In fact, there is no polynomial-time solution available for this problem as the problem is a known NP-Hard problem. We model this problem as a Markov decision process. Using the above recurrence relation, we can write dynamic programming based solution. If salesman starting city is A, then a TSP tour in the graph is-A → B → D → C → A . This problem can be related … Finally, we return the minimum of all [cost(i) + dist(i, 1)] values. This paper presents exact solution approaches for the TSP‐D based on dynamic programming and provides an experimental comparison of these approaches. Travelling Salesman problem in dynamic programming. It has been studied by researchers working in a variety of elds, including mathematics, computer science, and operations research. For more details on TSP please take a look here. The total travel distance can be one of the optimization criterion. Here problem is travelling salesman wants to find out his tour with minimum cost. In this article, we will discuss how to solve travelling salesman problem using branch and bound approach with example. 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”. For example, consider the graph shown in figure on right side. 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. Literature review. 4) Return the permutation with minimum cost. The time complexity is much less than O(n! This means you're free to copy and share these comics (but not to sell them). The exact problem statement goes like this, I am really sorry for not writing any tutorial for last 3 days. Travelling Salesman Problem using Dynamic Programming - Easiest Approach with Code. Featured on Meta Feature Preview: New Review Suspensions Mod UX 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. Dynamic travelling salesman problems (DTSPs) are categorised under DOPs. let see how to slove. i is a Starting point of a tour and S a subset of cities. 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 … So this approach is also infeasible even for slightly higher number of vertices. Program to find whether a no is power of two, Cyclic Redundancy Check and Modulo-2 Division, Write Interview 4. 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. 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. Problem Statement 4) Return the permutation with minimum cost. How about we watch that. We start with all subsets of size 2 and calculate. 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. Note that 1 must be present in every subset. cities) are very large. 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. 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 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). ), but still exponential. 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 For more details on TSP please take a look here. 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. So, in this tutorial, I am going to discuss a really famous problem – Travelling Salesman. An error occurred while retrieving sharing information. For the classic traveling salesman problem (TSP), dynamic programming approaches were first proposed in Held and Karp [10] and Bellman [3]. Voyaging Salesman Problem (TSP) Using Dynamic Programming. Permutations of cities. Please use ide.geeksforgeeks.org, generate link and share the link here. How to solve a Dynamic Programming Problem ? 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 … The Hamiltonian cycle problem is to find if there exists a tour that visits every city exactly once. Experience. Numerical examples are presented that indicate that the value of using current … 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 However, its time complexity would exponentially increase with the number of cities. Given a set of cities(nodes), find a minimum weight Hamiltonian Cycle/Tour. Writing code in comment? Get more help from Chegg Get 1:1 help now from expert Computer Science tutors 2) Generate all (n-1)! The arrival time of a parcel to the depot is called its release date. Travelling Sales Person Problem. If a travelling salesman problem is solved by using dynamic programming approach, will it provide feasible solution better than greedy approach?. Permutations of cities. Dynamic Programming: There are at most O(n*2n) subproblems, and each one takes linear time to solve. Now the question is how to get cost(i)? Both of the solutions are infeasible. n2" nlgn 2 n2 Ign None of these n! 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. The right approach to this problem is explaining utilizing Dynamic Programming. We need to start at 1 and end at k. We should select the next city in such a way that. Here we can observe that main problem spitted into sub-problem, this is property of dynamic programming. This looks simple so far. More details. 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. Using dynamic programming to speed up the traveling salesman problem! The problem is a famous NP hard problem. Travelling Salesman Problem with Code. TSP is an extension of the Hamiltonian circuit problem. 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. Traveling Salesman Problem - Dynamic Programming - Explained using Formula PATREON The video depicts four metaheuristic algorithms applied to the travelling salesman problem: local search, tabu. The traditional lines of attack for the NP-hard problems are the following: Following are different solutions for the traveling salesman problem. Attention reader! 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. The dynamic programming or DP method guarantees to find the best answer to TSP. The Scientific World Journal, Vol. 1.2. 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. We introduced Travelling Salesman Problem and discussed Naive and Dynamic Programming Solutions for the problem in the previous post. 