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--- title: "dynamic programming" type: concept related: [Greedy Programming, Relaxation Algorithums, Dynamic Programming] source: https://www.jemoka.com/posts/kbhdynamic_programming/ confidence: high status: active --- dynamic programming is a three-step algorithm to tackle large, multi-step problems; high level idea: guessing + caching + recursion. dynamic programming can sometimes not be good enough, and it doesn’t really give us fast enough to get what we need to use. That’s when we need to deal with relaxation, or possibly greedy programming. constituents optimal sub-structure big problems break up into sub-problems optimal solutions can be explained in terms of sub-problems solution overlapping sub-problems many different entries of each solution use the same sub-problem answer …meaning, requirements bottom-up approach initialize a table fill it top-down approach do the recursive approach memoize problem solutions during recursive call additional information main steps of dynamic programming Break a hard problem into sub-problems Guess what sub-problem to solve Solve the sub-problem and store the solution Repeat #2 and #3 Combine sub-problem solutions to solve the hard problem analyzing runtime of dynamic programming To analyze runtime of dynamic programming problems, you ask: How many sub-problems are there? How long does it take to solve each sub-problem? How long does it take to combine sub-problems? proofing correctness Typically the proof strategy is to use induction on each subproblem, showing that the subproblems are correct. examples fibonacchi numbers: dynamic programming here’s an example top-down dynamic programming problem: There are \(n\) sub-problems: \(F_1, F_2, \ldots, F_{n-1}\). Solve a sub-problem, then store the solution \(F_{n-1} = F_{n-2}+F_{n-3}\) Continue until \(F_1 =1\). Now, we can recurs back up (popping the call stack) and cache all calculated results So then we can just look up any \(F_k\). shortest path: dynamic programming here’s a graph! how do we get to node \(6\)? Guess that the shortest path goes through 10 Go recursively until you get to root, cache the solution Do it again until you got to all subproblems Look up cached result