1.2. Dahan F., El Hindi K., Mathkour H., AlSalman H.Dynamic flying ant colony optimization (DFACO) for solving the traveling salesman problem Sensors, 19 (8) (2019), p. 1837 Google Scholar 2013. The right approach to this problem is explaining utilizing Dynamic Programming. Travelling salesman problem Solution for the famous tsp problem using algorithms: Brute Force (Backtracking), Branch And Bound, Dynamic Programming, … We present a self-learning approach that combines deep reinforcement learning and Monte Carlo tree search to solve the traveling salesman problem. See Solomon and Desrosiers (1988) that describe early papers to … The Held–Karp algorithm, also called Bellman–Held–Karp algorithm, is a dynamic programming algorithm proposed in 1962 independently by Bellman and by Held and Karp to solve the Traveling Salesman Problem. Here we can observe that main problem spitted into sub-problem, this is property of dynamic programming. I am really sorry for not writing any tutorial for last 3 days. The Hamiltonian cycle problem is to find if there exists a tour that visits every city exactly once. Both of the solutions are infeasible. Problem Statement. One example is the traveling salesman problem mentioned above: for each number of cities, there is an assignment of distances between the cities for which the nearest-neighbor heuristic produces the unique worst possible tour. Understanding The Coin Change Problem With Dynamic Programming, Bitmasking and Dynamic Programming | Set 1 (Count ways to assign unique cap to every person), Compute nCr % p | Set 1 (Introduction and Dynamic Programming Solution), Bitmasking and Dynamic Programming | Set-2 (TSP), Dynamic Programming vs Divide-and-Conquer, Dynamic Programming | Wildcard Pattern Matching | Linear Time and Constant Space, Overlapping Subproblems Property in Dynamic Programming | DP-1, Optimal Substructure Property in Dynamic Programming | DP-2, Top 20 Dynamic Programming Interview Questions. The dynamic traveling salesman problem with stochastic release dates (DTSP-srd) is a problem in which a supplier has to deliver parcels to its customers. Voyaging Salesman Problem (TSP) Using Dynamic Programming. Permutations of cities. Dynamic travelling salesman problems (DTSPs) are categorised under DOPs. 2013 . In simple words, it is a problem of finding optimal route between nodes in the graph. It has been studied by researchers working in a variety of elds, including mathematics, computer science, and operations research. The travelling salesman problem follows the approach of the branch and bound algorithm that is one of the different types of algorithms in data structures. Travelling Salesman Problem with Code. Travelling Salesman problem in dynamic programming. http://www.cs.berkeley.edu/~vazirani/algorithms/chap6.pdf, Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. Now, it’s time to calculate your own optimal route. We can use brute-force approach to evaluate every possible tour and select the best one. Hello guys, welcome back to “code with asharam”. The challenge of the problem is that the traveling salesman needs to minimize the total length of the trip. The traditional lines of attack for the NP-hard problems are the following: acknowledge that you have read and understood our, GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Travelling Salesman Problem | Set 2 (Approximate using MST), Travelling Salesman Problem | Set 1 (Naive and Dynamic Programming), Traveling Salesman Problem (TSP) Implementation, Travelling Salesman Problem implementation using BackTracking, Printing all solutions in N-Queen Problem, Warnsdorff’s algorithm for Knight’s tour problem, The Knight’s tour problem | Backtracking-1, Count number of ways to reach destination in a Maze, Count all possible paths from top left to bottom right of a mXn matrix, Print all possible paths from top left to bottom right of a mXn matrix, Unique paths covering every non-obstacle block exactly once in a grid, Tree Traversals (Inorder, Preorder and Postorder). The travelling salesman problem follows the approach of the branch and bound algorithm that is one of the different types of algorithms in data structures. g(2, Φ ) = C21 = 5g(3, Φ ) = C31 = 6g(4, Φ ) = C41 = 8, g(3,{2}) = c32 + g(2, Φ ) = c32 + c21 = 13 + 5 = 18g(4,{2}) = c42 + g(2, Φ ) = c42 + c21 = 8+ 5 = 13, g(2,{3}) = c23 + g(3, Φ ) = c23 + c31 = 9 + 6 = 15g(4,{3}) = c43 + g(3, Φ ) = c43 + c31 = 9+ 6 = 15, g(2,{4}) = c24 + g(4, Φ ) = c24 + c41 = 10 + 8 = 18g(3,{4}) = c34 + g(4, Φ ) = c34 + c41 = 12 + 8 = 20, g {2,{3,4}} = min {c23 + g(3,{4}) , c24 + g(4,{3})} = min { 9 + 20 , 10 + 15} = min { 29, 25} = 25, g {3,{2,4}} = min {c32 + g(2,{4}), c34 + g(4,{2})} = min { 13+ 18, 12 + 13} = min { 31, 25} = 25, g(4,{2,3}) = min {c42 + g(2,{3}), c43 + g(3,{2})} = min { 8 + 15 , 9 + 18} = min { 23, 27} = 23, g { 1, {2,3,4}} = min{ c12 + g(2,{3,4}), c13 + g(3,{2,4}), c14 + g(4,{2,3})} = min { (25 + 10 ) , (25 + 15) , (23 + 20) } = min { ( 35), (40), (43)} = 35. We introduced Travelling Salesman Problem and discussed Naive and Dynamic Programming Solutions for the problem in the previous post. 4) Return the permutation with minimum cost. How about we watch that. Literature review. Cost of the tour = 10 + 25 + 30 + 15 = 80 units . Graphs, Bitmasking, Dynamic Programming We use cookies to ensure you have the best browsing experience on our website. What is the time complexity of the Dynamic Algorithm for the Traveling Salesman Problem? This looks simple so far. Let us consider a graph G = (V, E), where V is a set of cities and E is a set of weighted edges. i is a Starting point of a tour and S a subset of cities. Java Model Let the given set of vertices be {1, 2, 3, 4,….n}. 4. Experience. There are approximate algorithms to solve the problem though. Journal of Applied Mathematics, Vol. A Hybrid Approach of Bundle and Benders Applied Large Mixed Linear Integer Problem. We can say that salesman wishes to make a tour or Hamiltonian cycle, visiting each city exactly once and finishing at the city he starts from. Instead of brute-force using dynamic programming approach, the solution can be obtained in lesser time, though there is no polynomial time algorithm. Note the difference between Hamiltonian Cycle and TSP. Videos you watch may be added to the TV's watch history and influence TV recommendations. For many other problems, greedy algorithms fail to produce the optimal solution, and may even produce the unique worst possible solution. 4. Say it is T (1,{2,3,4}), means, initially he is at village 1 and then he can go to any of {2,3,4}. Effectively combining a truck and a drone gives rise to a new planning problem that is known as the traveling salesman problem with drone (TSP‐D). cpp analysis sort insertion-sort sorting-algorithms dijkstra prim knapsack-problem radix-sort cplusplus-11 heuristic-search-algorithms alogrithms a-dynamic-programming travelling-salesman-problem clique-aqui minimum-spanning-tree greedy-programming This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. In the TSP, a salesman departs … 2) Generate all (n-1)! The problem can be described as: find a tour of N cities in a country, the tour should visit every city just once, return to the … For more details on TSP please take a look here. Writing code in comment? Inorder Tree Traversal without recursion and without stack! Travelling Sales Person Problem. Next Article: Traveling Salesman Problem | Set 2, References: Traveling Salesman Problem • Problem Statement – If there are n cities and cost of traveling from any city to any other city is given. The total travel distance can be one of the optimization criterion. Attention reader! … February 26, 2020 March 17, 2020 / Dynamic programming / Leave a Comment. Improving the runtime of the Travelling Salesman Problem with Dynamic Programming In this problem we shall deal with a classical NP-complete problem called Traveling Salesman Problem. Each sub-problem will take O (n) time (discovering way to outstanding (n-1) hubs). The traveling salesman problems abide by a salesman and a set of cities. From there to reach non-visited vertices (villages) becomes a new problem. ), but still exponential. Before solving the problem, we assume that the reader has the knowledge of . With or without time windows, traveling salesman problems are NP-hard in deterministic settings. We introduced Travelling Salesman Problem and discussed Naive and Dynamic Programming Solutions for the problem in the previous post,.Both of the solutions are infeasible. The time complexity is much less than O(n! In the traveling salesman Problem, a salesman must visits n cities. Travelling salesman problem - Simple English Wikipedia, the free encyclopedia. The Travelling Salesman Problem (TSP) is the most known computer science optimization problem in a modern world. What is the problem statement ? The traveling salesman's problem is one of the most famous problems of combinatorial optimization, which consists in finding the most profitable route passing through these points at least once and then returning to the starting point. The traveling salesman problem(TSP) is an algorithmic problem tasked with finding the shortest route between a set of points and locations that must be visited. We start with all subsets of size 2 and calculate C(S, i) for all