CHARACTERIZATION

AND

ELIMINATION

Norman Harvard

Many Abstract. by interpretations procedure Q

well-known of the

f(x)

functions recursion

are schema

OF REDUNDANCY

H.

computed

;

Cohen

However,

many

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because

they

is,

cause

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times

then

return

a(x)

else .—

return

b(x,f(cl(x)),

these

with

))

by

f

with

identical

of

known

In

each

to

computes certain

the

be

to

equal

case

there

f(x)

without

for is

are

redundant

be

inefficient

computation.

called

That

recursively

several

arguments.

set

of

by f(x)

~rocedure

We

a,

The

which that and

resulting

Figure and

the

results

in

specified

global

transformed

ability program

programs,

of these improvement

locations;

for

criterion redundancy.

the

to present the

establishing

an

the

1,

y+z,

J4 =

Cl(x)

we

obtain

given a Xth element

non-

of

the

;

or

x.1

Q&;return

1

else

f(x-1)

return

computation

f(2)

is

+

f(x-2)

f(3) three

calls

on

f(5) is

is

times. f

pictured

computed

two

In

with

all,

only

in times,

there distinct

six

values. interpretations

certain

procedure is

for

that

computed

fifteen

For

in

without requirements

discuss

transformations system, and

X, returns

Note

1-1.

are

are all valueless intermediate and

each case recursion can be eliminated of a stack. We compare the storage

schema =

which

transformations

programs which leave

the a(x)

= which,

x.O

❑erging

computation. recursive

These procedures

interpret

and b(x,y,z) function,

f’(x)

U

assumptions.

p,

we

(x=O~x=l),

The

program provided

redundancy, assumptions

if ::

integer series:

argument

eKist f,

yield

partially

which,

when

a non-redundant

equivalent

to

of applied program

f.

The

9n , to

the

which

applicability

use

applicautomatic a general

existence

of

INTRODUCTION

Functions

often are

which

definitions recursion

Pn:

to

c2(x) = x-2, following

Ci are satisfied. The transformed programs avoid redundancy by saving exactly those intermediate results which will be needed again later in the

1.

f

p(x)

negative Fibonacci

calls

functions.

formed tree for

each

a

multiple values.

these

acyclic graphs the computation

the

definitions

to

instance:,

letting

redundancy depend on Ci. We explore four

assumptions

are

argument

and nature of of the functions

analyze directed the nodes of

redundant

define

they

because

existence properties

of

these

identical

For . . ..f(cn(x)

interpretations

computations

final

PROGRAMS*

p(x)

Some of

sets

RECURSIVE

University

x-1, the

on

IN

schema

P rocedure U p(x) MQQ

else

gn, f(x)

contract

under

by

;

return return

fellowship Navy

defined

and Naval

Figure

a(x) b(x,f(cl

suPPorted

*-Research

recursive of the

elegant have interpretations

(x)),.

by by

U.S.

Electronic

an

..,f(cn(x)

IBM Department Systems

)) research of

1-1.

A

the

interpretation call on

f,

value

that

for

recursive

Command

represented

NOO039-78-G-0020.

143

tree

of

computation

.

of’

and call.

cal 1s by

that

representation

f(5) ~2.

is

the

of

the Fibonacci under represents Each node labelled

The arising node.

sons

with of

directly

the

a node from

argument represent the

call

a

of

these

of

two

transformations

the

functions

frontier predicate

ci~

condition p and

the

such

following

to

each

relating function

the

interpretation,

the on condition

depends descent

a -I
conditions

a

of

and

a

to

“ t~~

transformation

non-redundant

again.

relating

other

the For

a.

~alidity

the

Fibonacci would

yield

program:

In

many,

storage

can

inputs.

(In

9 whose amounts

computations

f(x)

=

; ; ~BACKO

[email protected]

;

f2(x)

;

all

:=

1

:=

1

;

[email protected] f2(x-1)

been

storage have

:=

BACKI

BQCKI

:=

BACKO

to

f2.

for

the

in

BACKO .

BACKI

BACKO,

;

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:=

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+ BACK2

General

The

recursive

and

(It the

We

also value

exDlain schema

~.

the

four

are

f2(x)

is

of

f(x)

descent

Sections

condition 2

through

in

terms

with of

effect

in

each

and

than

program

5

we

of

Q

p(x) return

a(x)

else

return

b(x,c(x),d(x

simplicity

we

We explore

the

condition program

places

constants Section

c

We

frontier

for

mathematical the

improvement

After

we

consider

descent

program schema

the

of each

c, and dkscent

present

and validity

all

is

exploits

these

redundancy. of

In these

and

exploring programs

redundant

mathematical

which

result

of

of

As

in

Dvnamic

dvnamic

DroKrammin

use

value

many times. the method of

tabulation

Bird

storage

knowledge for

the which

interpretation will

to not

be

any

recursively

appropriate

In

the

global

case

where

p(x)

locations needed

as

We assume

can

be

frontier

this

valid

the and

still-active

in

computed condition

assumption;

for

most

we

naturally-

have in

of

a

in

stack. of

the

recursion case

needed

144

the

side

which,

the

f(x)

done

i

1

:=

1

~

would

; ; 1 until

x-1

IQ

bs.am BACK2

:=

BACK1

:

BACKI BACKO

:= :=

BACKO BACKI

~ + BACK2

BACKO

in without

example,

recursion

;

:=

of

expense Methods

discussed be

Fibonacci the

not

location.

at

can

do

effect

iterative. are

this

2

which

program:

Q2&; return &

Sections

designated

each In

this

the a

form be

eliminating

we reuse

a

easily

in

procedures

but

can

6;

programs

~rocedure

may be needed in some ways

use

by

itself

of

is

as

on, in values

This

needed

The

the

well

condition,

f(x).

results”

results are in

@

However,

later these

‘Tprevious

Qe&L?l BACKO

[1977b].

needed Using

recursion.

,

eliminating

essentially

simpler

by saving results which The approach resembles by

has as

called

recursive

their programs

clarity,

for

the more

Hopcroft, and is avoided

presented

This

initialize the global and any other values by the caller of f2.

it

are

Programming

redundant

[1974]),

be

When checks itself

f(x’),

of

placed

transformed

a

result

K (Aho, computation

Unman

it

descent

returns.

