on philosophically shaky ground with randomized constructions. Thus, in many contexts, such constructions are accorded only the status of a pure existence proof and, almost always, they are seen as lacking at least some of the pristine virtue of deterministic constructions.
The classical way to atone for the easy virtue of probabilistic constructions has been to supply deterministic replacements, and a famous illustration of this tradition arises in the discussion of expander graphs in Chapter 11. Though it is shockingly easy to show the existence of all sorts of expander graphs via probability, many researchers seemed to breathe a sigh of relief when deterministic constructions for good expanders were developed, even when the new constructions called on methods as sophisticated as those used by Lubotzky et al. (1988).
Although most of the discoveries provoked by the urge to replace a randomized construction with a deterministic one have turned out to be rather specialized, there are recent developments that change this situation and offer enticing suggestions of an important new theory of derandomization. Instead of pursuing clever ad hoc constructions, the new idea is to look for algorithmic procedures that replace the very steps employed in the randomized construction. This shift in perspective turns out to be very fruitful, and a good sense of its power can found in Chapter 15 of the important new book The Probabilistic Method, by Alon et al. (1992). Further, because of the remarkable utility of the Lovász Local Lemma in the more subtle probabilistic constructions, the recent algorithmization of the Local Lemma by Beck (1992) and Alon (1992) offers a compelling validation of the derandomization concept.
In addition to its useful discussion of derandomization, the volume of Alon et al. (1992) also provides charming introductory treatments of at least two other topics that may seem underrepresented in this survey, graph algorithms and random graphs. The latter topic is also well covered in the treatise Random Graphs by Bollobás (1985), which is a bible for any serious student of random graphs. From the probabilists' point of view, two of the most important recent links to these areas are to percolation theory and to the Chen-Stein method of Poisson approximation. Much of the recent progress in percolation theory is beautifully introduced by Grimmett (1989), and the recent treatise on Poisson approximation of Barbour et al. (1992) surely belongs on the bookshelf of anyone interested in probability, graphs, or discrete algorithms.
Another important sphere of probability in the service of algorithms that some may see as underrepresented here is the analytical probabilistic