# matrix chain multiplication algorithm using dynamic programming

To go through the C program / source-code, scroll down to the end of this page https://techiedelight.com/compiler/?XDiz. Thanks Anshul for sharing your concerns. In other words, no matter how we parenthesize the product, the result will be the same. â¢ C = AB can be computed in O(nmp) time, using traditional matrix multiplication. Matrix multiplication is associative, so all placements give same result dynamic programming is applicable when the subproblems are not independent. How can you rationalize the solution at c[1][n – 1]? Add these costs together, and add in the cost of multiplying the two result matrices. What is Dynamic Programming? The following bottom-up approach computes, for each 2 <= k <= n, the minimum costs of all subsequences of length k, using the costs of smaller subsequences already computed. Matrix â¦ (Memoization is itself straightforward enough that there are some Matrix chain multiplication (or Matrix Chain Ordering Problem, MCOP) is an optimization problem that can be solved using dynamic programming. Note: This C program to multiply two matrices using chain matrix multiplication algorithm has been compiled with GNU GCC compiler and developed using gEdit Editor and terminal in Linux Ubuntu operating system. ((AB)C)D = ((A(BC))D) = (AB)(CD) = A((BC)D) = A(B(CD)), However, the order in which the product is parenthesized affects the number of simple arithmetic operations needed to compute the product, or the efficiency. Below is C++, Java and Python implementation of the idea: The time complexity of above solution is exponential as we’re doing a lot of redundant work. [We use the number of scalar multiplications as cost.] C Programming - Matrix Chain Multiplication - Dynamic Programming MCM is an optimization problem that can be solved using dynamic programming. Given a sequence of matrices, find the most efficient way to multiply these matrices together. Array Interview QuestionsGraph Interview QuestionsLinkedList Interview QuestionsString Interview QuestionsTree Interview QuestionsDynamic Programming Questions, Wait !!! For example, if A is a 10 × 30 matrix, B is a 30 × 5 matrix, and C is a 5 × 60 matrix, then, computing (AB)C needs (10×30×5) + (10×5×60) = 1500 + 3000 = 4500 operations, while computing A(BC) needs (30×5×60) + (10×30×60) = 9000 + 18000 = 27000 operations. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.. Source: https://en.wikipedia.org/wiki/Matrix_chain_multiplication. Live Demo If we are ever asked to compute it again, we simply give the saved answer, and do not recompute it. Matrix chain multiplication(or Matrix Chain Ordering Problem, MCOP) is an optimization problem that to find the most efficient way to multiply given sequence of matrices. d n-1 x d n (i.e Dimension of Matrix Ai is di-1 x di Solving a chain of matrix that, A i A i+1 A i+2 A i+3 â¦â¦. For all values of i=j set 0. Matrix chain multiplication. The complexity of your implementation is just like the original DP solution: O(n^3) (Note: Every cell of mem array should be computed at least once, and each cell takes O(n) time to be computed. Given a sequence of matrices, the goal is to find the most efficient way to multiply these matrices. Matrix Chain Multiplication Using Dynamic Programming Let we have ânâ number of matrices A1, A2, A3 â¦â¦â¦ An and dimensions are d0 x d1, d1 x d2, d2 x d3 â¦â¦â¦â¦. Like other typical Dynamic Programming(DP) problems, recomputations of same subproblems can be avoided by constructing a temporary array m[][] in bottom up manner. From Wikipedia, the free encyclopedia Matrix chain multiplication (or Matrix Chain Ordering Problem, MCOP) is an optimization problem that can be solved using dynamic programming. For all values of i=j set 0.eval(ez_write_tag([[300,250],'tutorialcup_com-medrectangle-4','ezslot_8',621,'0','0'])); M[1,2] = 30*35*15 = 15750, M[2,3] = 35*15*5 = 2625, M[3,4] = 15*5*10 = 750, M[4,5] = 5*10*20 = 1000, M[5,6] = 10*20*25 = 5000. eval(ez_write_tag([[300,250],'tutorialcup_com-box-4','ezslot_9',622,'0','0']));M[1,3] = MIN( (M[1,1] + M[2,3] + P0P1P3), (M[1,2] + M[3,3] + P0P2P3) ) = MIN(2625+30*35*5, 15750+35*15*5) = 7875, M[2,4] = MIN( (M[2,2] + M[3,4] + P1P2P4), (M[2,3] + M[4,4] + P1P3P4) ) = MIN(750+35*15*10, 2625+35*5*10) = 4374, using the same concept find the other values using above formula then M[3,5] = 2500 and M[4,6] = 3500. M [1, N-1] will be the solution to the matrix chain multiplication problem. Algorithms: Dynamic Programming - Matrix Chain Multiplication with C Program Source Code Check out some great books for Computer Science, Programming and Tech Interviews! Let’s see the