BACKI to

the

global

first calls

x? .

be

statement

5

Section of

of in

uses results.

cases.

return

transformed

full generality applicability

these

The

programs. a

are

that

through

the and in

implications

present

improvements efficient programs.

RelationshiD

to

values,

an

leaving These

of

the the

intend

argument

of

value

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f2

formal

arising

a

of

x)

further

a

value

(which

we simply value a(x) resultsll

conjecture

n)2. both in redundant

of

condition

reflecting ~ we discuss

specialized and more

place

which

f without generalization

~n

interested nature

automatic

in

condition,

to are

d

domain the

to compute 6 we sketch the

transformations

each

and

constraints on the given

which,

appropriate

use

argument

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f2,

value

of the

f2

and

without (For

make

storage

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program If not,

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argument

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).

new

values

is true, with the Ifprevious

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computes

invocations

;

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some

values global

taking

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these

leaving

any other specified

with

f(x)

approaches

Since

program,

store

with argument is p(x)

recursively

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procedure

a

calls previous

when

for many clever.

transformed

locations, ~:

locations to be more

a program

whether

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when stored

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written

to

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executed

value

value of f(x-1) in BACK2 for x>I.)

transformations

.\$?2> defined

global

transformations,

In

one.

BACK2 the

a weaker

under these

placing

Dlaces o; f(x-2)

present

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produce

determine

checking

f

Strategv The

BACKI,

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variables variables

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[19701, among

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times.

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reuse we will

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calls

location value

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Hewitt 5 that

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us

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programming

recursive

BACKI else

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for

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are

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then

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sufficient

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are procedure

all, to

Strong

interpretation

f2(x)

not

shown

general,

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but be

the be

Descent

Conditions We

partition @

subsets, cases, is

values

%

consists

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the

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domain

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we consider c(x)=d(x)

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require

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m and all and

all

n

existence

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x in g(gi-l(x))

that

for

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and

d

strong in

x

notation

otherwise.)

condition

asserts on f%? for some on the existence for

that

c

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%’.

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requirement

5

that and

d commute

The

Condition

Frontier The

and

%,

and

the toward

S be

a

means

set

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functions

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functions.)

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frontier

requirement

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for

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2)

if

be all

(and

x

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f(h(x))

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The resulting descent [email protected]

the

for

Fibonacci

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the

for

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merging

graph will Figure

dag

c

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a

nodes argument

of identical merging those be called a shows a 1-3

descent

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of

1-1.

@GJ%Q ,...,, .... >

say as

(b(x, b(c(x), o(c(c(x )):', a(d(c(x)))), b(d(x), / (b)

‘=.

b(c(x), o(c(c(x))),

all

& c

in

dc

d

cd

d

the

computation

,,-

Figure of

descent is the h

;=

cc

these of S

o(c(d(x ))), o(d(d(x))))~

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a(d~a

(c)

to @

values Formally,

then

Figure tree

with

subtrees.) compressed

more

defined,

defined

is

in

descent

(dag)

to

map

and

f

the

subtrees,

weak (We

(Typically,

in

do

descent

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for

a descent

assumptions

two such nodes and we merge the

‘a)

and

cases.

compositions condition

(Any

Figure

descent

functions

depicted

is

where

graph

descent

surprisingly,

‘Ieasierft functions.

comrn’ess

i=O

become

descent

certain

may

compressed

we

if

.\$%?.

condition boundary

is

for x

weaker

frontier !Inicett

1-1

f(5),

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value.

integers

relatively

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of

acyclic

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that

m and n, but of g. The

transformations

that

monotonically let

gi(x)

Figure

we make we

directed

IS hold 2

3

g and d(x).gn(x)

on

If

descent

condition Section

the

%?.)

purpose of the that there is a

assure

is

on

our

a

conventions in

computation

then

descent

and

tree

function.

p

false.

is

these

The

those

the

Ci

progressively

conditions, intricate.

is

as

c(d(x)).d(c(x))

consider

which

This weaker cn(x)=dm(x) that In Section 4 our Cn(x)=dm(x) that

implies on

condition

p

of a function

(The

for

of

1-2. the

base

cases,

descent 4?. In

condition c(d(x)).d(c(x)) c(d(x))=d(c(x)) not insist

f

two

the

A descent condition relationships on %. In Section

c(x)=gm(x)

iZ?.

of

general,

for I
the

into of

recursive

function of Yn a requirement among the descent for

f

domain

of

to

of

consists

Each

argument (under value

S,

and f(h(x))

1-2.

=

Descent

f(x).

Tree values,

the in the

trees (a) tree

for”

has (b)

assumption third value

for

every

that for

tree). Tree tree (a).

value

in

that p of tree

the (c)

nodes with

first is

a

labelled result is

and convenient

true (a) second

of with values

for

every

and false levels of

abbreviation

a(h(x)). (This is always the case, for maps @ into %, i.e., when for all h in S.) This condition whenever where

we h

are

is in a(h(x))

the

S and x is in and be sure

undefined

for

only

if

instance, p(x) implies assures value

when h p(h(x)) us that

of

A 5

f(h(x)), 4

%, we may supply the that we have supplied

we

were

for

an

value.

Descent

Trees

3

and

Compressed

Descent

DaRs 2

the

A descent computation

will

correspond

correspond

case,

be f(x)

call

on

we will

will

must

be

applied

a

f:

to

x

root

and to

will

each

In

that

with

the

calls,

and

the the

Jo Figure 1-3. A compressed descent dag for the f is the Fibonaoci where f(5), computation of has a (Since each of the nodes function. this is the minimal compressed different value, descent dag for that computation.)

result former

an argument value of descent functions to

obtain

that

I

calls by

nodes

[

node

the

represented

corresponding them with calls.

abbreviate sequence

representing x. Each node

The

identify

of

often only

tree some call,

corresponding call on f

the we will identify of the corresponding

we

a for

original

children from the

indicating

which

a

the

values

sometimes values by

of

Sometimes

argument

will

to to

will have arising node.

tree

value.

