom-right entry to the top-left entry along the smallest values to obtain our alignment shown in the first figure in this lecture. 4 Finding the optimal solution to the linear programming problem by the simplex method. According to the Dynamic Programming Introduction. Jan 9, 2025 · Introduction to Dynamic Programming¶ The essence of dynamic programming is to avoid repeated calculation. During coding, usually memoization is implemented by recursion while dynamic programming does calculation by iteration. The idea is to calculate the sum of all elements in the set, say sum. Inconceivable! No, easily conceived! Feb 7, 2024 · Grid problems involve a 2D grid of cells, often representing a map or graph. Dynamic programming breaks the problem into subproblems. This is one of the most helpful visualization techniques for designing bottom-up DP algorithms when the problem is a multi-prefix/multi-suffix or subsequence problem type. Allowed moves into a given cell are from above, from the left, or diagonally from the upper-left. Dynamic Programming 12. Solve linear programming tasks offline! The decision of problems of dynamic programming. Question: In what order is the table lling up? Where are the base-cases in the table? Answer: Table is lling up left to right and top-down. Dynamic Programming: (a) In class, we discussed a table-filling algorithm for the matrix chain multiplication problem. The algorithm uses dynamic programming to calculate a table of possible states using the characters of the word and the given grammar. for i = 1 to dim. The key idea behind dynamic programming is to avoid Rod Cutting Prices. The cheapest method to compute ABCDEFGHIJKLMN is (((AB)((C(D(E(FG))))(((((HI)J)K)L)M)))N) with cost 251: A * BCDEFGHIJKLMN: A is a 3 x 8 matrix Jun 18, 2018 · Dynamic Programming- Primitive Calculator. » One per C(i,j) calculation: only O(mn)different ones ! Two different ways to implement: » memoization: after computing sub-problem answer, remember it. Let’s go step by step: Step 1: Create the table We create a 4×4 table to store the minimum number of scalar multiplications needed for each subchain. For fibonacci, initialize the table to zeros. References: Oct 26, 2024 · 1. In this beginner‘s guide, I‘ll walk through the fundamental […] Feb 20, 2023 · The Best Article to Understand What Is Dynamic Programming Lesson - 44. CYK Calculator. A Holistic Guide to Learn Stop Solution Using Dynamic Programming Lesson - 46. In addition to this advantage, the use of dynamic programming mentioned above can reduce the function call stack generated by recursion, so the performance is better. Complete, detailed, step-by-step description of solutions. Bottom Up Algorithm to Calculate Minimum Number of Multiplications; n -- Number of arrays ; d -- array of dimensions of arrays 1 . 4: Dynamic Programming Before proceeding to a solution of the sequence alignment problem, we first discuss dynamic programming, a general and powerful method for solving problems with certain types of structure. Loop through each action and each possible new state, and accumulate (mean). Solve LP using zero-one Integer programming problem method Max Z = 300x1 + 90x2 + 400x3 + 150x4 subject to 35000x1 + 10000x2 + 25000x3 + 90000x4 ≤ 120000 4x1 + 2x2 + 7x3 + 3x4 ≤ 12 x1 + x2 ≤ 1 and x1,x2,x3,x4 ≥ 0 2. In which order can the entries be computed such that dependencies are fulfilled? 4 Read-out the solution Apr 23, 2023 · METHOD 1: (Using Dynamic Programming) A Simple Solution is to first compute n C r , then compute n C r % p. Each computation requires looking up at most 3 entries of the table. The table initially looks like this: Dec 28, 2022 · Solve more challenging problems using dynamic programming; 如果使用 fib_table(10000) 就不會遇到 stack overflow 的問題。 Prefix Sum is a technique used to efficiently calculate Mar 17, 2021 · Using this framework, we can use dynamic programming (DP) to calculate the optimal structure because 1) this scoring scheme is additive 2) we disallowed pseudo knots, which means we can divide the RNA into two smaller ones which are independent, and solve the problem for these smaller RNAs. The value of each entry is calculated according to a new basis and recurrence for / $ Jan 13, 2000 · The dynamic programming algorithm for optimal local alignment is similar to the dynamic programming algorithm for optimal global alignment given in Section 4. Dynamic Programming table for Binomial Coefficients nCr = (n-1)C(r-1) + (n-1)Cr. Fill the table, re-using solutions to previous sub-problems. Ask Question Asked 6 years, 6 months ago. By leveraging these calculations, you can construct a tree that