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Algorithm: recursion
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2015-03-10
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伊利诺伊大学厄本那香槟分校(University of Illinois at Urbana-Champaign)算法课(CS374)讲义,主要讲述递归(recursion),由Jeff Erickson撰写
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Algorithms Lecture 1: Recursion [Fa’14]
The control of a large force is the same principle as the control of a few men:
it is merely a question of dividing up their numbers.
— Sun Zi, The Art of War (c. 400 C.E.), translated by Lionel Giles (1910)
Our life is frittered away by detail. . . . Simplify, simplify.
— Henry David Thoreau, Walden (1854)
Nothing is particularly hard if you divide it into small jobs.
— Henry Ford
Do the hard jobs first. The easy jobs will take care of themselves.
— Dale Carnegie
1 Recursion
1.1 Reductions
Reduction is the single most common technique used in designing algorithms. Reducing one
problem
X
to another problem
Y
means to write an algorithm for
X
that uses an algorithm for
Y
as a black box or subroutine. Crucially, the correctness of the resulting algorithm cannot depend
in any way on how the algorithm for
Y
works. The only thing we can assume is that the black
box solves
Y
correctly. The inner workings of the black box are simply none of our business;
they’re somebody else’s problem. It’s often best to literally think of the black box as functioning
by magic.
For example, the Huntington-Hill algorithm described in Lecture 0 reduces the problem
of apportioning Congress to the problem of maintaining a priority queue that supports the
operations Insert and ExtractMax. The abstract data type “priority queue” is a black box; the
correctness of the apportionment algorithm does not depend on any specific priority queue data
structure. Of course, the running time of the apportionment algorithm depends on the running
time of the Insert and ExtractMax algorithms, but that’s a separate issue from the correctness of
the algorithm. The beauty of the reduction is that we can create a more efficient apportionment
algorithm by simply swapping in a new priority queue data structure. Moreover, the designer of
that data structure does not need to know or care that it will be used to apportion Congress.
Similarly, if we want to design an algorithm to compute the smallest deterministic finite-state
machine equivalent to a given regular expression, we don’t have to start from scratch. Instead
we can reduce the problem to three subproblems for which algorithms can be found in earlier
lecture notes: (1) build an NFA from the regular expression, using either Thompson’s algorithm
or Glushkov’s algorithm; (2) transform the NFA into an equivalent DFA, using the (incremental)
subset construction; and (3) transform the DFA into the smallest equivalent DFA, using Moore’s
algorithm, for example. Even if your class skipped over the automata notes, merely knowing that
those component algorithms exist (Trust me!) allows you to combine them into more complex
algorithms; you don’t need to know the details. (But you should, because they’re totally cool.
Trust me!) Again swapping in a more efficient algorithm for any of those three subproblems
automatically yields a more efficient algorithm for the problem as a whole.
When we design algorithms, we may not know exactly how the basic building blocks we use
are implemented, or how our algorithms might be used as building blocks to solve even bigger
problems. Even when you do know precisely how your components work, it is often extremely
useful to pretend that you don’t. (Trust yourself!)
© Copyright 2014 Jeff Erickson.
This work is licensed under a Creative Commons License (http://creativecommons.org/licenses/by-nc-sa/4.0/).
Free distribution is strongly encouraged; commercial distribution is expressly forbidden.
See http://www.cs.uiuc.edu/~jeffe/teaching/algorithms/ for the most recent revision.
1
Algorithms Lecture 1: Recursion [Fa’14]
1.2 Simplify and Delegate
Recursion is a particularly powerful kind of reduction, which can be described loosely as follows:
• If the given instance of the problem is small or simple enough, just solve it.
• Otherwise, reduce the problem to one or more simpler instances of the same problem.
If the self-reference is confusing, it’s helpful to imagine that someone else is going to solve
the simpler problems, just as you would assume for other types of reductions. I like to call
that someone else the Recursion Fairy. Your only task is to simplify the original problem, or to
solve it directly when simplification is either unnecessary or impossible; the Recursion Fairy will
magically take care of all the simpler subproblems for you, using Methods That Are None Of Your
Business So Butt Out.
¹
Mathematically sophisticated readers might recognize the Recursion Fairy
by its more formal name, the Induction Hypothesis.
There is one mild technical condition that must be satisfied in order for any recursive method
to work correctly: There must be no infinite sequence of reductions to ‘simpler’ and ‘simpler’
subproblems. Eventually, the recursive reductions must stop with an elementary base case that
can be solved by some other method; otherwise, the recursive algorithm will loop forever. This
finiteness condition is almost always satisfied trivially, but we should always be wary of “obvious”
recursive algorithms that actually recurse forever. (All too often, “obvious” is a synonym for
“false”.)
1.3 Tower of Hanoi
The Tower of Hanoi puzzle was first published by the mathematician François Éduoard Anatole
Lucas in 1883, under the pseudonym “N. Claus (de Siam)” (an anagram of “Lucas d’Amiens”).
The following year, Henri de Parville described the puzzle with the following remarkable story:
²
In the great temple at Benares beneath the dome which marks the centre of the world, rests a brass
plate in which are fixed three diamond needles, each a cubit high and as thick as the body of a bee.