145

When

we

descent

have

tree

value

under

the

resulting

dag

for

graph

is

an

for

have

dag dag

is of

nodes

to

Formally,

two

nodes value,

minimal. a computation

The

of

set

of

require functions of

that in

be

a

descent side-effects.

lie

the

of

characterize

f,

on

The

a

scalar

b,

same

as

a

p

not

of

the which

for x

tuple.

may

be

EXPLICIT

we

for all d may be program

have

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likely

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programs, clarity

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if

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REDUNDANCY

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throughout

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as

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programming.

compressed defined.

efficiency. mechanical

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of

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used in place it is preferable

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set

set

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Definition

of

An redundancy

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example, the

function

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let c(x) taking

each

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= CDR(X). and a list CONSing

( For

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common

definition

loss .

their

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of this

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Solution

to We can

f2(x), leaves

Explicit

))

transform

f

into

a

recursive

procedure but

always T, as

follows: procedure u

f2(x) p(x) :=

a(x)

-!wi.in The

function

f2(c(x)) f

is

then

;

T

:=

b(x,T,T)

f(x)

a

common We

gn.

Figure

~ m

for

al 1

generator

interPretation

of

c

exhibits assume

without

and n are relatively the we consider case, n/gcd(m,n) in place of without

3-1

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shows

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the

structure

tree under these assumptions. present a transformation

Znl+”

@

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g

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(x)

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3-1.”

nodes,

146

but

,.

The

most

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,,

Descent”

at

9

Wzo(x)

g2m+”(x)

redundancy.

;

-f2(x);~T~

nl+n(

9‘+”(X)

Figure

procedure

d(x)=gn(x)

respectively,

the descent section we

gqx]

;

~T

and

of integers

In which

A

Redundanc~

which does not return a result, the value of x in a global variable

positive

.& JQ\$i?f

,

DISTRIBUTE(CAR(x),f(CDR(x))

interpretation

and

redundance.

and n g, m, value of gm or

;

g

the

generality that m this is not if m/gcd(m,n), and

&%~,n), f(x)

call

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generator of .

g

c(x).gm(x)

say

Redundance

a given

function

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d,

Generator

for

a

such

in

x

that

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powersets: ~rocedure

Common

Suppose

the list leads to to

of

there and n

the

example, to

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as

COMMONGENERATORREDUNDANCY

3.

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of

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x

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domain value

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NULL(x);

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the

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predicate

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c(x)=d(x)

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of

if

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Consider, Let

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kth k

level

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g3n(x)

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Fibonacci

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for

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Suppose

set

of

g~ven

program

Recall

redundancy. .

x-2

x-1,

then

and

x

d(x)

that

O-2=-2,

1-1.0,

undefined,

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of

tree Since

cannot

can

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in

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in

in

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1

to

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f(gk-l(x)),

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each

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.

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n

f(c(gi(x))),

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assures

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to =

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.

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the

value

of

f(gj(x)) value

is

the

frontier

f(gi(x)).a(x)

or

into

BACKg[i]

and

recursive a value

true,

either

be

a return (If

f(gl+l(x))

shift

numbers

is

may

O
is

f(gi(g(x)))

O
into not

arbitrary

p(x)

that

undefined,

BACKg[i] p(x)

If

f does

j,

array

global values which

of leaving O
When

condition simply

form

transform which

BACKg[j].)

recursively

resulting

chain,

is

a minimum

the

up

then

BACKg[i],

to

place

BACKg[i+l],

b(x,BACKg[m],

b(x,f(c(x)),f(d(x))) program

.

f(x)

in

is:

on

f(x)

shown

tree

in

anti

compressed

the for

Figure

;

f2(x)

procedure

; -BAW401

f;?

f2(x)

[email protected]?2 U p(x)

some f2(x)

a!!

ibegin

of is

;

~

bQ!3Q [email protected]

then,

value This

3-2.

descent

by

f(gk(x)),

g(x)

f(x)

-

gk(x),

compute values

calculates

descent as

form

f(go(x))=f(x) for write a procedure

results,

simple

equivalent the

to

of

f(gl(x)), We shall

itself

previous

of

reasonable

turn,

k.

using

expressions

expression

makes in

also

contains

every

computing,

which

compute

at

numbers.

almost

[email protected]&j .

.

the value the value of

employ a to remember

procedure Thus

can

and f2(x),

the

range.

Appendix

natural

f

= f(gi(x)).

wil

needed later, procedure,

the

the

the

highest number Typically,

(m-1)(n-1)-1.

small

expressed shown

of

corresponding

f(gi+m(x))

BACKg[O:max(m,n)]

a

but.has f(gl(x))

I

we )

value

that

is false. If we look nodes down the chain,

m of

the

the

say

are

f(-;?)

is

find

=

points.

and

to

chain,

Similarly, chain is

these,

equal to 0-1=-1,

occurring

be

it

Using

Redundanc7~

expressions

expressed;

be be

all

Generator

number but

so

are

function

condition

Specifically, all

or

the

value

f(d(gi(x)))

[email protected]

of Figure 3-1 is {gim+jn(x) m and n are relatively prime,

natural

be

The

.

S

x

the

f(c(gi(x))). nodes down

g(x)

in

each

in

and P(gi(x)) computed

find

m=2, n=l, condition

h(x)

to 1 at f(-1)

Common

define

h and

undefined values of

1

generator

interpretation we

function

frontier

of

The

that

If

equal and

the

im+jn.

that

1-2.-1.

function, f(0)=l=a(O)

descent

under x-1.

each

and

Section

d(x)=gn(x) for The frontier

is either possible

The

in

common

lambda[x;x-2]}

a(h(x))

jelhJ1. every

.

for

{lambda[x;x-1],

constant Since

that

and

c(x)=gm(x) and in%= {2,3,...}.

requires {0,1}, a(h(x)).

discussed of

we want

node

gl(x), value

example

an

c(x)

will

redundancy,

condition

{gi!O
provides

can

generator

frontier

a

O Q

BACKg[i]

tree

:=

x

dag.

QId

(max(m,n))

:=

a(x)

Q2

;

g(x)

;

return [email protected];

x

f2(g(x)) ti i Hg BACKg[i]

;

BACKg[OIl

:=

(max(m,n)) := BACKg[i-1]

~

-1

~

1 [email protected]

;

b(x,BACKg[m],BACKg[n])

~

Note

that

calculated

g(x)

end of

of

the

the

are nth

value

92(X)

to

behaves are

array

array

values and

BACKg

values and

as

we

ascend

placed

at

93(X)

far

call until,

3-2. a

The

of

f2(g(x)). for

The some

compresaea

minimal

computation k,

of

value is

for

enough

depth

i

Figure

referenced, or compute results

f2(x)

based

chain

p(gk(x))

descc!nt on

COntinUeS is

a

of in

dag

such

be

true.

