is a powerful technique for solving optimization problems that can be divided into smaller sub-problems. The main idea of dynamic programming is to avoid recomputing the same sub-problems over and over again, by storing and reusing their solutions. This way, dynamic programming can reduce the time and space complexity of recursive algorithms and often find the optimal solution to a problem. Dynamic programming In this article, we will explore some real-world examples and applications of dynamic programming, and see how it can be used to solve various kinds of problems in different domains. Knapsack Problem The knapsack problem is a classic example of dynamic programming. The problem is as follows: 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 does not exceed a given limit and the total value is as large as possible. This problem can be solved by using a two-dimensional array to store the optimal value for each sub-problem, defined by the number of items considered and the remaining capacity of the knapsack. At each step, we can either include or exclude the current item and choose the option that maximizes the value. The final answer is the value at the bottom-right corner of the array. The knapsack problem has many practical applications, such as packing a suitcase, selecting investments, allocating resources, etc. All Pair Shortest Path Another common problem that can be solved by dynamic programming is finding the shortest path between any two nodes in a weighted graph. This problem is known as the all pair shortest path problem, and it can be solved by using algorithms such as Floyd-Warshall or Bellman-Ford. These algorithms use a three-dimensional array to store the shortest distance between any pair of nodes for each intermediate node considered. At each step, they update the distance by comparing the current distance with the distance obtained by going through the intermediate node. The final answer is the distance matrix at the last iteration. The all pair shortest path problem has many applications in network analysis, routing, navigation, social network analysis, etc. Seam Carving Seam carving is an interesting application of dynamic programming in image processing. The problem requires you to resize an image without distorting its important features, such as faces, objects, or text. This can be done by removing or adding pixels along seams, which are paths of low energy in the image. To find the optimal seams to remove or add, we can use dynamic programming to compute the cumulative energy of each pixel in the image, based on its gradient and its neighbors. Then, we can find the seam with the lowest energy by tracing back from the bottom to the top of the image. We can repeat this process until we reach the desired size. Seam carving can be used to create content-aware image resizing, cropping, retargeting, etc. Machine Learning and Genomics Dynamic programming can also be used to solve problems in and , such as sequence alignment, hidden Markov models, and phylogenetic trees. machine learning genomics Sequence alignment is the process of finding similarities between two or more sequences of symbols, such as DNA or protein sequences. This can help us understand their evolutionary relationships, functional roles, or structural properties. Dynamic programming can be used to find optimal alignments between sequences by scoring their matches and mismatches according to some criteria. Hidden Markov models are probabilistic models that describe sequences of events or observations that depend on some hidden states. They can be used to model complex phenomena such as speech recognition, natural language processing, bioinformatics, etc. Dynamic programming can be used to find the most likely sequence of hidden states given a sequence of observations by using algorithms such as Viterbi or Forward-Backward. Phylogenetic trees are graphical representations of evolutionary relationships among organisms or genes. They can help us infer their common ancestors, divergence times, or evolutionary events. Dynamic programming can be used to construct optimal phylogenetic trees from sequence data by using algorithms such as Fitch or Sankoff. Cryptography Dynamic programming can also be used to solve problems in cryptography, which is the science of secure communication. Cryptography involves techniques such as encryption, decryption, digital signatures, authentication, etc. Encryption is the process of turning readable information into unreadable information that can only be accessed with an encryption key. Decryption is the reverse process of turning unreadable information back into readable information with the same or a different key. Digital signatures are methods of verifying the authenticity and integrity of a message or document. Authentication is the process of verifying the identity of a sender or receiver. Dynamic programming can be used to implement various cryptographic algorithms, such as dynamic key cryptography, code-based cryptography, and dynamic programming-based cryptography. Dynamic key cryptography is a technique that uses dynamic keys to encrypt and decrypt messages. Dynamic keys are keys that change over time or according to some criteria. This makes them more secure and resistant to attacks than static keys. Dynamic key cryptography can be implemented using dynamic programming to generate and update the keys. Code-based cryptography is a technique that uses error-correcting codes to encrypt and decrypt messages. Error-correcting codes are codes that can detect and correct errors in transmission. Code-based cryptography is considered to be quantum-resistant, which means it can withstand attacks from quantum computers. Code-based cryptography can be implemented using dynamic programming to encode and decode the messages. Dynamic programming-based cryptography is a technique that uses dynamic programming algorithms to encrypt and decrypt messages. Dynamic programming algorithms are algorithms that solve optimization problems by breaking them into smaller subproblems. Dynamic programming-based cryptography can use various dynamic programming algorithms, such as knapsack, shortest path, or seam carving, to perform encryption and decryption. Conclusion Dynamic programming is a useful technique for solving problems that have overlapping sub-problems and optimal substructure. By storing and reusing sub-problem solutions, dynamic programming can improve the efficiency and optimality of recursive algorithms. Dynamic programming has many real-world examples and applications in various domains, such as combinatorial optimization, graph theory, image processing, machine learning, genomics, and cryptography. Sources: The complete beginners guide to dynamic programming - Stack Overflow Blog Dynamic Programming: Examples, Common Problems, and Solutions - MUO Real-world dynamic programming: seam carving - Medium Dynamic Programming: Applications In Machine Learning and Genomics - edX Dynamic Programming for Machine Learning - Towards Data Science Dynamic key cryptography and applications - Monash University The lead image for this article was generated by HackerNoon's AI Image Generator via the prompt "A computer screen with a running program"

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