This section introduces algorithms as abstractions of programs, and motivates why we need to do analysis of algorithms rather than just run empirical tests of programs. It then introduces some mathematical and conceptual tools for doing analysis. Two useful sorting algorithms are used for illustration; we’ll return to sorting later.
In this section we introduce problem solving and analysis methods (chapters 3-5), some of which have been covered in ICS 241, and also review review basic data structures and algorithms (chapters 10-12) that were introduced in ICS 211. The chapters from the text are interleaved and paired up in a manner that uses the basic dictionary and set data structures and algorithms to illustrate the problem solving and analysis methods. Hopefully this is mostly review of the two prerequisite courses with some added depth.
We continue into more advanced applications of trees (chapters 6 and 13), also providing the basis for another efficient sorting algorithm. We compare additional sorting algorithms to those from Chapter 2. Sorting is one of the most fundamental and common applications of computers, so efficiency is very important. We consider the broader question of how fast any sort algorithm can be. This is an example of a powerful method of computer science: reasoning about sets of possible algorithms rather than specific algorithms.
This part introduces two further problem solving methods, dynamic programming and greedy algorithms, with example applications. We then cover another important analytic method, _amortization _, with examples in the analysis of the union-find representation of sets. (Chronologically, #14 graphs will be introduced in this sequence to help you get started on a programming assignment, but conceptually they belong in the next section.)
Graphs are a very flexible data structure for which many applications exist. Equipped with various problem solving and analytic tools, we examine the most important algorithms on graphs – including some of the most classic work in computer science – and their applications.
There are a number of other important or common application areas that have their own specialized algorithms of interest: multithreading (parallel algorithms), matrix operations, linear programming, operations on polynomials, number-theory, string matching, and computational geometry. These are covered in the text, but we can’t fit them all in. Linear Programming will be examined because it fills out an area not covered well above (numeric algorithms), and also has interesting connections to previous material (e.g., you can solve flow problems with a linear program). Two other topics will be covered more lightly in class (no homework).
Finally, abstracting further from algorithms to problems, we deal with the important question of whether an efficient algorithm is known (or even possible) for a given problem, and what to do if none are known. We encounter the most important open problem in theoretical computer science (indeed in all of mathematics), which is of practical interest because awareness of “intractable” problems and approximation algorithms could save you considerable trouble if you encounter one of these very common problems on the job! The seminal book on this topic is the most cited reference in computer science.