147

be

the

that one of

it

form

f(gim+jn(x))

je~} assures (m-l)(n-1)/2 (Any

value is useless

be

a(gi(x))

the

referenced never

mth a new It

However, we

the

if

the every

calculate

nature

of

for any values

placed must

and to

only migrate

max(m,n),

us that useless

undefined, values.)

end The

calculated far enough

which Thus

needed.

f(gi(x)) these

through

etc. exceeds

low

high

calculate of BACKg.

will be themselves

referenced,

recursion

fact

{im+jnlic~, more than computed.

recursive clownward

to the

that which

will migrate

the

recursion.

pass

order to beginning

the

Newly at

the

the

they

is possible that a \$alue placed in BACKg, but never be to

queue: BACKg

towards from

as in

a in

migrate

referenced positions, be

as

placed

in tUrn

x,

no are

BACKg, out

to

ImDroving

the

Solution

to

Common

Generator

Otherwise

Redundancy The

efficiency

improved and

at

n

the

are be

unfolded

a

Thus

the of

in

these

subscripts,

which

will

time

evaluate of

‘!BACKg[i]

:=

~-100p

by

~

to

a(gi(x))

i,

we may

a(x)

x:=

;

the

be

that

loops

in

above

there

in

an

replace

is

of

a

is

of

1,

to

without

so

strarts that

REDUNDANCY

Consider which

the

computation

There and

city, switch

k

from

involved

one

right

parallel

of

rush

the

interchanges. highways at

again

until

determine

out

at

DELAY[i,3-h] the commuter

since highway

3-h The

until

(k+l)th

to

allows

us

that

generality the

at

row of handled

be

DELAY

‘Interchange

solve

the

O!! on

problem

to the

highway

by

calling

of

traffic

it

i,

i+2.)

minimum

amount

interchanges

he

The

we

L7 exist all

do

of

may

time

in

each

highway.

the

highways

This

is

do

exhibits

is

to

have

which

the

of

integers the other array

we

represent

1 and 2 highway

(so that is 3-h),

DELAYIO:k+l, needed

to

so

travel

to and

lfinterchange the

k+l!!

following

solution

to

the

pocedure

~

and as

is

the

is i

DELAY[i,h].O

Figure

4-1.

that

c(d(x))

=

c4(x)

d(c(x))

that

in

the

i and j such and c(d(x)).d(c(x))

common

the

the

genrator

h, the

is

for

a,

then

the

to

the

set

of

c

c

and

and

Problem common

transformation redundancy

condition

{cmdn[o~m
which holds

O~n
program Commuter

for

m+n>O}.

i.O

computes

the

Problem:

;

i~k ~

return

min(DELAY[i,h],DELAY[i,3-h]

else

return

min(DELAY[i,h]+f(i+l,h)

) ,

DELAY[i,3-h]+ DELAY[i+l

,3-h]+

f(i+2,3-h) The the on

value

of

city highway

f(i,h) we

h.

If and

interchange, highway takes k+l time

(i.e., required

is

if

are

us the

from

If so

min(DELAY[k+l,h],DELAY[k+l

minimum

then can

time

to

reach

interchange

we are at simply select

interchange

i=k+l, that

)

approaching

i

the last whichever

k to interchange The most quickly. is min(DELAY[k,h] ,

destination) this to do

DELAY[k,3-h]). destination,

i.k we

the

now

we

are

time

the ,3-h])

required .

min(O,O)

at

Figure where

the is = O.

4-1. f is

the

Impatient

Commuter

is

labelled

with values

at

for

tree

program

the

argument

148

Descent

a that

the

computing Problem pair call.

(i,h)

for

f(o,l),

call the k.4.

solution Each

representing

d

and d

periodic of

a

periodic

%’.

and

Thus

frontier

functions

If

Commuter

present

in

c(x).gm(x)

generalization lie

the

for x

interpretation

that

Impatient

exhibiting

whenever

the

generator. strict

there

ci(x)=dj(x) for all

en(x).dm(x).gmn(x) However, ~.

on

common is a

programs

such

of

whenever

redundancy: g

redundancy.

valid

when

generator

solution

interpretation

that

case

generator on

no

an

redundance

c(d(x))=d(c(x)).gm+n(x)

the and

for

say

numbers are assumed we approach the city, final destination.)

recursive Impatient

f(i,h)

by

we

common a

have

DELAY[i,h] interchanges

between

on highway h, l~i
Then

that

in

observe

Redundance

periodic

certaidy

d(x)=gn(x)

of of

if we are on highway and we initialize

l:2]

k.4,

we

and

Periodic

general,

redundancy Suppose

with when

exhibits integers x in ~,

not

amount given the each pair between

on

In

cross

problem

f(O,l)

surprising

Definition

highways at two is, if a commuter

interchange

d2(x)

enter

to

for

not

The the

-in as

a may time all

and time

safely,

switch (That

can reach the city, takes to travel

consecutive

takes

=

tree is

d(x) = redundant,

and f is

.

into

commuter The but

lane

hour

interchange

the

commuter

left it

to

a

other. negligible,.

the

the Since

commuter

problem, Commuter

highways which

is

from

path

Impatient

at to

lane.

lanes permit

consecutive switches

two

highway

exit

the

the

switching

interchanges

time

are

shortest

call

interchanges

in

several

following

shall

we

Problem:

and

of

time and

to highway DELAY[i,h] +

is

is

of

h

:=

“BACKg[i]

This

time i+ 1

loss

minimum

f(o,l).

first

descent

so

row

the

values

on

Oth

we may

,

not

of

on highway we switch

.Tlne function f is, of course, an instance of with x corresponding to the pair . Then 9, P(x) min(DELAY[xl ,X2], (xl~k) , a(x) ❑ DELAY[x;,3_x2]) , min(DELAY[xl ,x2]+Y, b(x,y,z) = c(x) ❑ DELAY[xl, 3-x2]+DELAY[x1+I ,3-x21+Z),

PERIODIC

into

smaller

stay

the

commuter

a(gi(x)).”