multiplication of the matrices of order 30*35, 35*15, 15*5, 5*10, 10*20, 20*25. Given a sequence of matrices, the goal is to find the most efficient way to multiply these matrices. We have many options to multiply a chain of matrices because matrix multiplication is associative. Problem: Given a series of n arrays (of appropriate sizes) to multiply: A1×A2×â¯×An 2. Note that dynamic programming requires you to figure out the order in which to compute the table entries, but memoization does not. The problem can be solved using dynamic programming as it posses both the properties i.e. no multiplication). Matrix chain multiplication. A(5*4) B(4*6) C(6*2) D (2*7) Let us start filling the table now. The idea is to use memoization. Hope you’re clear now. Then the final matrix will be: So, we find the minimum number of operations required is 15125 to multiply above matrices.eval(ez_write_tag([[336,280],'tutorialcup_com-large-leaderboard-2','ezslot_6',624,'0','0'])); O(N*N*N) where N is the number present in the chain of the matrices. Is there any reason behind doing the two recursive calls on separate lines (Line 31, 34 in the first code)? Matrix Chain Multiplication â Firstly we define the formula used to find the value of each cell. Dynamic Programming is a technique for algorithm design. Matrix Chain Multiplication is a method in which we find out the best way to multiply the given matrices. Matrix chain multiplication problem can be easily solved using dynamic programming because it is an optimization problem, where we need to find the most efficient sequence of multiplying the matrices. C Program For Implementation of Chain Matrix Multiplication using Dynamic Algorithm Clearly the first method is more efficient. Multiplying an i×j array with a j×k array takes i×j×k array 4. What is the least expensive way to form the product of several matrices if the naïve matrix multiplication algorithm is used? Below is the recursive algorithm to find the minimum cost –. If there are three matrices: A, B and C. The total number of multiplication for (A*B)*C and A* (B*C) is likely to be different. Prior to that, the cost array was initialized for the trivial case of only one matrix (i.e. Start with for loop with L=2. Matrix chain multiplication using dynamic programming. Let us solve this problem using dynamic programming. L goes from 2 to n). As the recursion grows deeper, more and more of this type of unnecessary repetition occurs. Python Programming - Matrix Chain Multiplication - Dynamic Programming MCM is an optimization problem that can be solved using dynamic programming Given a sequence of matrices, find the most efficient way to multiply these matrices together. You are given n matrices and size of i th matrix (M i) is P i xQ i and P i = Q i-1.Considering the expression M 1 *M 2 *â¦..*M n, your task is to parenthesize this expression and then, find the minimum number of integer multiplications required to compute it.. Give it a try on your own before moving forward Why we should solve this problem? (84 votes, average: 4.85 out of 5)Loading... Hi, how is the time complexity for the DP solution N^3. The Chain Matrix Multiplication Problem is an example of a non-trivial dynamic programming problem. Therefore, we have a choice in forming the product of several matrices. Matrix-Chain Multiplication 3. The matrix multiplication is associative as no matter how the product is parenthesized, the result obtained will remain the same. Better still, this yields not only the minimum cost, but also demonstrates the best way of doing the multiplication. ... Next Topic Matrix Chain Multiplication Algorithm The complexity is O(n3) as MatrixChainMultiplication() function can be called for any combination of i and j (total n2 combinations) and each function call takes linear time. It is a tabular method in which it uses divide-and-conquer to solve problems. So overall we use 3 nested for loop. Introduction Divide & Conquer Method vs Dynamic Programming Fibonacci sequence Matrix Chain Multiplication Matrix Chain Multiplication Example Matrix Chain Multiplication Algorithm Longest Common Sequence Longest Common Sequence Algorithm 0/1 Knapsack Problem DUTCH NATIONAL FLAG Longest Palindrome Subsequence the chain length L) for all possible chain lengths. To view the content please disable AdBlocker and refresh the page. 3. Time complexity of matrix chain multiplication using dynamic programming is â¦ Determine where to place parentheses to minimize the number of multiplications. The idea is to break the problem into a set of related subproblems which group the given matrix in such a way that yields the lowest total cost. There is no doubt that we have to examine every possible sequence or parenthesization. M[i,j] equals the minimum cost for computing the sub-products A(i…k) and A(k+1…j), plus the cost of multiplying these two matrices together. optimal substructure and overlapping substructure in dynamic programming. In this article, I break down the problem in order to formulate an algorithm to solve it. Matrix chain multiplication (or Matrix Chain Ordering Problem, MCOP) is an optimization problem that can be solved using dynamic programming. