Question

Whenever I look at solutions to computer contests, I always see the term “dynamic programming”. I Googled the term and read a few articles, but none of them provide a simple example of programming VS “dynamic” programming. So how is “dynamic” programming different than “normal” programming? (simple terms please!)

Answer 1

Dynamic Programming uses programming more in the sense used with Linear Programming — a mechanism of solving a problem.

One description I recently read (but can no longer recall the source — [citation needed]) suggested that the usual approach of divide and conquer used in recursion is a top-down approach to solving problems, while dynamic programming is a bottom-up approach to solving problems.

The Wikipedia article suggests computing the Fibonocci sequence is an excellent use of dynamic programming — you memoize results as you compute them for further use in the algorithm, to avoid re-computing similar results.

Knuth’s algorithm for line-breaking paragraphs is another good example of dynamic programming: if you consider the possibility of inserting line breaks between every word (and even breaking lines inside words, at hyphenation points), it feels like the only algorithms will be exponential — or worse. However, by keeping track of the “badness” associated with previous line breaks, Knuth’s algorithm actually runs in linear time with the size of the input. (I must admit that I don’t fully understand Knuth’s algorithm — only that it is supremely clever.)