minimizes the expected search cost. length; i++); for (var j = 1; j ; = text1. We then used the value 0! to calculate 1!, the value of 1! to calculate 2! and so on. The purpose of this app is to visually illustrate how the alignment matrix is constructed and how the Needleman-Wunsch dynamic programing algorithm fills this matrix based on user defined Match, Mismatch and Gap Scores. It proceeds by filling in a table with the values of $ , with / increasing. The solution to each subproblem is stored and used to solve other, larger, subproblems. Bellman-Ford's Shortest Path Catalan Number Fibonacci Sequence Floyd-Warshall's Shortest Path Integer Partition Knapsack Problem Knuth-Morris-Pratt's String Search Levenshtein's Edit Distance Longest Common Subsequence Longest Increasing Subsequence Longest Palindromic Subsequence Maximum Subarray Maximum Sum Path Nth Factorial Pascal's Triangle Shortest Common Supersequence Sieve of Sep 14, 2022 · Practice this problem. Jan 31, 2015 · I'm having trouble figuring out my last section of code for a Dynamic Coin Changing Problem. Even though the method was tested in solving economic problems by Richard Belman for the first time, Belman (mathematician) formulated this approach to mathematical optimization and all the necessary conditions for applicability in difficulties. LCS - DP Table(s) Example table(s) for BREATHER and CONSERVATIVES: Stare at the table a while - what do you notice ; Make up your mind: Is it "table" or "tables" This one table shows two arrays . length; j++) if (text1[j - 1] == text2[i - 1]) DP[i][j] = DP[i - 1][j - 1] + 1; else; DP[i][j] = max(DP[i - 1][j], DP[i][j - 1]); Knapsack Calculator Given a set of items, each with a weight and a value. Dimension of the entries? Semantics of the entries? 2 Computation of the base cases Which entries do not depend on others? 3 Determine computation order. Many programs in computer science are written to optimize some value; for example, find the shortest path between two points, find the line that best fits a set of points, or find Apr 1, 2016 · Dynamic programming problems can be solved in a top down or bottom up fashion. The solution to the subset sum problem can be derived from the optimal solutions of smaller subproblems. Interactive Examples: Use pre-defined examples to see how different key distributions affect tree efficiency. So if you have carefully calculate the space and time complexity 22-12: Dynamic Programming •To calculate Fibonacci(100),only need to calculate 101 values •Fibonacci(n)can be calculated in time O(1) •Assuming we have values for Fibonacci(n-1)and Fibonacci(n-2) 22-13: Dynamic Programming •Create a table: FIB[] •FIB[n] = nth Fibonacci number •Fill the table from left to right •Use old values in May 16, 2020 · When doing dynamic programming, I would usually fill the table eagerly (from bottom to top, instead of using top-down recursion). 1- find a recursive algorithm. The Cocke-Younger-Kasami algorithm (also called as CYK algorithm) is an efficient parsing algorithm for context free grammars in the Chomsky Normal Form (CNF). » dynamic programming: compute the table bottom-up. Dynamic programming is both then one can easily memoize or store the solutions to the sub-problems in a table Using dynamic programming in the calculation of Dynamic Programming Overview Dynamic programming. , instead of knapsack(n-1, KW), we will use memo-table[n-1, KW]. Apr 25, 2019 · The Chain Matrix Multiplication Problem is an example of a non-trivial dynamic programming problem. ” Avoiding the work of re-computing the answer every time the sub problem is encountered. And most dynamic programming problems are accompanied by space optimization (rolling array), which is the advantage of dynamic programming over traditional memoized recursion. Calculate p * (r + γ 𝓥(s’)). Argue that the algorithm runs in Θ(n3) time. Hungarian method, dual simplex, matrix games, potential method, traveling salesman problem, dynamic programming Dynamic Programming: Description 1 Use a DP-table with information to the subproblems. In this case that is O(n)×O(1) = O(n) 3 Example 2 Suppose you had two operations: 9: Dynamic Programming-2 Jan 11, 2010 · First the table must be filled in with "unsolved" marks. Problem 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. So to solve problems with dynamic programming, we do it by 2 steps: Find out the right recurrences(sub-problems). Dec 3, 2020 · Learn how to use Dynamic Programming in this course for beginners. One common approach to implementing dynamic programming is Aug 18, 2024 · Implementing an Optimal Binary Search Tree involves setting up dynamic programming tables to calculate the cost of different subtrees and their optimal roots. Many programs in computer science are written to optimize some value; for example, find the shortest path between two points, find the line that best fits a set of points, or find the smallest set of objects that satisfies some criteria. 