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. Concepts Used:. Following are different solutions for the traveling salesman problem. 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. A Hybrid Approach of Bundle and Benders Applied Large Mixed Linear Integer Problem. There is a non-negative cost c (i, j) to travel from the city i to city j. Cost of the tour = 10 + 25 + 30 + 15 = 80 units . In the TSP, a salesman departs … 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. We will soon be discussing approximate algorithms for travelling salesman problem. 14 May 2020. the principle problem can be separated into sub-problems. We present a self-learning approach that combines deep reinforcement learning and Monte Carlo tree search to solve the traveling salesman problem. Ask Question Asked 6 months ago. Home ACM Journals Journal of the ACM Vol. The Hamiltoninan cycle problem is to find if there exist a tour that visits every city exactly once. An edge e(u, v) represents th… 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. Note the difference between Hamiltonian Cycle and TSP. In simple words, it is a problem of finding optimal route between nodes in the graph. Travelling 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. 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. What is Travelling Salesman Problem? This paper presents exact solution approaches for the TSP‐D based on dynamic programming and provides an experimental comparison of these approaches. 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. For the general TSP without additional assumptions, this is the exact algorithm with the best known worst-case running time to this day [2]. Let us consider 1 as starting and ending point of output. Genetic Algorithm, Dynamic Programming and Branch and Bound Algorithm Regarding Traveling Salesman Problem. Ask Question Asked 6 months ago. The total travel distance can be one of the optimization criterion. Service requests are generated according to a Poisson process which is 1 Dynamic Programming Treatment of the Travelling Salesman Problem article Dynamic Programming Treatment of the Travelling Salesman 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). Next Article: Traveling Salesman Problem | Set 2, References: How to swap two numbers without using a temporary variable? From there to reach non-visited vertices (villages) becomes a new problem. If playback doesn't begin shortly, try restarting your device. 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 … Efficient DPSO Neighbourhood for Dynamic Traveling Salesman Problem. This algorithm falls under the NP-Complete problem. Solution for the famous tsp problem using algorithms: Brute Force (Backtracking), Branch And Bound, Dynamic Programming, … The Travelling Salesman Problem (TSP) is the most known computer science optimization problem in a modern world. Dynamic Programming. 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 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). 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. 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. Actually, I took part in a hackathon and was pretty busy. This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. 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. In simple words, it is a problem of finding optimal route between nodes in the graph. Before solving the problem, we assume that the reader has the knowledge of . The nature of the problem makes it a stochastic dynamic traveling salesman problem with time windows (SDTSPTW). Naive Solution: 1) Consider city 1 as the starting and ending point. Space required is also exponential. Hello guys, welcome back to “code with asharam”. There is a non-negative cost c (i, j) to travel from the city i to city j. 4. 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. 2) Generate all (n-1)! Note the difference between Hamiltonian Cycle and TSP. 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). 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. 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. 3) Calculate cost of every permutation and keep track of minimum cost permutation. Traveling Salesman Problem Aulia Rahma Amin1, Mukhamad Ikhsan2, Lastiko Wibisono3 Departemen Teknik Informatika, Institut Teknologi Bandung Jl. We assume that every two cities are connected. By using our site, you Dynamic programming … Videos you watch may be added to the TV's watch history and influence TV recommendations. February 26, 2020 March 17, 2020 / Dynamic programming / Leave a Comment. Travelling salesman problem - Simple English Wikipedia, the free encyclopedia. To calculate cost(i) using Dynamic Programming, we need to have some recursive relation in terms of sub-problems. 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. Journal of Applied Mathematics, Vol. There is no polynomial time know solution for this problem. Though I didn’t win it, yet I learned a lot from it. 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}. Active 6 months ago. Instead of brute-force using dynamic programming approach, the solution can be obtained in lesser time, though there is no polynomial time algorithm. For n number of vertices in a graph, there are (n - 1)!number of possibilities. Active 6 months ago. 2013. 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). This problem falls under category of NP-Hard problems. The moving-target traveling salesman problem ... based on a mixed integer linear programming formulation and dynamic programming [9,10,12]. Solve Traveling Salesman Problem by Monte Carlo Tree Search and Deep Neural Network. Dynamic Programming can be applied just if. Don’t stop learning now. This problem is really interesting as it has been bothering computer scientists for a long time. 