Memoization and Fibonacci Numbers for Dynamic Programming

Question: Discuss about theMemoization and Fibonacci Numbers for Dynamic Programming. Answer: Introduction Dynamic programming involves breaking down complex problem into sub-programs that can be solved easily. Once the sub-problem is solved, the answer is combined to get solution to complex a problem. The main problem in this project would be using memoization and dynamic programing concepts in Fibonacci numbers. In many cases, Fibonacci numbers calculation makes use of recursion which is quite iterative in nature. Important to note is that, dynamic programming application in Fibonacci numbers is used to avoid multiple sub-program calculations experienced in recursive algorithms. Memoization in dynamic programming takes both Bottom-pup and Top-down approach in solving the subject problem (Moerkotte Neumann, 2008). The Top-down approach breaks complex problem into sub-optimal problems while Bottom-Up approach combines sub-optimal solutions to desirable solution. The process starts by selecting a problem. Once problem has been identified, the best approach is chosen, Top-down or Bottom-up . Generally, dynamic problem works in cases where problems have right-left inherent order such as sequence of integers, strings ad trees graphs. Memoization involves concepts of storing results from previously computed functions and calling them on demand. On the other hand, recursion takes place when a program function calls itself several times while giving similar results from provided inputs. When results from integers are computed from provided inputs, they are stored in a buffer waiting to be conjoined to one desirable but complex optimal solution. The process might look similar to recursion but dynamic programming does not need recursion in order to work. Dynamic programing has its power on being able to understand which partial results would be required in building up the final answer (Dai, Chen Zheng, 2018). Therefore, the goal of this project would be to implement dynamic programming concepts when calculating an nth value in Fibonacci numbers through memoization. Typical problems There are many cases where dynamic programming has been applied but it is very important to evaluate which approach would work best. To understand the concept of memoization, dynamic programming and its application in Fibonacci numbers, some case studies would feature in the discussion. This section would be analyzed intensively by breaking it down into overview of memoization from inception to present. The background information would give detailed concepts of memoization and its application in dynamic programming. Similarly, it will involve evaluation of the problem, its importance and relevance to the study. It is at this section where key important aspect of memoization and dynamic programming are incorporated. It is at these two levels where implementation of memoization as it has been conjoined in the dynamic programming is done. Background information Dynamic programming date back 1950s when its concept was first introduced with an objective of making complex calculation simple (Cormen et al, 2009). Its operation is based on common phenomenon of principle of optimality. The principle implies that, the general optimal solution is a mere combination of sub-optimal solutions to some of its sub-problems. An evaluation of matrix chain multiplication problem shows that, it is quite wrong to assume the only value of interest is optimal. All values in the matrix table serves as a representation of optimal solution in the problem domain. It is important to note that Fibonacci numbers starts with only two set of values; either integer 1 and 1 or 0 and 1 in relation to chosen starting point. According to Stivala et al (2010), memoization and dynamic programming is applicable in Fibonacci numbers due to the fact that, it can be expressed in a finite sequence of decisions at several stages. The combination of both recursive and memoization was meant to come up with more reliable methodology to increase the performance of program execution. It is very clear from various evaluations of research that, dynamic programming through memoization has a wide array of applications. Finally, though it is highly recommended in many projects, it presents several challenges. However, it has been successfully implemented in various projects. Problem relevance and importance The problem is quite relevant to the study in that, with dynamic programming, the recursive nature of the problem is eliminated in the program. A good example depicts itself when a program to find for nth value such as 100 is run. In this case, instead of generating an array of numbers recursively, the entire set of 99 arrays is generated once and stored in order to be used in capturing desired results (Dai, Chen Zheng, 2018). Similarly, when dynamic programming is used in a program, memoization is the critical idea that improves program execution performance by eliminating the recursive nature of execution. Dynamic programming makes use of recursion and memoization to come up with more improved performance of generating and locating a given set of Fibonacci value (Fender, 2014). Therefore, the most important aspect would be to implement memoization in a program that generates a given value in Fibonacci numbers to improve its performance index. Timeline and milestones Period Milestones 1st Week 2nd Week 3rd Week 4th Week Planning Resources acquisition Coding Testing and deployment References Cormen, T. H., Leiserson, C. E., Rivest, R. L., Stein, C. (2009). Introduction to algorithms. Cambridge: MIT Press. Dai, H. P., Chen, D. D., Zheng, Z. S. (2018). Effects of Random Values for Particle Swarm Optimization Algorithm. Algorithms, 11(2), 23. Fender, p. I. T. (2014). Efficient memoization algorithms for query optimization: top-down join enumeration through... Memoization on the basis of hypergraphs: anchor academic publishing. Jaffar, J., Santosa, A. E., Voicu, R. (2008). Efficient Memoization for Dynamic Programming with Ad-Hoc Constraints. In AAAI (Vol. 8, pp. 297-303). Moerkotte, G., Neumann, T. (2008). Dynamic programming strikes back. In Proceedings of the 2008 ACM SIGMOD international conference on Management of data. (pp. 539- 552). ACM. Stivala, A., Stuckey, P. J., de la Banda, M. G., Hermenegildo, M., Wirth, A. (2010). Lock-free parallel dynamic programming. Journal of Parallel and Distributed Computing, 70(8), 839-848.