5: The Needleman-Wunsch Algorithm; 2. Do I always have to fill up the entire table? Example DP task: Calculate the distance between the strings hello and world: Oct 12, 2023 · In this method, we create a lookup table. Specifically, for any given n (the number of elements considered) and a target sum, we can express the recursive relation as follows: Nov 17, 2022 · Method 02) Using Dynamic Programming. The LCS is: Dynamic Programming; for (var i = 1; i ; = text2. This results in a nested loop filling in all values in the table. We also showed that this algorithm runs in O(n3) time, where n is the number of matrices in the chain. We created the lookup table till 20 only because it is the largest number whose factorial long can store. In any case, dynamic programming is an important concept to learn and perhaps Fibonacci numbers is used since it is a simple example and it makes dynamic programming easier to understand. Dynamic programming (DP) is an algorithm design technique for solving some of the seemingly most nasty problems you might face in an interview. Optimal Substructure: . The table reads as follows: to calculate M i * M j, we set k = K ij given by the table then we calculate M i *… * M k and M (k + 1) *… * M j that we then multiply between them. 1 Overview Dynamic Programming is a powerful technique that allows one to solve many different types of problems in time O(n2) or O(n3) for which a naive approach would take exponential time. Solve LP using 0-1 Integer programming problem method MAX Z = 650x1 + 700x2 Dynamic Programming (Making Change) - University of San Francisco explains how to use dynamic programming to solve the coin change problem. Chasnov via source content that was edited to the style and standards of the LibreTexts platform. The key idea behind dynamic programming is to avoid Enter two texts and choose an operations. We can apply Dynamic Programming on Grids when the solution for a cell is dependent on solutions of previously traversed cells like to find a path or count number of paths or solve an optimization problem across the grid, with certain constraints on movement or cost. – sub-problems are not independent – save solutions to repeated sub-problems in table Recipe. Therefore the complexity of this algorithm is \(O(mn)\). Dynamic programming is a powerful technique that can greatly improve the efficiency and speed of algorithms by breaking down complex problems into smaller, overlapping subproblems. In this case, an in-memory table stores the max value at different weights and number 2. Should I just use the greedy algorithm at that point or can I calculate the answer from values already in the table? Mar 18, 2024 · In this article, we showed how to multiply a chain of matrices using dynamic programming. 7: Tools and Techniques; 2. The video explains the dyn Jan 13, 2000 · The dynamic programming algorithm for optimal local alignment is similar to the dynamic programming algorithm for optimal global alignment given in Section 4. Note: you could also solve this problem using a heap search, which wouldn't hit every node and I would imagine is pretty close to this in terms of timing and presumably less space. The method of solving this question using Recursion will work perfectly fine, but what if, your interviewer asks you to optimize it further? Don't worry we got you covered! The time taken by the previous method was exponential so, we will use Dynamic Programming to bring its complexity down further. What is the Principle of Optimality? The dynamic programming algorithm obtains the solution using the principle of optimality. But, what is the subproblem here? Jun 30, 2020 · Retrieve the state value 𝓥(s’) for the new state s’ from the current value table. 2. Compute and memorize all result of sub-problems to “re-use”. Often, dynamic programming problems are naturally solvable by recursion. May 25, 2023 · Dynamic programming algorithm is designed in a way to optimize the given problem to get output by combining the solutions of sub-problems and appearing to the “principle of optimality”. The problem will often take the form: Given a set of rules, find the optimal cost of some objective that is governed by those rules. Filling the table: It consist of two stages, table initialization and building the solution from the smaller subproblems: Dec 22, 2024 · Dynamic programming is a powerful algorithmic technique used to solve optimization problems that can be broken down into smaller subproblems. Efficiency Analysis: Compare the efficiency of the constructed tree with traditional binary search tree methods. Similar to divide-and-conquer. As with all dynamic programming solutions, at each step, we will make use of our solutions to previous sub-problems. 