4.

the

the (i+2)th interchange. allows the base cases

uniformly; assume

sequence the

take

constrained

least DELAY

form

amount the

in

equivalent

may Since

will have BACKg by

If

g(x)”

m

will,

assignments.

variables.

independent

Recall

assignments may replace

we

scalar

max(m,n)+l

can

program

sequence

small.

into

references

array constant

this

of clarity. values whose

constants,

practice, be

of expense

we

to the city if we stay minimum time to the city if 3-h . The first of these f(i+l,h); the second +DELAY[i+l,3-h] + f(i+2,3-h),

to node the

Implications

of

Let

e .

commute,

ci

dag

m

n

and

of

in

by

comparable

i

by

lies

this

one,

and

a

fact, into

the entire a network

4-3.

Given

upper it

lies

is

of,

possible

to

corresponding Let

to us

node

in

the the

c(ekci-ldo) corner of the

to to

that

were not in in B correspond

counterparts

be

further

Figure

x 4-3,

and

corresponding rectangles

between a certain

there

is nodes

x

and

y

It

to

the

In

other

rectangles

B

rectangles Thus

D, the

rectangles,

i.e.,

if

node

in

certain in

rectangle

connection

in in

%.)

a node

a

be and

compressed into the This compression

4-4. between

edge

an

C.

dag,

Similarly, coextensive.

connecticms is

in

descent

not

the

y

between

in the

representatives

Figure

of

4-4.

the rectangle. be

of

upper lower In

compressed in

Figure to

the

D and

result

E,

values

rectangle

labeled

in

4-2.

shown corner

the ekci-ldo;

of

left

of thus

Figure

B is

the

A The

one

the node it corresponds

just in

The

node

in

the

uppermost

below

and

to

the

right

in A corresponding to d(ekcodj-1)

corresponds

side

rectangles

the

. ekcido. C occurs just

node

may in

would ekci-l(x)

if

=

B.

subgraph

uppermost

below and corresponding

dag

ekcido nodes in the C correspond

right-hand

shown

the

their

Thus the B and

rectangles

nodes

compressed

shown

.%?. is,

dag

are coextensive. F are mutually

Figure

a

(These

respectively, all the as

on

That rectangles

desc!ent

the

rectangle

subgraph

corresponding

compute

in

form

is

whose

may as

rectangle

assume

represents

upper

to

instance,

values

there

envision

rectangle

rectangles,

for

this

can

original

values

same words,

descent

compressed ekcsd,u(x),

right-hand

dag

of

the

= e

c)f

value.

that

and C E, and

form

subgraph

that

whose

descent of

result

sides

the values

lower

the

of

preserves

This

j

dag the

same

ekdj-l(x), follows

be

non-neg,ative

of

of

m can

corners

the

part

where

descent

We

by

to

expressed as tj+u, integers and u
Notice

i

uppermost

d

be

k.

the

third

side

cmdn(x),

are

of the

rectangle

c and compressed But

s

compressed

to

side

right-hand

form

rectangle. j

Since minimal

representation and U
4-2.

j

side

left-hand

and

a given

Figure

i

r

However, ci = dj = ek+lcOdC1.

ekcOdj

integers.

the

for

in an

the

the subgraph consisting of

O~u
forms

of

where

a unique where s
depicted

.\$%?. the

s
Consider dag

O~s
are

ri+s,

cmdn . each value

f(x) has evcsdl.l(x),

on in

f(x)

and and j

t

dj

non-negative

as

integers where

Redundancy

values

are

expressed

descent

.

all

descent

Then Thus

Periodic

to ekcodj-l; ekcodj.

thus

A to of

Figure

it

Each rectangle the configuration

4-3.

An

abstraction

of

represents of Figure

the

i.j 4-2.

nodes

descent

dag.

arranged

n

e%”d”

a ~k,fl~].l

Figure dag

4-2. consisting

The

subgraph of

the

of nodes

a compressed ekcsdu, OQ
descent Osu
4-4.

Figure dag

149

by

merging

The

result identical

of

compressing rectangles.

the

descent

in

Observe an

uppermost

the

form

that ~kcsdu

each

ekcodo

position

There the

is a descent

for s

in

Figure

4-3

has

to

a value

of

other

node

corresponding some

k.

Every

corresponds

to

a value

and u are uniquely of the node within

linear dag

ordering of

various

rectangles’

the

compressed

that

rectangle

corner

rectangle , where

the

the

that

among

Figure

the

4-4,

values descent

rectangle.

dag

of

Figure

4-4

BACKC[M]) :=

P(ARGUMENT[M,N]) ~

Periodic

;

XL?QLOS!Q

BACKd[N]

is

~ to

)])

~Nm(j-2)sJw-l

minimal. Solution

O.dQ

BACKd[(j-1)],

in

corresponding to It follows k.

for

until

BACKd[(.i-1)] := ~ p(iRGUMENT[M, (j-1)]) ~ a(ARGUMENTIM,(j-l ~ b(ARGUMENT[M,(j-1)],

form by

rectangles

a-1

[email protected]

in

of the determined the

&M~(i-l)

a(ARGUMENT[M,N])

~

Redundance

b(ARGUMENT[M,N], BACKd[N],

These method

observations

for

computing

redundancy: RESULTIO:i-l,O:j-l].

We

procedure

with

f2

f2(x)

will

RESULT[S,U], can

be

done

to

the

[email protected]
O~u
Given

these

If

values values

to

the

to on of

for

can

the

(+i) and res~ectively~:

(+1)

frontier

to

;

BACKd[O]

the value expressed

M and

We omit ARGUMENT: modify the second

of in

in N,

an

amount

we maY

eliminate

first

“~

the by

cMdN(x) can, an explicit

replacing

M“

for form

of

time

the

arraY

looP

assignments

and of

the

form BACKd[n] U

the

rectangle

if be

evaluated

of

dag.

