What is the difference between memoization and dynamic programming?

Question

What is the difference between memoization and dynamic programming? I think dynamic programming is a subset of memoization. Is it right?

Solution 1

Relevant article on Programming.Guide: Dynamic programming vs memoization vs tabulation

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What is difference between memoization and dynamic programming?

Memoization is a term describing an optimization technique where you cache previously computed results, and return the cached result when the same computation is needed again.

Dynamic programming is a technique for solving problems of recursive nature, iteratively and is applicable when the computations of the subproblems overlap.

Dynamic programming is typically implemented using tabulation, but can also be implemented using memoization. So as you can see, neither one is a "subset" of the other.

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A reasonable follow-up question is: What is the difference between tabulation (the typical dynamic programming technique) and memoization?

When you solve a dynamic programming problem using tabulation you solve the problem "bottom up", i.e., by solving all related sub-problems first, typically by filling up an n-dimensional table. Based on the results in the table, the solution to the "top" / original problem is then computed.

If you use memoization to solve the problem you do it by maintaining a map of already solved sub problems. You do it "top down" in the sense that you solve the "top" problem first (which typically recurses down to solve the sub-problems).

A good slide from here (link is now dead, slide is still good though):

• If all subproblems must be solved at least once, a bottom-up dynamic-programming algorithm usually outperforms a top-down memoized algorithm by a constant factor
  • No overhead for recursion and less overhead for maintaining table
  • There are some problems for which the regular pattern of table accesses in the dynamic-programming algorithm can be exploited to reduce the time or space requirements even further
• If some subproblems in the subproblem space need not be solved at all, the memoized solution has the advantage of solving only those subproblems that are definitely required

Additional resources:

• Wikipedia: Memoization, Dynamic Programming
• Related SO Q/A: Memoization or Tabulation approach for Dynamic programming

Solution 2

Dynamic Programming is an algorithmic paradigm that solves a given complex problem by breaking it into subproblems and stores the results of subproblems to avoid computing the same results again.

http://www.geeksforgeeks.org/dynamic-programming-set-1/

Memoization is an easy method to track previously solved solutions (often implemented as a hash key value pair, as opposed to tabulation which is often based on arrays) so that they aren't recalculated when they are encountered again. It can be used in both bottom up or top down methods.

See this discussion on memoization vs tabulation.

So Dynamic programming is a method to solve certain classes of problems by solving recurrence relations/recursion and storing previously found solutions via either tabulation or memoization. Memoization is a method to keep track of solutions to previously solved problems and can be used with any function that has unique deterministic solutions for a given set of inputs.

Solution 3

Both Memoization and Dynamic Programming solves individual subproblem only once.

Memoization uses recursion and works top-down, whereas Dynamic programming moves in opposite direction solving the problem bottom-up.

Below is an interesting analogy -

Top-down - First you say I will take over the world. How will you do that? You say I will take over Asia first. How will you do that? I will take over India first. I will become the Chief Minister of Delhi, etc. etc.

Bottom-up - You say I will become the CM of Delhi. Then will take over India, then all other countries in Asia and finally I will take over the world.

Solution 4

Dynamic Programming is often called Memoization!

1. Memoization is the top-down technique(start solving the given problem by breaking it down) and dynamic programming is a bottom-up technique(start solving from the trivial sub-problem, up towards the given problem)

2. DP finds the solution by starting from the base case(s) and works its way upwards. DP solves all the sub-problems, because it does it bottom-up

   Unlike Memoization, which solves only the needed sub-problems

3. DP has the potential to transform exponential-time brute-force solutions into polynomial-time algorithms.

4. DP may be much more efficient because its iterative

   On the contrary, Memoization must pay for the (often significant) overhead due to recursion.

To be more simple, Memoization uses the top-down approach to solve the problem i.e. it begin with core(main) problem then breaks it into sub-problems and solve these sub-problems similarly. In this approach same sub-problem can occur multiple times and consume more CPU cycle, hence increase the time complexity. Whereas in Dynamic programming same sub-problem will not be solved multiple times but the prior result will be used to optimize the solution.

Solution 5

(1) Memoization and DP, conceptually, is really the same thing. Because: consider the definition of DP: "overlapping subproblems" "and optimal substructure". Memoization fully possesses these 2.

(2) Memoization is DP with the risk of stack overflow is the recursion is deep. DP bottom up does not have this risk.

(3) Memoization needs a hash table. So additional space, and some lookup time.

So to answer the question:

-Conceptually, (1) means they are the same thing.

-Taking (2) into account, if you really want, memoization is a subset of DP, in a sense that a problem solvable by memoization will be solvable by DP, but a problem solvable by DP might not be solvable by memoization (because it might stack overflow).

-Taking (3) into account, they have minor differences in performance.

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