On one of these needles, at the creation, God placed sixty-four discs of pure gold, the largest disc
resting on the brass plate, and the others getting smaller and smaller up to the top one. This is the
Tower of Bramah. Day and night unceasingly the priests transfer the discs from one diamond needle
to another according to the fixed and immutable laws of Bramah, which require that the priest on
duty must not move more than one disc at a time and that he must place this disc on a needle so that
there is no smaller disc below it. When the sixty-four discs shall have been thus transferred from the
needle on which at the creation God placed them to one of the other needles, tower, temple, and
Brahmins alike will crumble into dust, and with a thunderclap the world will vanish.
Of course, as good computer scientists, our first instinct on reading this story is to substitute the
variable
n
for the hardwired constant
64
. And following standard practice (since most physical
instances of the puzzle are made of wood instead of di amonds and gold), we will refer to the
three possible locations for the disks as “pegs” instead of “needles”. How can we move a tower
of
n
disks from one peg to another, using a third peg as an occasional placeholder, without ever
placing a disk on top of a smaller disk?
The trick to solving this puzzle is to think recursively. Instead of trying to solve the entire
puzzle all at once, let’s concentrate on moving just the largest disk. We can’t move it at the
¹
When I was a student, I used to attribute recursion to “elves” instead of the Recursion Fairy, referring to the
Brothers Grimm story about an old shoemaker who leaves his work unfinished when he goes to bed, only to discover
upon waking that elves (“Wichtelmänner”) have finished everything overnight. Someone more entheogenically
experienced than I might recognize them as Terence McKenna’s “self-transforming machine elves”.
²
This English translation is from W. W. Rouse Ball and H. S. M. Coxeter’s book Mathematical Recreations and Essays.
2
Algorithms Lecture 1: Recursion [Fa’14]
The Tower of Hanoi puzzle
beginning, because all the other disks are covering it; we have to move those
n −1
disks to the
third peg before we can move the
n
th disk. And then after we move the
n
th disk, we have to
move those n −1 disks back on top of it. So now all we have to figure out is how to. . .
STOP!!
That’s it! We’re done! We’ve successfully reduced the
n
-disk Tower of Hanoi
problem to two instances of the
( n −1)
-disk Tower of Hanoi problem, which we can gleefully
hand off to the Recursion Fairy (or, to carry the original story further, to the junior monks at the
temple).
recursion
recursion
The Tower of Hanoi algorithm; ignore everything but the bottom disk
Our recursive reduction does make one subtle but important assumption: There is a largest
disk. In other words, our recursive algorithm works for any
n ≥ 1
, but it breaks down when
n = 0
. We must handle that base case directly. Fortunately, the monks at Benares, being good
Buddhists, are quite adept at moving zero disks from one peg to another in no time at all.
The base case for the Tower of Hanoi algorithm. There is no spoon.
While it’s tempting to think about how all those smaller disks get moved—or more generally,
what happens when the recursion is unrolled—it’s not necessary. For even slightly more
complicated algorithms, unrolling the recursion is far more confusing than illuminating. Our
only
task is to reduce the problem to one or more simpler instances, or to solve the problem
directly if such a reduction is impossible. Our algorithm is trivially correct when
n = 0
. For any
n ≥ 1
, the Recursion Fairy correctly moves (or more formally, the inductive hypothesis implies
3
Algorithms Lecture 1: Recursion [Fa’14]
that our recursive algorithm correctly moves) the top
n −1
disks, so (by induction) our algorithm
must be correct.
Here’s the recursive Hanoi algorithm in more typical pseudocode. This algorithm moves a
stack of
n
disks from a source peg (src) to a destination peg (dst) using a third temporary peg
(tmp) as a placeholder.
Hanoi(n, src, dst, tmp):
if n > 0
Hanoi(n −1, src, tmp, dst)
move disk n from src to dst
Hanoi(n −1, tmp, dst, src)
Let
T (n)
denote the number of moves required to transfer
n
disks—the running time of
our algorithm. Our vacuous base case implies that
T (0) = 0
, and the more general recursive
algorithm implies that
T (n) = 2T (n −1) + 1
for any
n ≥ 1
. The annihilator method (or guessing
and checking by induction) quickly gives us the closed form solution
T (n) = 2
n
− 1
. In particular,
moving a tower of 64 disks requires
2
64
−1 =
18,446,744,073,709,551,615 individual moves. Thus,
even at the impressive rate of one move per second, the monks at Benares will be at work for
approximately 585 billion years before tower, temple, and Brahmins alike will crumble into dust,
and with a thunderclap the world will vanish.
1.4 Mergesort
Mergesort is one of the earliest algorithms proposed for sorting. According to Donald Knuth, it
was proposed by John von Neumann as early as 1945.
1. Divide the input array into two subarrays of roughly equal size.
2. Recursively mergesort each of the subarrays.
3. Merge the newly-sorted subarrays into a single sorted array.
Input: S O R T I N G E X A M P L
Divide: S O R T I N G E X A M P L
Recurse: I N O S R T A E G L M P X
Merge: A E G I L M N O P R S T X
A mergesort example.
The first step is completely trivial—we only need to compute the median array index—and
we can delegate the second step to the Recursion Fairy. All the real work is done in the final step;
the two sorted subarrays can be merged using a simple linear-time algorithm. Here’s a complete
description of the algorithm; to keep the recursive structure clear, we separate out the merge
step as an independent subroutine.
4
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