subsets where S is the subset, then we calculate C(S, i) for all subsets S of size 3 and so on. The salesman has to visit every one of the cities starting from a certain one (e.g., the hometown) and to return to the same city. Code for the paper 'An Efficient Graph Convolutional Network Technique for the Travelling Salesman Problem' (arXiv Pre-print) deep-learning pytorch combinatorial-optimization travelling-salesman-problem geometric-deep-learning graph-neural-networks Updated Nov 13, 2020; Python; rhgrant10 / acopy Star 71 Code Issues Pull requests A Python implementation of the Ant Colony … TSP is an extension of the Hamiltonian circuit problem. The Travelling Salesman Problem (TSP) is the most known computer science optimization problem in a modern world. What is the shortest possible route that he visits each city exactly once and returns to the origin city? Dynamic traveling salesman problem (DTSP), as a case of dynamic combinatorial optimization problem, extends the classical traveling salesman problem and finds many practical importance in real-world applications, inter alia, traffic jams, network load-balance routing, transportation, telecommunications, and network designing. The Multi-objective Dynamic Traveling Salesman Problem: Last Mile Delivery with Unmanned Aerial Vehicles Assistance Ben Remer, Andreas A. Malikopoulos, Senior Member, IEEE Abstract—In this paper, we present an approach to optimiz-ing the last-mile delivery route of a truck using coordination with unmanned aerial vehicles (UAVs). By using dynamic programming, we’ve made our solution for the traveling salesman problem just a little bit better by choosing to smartly enumerate … We start with all subsets of size 2 and calculate. The idea is to compare its optimality with Tabu search algorithm. Dynamic programming … Dynamic Traveling Salesman Problem: Value of Real-Time Traffic Information Abstract: We investigate the value of choosing the next stop to visit in a multistop trip based on current traffic conditions to minimize the expected total travel time of the tour. By using our site, you We will soon be discussing approximate algorithms for travelling salesman problem. Concepts Used:. It is also popularly known as Travelling Salesperson Problem. Though I didn’t win it, yet I learned a lot from it. In this post, we will be using our knowledge of dynamic programming and Bitmasking technique to solve one of the famous NP-hard problem “Travelling Salesman Problem”. 9, No. The travel costs are symmetric from the travel of view that travelling from city X to city Y costs just as much as travelling from Y to X - the manner of visiting all the researches is simply the order in which the cities are visited. Let the cost of this path be cost(i), the cost of corresponding Cycle would be cost(i) + dist(i, 1) where dist(i, 1) is the distance from i to 1. Get more help from Chegg Get 1:1 help now from expert Computer Science tutors Please write to us at contribute@geeksforgeeks.org to report any issue with the above content. Here we know that Hamiltonian Tour exists (because the graph is complete) and in fact many such tours exist, the problem is to find a minimum weight Hamiltonian Cycle. Example Problem An edge e(u, v) represents th… Algorithms Travelling Salesman Problem (Bitmasking and Dynamic Programming) In this article, we will start our discussion by understanding the problem statement of The Travelling Salesman Problem perfectly and then go through the basic understanding of bit masking and dynamic programming. Dynamic programming(DP) is the most powerful technique to solve a particular class of problems.DP is an algorithmic technique for solving an optimization problem by breaking it down into simpler sub-problems and utilizing the fact that the optimal solution to the overall problem depends upon the optimal solution to its sub-problems. This paper presents exact solution approaches for the TSP‐D based on dynamic programming and provides an experimental comparison of these approaches. Active 6 months ago. 2) Generate all (n-1)! This problem is really interesting as it has been bothering computer scientists for a long time. This problem falls under category of NP-Hard problems. There is a non-negative cost c (i, j) to travel from the city i to city j. A traveler needs to visit all the cities from a list, where distances between all the cities are known and each city should be visited just once. The arrival time of a parcel to the depot is called its release date. The travelling salesman problem was mathematically formulated in the 1800s by the Irish mathematician W.R. Hamilton and by the British mathematician Thomas Kirkman.Hamilton's