master-algorithms-py/dynamic_programming

dynamic programming

• dynamic programming is the process of taking a recursive algorithm and cache overlapping problems (repeated calls).
• the runtime is given by the number of calls.
• top-down: how can we divide the problem into sub-problems?
  • top-down dynamic programming is called memoization.
• bottom-up: solve for a simple case, then figure out for more elements.

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recursion

• recursion is an approach to solving problems using a function that calls itself as a subroutine. every time the function calls itself, it reduces the problem into subproblems. the recursion calls continue until it reaches a point where the subproblem can be solved without further recursion.
• a recursive function should have the following properties so it does not result in an infinite loop:
  • one or more base cases (a terminating scenario that does not use recursion to produce an answer)
  • a set of rules, also known as recurrence relation, that reduces all other cases towards the base case.
• there can be multiple places where the function may call itself.
• any recursion problem can be solved iteratively and vice-versa.

vizualing the stack

• to visualize how the stack operates during recursion calls, check the example below where we reverse a string:

def reverse(s):
   if len(s) == 0:
      return s
   else:
      return reverse(s[1:]) + s[0]

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memoization

• memoization is an optimization technique that avoids recursion's duplicate calculations.
• it's primarily used to speed up code by storing the intermediate results in a cache so that it can be reused later.
• for example, a hash table can be used as a cache and should be passed along each subroutine call.