Algorithm Design

794
Chapter 13 Randomized Algorithms
technique in the area of combinatorics; the book by A_lon and Spencer (2000)
covers a wide range of its applications.
Hashing is a topic that remains the subject of extensive study, in both
theoretical and applied settings, and there are many variants of the basic
method. The approach we focus on in Section 13.6 is due to Carter and Wegman
(1979). The use of randomization for finding the closest pair of points in the
plane was originally proposed by Rabin (1976), in an influential early paper
that exposed the power of randomization in many algorithmic settings. The
algorithm we describe in this chapter was developed by Golin et al. (1995).
The technique used there to bound the number of dictionary operations, in
which one sums the expected work over all stages of the random order, is
sometimes referred to as backwards analysis; this was originally proposed
by Chew (1985) for a related geometric problem, and a number of further
applications of backwards analysis are described in the survey by Seidel (1993).
The performance guarantee for the LRU caching algorithm is due to Sleator
and Tarjan (1985), and the bound for the Randomized Marking algorithm is
due to Fiat, Karp, Luby, McGeoch, Sleator, and Young (1991). More generally,
the paper by Sleator and Tarian highlighted the notion of online algorithms,
which must process input without knowledge of the future; caching is one
of the fundamental applications that call for such algorithms. The book by
Borodin and E1-Yaniv (1998) is devoted to the topic of online algorithms and
includes many further results on caching in particular.
There are many ways to formulate bounds of the type in Section 13.9,
showing that a sum of 0-1-valued independent random variables is unlikely to
deviate far from its mean. Results of this flavor are generally called Ctzeraoff
bounds, or Chernoff-Hoeffding bounds, after the work of Chernoff (1952)
and Hoeffding (1963). The books by A!on and Spencer (1992), Motwani and
Raghavan (1995), and Mitzenmacher and Upfa! (2005) discuss these kinds of
bounds in more detail and provide further applications.
The results for packet routing in terms of congestion and dilation are
due to Leighton, Maggs, and Rao (1994). Routing is another area in which
randomization can be effective at reducing contention and hot spots; the book
by Leighton (1992) covers many further applications of this principle.
Notes on the Exercises Exercise 6 is based on a result of Benny Chor and
Madhu Sudan; Exercise 9 is a version of the Secretary Problem, whose popu-
larization is often credited to Martin Gardner.
Epilogue: Algorithms That Run
Forever
Every decade has its addictive puzzles; and if Rubik’s Cube stands out as the
preeminent solita~e recreation of the early 1980s, then Tetris evokes a similar
nostalgia for the late eighties and early nineties. Rubik’s Cube and Tetris have a
number of things in common--they share a highly mathematical flavor, based
on stylized geometric forms--but the differences between them are perhaps
more interesting.
Rubik’s Cube is a game whose complexity is based on an enormous search
space; given a scrambled configuration of the Cube, you have to apply an
intricate sequence of operations to reach the ultimate goal. By contrast, Tetris-in
its pure form--has a much fuzzier definition of success; rather than aiming
for a particular endpoint, you’re faced with a basically infinite stream of events
to be dealt with, and you have to react continuously so as to keep your head
above water.
These novel features of Tetris parallel an analogous set of themes that has
emerged in recent thinking about algorithms. Increasingly, we face settings in
which the standard view of algorithms--in which one begins with an input,
runs for a finite number of steps, and produces an output--does not ready
apply. Rather, if we think about Internet touters that move packets while
avoiding congestion, or decentralized file-sharing mechanisms that replicate
and distribute content to meet user demand, or machine learning routines
that form predictive models of concepts that change over time, then we are
dealing with algorithms that effectively are designed to ran forever. Instead
of producing an eventual output, they succeed if they can keep up with an
environment that is in constant flux and continuously throws new tasks at
them. For such applications, we have shifted from the world of Rubik’s Cube
to the world of Tetris.