icosian game was a recreational puzzle based on finding a Hamiltonian cycle. The cost of the tour is 10+25+30+15 which is 80. How to solve a Dynamic Programming Problem ? Analysis of the Dynamic Travelling Salesman Problem with Di erent Policies Santiago Ravassi We propose and analyze new policies for the traveling salesman problem in a dynamic and stochastic environment (DTSP). ABSTRACT In this paper we examine a version of the dynamic traveling salesman problem in which a single mobile server provides service to customers whose positions are known. Space required is also exponential. Keywords: Traveling Salesman Problem, time windows, time dependent travel times, dynamic discretization discovery 1 Introduction The Traveling Salesman Problem (TSP) is a classical combinatorial optimization problem. Above we can see a complete directed graph and cost matrix which includes … The goal is to find a tour of minimum cost. Traveling Salesman Problem - Dynamic Programming - Explained using Formula PATREON The video depicts four metaheuristic algorithms applied to the travelling salesman problem: local search, tabu. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready. The nature of the problem makes it a stochastic dynamic traveling salesman problem with time windows (SDTSPTW). Program to find whether a no is power of two, Cyclic Redundancy Check and Modulo-2 Division, Write Interview We can say that salesman wishes to make a tour or Hamiltonian cycle, visiting each city exactly once and finishing at the city he starts from. In this manner all-out time unpredictability is O (n2n) * O (n) = O (n22n) Space multifaceted nature is likewise number of sub-problems which is O (n2n) Program for Traveling Salesman Problem in C Service requests are generated according to a Poisson process which is In simple words, it is a problem of finding optimal route between nodes in the graph. For a set of size n, we consider n-2 subsets each of size n-1 such that all subsets don’t have nth in them. Naive Solution: The traveling salesman problem (TSP) is an algorithmic problem tasked with finding the shortest route between a set of points and locations that must be visited. For example, consider the graph shown in figure on right side. The optimal tour route is, 1 -> 2 -> 4 -> 3 -> 1 . Dynamic Programming: The moving-target traveling salesman problem ... based on a mixed integer linear programming formulation and dynamic programming [9,10,12]. Ask Question Asked 6 months ago. Travelling Salesman Problem using Dynamic Programming - Easiest Approach with Code. Java Model the principle problem can be separated into sub-problems. To avoid this, cancel and sign in to YouTube on your computer. We need to start at 1 and end at k. We should select the next city in such a way that. Here problem is travelling salesman wants to find out his tour with minimum cost. A Heuristic Approach Based on Clarke-Wright Algorithm for Open Vehicle Routing Problem. This paper presents exact solution approaches for the TSP‐D based on dynamic programming and provides an experimental comparison of these approaches. Traveling salesman problem 1. Using dynamic programming to speed up the traveling salesman problem! For every other vertex i (other than 1), we find the minimum cost path with 1 as the starting point, i as the ending point and all vertices appearing exactly once. Effectively combining a truck and a drone gives rise to a new planning problem that is known as the traveling salesman problem with drone (TSP‐D). I know that in terms of optimal solution, greedy algorithms are used for solving TSPs, but it becomes more complex and takes exponential time when numbers of vertices (i.e. Let us define a term C(S, i) be the cost of the minimum cost path visiting each vertex in set S exactly once, starting at 1 and ending at i. The Scientific World Journal, Vol. The problem is a famous NP hard problem. 4) Return the permutation with minimum cost. The task is to print minimum cost in TSP cycle. This means you're free to copy and share these comics (but not to sell them). This problem can be related … NP-Hard problems are the ones which don’t have any known polynomial time algorithms. The exact problem statement goes like this, "Given a set of cities and distance between every pair of cities, the problem is to find the shortest possible route that visits … Traveling Salesman Problem Aulia Rahma Amin1, Mukhamad Ikhsan2, Lastiko Wibisono3 Departemen Teknik Informatika, Institut Teknologi Bandung Jl. Problem Statement The traveling salesman problem I. Travelling salesman problem. Travelling Salesman Problem | Greedy Approach Last Updated: 18-11-2020 