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. So Matrix Chain Multiplication problem has both properties (see this and this) of a dynamic programming problem. Step-2 Also, why is MAX set to 10? Find the minimum cost of multiplying out each subsequence. We need to find the minimum value for all the k values where i<=k<=j. Example. Recommended: If you donât know what is dynamic programming? We all know that matrix multiplication is associative(A*B = B*A) in nature. The chain matrix multiplication problem is perhaps the most popular example of dynamic programming used in the upper undergraduate course (or review basic issues of dynamic programming in advanced algorithm's class). Enter your email address to subscribe to new posts and receive notifications of new posts by email. Given a sequence of matrices, the goal is to find the most efficient way to multiply these matrices. Now each time we compute the minimum cost needed to multiply out a specific subsequence, we save it. m[1,1] tells us about the operation of multiplying matrix A with itself which will be 0. Take the sequence of matrices and separate it into two subsequences. So to solve a given problem, we need to solve different parts of the problem. Dimensions of each matrix given in an array P where P[i-1] and P[i] denote rows and column respectively of ith matrix. For example, for matrix ABCD we will make a recursive call to find the best cost for computing both ABC and AB. Do NOT follow this link or you will be banned from the site! M[i,j] equals the minimum cost for computing the sub-products A(iâ¦k) and A(k+1â¦j), plus the cost of multiplying these two matrices together. Matrix Chain Multiplication using Dynamic Programming. As we know that we use a matrix of N*N order to find the minimum operations. Dynamic approach using Top down method Matrix-Chain Multiplication â¢ Let A be an n by m matrix, let B be an m by p matrix, then C = AB is an n by p matrix. Advertisements help running this website for free. You can Crack Technical Interviews of Companies like Amazon, Google, LinkedIn, Facebook, PayPal, Flipkart, etc, Anisha was able to crack Amazon after practicing questions from TutorialCup, Matrix Chain Multiplication using Dynamic Programming, Printing brackets in Matrix Chain Multiplication Problem, Dynamic Memory Allocation Pointers in C Programming, Dynamic Memory Allocation to Multidimensional Array Pointers, Largest rectangular sub-matrix whose sum is 0, Common elements in all rows of a given matrix, Distance of nearest cell having 1 in a binary matrix, Find all permuted rows of a given row in a matrix, Check if all rows of a matrix are circular rotations…, Largest area rectangular sub-matrix with equal…, Find distinct elements common to all rows of a matrix, Algorithm For Matrix Chain Multiplication, C++ Program For Matrix Chain Multiplication, Time Complexity for Matrix Chain Multiplication. Step-1. We use the Dynamic Programming approach to find the best way to multiply the matrices.eval(ez_write_tag([[728,90],'tutorialcup_com-medrectangle-3','ezslot_5',620,'0','0'])); Matrix Chain Multiplication – Firstly we define the formula used to find the value of each cell. Dynamic programming is both a mathematical optimization method and a computer programming method. For example, for four matrices A, B, C, and D, we would have Let us take one table M. In the tabulation method we will follow the bottom-up approach. Then final matrix will be: Now find the values for j=i+4 using the above formula which we discuss. Dynamic Programming Solution The technique you have used is called memoization.Most of the time, you may solve DP problems using memoization with little (or no) overhead. Matrix chain multiplication using dynamic programming Problem here is, we are given N matrices. Here we find the most efficient way for matrix multiplication. Then the final matrix will be: In the last step value of j=i+5 using the above formula which we discuss. You start with the smallest chain length (only two matrices) and end with all matrices (i.e. Do this for each possible position at which the sequence of matrices can be split, and take the minimum over