The.Algorithm.Design.Manual.Springer-Verlag.1998

Breaking Problems Down
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Breaking Problems Down
One of the most powerful techniques for solving problems is to break them down into smaller, more
easily solved pieces. Smaller problems are less overwhelming, and they permit us to focus on details that
are lost when we are studying the entire problem. For example, whenever we can break the problem into
smaller instances of the same type of problem, a recursive algorithm starts to become apparent.
Two important algorithm design paradigms are based on breaking problems down into smaller problems.
Dynamic programming typically removes one element from the problem, solves the smaller problem,
and then uses the solution to this smaller problem to add back the element in the proper way. Divide and
conquer typically splits the problem in half, solves each half, then stitches the halves back together to
form a full solution.
Both of these techniques are important to know about. Dynamic programming in particular is a
misunderstood and underappreciated technique. To demonstrate its utility in practice, we present no
fewer than three war stories where dynamic programming played the decisive role.
The take-home lessons for this chapter include:
● Many objects have an inherent left-to-right ordering among their elements, such as characters in a
string, elements of a permutation, points around a polygon, or leaves in a search tree. For any
optimization problem on such left-to-right objects, dynamic programming will likely lead to an
efficient algorithm to find the best solution.
● Without an inherent left-to-right ordering on the objects, dynamic programming is usually
doomed to require exponential space and time.
● Once you understand dynamic programming, it can be easier to work out such algorithms from
scratch than to try to look them up.
● The global optimum (found, for example, using dynamic programming) is often noticeably better
than the solution found by typical heuristics. How important this improvement is depends upon
your application, but it can never hurt.
● Binary search and its variants are the quintessential divide-and-conquer algorithms.
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