Given a 2D matrix tsp [] [], where each row has the array of distances from that indexed city to all the other cities and -1 denotes that there doesn’t exist a path between those two indexed cities. Travelling Salesman Problem is based on a real life scenario, where a salesman from a company has to start from his own city and visit all the assigned cities exactly once and return to his home till the end of the day. Genetic Algorithm, Dynamic Programming and Branch and Bound Algorithm Regarding Traveling Salesman Problem. Using this formula we are going to solve a problem. A TSP tour in the graph is 1-2-4-3-1. Print Postorder traversal from given Inorder and Preorder traversals, Construct Tree from given Inorder and Preorder traversals, Construct a Binary Tree from Postorder and Inorder, Compute the integer absolute value (abs) without branching, Left Shift and Right Shift Operators in C/C++, http://www.lsi.upc.edu/~mjserna/docencia/algofib/P07/dynprog.pdf, http://www.cs.berkeley.edu/~vazirani/algorithms/chap6.pdf, Traveling Salesman Problem using Genetic Algorithm, Proof that traveling salesman problem is NP Hard, Vertex Cover Problem | Set 2 (Dynamic Programming Solution for Tree), Dynamic Programming | High-effort vs. Low-effort Tasks Problem. Viewed 392 times 0. i am trying to resolve the travelling salesman problem with dynamic programming in c++ and i find a way using a mask of bits, i got the min weight, but i dont know how to get the path that use, it would be very helpful if someone find a way. Time Complexity: Θ(n!) If playback doesn't begin shortly, try restarting your device. The original Traveling Salesman Problem is one of the fundamental problems in the study of combinatorial optimization—or in plain English: finding the best solution to a problem from a finite set of possible solutions. Please use ide.geeksforgeeks.org, generate link and share the link here. It is also popularly known as Travelling Salesperson Problem. Travelling Salesman Problem (TSP) : Given a set of cities and distances between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point. Travelling Salesman | Dynamic Programming | Part 18. Home ACM Journals Journal of the ACM Vol. Solve Traveling Salesman Problem by Monte Carlo Tree Search and Deep Neural Network. There is a non-negative cost c (i, j) to travel from the city i to city j. In fact, there is no polynomial time solution available for this problem as the problem is a known NP-Hard problem. Naive Solution: 1) Consider city 1 as the starting and ending point. Actually, I took part in a hackathon and was pretty busy. Now the question is how to get cost(i)? The paper presents a naive algorithms for Travelling salesman problem (TSP) using a dynamic programming approach (brute force). In the traveling salesman Problem, a salesman must visits n cities. Travelling Salesman problem in dynamic programming. The Multi-objective Dynamic Traveling Salesman Problem: Last Mile Delivery with Unmanned Aerial Vehicles Assistance Ben Remer, Andreas A. Malikopoulos, Senior Member, IEEE Abstract—In this paper, we present an approach to optimiz-ing the last-mile delivery route of a truck using coordination with unmanned aerial vehicles (UAVs). More details. The time complexity with the DP method asymptotically equals N² × 2^N where N is the number of cities. If a travelling salesman problem is solved by using dynamic programming approach, will it provide feasible solution better than greedy approach?. Google Maps and the Traveling Salesman Problem Known and loved as the de facto standard for finding directions from point A to point B, the Google Maps Platform Directions API can do so much more than just find simple directions. 1 Dynamic Programming Treatment of the Travelling Salesman Problem article Dynamic Programming Treatment of the Travelling Salesman Problem Travelling Salesman Problem (TSP): Given a set of cities and distance between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point. The exact problem statement goes like this, cities) are very large. Active 6 months ago. We model this problem as a Markov decision process. Following are different solutions for the traveling salesman problem. 