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. These parcels are delivered to its depot while the distribution is taking place. Inorder Tree Traversal without recursion and without stack! Now, it’s time to calculate your own optimal route. 3) Calculate cost of every permutation and keep track of minimum cost permutation. Traveling Salesman Problem • Problem Statement – If there are n cities and cost of traveling from any city to any other city is given. The cost of the tour is 10+25+30+15 which is 80. The task is to print minimum cost in TSP cycle. The traveling salesman problem I. Naive Solution: 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. Let the given set of vertices be {1, 2, 3, 4,….n}. In fact, there is no polynomial time solution available for this problem as the problem is a known NP-Hard problem. The total running time is therefore O(n2*2n). See Solomon and Desrosiers (1988) that describe early papers to … 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. In this problem, we approach the Bottom-Up method. 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. 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. 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 … In the traveling salesman Problem, a salesman must visits n cities. The paper presents a naive algorithms for Travelling salesman problem (TSP) using a dynamic programming approach (brute force). A Heuristic Approach Based on Clarke-Wright Algorithm for Open Vehicle Routing Problem. It is also popularly known as Travelling Salesperson Problem. 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. We can use brute-force approach to evaluate every possible tour and select the best one. The traveling salesman problems abide by a salesman and a set of cities. NP-Hard problems are the ones which don’t have any known polynomial time algorithms. For many other problems, greedy algorithms fail to produce the optimal solution, and may even produce the unique worst possible solution. Dynamic Programming: The time complexity with the DP method asymptotically equals N² × 2^N where N is the number of cities. Example Problem 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. 1) Consider city 1 as the starting and ending point. Problem Statement. DP and formation of DP transition relation; Bitmasking in DP; Travelling Salesman problem What is the problem statement ? What is the shortest possible route that he visits each city exactly once and returns to the origin city? In fact, even the feasibility problem with time window is NP-complete (Savelsbergh, 1984). To avoid this, cancel and sign in to YouTube on your computer. 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. A TSP tour in the graph is 1-2-4-3-1. Travelling Salesman | Dynamic Programming | Part 18. It is also popularly known as Travelling Salesperson Problem. Travelling salesman problem Solution for the famous tsp problem using algorithms: Brute Force (Backtracking), Branch And Bound, Dynamic Programming, … The Travelling Salesman Problem (TSP) is the most known computer science optimization problem in a modern world. 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. Let us consider a graph G = (V, E), where V is a set of cities and E is a set of weighted edges. Java Model This algorithm falls under the NP-Complete problem. 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. 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. 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. 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). 9, No. Each sub-problem will take O (n) time (discovering way to outstanding (n-1) hubs). With or without time windows, traveling salesman problems are NP-hard in deterministic settings. Using this formula we are going to solve a problem. The challenge of the problem is that the traveling salesman needs to minimize the total length of the trip. 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. Travelling Salesman problem in dynamic programming. The idea is to compare its optimality with Tabu search algorithm. There are approximate algorithms to solve the problem though. 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. 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. Please write to us at contribute@geeksforgeeks.org to report any issue with the above content. 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. Above we can see a complete directed graph and cost matrix which includes … The goal is to find a tour of minimum cost. By using dynamic programming, we’ve made our solution for the traveling salesman problem just a little bit better by choosing to smartly enumerate … What is the time complexity of the Dynamic Algorithm for the Traveling Salesman Problem? The optimal tour route is, 1 -> 2 -> 4 -> 3 -> 1 . Browse other questions tagged algorithms complexity-theory algorithm-analysis space-complexity traveling-salesman or ask your own question. … – Then we have to obtain the cheapest round-trip such that each city is visited exactly ones returning to starting city, completes the tour. Travelling salesman problem is the most notorious computational problem. http://www.lsi.upc.edu/~mjserna/docencia/algofib/P07/dynprog.pdf In the traveling salesman Problem, a salesman must visits n cities. Java Model Traveling salesman problem 1. 2013 . Time Complexity: Θ(n!) We use cookies to ensure you have the best browsing experience on our website. Find tour of traveling salesman problem using dynamic programming. Graphs, Bitmasking, Dynamic Programming Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready. 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. Discussed Traveling Salesman Problem -- Dynamic Programming--explained using Formula.