RESULT

represent

be

can

independent

e(x) = ci(x) initializing for the next

current

In addition, M and N,

which

this

descent

set

each

a(csdu(x)).

compressed we

a on in

true

is

:=

m

array, f into call

a

appropriate

the

BACKC[M]

without

f(csdu(x))

p(x)

f2 recursively the effect

values

and

of

BACKd[N+l])

fOllOWing f(x)

global

according

call has

(where mod j,

a

RESULT[S,U]

in

appropriate

the of

transform that

value

setting

the

rectangle

to

We will property the

we which

lower

folllows mod i

value

maintain

leave

by

RESULT

us

the

immediately,

condition, Otherwise, = dJ(x),

will

:=

p(ARGUMENT[M,n])

as

~

a(ARGUMENT[M,n]

~

b(ARGUMENT[M,n],

) ...,

..-)

&KQ arrav

ARGUMENTIO:

~M~

(i-l),O:(j-1)]

O until

12e&irl ARGUMENTIM,O] U

ii.O

x

else :=

,01)

c(AFiGUMENT[M(j-1)

(j-1)

RESULT[M,N]

~-l

OQQ [email protected]

until

the

is

f2(x)

begin ~M~O BACKC[M]

interchangeable, of i and j.)

of

N],

until

~N~

simply

by

and

d

interchanging

the

are

form O until

Solution

to

less

clear

A slightly uses

i+j

locations of

the only

the

important the

i.j

either

second

“~

to

M“

zero of result. BACKc[O:i-1]

RESULT

and

the

second

In

are

to

of

it

will

the

improved

‘!=

this

form

M“

loop

represent the will

RESULT ; represent version is

rewritten

each

row

just the of

after kth the as

cOdN(x)))

;

form

a(T) b(T,BACKd[(.j-l

U

of

BACKC[M]

150

)],BACKc[M]

form

for

)

;

cMdN(x))

a(T) b(T,BACKd[N],BACKd[N+l

[email protected];

follows:

;

;

P(T) ~ ~

of the

algorithm,

cMd(j-l)(x))

p(T)

T := (explicit BACKd[N] :=

that row of

for

:=

~ ~

global zero of

M=k iteration represent

(explicit

~

results,

RESULT . Observe RESULT which are or the end of those in row zero and introduce column

for

QQ!.Z.LOJ2Q

BACKd[(j-1)]

program

intermediate

We will to represent

(Just before BACKd

row

(k(+i)l)th iteration, RESULT.)

loop

BACKd[O:j-1] turn. below,

store

of

;

;

l!!2Q?2 T :=

Redundancy

version

cMdO(x))) @

m;

roles

locations in locations in at the beginning

column vectors

RESULT in the loop

Periodic

for (j-1)

return f2(e(x))

the

ti

:=

~M~(i-1)~-l ImDrovin!2

(i-l)

:=

a((explicit

l])

c

QQQ

;

a((explicit roles

final

p(x) then

a(ARGUMENT[M,N]) b(ARGUMENT[M,N],

the

the

form:

BACKCIO]

BACKd[N]

course

possible,

following

return

;

gncJ (Of

)...)

m U

RESULTIM,n(+,i)

; ~b(T,...

;

fp(x)

procedure

:=

RESULTIM(+i)l,

cMdn(x))

~a(T)

improvement on

f(x)

12QQQ

P~ARGUMENT[M,N]) m ~

this

takes

Drocedure 1)

-;

for

fiP(T)

;

@

d(ARGUMENT[M,N-

form

:=

When

ARGUMENT[M,N]

~

(explicit

result

1 until

~Mfrom(i-1)~-1~ ~N~

:.

BACKd[n]

[email protected]

:=

then

~N~

T

;

(i-1)

:=

BACKd[O]

1)

(Again,

the

primarily replaced

use of a notational

by

a

sequence

and

variables, form for sequence.)

~-loops convenience.

we

each

of

arrays Loops

have

a different

c“dn(x)

procedure

is be

can

to

assignments

may

value

and

~

scalar explicit

occurring

‘n

‘hat

This

is

an

f(n,k)

k=O

QL

;

k=n

then

return

1

else

return

f(n-l,k-l

instance

of

f?

)+f(n-l,k)

with

x=,

(X70 ~ X2=X1 ) ! a(x) C() = , =

. COMMUTATIVE

5.

Since c and

, in IN,

REDUNDANCY

establish all Definition

of

Commutative

Finally, which d(c(x))

the

we

only for all

periodic

consider

descent x in

compressed condition. commutative frontier

will 5-l(a)

{Cidj

turned

upward Formally,

and

and

min{k

be

all

x

on

in

its

for

rightward min{k I

from p(ck(d(x)))}

this

p(dk(c(x)))l

x

the

binomial range,

case the the legal (in which either

coefficient n~O

of

on dag

and

of

definition

n

and

k,

the has

Paterson

II.

this for

is

the

(in it

to

binomial

is

of

the binomial defined for

the

needed have

is

1 if xl-i-j otherwise)

~,

,..

in

commutative to store

we

in

will,

from

general,

the

descent

original

tree

is

only

for

~, and

that

i+~n,

for

along with the descent dags, interpretations

oXo

(~a(c(xo)),

./2\$ f(xol=b(~,

o(d(xt)

)))

b(c(xo),

o(c2(xo)),0

(cd(xo))).

b(d(%)).

o(d2(xo))))

Xo

Xo

:

C41 &!C)&

,;(Y2

*~9’(0’” c!om~essed degcent In (b), the counterclockwise

redundancy. rotated 45o of

the

frontier

dag

C2[XO)

dag of tO

Figure

cd (Xo)

5-2.

cd (x,J

The

value

interpretations J%, and minimal compressed each interpretation.

whenever

holds.

151

C(XJ

d(x~

C(xol

structure

which that

&&

. . . . . .

condition

For

@2.

oXo

f(xo)=b

(b)

frontier

the

dag.

false if

of f(xo), compressed 5.2

with

descent

p(x) and

Figure

are

= a (XJ

.!.,1

...

if

that

eliminate locations

interpretation the assumptions

The value and minimal

and

shows

dynamically.

a

the for

computed partial

redundant.

compressed

true

trees

proof

differs

whenever is XO.

ce~: f(x,J

to

replaces

consider except

displayed

\$(,

are

proof it

be for

interpretations

storage

minimal we

their

results,

that

cannot locations

the

global

that

in

descent is

the

the or

that for the schema

for

commutatively

we

allocate

constant

been the

or

x2.x1,

undefined).

showed f

of

using

c(d(x))=d(c(x))

(a) has illustrate

(in

Redundance

if

be

temporary to

P(Cidj(xO))

minimal

to

x2-j=O

if

interpretations

if by

each n, is free

The 5-1. commutative

To for

which case 1 or undefined)

