Concise representation of subsets of small integers {0, 1, . . .} – Does this make sense now? Remember the three steps!

In this lecture, we discuss this technique, and present a few key examples. Topics in this lecture include: The basic idea of Dynamic Programming. Example: Longest Common Subsequence. Example: Knapsack. Example: Matrix-chain multiplication.

This chapter discusses dynamic programming, a method to solve optimization problems that in-volve a dynamical process. This is in contrast to our previous discussions on LP, QP, IP, and NLP, where the optimal design is established in a static situation.

Dynamic Programming 101 • DP = recursion (divide-n-conquer) + caching (overlapping subproblems) • the simplest example is Fibonacci 1 naive recursion without memoization: O(1.618...n) def fib(n): if n <= 2: return 1 return fib(n-1) + fib(n-2) f (n)=f (n 1)+ f (n 2) f (1) = f (2) = 1

Dynamic Programming is a powerful technique that often allows you to solve problems that seem like they should take exponential time in polynomial time. Sometimes it allows you to solve exponential time problems in slightly better exponential time. It is most often used in

Dynamic programming (DP) involves solving problems incrementally, starting with instances of size one and working up to instances of generic size n. It is similar to the method of induction in proofs. A key step in DP is to identify a recursive (or inductive) structure that helps reduce one instance of size ninto several instances of size at ...

Dynamic programming is a very powerful algorithmic paradigm in which a problem is solved by identifying a collection of subproblems and tackling them one by one, smallest rst, using the answers to small problems to help gure out larger ones, until the whole lot of them is solved. In dynamic programming we are not given a dag; the dag is ...

The longest increasing subsequence (LIS) problem is a classic dynamic programing problem specified as follows. You are given an array n of values and want to find the longest subsequence of that array where the values are in strictly increasing order. For example, given an array holding 3, 1, 4, 1, 5, 9, 2, 6, 5, 3 , one LIS is 3, 4, 5, 9 .

Dynamic Programming Many programs in computer science are written to optimize some value: Find the shortest path between two points, Find the line that best fits a set of points Find the smallest set of objects that satisfies some criteria

Dynamic Programming Overview Dynamic programming. Similar to divide-and-conquer. – solves problem by combining solution to sub-problems Different from divide-and-conquer. – sub-problems are not independent – save solutions to repeated sub-problems in table Recipe. Characterize structure of problem. – optimal substructure property