\n Dynamic programming is a technique for solving problems with\n overlapping sub problems. Typically, these sub problems arise\n from a recurrence relating a given problem’s solution to\n solutions of its smaller sub problems.\n
\n\n Rather than solving overlapping sub problems again and again,\n dynamic programming suggests solving each of the smaller\n sub problems only once and recording the results in a table\n from which a solution to the original problem can then be\n obtained.\n
\n\n The difference between Dynamic Programming and straightforward\n recursion is in memoization of recursive calls. If the sub\n problems are independent and there is no repetition then\n memoization does not help.\n
\n\n In this method, the recursive call is made only if n is\n neither 1 not 0. So there occurs a situation where recursive\n calls has to be made for same value of n but at different\n instance.\n
\n Suppose we calculate Fib(4): (i) The\n recursive calls made in root function are - Fib(3) and\n Fib(2). (ii) The recursive calls made in\n Fib(3) are - Fib(2) and Fib(1). (iii) The\n recursive calls made in Fib(2) are - Fib(1) and Fib(0).\n
\n Notice that the the recursive call for n=2 is made twice\n here and this situation occurs for other values of n also.\n The simple solution is to memoize the newly calculated value\n for n as shown in next step.\n
\n Here, we see that if memo[n] is not null then it returns\n the value stored in it which prevents from making a new\n recursive call.\n
\n It is top-down dp because the problem is broken into sub\n problems, each of these sub problems is solved, and the\n solutions are remembered in case they need to be solved\n again.\n
\n Here, we see that if memo[n] is calculated just by\n adding the previous two integers stored in that array.\n
\n It is bottom-up dp because these methods start with\n lower values of input and keep building the solutions\n for higher values.\n
\n Dynamic programming is a technique for solving problems with\n overlapping sub problems. Typically, these sub problems arise\n from a recurrence relating a given problem’s solution to\n solutions of its smaller sub problems.\n
\n\n Rather than solving overlapping sub problems again and again,\n dynamic programming suggests solving each of the smaller\n sub problems only once and recording the results in a table\n from which a solution to the original problem can then be\n obtained.\n
\n\n The difference between Dynamic Programming and straightforward\n recursion is in memoization of recursive calls. If the sub\n problems are independent and there is no repetition then\n memoization does not help.\n
\n\n In this method, the recursive call is made only if n is\n neither 1 not 0. So there occurs a situation where recursive\n calls has to be made for same value of n but at different\n instance.\n
\n Suppose we calculate Fib(4): (i) The\n recursive calls made in root function are - Fib(3) and\n Fib(2). (ii) The recursive calls made in\n Fib(3) are - Fib(2) and Fib(1). (iii) The\n recursive calls made in Fib(2) are - Fib(1) and Fib(0).\n
\n Notice that the the recursive call for n=2 is made twice\n here and this situation occurs for other values of n also.\n The simple solution is to memoize the newly calculated value\n for n as shown in next step.\n
\n Here, we see that if memo[n] is not null then it returns\n the value stored in it which prevents from making a new\n recursive call.\n
\n It is top-down dp because the problem is broken into sub\n problems, each of these sub problems is solved, and the\n solutions are remembered in case they need to be solved\n again.\n
\n Here, we see that if memo[n] is calculated just by\n adding the previous two integers stored in that array.\n
\n It is bottom-up dp because these methods start with\n lower values of input and keep building the solutions\n for higher values.\n