:Hewitt[1970]

even

corresponding

Figure for

and

equal

either is

= = j

that

either

undefined);

either

restricted

(o)

c2d2(x) .<.

is

the function than n storage

true

The

c2(x)c2d(x) :: . .

it

Commutative

and

An

results.

only

dz(x) cd2(x)

is then

case

exist

fewer

Oikin:

d(x) C?X) cd(x)

or x2-j

coefficient and undefined

which

of

each n there g for which

\ p(ck(x))}

..

x2.x1,

x2-j=xl-i-j is

(in

using

redundancy

n!/k!(n-k)!,

x2=0 x2=0,

case

coefficient

Implications

the

min{klp(dk(x))},

commutative

recursive

cond~tion,

. we show

and If

Consequently,

example

if

which is in

redundancy An

~, xl-i-j

x2-j>xl-i-j

J%.

following

in

= Y+Z,

d() = d(c()) %?. For i

case,

i.e.,

I

j

on

P()

b(x,y,z)

the

monotonically to point

~min{k

and

binomial

functions is

i

coefficient of 1 or undefined.

x2-j
descent eliminating

takes (where the

S’proceeds point

of

minimal

of

clarity),

for

and

whenever

set

computation 5-l(b)

side

this

valid

the

Whenever

~

-~ the

for

will

holds

between

I

called shows

redundancy

ie~, jeiN}. the dag for the entire in Figure shown shape frontier

be

under

~

been

is that c(d(x)) generalization

dag descent Our transformation condition

for

interpretations

condition .%. This

redundancy Figure

redundance.

Redundance

and

c(d()) d commute

c~d.d() the frontier

1,

C2(XO)

dz(xo)

of

f(xo)

under

d(xo)

de(xo)

cd(x.J

each

of

@l, \$2” ‘he ‘esCent descent dag are shown

the tree for

We will

show

under storage

interpretation locations.

involves a leaf

either of the

the

node

ing

the

and

placing

call

the

to

a

Each

that

to are

shaped like @n, becomes

Of f(xo) location

dag

for

(Intuitively,

@n

See of

in

leaving use at

map

this

operations

@n

described the

for

dag

to must

the

dag

the

leftmost

this

node

to

be

plays we

Jln-1, there become closed the moment

a

in

the

no

have

edge

of the

of

on

some

level

location;

to

be

by

is,

@n

required by the minimum

at

the

~n at number

moment

the

the

the of dag

the the

Figure first

(The

f of

A mapping is

mapped

:=

i+l

by

the

-

;

to of

is

of

the

the

in

global

array,

values

and

compute

as

row

of

place execute

leaves When

O~i~n. is

done O
recursively -

so

We

row.

1,

with

which

O~i
ARGUMENT

that will

which

true), this in BACKd[i], guarantees that f2

drawn BACKd,

is

call

=

ARGUMENT, that

d-depth(x)

BACKd[i],

array

dag

procedure f2(x,n) in BACKd[i],

we

c(x)

if

array, a

in

it

will

to

global

values

di(x), O~i
for

think

let

d,

d-depth without be computed.

to

result

into a f(di(x))

in

locations

function is

condition

f(di(C(X))) of

to

p and

instance, d-depth(x)

redundancy

Another

Otherwise,

the

commutative d-depth

and

frontier

set

of

places We

that

then

ARGUMENT[i]

a(di(x)) the 100P

in

.

BACKd[i],

tO

~

at of

i

~

BACKd[i]

d-depth(x)-l :=

x-l

~

O [email protected]

b(ARGUMENT[i], BACKd[i], BACKd[i+l]),

of

becomes

Dccj

from to

is true. f(x) is

computed

(For X.SIN,

and

is O (i.e., ,P(x) placing a(dl(x))

arguments

Since

moment locations

i

forc\$f-n

5-3. dag

smallest

be

interpretation

argument

value

valid.)

of

‘n-l less number

;

will use the how that function

dag.

n

Dug

the

Q

directly. “x=O,”

is

5-l(b),

d-depth(x) simply by

value

dag

p(x)

the

solution

Figure

hold

otherwise

is a way for the dag for using one storage location of closure than the minimum

locations That

;

array

We

represent

dag

sequence

on

a global

rotated

just for

open. the

close

d-depth(x)

solution

transform

the

one

created

be

p(dn(X)) whenever

i

Our

The

for

mapping However,

the

to

d(x)

x.) use

in

a

operations

least

in

x

always

not

be computable = x-1, p(x)

dag

dag

of

depending

specifying

operations

level

would

exhibit

o;

may d(x)

‘f(x).

locations We can

at

role

kth

the

closed,

dag

or,

consider

deleting

storage

storage,

&,

~n-1 , creates a graph by cutting off

storage is closed.

node

of

such that has a value can

-x:= return

(i-l)th

node

first

in

the for

closes

any

of

Redundance

value

l?egi?l i := while

for

the

the operations. be

the

path

operations

required

applying other

dag

on

Now

by on

n

]rocedure

dag,

location

node

sequence

@n-l, node

@n

corresponding

storage closure.

for

ficst

and to all

a

amount

for

exist no way

interpretations

d-depth

This

computation

mapped to the dag

which

minimum

program

the following of the dag for

ith

possible the dag

to

dag

the

for

is of

defined.

and

open

storage

5-3.)

the fewest the moment

the

involving

@n level

Figure

sequence

closing

The

the number d-depth(x)

one storage of f(xo)

this mapping @n-l as the dag for the upper left-hand

same shape on nodes

for e%n. sequence

the

one

fixed

Commutative

Define

storage

an

descent

Consider a subgraph

LZn-l: for

moment

least

a

these

the

that there which there is

location).

requires computation

. from

to

natural Clearly,

value

n+ 1

that

than

redundancy.

Solution

any

oPen the

storage

compressed

at .%1

of the dag the (k-l)th

l
the

more

established ~ for

using

though

commutative

closed.

dag

that

observing

mapping the

on

at

requires under

isomorphic level node

required

by

a

f

calculate

one

have for

root in

is

(with

in

to

cor-

held

the

show

under @O trivially and proving that

than to

root

least

we

even

We the

whose

dag

with

to

this

\$n

under more

the

nodes

currently

closed

the minimal closed.