196 Computing the table Fill the table starting from top-left corner, and going row-by-row: Tabulation is an approach where you solve a dynamic programming problem by first filling up a table, and then compute the solution to the original problem based on the results in this table. length; i++) for (var j = 1; j = text1. 8: Appendix; 2. The partition problem is a special case of the Subset Sum Problem, which itself is a special case of the Knapsack Problem. Given a rod of length 8, what is the maximum revenue: r i This video explains the concept about how to approach a dynamic programming problem to solve using tabular DP. The idea is to store the solutions of the repetitive subproblems into a memo table (a 2D array) so that they can be reused i. length - l k = i + l - 1 opt[i,k] = ∞ for j = i to k-1 prev = best[i,j] + best[j+1 Apr 17, 2018 · DP just means using sub-problem results to shorten the time/space complexity, so it often builds up but doesn't necessarily mean every value. the table. Note that dynamic programming requires you to figure out the order in which to compute the table entries, but memoization does not. Alignment Visualizer is a web app which will show you the dynamic programming matrix for an alignment of two DNA sequences. The value of each entry is calculated according to a new basis and recurrence for / $ In memoization, we keep a global arrary—called the dynamic programming table—to maintain a record of our previous computations. The main difference between the tabular approach and memoization is the order in which the sub-problems are solved. This is not an in-place algorithm, since the table requires Θ(n x W) cells and this is not linear in n. 9: Bibliography Since the time to calculate each entry in the table V[k,w] is constant, the time complexity is Θ(n x W). Dynamic Programming Table. 1. getMaxNumActions()` returns an integer with max number of actions in Jul 29, 2024 · # A Dynamic Programming based # solution that uses # table P[][] to calculate the # Permutation Coefficient # Returns value of Permutation # Coefficient P(n, k) def permutationCoeff (n, k): P = [[0 for i in range (k + 1)] for j in range (n + 1)] # Calculate value of Permutation # Coefficient in # bottom up manner for i in range (n + 1): for j 0-1 Knapsack Solution using Dynamic Programming. In such cases, it's easiest to write the recursive solution, then save repeated states in a lookup table. The matrix chain multiplication problem is to find the minimum number of multiplications required to multiply a chain of matrices. The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. Nov 8, 2023 · To solve this problem, we can apply the Matrix Chain Multiplication algorithm using dynamic programming. Given the rod values below: Given a rod of length 4, what is the maximum revenue: r i 5 + 5 > 1 + 8 = 0 + 9 ⇒ 10 . The space-optimized approach reduces the space complexity by using only two rows of the DP table instead of the entire table. n Jun 12, 2020 · Array created in this solutions is known is dynamic programming table also known as memoization or bottom up approach to DP. In practice, dynamic programming likes recursive and “re-use”. Create a table (of the right dimensions to describe our problem. The full Oct 28, 2024 · Dynamic programming is an immensely powerful algorithm design technique for solving complex optimization problems by breaking them down into simpler subproblems. If not, calculate fib(i) and place the result in the table for future reference. days_needed = days / 2 # An inner function to do the calculation. length; j++); if Mar 22, 2022 · As one definition of dynamic programming explains, dynamic programming is designed such that “the optimal solution to the overall problem depends upon the optimal solution to its subproblems. Brute Force: try all tree configurations ; Ω(4 n / n 3/2) different BSTs with n nodes ; DP: bottom up with table: for all possible contiguous sequences of keys and all possible roots, compute optimal subtrees Dec 17, 2023 · The first association with dynamic programming is olympiad programming. It uses Dynamic Programming. It can help you solve complex programming problems, such as those often seen in programmin 1. 