14 May 2020. Note the difference between Hamiltonian Cycle and TSP. This algorithm falls under the NP-Complete problem. Finally, we return the minimum of all [cost(i) + dist(i, 1)] values. Dynamic Programming: Solution for the famous tsp problem using algorithms: Brute Force (Backtracking), Branch And Bound, Dynamic Programming, … Let us consider 1 as starting and ending point of output. It also can tackle what’s known as the traveling salesman problem (TSP)—the need to determine the most cost-efficient route across multiple destinations. To calculate cost(i) using Dynamic Programming, we need to have some recursive relation in terms of sub-problems. Travelling Salesman Problem is based on a real life scenario, where a salesman from a company has to start from his own city and visit all the assigned cities exactly once and return to his home till the end of the day. We assume that every two cities are connected. Featured on Meta Feature Preview: New Review Suspensions Mod UX For more details on TSP please take a look here. The total travel distance can be one of the optimization criterion. There is no polynomial time know solution for this problem. This algorithm falls under the NP-Complete problem. let see how to slove. Following are different solutions for the traveling salesman problem. In fact, even the feasibility problem with time window is NP-complete (Savelsbergh, 1984). Travelling salesman problem is the most notorious computational problem. These parcels are delivered to its depot while the distribution is taking place. In this problem, we approach the Bottom-Up method. An error occurred while retrieving sharing information. Keywords: Traveling salesman problem, Vehicle routing, Drones, Dynamic Programming 1 Introduction Several Internet retailers and logistics service providers including Amazon, Singapore post and DHL are experimenting with the use of drones to support the delivery of parcels and mail. 1) Consider city 1 as the starting and ending point. Efficient DPSO Neighbourhood for Dynamic Traveling Salesman Problem. 3) Calculate cost of every permutation and keep track of minimum cost permutation. In fact, there is no polynomial-time solution available for this problem as the problem is a known NP-Hard problem. Note that 1 must be present in every subset. Ganesha 10, Bandung E-mail : if13009@students.if.itb.ac.id1, if13033@students.if.itb.ac.id2, if13051@students.if.itb.ac.id3 Abstrak Permasalahan TSP (Traveling Salesman Problem ) adalah permasalahan dimana seorang salesman … n2" nlgn 2 n2 Ign None of these n! How to swap two numbers without using a temporary variable? Discussed Traveling Salesman Problem -- Dynamic Programming--explained using Formula. For the classic traveling salesman problem (TSP), dynamic programming approaches were first proposed in Held and Karp [10] and Bellman [3]. Given a set of cities(nodes), find a minimum weight Hamiltonian Cycle/Tour. Numerical examples are presented that indicate that the value of using current … Don’t stop learning now. – Then we have to obtain the cheapest round-trip such that each city is visited exactly ones returning to starting city, completes the tour. So, in this tutorial, I am going to discuss a really famous problem – Travelling Salesman. In this article, we will discuss how to solve travelling salesman problem using branch and bound approach with example. 3) Calculate cost of every permutation and keep track of minimum cost permutation. There are at most O(n*2n) subproblems, and each one takes linear time to solve. However, its time complexity would exponentially increase with the number of cities. DP and formation of DP transition relation; Bitmasking in DP; Travelling Salesman problem Linear Algebra 5 | Orthogonality, The Fourth Subspace, and General Picture of Subspaces, THE LORENTZ TRANSFORMATIONS AND THE TEMPORAL EXPANSION, Richard Feynman’s Distinction between Future and Past, Everything You Always Wanted to Know About Derivatives. Dynamic Programming. For n number of vertices in a graph, there are (n - 1)!number of possibilities. The Hamiltoninan cycle problem is to find if there exist a tour that visits every city exactly once. So this approach is also infeasible even for slightly higher number of vertices. For the general TSP without additional assumptions, this is the exact algorithm with the best known worst-case running time to this day [2]. Dynamic Programming can be applied just if. Permutations of cities. Browse other questions tagged algorithms complexity-theory algorithm-analysis space-complexity traveling-salesman or ask your own question. If salesman starting city is A, then a TSP tour in the graph is-A → B → D → C → A . http://www.lsi.upc.edu/~mjserna/docencia/algofib/P07/dynprog.pdf Find tour of traveling salesman problem using dynamic programming. What is Travelling Salesman Problem? Using the above recurrence relation, we can write dynamic programming based solution. The dynamic programming or DP method guarantees to find the best answer to TSP. The total running time is therefore O(n2*2n). Ask Question Asked 6 months ago.

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