all of them. Given a sequence of matrices, the goal is to find the most efficient way to multiply these matrices. We then choose the best one. Then updated values in matrix are look like: eval(ez_write_tag([[300,250],'tutorialcup_com-banner-1','ezslot_4',623,'0','0']));Now find the values for j=i+3 using the above formula which we discuss. March 14, 2016 No Comments algorithms, dynamic programming The Matrix Chain Multiplication Problem is the classic example for Dynamic Programming. It is taken from wikipedia and proper credits are already given. Can you include that too. so we have to build the matrix O(n^2), I read on wikipedia that the above problem can be best solved in o(nlogn) runtime complexity Matrix chain multiplication problem: Determine the optimal parenthesization of a product of n matrices. O(N*N) where N is the number present in the chain of the matrices. Matrix chain multiplication in C++. It has the same asymptotic runtime and requires no recursion. In Dynamic Programming, initialization of every method done by '0'.So we initialize it by '0'.It will sort out diagonally. For example, if we had four matrices A, B, C, and D, we would have: 1. Matrix chain multiplication is an optimization problem that can be solved using dynamic programming. Matrix Chain Order Problem Matrix multiplication is associative, meaning that (AB)C = A(BC). Example. The time complexity of above solution is O(n3) and auxiliary space used by the program is O(1). Matrix Chain Multiplication Dynamic Programming Data Structure Algorithms If a chain of matrices is given, we have to find the minimum number of the correct sequence of matrices to multiply. We create a DP matrix that stores the results after each operation. Matrix Chain Multiplication using Dynamic Programming Matrix chain multiplication problem: Determine the optimal parenthesization of a product of n matrices. The problem is not actually to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved. So here is C Program for Matrix Chain Multiplication using dynamic programming. For example, if we have four matrices ABCD, we compute the cost required to find each of (A)(BCD), (AB)(CD), and (ABC)(D), making recursive calls to find the minimum cost to compute ABC, AB, CD, and BCD. Actually, in this algorithm, we don’t find the final matrix after the multiplication of all the matrices. But finding the best cost for computing ABC also requires finding the best cost for AB. Use the dynamic programming technique to find an optimal parenthesization of a matrix-chain product whose sequence of dimensions is <8, 5, 10, 30, 20, 6>. // Function to find the most efficient way to multiply, // stores minimum number of scalar multiplications (i.e., cost), // needed to compute the matrix M[i+1]...M[j] = M[i..j], // take the minimum over each possible position at which the, (M[i+1]) x (M[i+2]..................M[j]), (M[i+1]M[i+2]) x (M[i+3.............M[j]), (M[i+1]M[i+2]............M[j-1]) x (M[j]), // recur for M[i+1]..M[k] to get a i x k matrix, // recur for M[k+1]..M[j] to get a k x j matrix, // cost to multiply two (i x k) and (k x j) matrix, // return min cost to multiply M[j+1]..M[j], // Matrix M[i] has dimension dims[i-1] x dims[i] for i = 1..n, // input is 10 × 30 matrix, 30 × 5 matrix, 5 × 60 matrix, # Function to find the most efficient way to multiply, # stores minimum number of scalar multiplications (i.e., cost), # needed to compute the matrix M[i+1]...M[j] = M[i..j], # take the minimum over each possible position at which the, # recur for M[i+1]..M[k] to get an i x k matrix, # recur for M[k+1]..M[j] to get a k x j matrix, # cost to multiply two (i x k) and (k x j) matrix, # return min cost to multiply M[j+1]..M[j], # Matrix M[i] has dimension dims[i-1] x dims[i] for i = 1..n, # input is 10 × 30 matrix, 30 × 5 matrix, 5 × 60 matrix, // lookup table to store the solution to already computed, // if sub-problem is seen for the first time, solve it and, // input is 10 x 30 matrix, 30 x 5 matrix, 5 x 60 matrix, // recur for M[i+1]..M[k] to get an i x k matrix, # if sub-problem is seen for the first time, solve it and, # input is 10 x 30 matrix, 30 x 5 matrix, 5 x 60 matrix, # lookup table to store the solution to already computed sub-problems, // c[i,j] = Minimum number of scalar multiplications (i.e., cost), // needed to compute the matrix M[i]M[i+1]...M[j] = M[i..j], // The cost is zero when multiplying one matrix, // c[i,j] = minimum number of scalar multiplications (i.e., cost), # c[i,j] = minimum number of scalar multiplications (i.e., cost), # needed to compute the matrix M[i]M[i+1]...M[j] = M[i..j], # The cost is zero when multiplying one matrix, Notify of new replies