2- memoization. Animation Speed: w: h: Algorithm Visualizations Dynamic Programming Solver The Dynamic Programming Solver add-in solves several kinds of problems regarding state based systems: Deterministic Dynamic Programming (DDP), Stochastic Dynamic Programs (MDP) and Discrete Time Markov Chains (DTMC). Algorithm for Location of Minimum Value . 3: Dynamic Programming is shared under a CC BY 3. In this lecture, we discuss this technique, and present a few key examples. This table stores the factorials of numbers from 0 to 20. Instead, dynamic programming is used to solve the problem. The seven steps in the development of a dynamic programming algorithm are as follows: Runtime: The runtime of a dynamic progrmaming algorithm is the size of the table multiplied by the work done to calculate each cell. (Memoization is itself straightforward enough that there are some Dynamic programming is a technique of breaking the main problem into smaller subproblems and then using those subproblems to construct the answer to the main problem. Characterize structure of problem. , independent of a particular dynamic programming algorithm, 1 we need a framework that (1) comprises a clearly defined and practically significant class of dynamic programming problems, (2) separates the issue of tabulation from the 1 We Tabulation in Dynamic Programming Basics of Dynamic Programming: Tabulation in Dynamic Programming. getNumStates()` returns an integer of total number of states - `env. In contrast to linear programming, there does not exist a standard mathematical for-mulation of “the” dynamic programming problem. This means that the subproblem of computing the Fibonacci number at index n-2 is used twice (note that the call for n – 1 will make two calls, one for n-2 and other for n-3) in the solution to the larger problem of computing the Apr 16, 2017 · Dynamic programming is a very powerful algorithmic design technique to solve many exponential problems. Choosing Correct size of Table: The range of parameters goes from 0 to m and 0 to n. Matrix Chain Multiplication Aug 5, 2022 · The idea is very simple, create a 2D array using dynamic programming which stores the value of nCr using the above property of Combination – nCr = (n-1)C(r-1) + (n-1)Cr, with nC0 = nCn = 1 as base case. May 31, 2011 · Dynamic Programming is a bottom-up approach during which you firstly calculate the answer of small cases and then use them to construct the answer of big cases. I can't figure out the last else. Consider all possible moves into a cell. Hungarian method, dual simplex, matrix games, potential method, traveling salesman problem, dynamic programming. Mar 28, 2019 · Step 3 (the crux of the problem): Now, we want to begin populating our table. One Stop Solution to All the Dynamic Programming Problems Lesson - 47. Global Alignment App. We initialized 0! as 1. Dynamic programming is a powerful technique used to solve optimization problems by breaking them down into smaller overlapping subproblems and solving each subproblem only once. The final solution is built up from these subproblems. Fill in the table by utilizing a series of “moves”. This is because of the 2D array of (m x n) dimensions. If the table location has a value, it is the value of fib(i) and it should be used. Dynamic Programming. In dynamic programming we are not given a dag; the dag is Jul 18, 2022 · This page titled 7. Viewed 2k times -3 . Step 3 is the most characteristic part of dynamic programming, but if you stopped at step 2, it would still qualify, I think, but it would not be as efficient as Dec 7, 2020 · Dynamic Programming is a tool that will help make your recursive code more efficient. ### REINFORCEjs API use of DP If you'd like to use the REINFORCEjs Dynamic Programming for your MDP, you have to define an environment object `env` that has a few methods that the DP agent will need: - `env. Dynamic Programming¶. While tricky to master at first, understanding dynamic programming can open the door to solving a wide variety of problems that might otherwise seem intractable. Dynamic Programming; for (var i = 1; i = text2. Nov 30, 2021 · The below Dynamic Programming table shows the calculation of Levenshtein distance between “kitten” and “sitting” using the DP approach: The time complexity of this algorithm is O(mn) where m and n are the input sizes of strings. Look at the code below: Aug 21, 2024 · So we need to construct a 2D table dp[][] to store the solution of the sub-problems. 9 0-1 Integer programming problem: 1. 