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Dynamic Programming (DP)


Dynamic Programming:

  • 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

Technique (Skiena):

  1. formulate answer as recurrence relation or recursive function
  2. show number of different parameter values taken on by recurrence is bounded by a hopefully small polynomial
  3. 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

Common Applications

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 diff does.

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.