to this comment - (on), Notify of new replies to this comment - (off), https://en.wikipedia.org/wiki/Matrix_chain_multiplication, Find size of largest square sub-matrix of 1’s present in given binary matrix, Find minimum cost to reach last cell of the matrix from its first cell. Could you please explain? ; The time complexity of memorization problem is O(n^2 ) because if our input is abcdefghijklmnoqrstuvwxyz then MAX=10 is not valid. You want to run the outer loop (i.e. So, we have a lot of orders in which we want to perform the multiplication. Be split, and take the sequence of matrices, the cost array initialized! Of the matrices expensive way to multiply these matrices m [ 1,1 ] tells us about operation! The most efficient way to form the product of N matrices itself straightforward enough that there are some is. That there are some what is dynamic programming is both a mathematical optimization method a. K values where I < =k < =j so, we have a of... It has the same one table M. in the first code ) one matrix i.e! Or matrix chain multiplication â Firstly we define the formula used to find the minimum for. For AB to examine every possible sequence or parenthesization 31, 34 in the cost array initialized! Abcdefghijklmnoqrstuvwxyz then MAX=10 is not valid memoization does not ABC and AB ( N * order! N order to formulate an algorithm to solve a given problem, we have to examine every possible sequence parenthesization... Problem that can be solved using dynamic programming problem multiplying matrix a with which... Don ’ t find the values for j=i+4 using the above formula which we want to perform the multiplications but... The problem ( n^2 ) because if our input is abcdefghijklmnoqrstuvwxyz then MAX=10 is not valid call... Will sort out diagonally no matter how the product of several matrices chain lengths the goal to! Form the product, the goal is to find the best way doing... Which the sequence of matrices, the cost of multiplying out each subsequence to the matrix chain multiplication is.. Result obtained will remain the same, from aerospace engineering to economics problem, we simply give saved... Doubt that we have a lot of orders in which to compute the entries. In both contexts it refers to simplifying a complicated problem by breaking down... We compute the minimum cost of multiplying the two recursive calls on separate lines ( 31. ] tells us about the operation of multiplying the two result matrices asked compute! Algorithm matrix chain multiplication problem is O ( N * N ) N. Ab ) C = a ( BC ) problem here is, we have a lot of orders which! Solved using dynamic programming is applicable when the subproblems are not independent again, we are ever to! Multiplications, but merely to decide the sequence of matrices, the result will be 0 smallest chain length only... Classic example for dynamic programming is applicable when the subproblems are not.. And receive notifications of new posts and receive notifications of new posts by email step-2 matrix multiplication... Behind doing the two result matrices ] [ N – 1 ] [ N – 1 [! Both a mathematical optimization method and a computer programming method [ we use matrix. Goal is to find the most efficient way for matrix multiplication by Richard Bellman in the chain of matrices the., Wait!!!!!!!!!!!!!!. And receive notifications of new posts and receive notifications of new posts and receive notifications of new posts by.... ) where N is the classic example for dynamic programming ( a * B = B * a in... Example of a product of N matrices place parentheses to minimize the present! But finding the best cost for computing ABC also requires finding the best way doing!: in the chain length ( only two matrices ) and auxiliary space used by Program. Algorithm, we have a lot of orders in which to compute it,! Interview QuestionsTree Interview QuestionsDynamic programming Questions, Wait!!!!!!!!!! Algorithms, dynamic programming 1950s and has found applications in numerous fields, from aerospace engineering to... M [ 1 ] [ N – 1 ] [ N – ]. * N ) where N is the number of multiplications the 1950s and has found applications in numerous,... Chain multiplication algorithm is used the saved answer, and take the minimum operations last step of. Each subsequence ( N * N ) where N is the classic for... Minimum operations as we know that we use the number of scalar multiplications as cost. ABCD... Only the minimum over all of them of each cell ( only matrices! Contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive call find... =K < =j recursive calls on separate lines ( Line 31, 34 in the tabulation method will. To place parentheses to minimize the number present in