5. Nov 21, 2022 · In this video, I will show you how to fill in the table for the Matrix Chain Multiplication problem. It provides a systematic procedure for determining the optimal com-bination of decisions. It is basically a way of drawing the DAG of computation when the computational structure of your problem is best explained in two dimensions. Modified 6 years, 6 months ago. – solves problem by combining solution to sub-problems Different from divide-and-conquer. This process is known as top-down dynamic programming with Dec 24, 2024 · Consider the problem of computing the Fibonacci series. Nov 25, 2024 · Dynamic programming is a powerful algorithmic technique used to solve optimization problems that can be broken down into smaller subproblems. 6: Multiple alignment; 2. It is an efficient way to solve problems and can significantly reduce the time complexity. A Guide to Implement Longest Increasing Subsequence Using Dynamic Programming Lesson - 45. Here we present an interactive example of the Needleman-Wunsch global alignment algorithm from BIMM-143 Class 2. Knapsack algorithm determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. e. But dynamic programming algorithms save time by storing results, so we wouldn't expect any dynamic programming solution to be in-place. In this article, I break down the problem in order to formulate an algorithm to solve it. See the below image of the DP table . Dynamic Programming Dynamic programming is a useful mathematical technique for making a sequence of in-terrelated decisions. So we choose dimensions as (m + 1)*(n + 1) 3. I’d really like to drill home the fact that I don’t think we should consider any given problem a “DP Dynamic Programming Approach: Implements dynamic programming to construct efficient binary search trees. The underlying idea of dynamic programming is: Avoid calculating the same stuff twice, usually by keeping a table of known results of sub problems. Then when the value of fib(i) needs to be consulted, look first in the table. Optimal BST - Algorithm and Performance. Dynamic programming: iterative solution It is possible to eliminate recursion by lling the table in the order in which recursion lls it. To compute the Fibonacci number at index n, we need to compute the Fibonacci numbers at indices n-1 and n-2. I developed it in the summer of 2016 as a teaching tool to help people understand how dynamic programming works in DNA sequence alignments. Aug 30, 2013 · Is that considered a dynamic programming technique ? or dynamic programming must solve a recursive function that is always computed or sort of ? Update: In dynamic programming the key is to have a memoization table, which is the solution for the sin,cos look up table, so what is really the difference in the technique ? Jan 6, 2025 · Dynamic Programming is an optimization technique that improves recursive solutions by storing results of subproblems to reduce time complexity from exponential to polynomial, applicable to various problems like Fibonacci numbers and the Longest Common Subsequence. length - 1 opt[i,i] = 0 for l = 2 to dim. Run time complexity of above solution is O(N) There is another way to solve these type of problems in O(Log N) time complexity, where solution can be described in terms of a linear recurrence relation. We memoize our algorithm to give MemoizedComputeWIS. Jul 16, 2024 · Time Complexity: O(m x n) Auxiliary Space: O(m x n) Method 4: Using Dynamic Programming (Space-Optimized). 0 license and was authored, remixed, and/or curated by Jeffrey R. . Jul 26, 2024 · The Edit Distance problem is a common dynamic programming problem in which we need to transform one string into another with the minimum number of operations Dynamic programming for global alignment of amino acid sequences (Simplified Needleman-Wunsch algorithm) Procedure Start in upper left corner. This solution works fine when the value of n C r is small. 3- make it iterative / bottom-up. I have included the code below. – optimal substructure property Sep 1, 2006 · The algebraic approach to dynamic programming In order to study the table design problem in general, i. Nov 29, 2020 · Second, our original solution had linear time complexity and constant space complexity (without recursion or dynamic programming). The key difference is that we maintain a 1D-dynamic programming tableW, initialized to all −1 values (as a bookkeeping measure). In our example, to calculate vs 16, we calculate c 13 *vs 46; to calculate vs 13, we calculate M 1 *vs 23; to calculate vs 46 we calculate c 45 * M 6. 12. length for i = 1 to dim. To my approach to dynamic programming, memoization is step 2 of 3. fgpbev laxvvl hurlh gatijw seaewe vfqfg qjivew lshu ahduap wgywm