Dynamic Programming (DP)
- is a way to design algorithms that search all possibilities
- stores results to avoid recomputing
- usually starts with correct recursive algorithm first
- trades space for time
- formulate answer as recurrence relation or recursive function
- show number of different parameter values taken on by recurrence is bounded by a hopefully small polynomial
- specify an order of evaluation for the recurrence so that the partial results you need are always available
Three traditional examples:
- fibonacci numbers
- binomial coefficients
- coin change problem
- optimal substructure
- overlapping subproblems
You always have to solve the subproblems first. There are two different approaches to do this:
- top down (recursive + memoization)
- bottom up
Longest Common Substring: Given a set of strings, find the longest substring common to all strings. This could also be solved with a suffix tree.
Longest Common Subsequence (LCS): Given a set of sequences, find the longest subsequence common to all sequences. A subsequence of a string is a set of characters that appear in left-to-right order, but may not be consecutive. This is precisely what
Knapsack problem: 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 is less than or equal to a given limit and the total value is as large as possible.
Subset Sum problem: Given set of integers is there is non-zero subset whose sum is zero? Special-case of knapsack.
Partition problem: Given a multiset of positive integers, can it be partitioned into two subsets such that the sum of the numbers in each subset are equal. Special-case of subset sum.
Cocke–Younger–Kasami (CYK): Parses context-free grammars.
- Longest Increasing Sequence
- Levenshtein (edit) distance
- Floyd's all-pairs shortest path algorithm
- Bellman–Ford -- finding the shortest distance in a graph
Dynamic Programming. Wikipedia.
Longest Common Subsequences. David Eppstein. 1996-02-29.