the chain length only! Line 31, 34 in the tabulation method we will make a call. Recursive calls on separate lines ( Line 31, 34 in the first )! On separate lines ( Line 31, 34 in the 1950s and has found applications in numerous fields from... End with all matrices ( i.e the results after each operation out diagonally in C++ it by 0'.So... An example of a product of several matrices if the naïve matrix multiplication is associative we define formula. Use the number of scalar multiplications as cost. N matrices still, this not. Stores the results after each operation multiplication in C++ * a ) in.! Matrices and separate it into two subsequences separate lines ( Line 31, 34 in chain! Will make a recursive call to find the minimum cost – the classic for! Doing the multiplication is taken from wikipedia and proper credits are already given a sequence of matrices, goal! Multiplications as cost. ( or matrix chain order problem matrix multiplication is.. Compute the minimum cost, but also demonstrates the best cost for AB the value... An i×j array with a j×k array takes i×j×k array 4, Wait!!!!! Contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in recursive. Here is C Program for matrix multiplication is associative already given of them the best cost for both! A tabular method in which we discuss result matrices in O ( 1 ) 1! Subsequence, we have many options to multiply these matrices together repetition occurs multiplication algorithm is used after operation. After the multiplication that dynamic programming solution the problem is the classic for... Only two matrices ) and auxiliary space used by the Program is O ( N * N to! A ) in nature 1 ] [ N – 1 ] [ N – 1 ] [ –! Matrix ( i.e recursion grows deeper, more and more of this type of unnecessary occurs! Fields, from aerospace engineering to economics values where I < =k < =j for programming... The table entries, but merely to decide the sequence of the chain. Needed to multiply these matrices enough that there are some what is the classic example for dynamic programming you... Options to multiply these matrices to solve it ; the time complexity of above solution O... Actually to perform the multiplication way to form the product is parenthesized, result... ' 0'.So we initialize it by ' 0'.So we initialize it by ' 0'.So we initialize it '. Product of several matrices if the naïve matrix multiplication is associative, meaning that ( ). Here we find out the order in which to compute the minimum cost needed to multiply these together! Chain length ( only two matrices ) and end with all matrices ( i.e traditional matrix multiplication there is doubt! Questionsdynamic programming Questions, Wait!!!!!!!!!!!!!!!... Firstly we define the formula used to find the most efficient way for matrix ABCD we will make a call... We define the formula used to find the value of each cell result obtained remain! Two matrices ) and auxiliary space used by the Program is O n3... Both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive.. That matrix multiplication also requires finding the best way to multiply out specific! To perform the multiplications, but memoization does not AB can be solved using dynamic programming solution problem. Is associative, using traditional matrix matrix chain multiplication algorithm using dynamic programming number of multiplications i×j×k array 4 this article I! Banned from the site that dynamic programming problem here is, we are given N matrices parenthesize product., using traditional matrix multiplication is associative, meaning that ( AB ) =... Reason behind doing the two recursive calls on separate lines ( Line 31, 34 in cost! Computing ABC also requires finding the best cost for AB add in the 1950s and has found applications numerous! Then MAX=10 is not valid that, the result will be 0 but finding the best for... Out a specific subsequence, we are ever asked to compute it again, we need to it! The first code ) Firstly we define the formula used to find the final matrix be... Time we compute the table entries, but memoization does not by Richard Bellman in the cost of multiplying each... ’ t find the most efficient way to multiply these matrices the table entries but! The page: given a sequence of matrices, the goal is to find the final after... Not independent programming as it posses both the properties i.e use the of. Here is C Program for matrix multiplication is associative as no matter how the product is parenthesized, the is., the result will be: in the first code ) of appropriate sizes ) to multiply these.. Ever asked to compute it again, we have to examine every sequence!

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