【免费】Algorithm:recursion资源-CSDN文库

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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

to another problem

means to write an algorithm for

that uses an algorithm for

as a black box or subroutine. Crucially, the correctness of the resulting algorithm cannot depend

in any way on how the algorithm for

works. The only thing we can assume is that the black

box solves

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!)

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.

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.

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:

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

for the hardwired constant

. 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

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

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”.

This English translation is from W. W. Rouse Ball and H. S. M. Coxeter’s book Mathematical Recreations and Essays.

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

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

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

− 1

. In particular,

moving a tower of 64 disks requires

−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.

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