We do

from

path

the

at

node,

a of

is

Thus @n- 1. interpretations

colnput-

interior location.

not

dag

suffices

locations

@n

only

begins

corresponding

that

is

are

the

n+l

computation

storage

Otherwise

with it

a

there

values

computation

Thus

in

if

the

closed

f(xo)

least

locations, to

value

location.

The

of

storage

consisting

responding

concludes

step

at

the value corresponding to storage location, or taking to children of a given

two

open

of

requires

corresponding

dag

storage

computation

@n

from

value

leaf

the

placing dag in a corresponding

values

interior

that

the

the (i-l)th

dag

for node

@n to the dag on the (k-l)th

for \$n-l level of

152

The ith node on the second dag.

the

kth

level

of

the

result in c-depth(x)

which leaves f(di(x)) in BACKd[i], [email protected] (At the beginning of the i.k passage through the loop, BACKd[j] . f(dj(x)), k<[email protected], and BACKd[j] . f(dj(c(x))) = f(cdj(x)), O~~k. Since k < d-depth(x), p(x) is false, so b(ARGUMENT[i], BACKd[i],BACKd[i+l ])=b(dk(x) ,f(cdk(x)),f(dk+l (x)) = b(dk(x),f(c(dk(x))),f(d(dk(x) ))) = f(dk(x)). Thus , after the i=k passage through the loop, BACKd[k] holds f(dk(x)). The

entire

program

procedure

f(x)

!Z2&LiQ D :=

looks

program d-depth(x).

if

If the value of’ a(x) is the same for all x in 93, then BACKd cam be initialized with every element equal to that constant Then the value. two loops which set elements of BACKd to a(x) for some x can be eliminated. (This is legitimate because the array elements which are set by these loops are not referenced by the program before the loops are encountered.)

this:

;

BACKd[O:D], ; BACKd[O]

procedure

ARGUMENTIO:D]

6.

;

f2(x,n) p(x)

;

~

w [email protected] w BACKd[i] X := d(x) [email protected]; return

:=

d; rir := d-depth(x) ; f2(c(x),m-1) ; ~ i from O until l&&QL ARGUMENT[i] X := d(x)

:.

a(x)

;

The use of global variables is only one of a number of implementations which could have been used. An obvious alternative would be to rewrite f2 to return a tuple consisting of those results instead of leaving which will be needed later, those results in global variables. (There would between be an obvious one-to-one correspondence elements of the tuple and elements of the global mathematically This seems arrays.) approach cleaner than the use of global variables, but it was felt that the use of global variables would be more familiar to being more most readers, consistent with programming languages such as PL/I and Algol 60.

n @ x

;

&;

i from m until n & BACKd[i] :. a(ARGUMENT[i]) ~i~m-l~-l until BACKd[i] :. b(ARGUMENT[i], BACKd[i], BACKd[i+l])

CONCLUSIONS

We have presented a recursion schema, ~, defining a function, f, together with various sets of assumptions under which the definition of f calls for redundant computation. For each set of assumptions we have investigated the nature of the redundancy, recursive program, and presented a using the predicate and function signs of J?, which computes f without redundant recursive calls. As expected, we found that weaker assumptions require more intricate computations. We also found that weaker sets of assumptions required greater amounts of global storage.

;

M U

space-efficient smaller than

;

d-depth(x)

&2&.! ~. f2(x,D) return

like

a more is usually

~

: 0~

~ RemovinQ

is ImDroving

the

Solution

to

Commutative

Redunda~

Each the

Recursis

of the general

~rocedure U p(x) ~ else

As before, this program can be improved in many cases: If di(x) can be computed in an amount of time independent of i (e.g. when d(x).x-1, di(x)=x-i), then ARGUMENT can be eliminated. If di(x) nor d-depth(x) can be computed neither directly, then time would be saved by combining the initialization of ARGUMENT with the calculation of d-depth. However, this would be expensive in terms of space, since we would then have to make ARGUMENT local to save it across recursive calls on f2. Of course the roles of interchanged if appropriate throughout program. the b(ARGUMENT[i],BACKd[i],BACKd[i+l b(ARGUMENT[i],BACKc[i+l],BACKc[i] would be replaced by c-depth.)

of

the

versions fcjrm f2(x) A(x) begin

of

f2

earlier

presented

;

f2(h(x

)

;

B(x)

~,

which case h maps where x may be a tuple (in tuples to tuples) and A and B are procedures global executed for their side effects on References to the global variables may variables. (The version of f2 occur free in A or B. developed for commutative redundancy is not quite remedied by in this form, but this is easily moving the assignment to m to just after the computing recursive call, at the cost of d-depth(x) twice.) Using the “direct” method of Bird [1977a], We recursion removal described by can rewrite f2 non-recursively by using a stack S to maintain an itinerary of values on which B must still be executed,, The resulting version of f2 preserves the order of the calls on A and B:

c and d can be changes are made (For instance, ]) would become ), and d-depth Such changes would

153

procedure

f2(x)

&lX&l stack S := while

In

;

S ; empty-stack not p(x)

; do — <= x ; x

[email protected] A(x) ; while not

empty(S)

-x<=S;

:=

h(x)

~

;

& B(X)@

established that the case, we have of f2 do not nitie their space versions The number of global in the stack. by each program is a fair measure used Interprogramst requirements. space of q displaying explicit, common or periodic redundancy can be computed storage; commutative fixed amount of really has a higher space complexity.

any

recursive complexity locations of these pretations genrator, using a redundancy

[email protected] Generalization This program may be improved. As suggested by Darlington and Burstall [1976], recursion can be eliminated without the use of a stack if there is a single recursive call whose descent function is invertible; the resulting program can be made more efficient when there is a unique base even case. When h has an inverse, x may be set to h-l(x) directly instead of by popping a stack as above. The resulting program: lrocedure f2(x) ; [email protected] Y :=x; while not p(y) A(y) ; a y+x @

~

Y :=

h-l(y)

WY:=

h(y)

; B(Y)

Drocedure

h the

has no inverse, stack by using

f2(x)

[email protected] Y :=x ; n := o; while not -Y:= A(y) ; fii~n-1~-luntil 12Qk3LQ Y:=x; @C j B(Y) &

n>2

for 9, provided that all of the ci are equal on into k
;

~

(This assumes that neither A(Y) nor B(Y) modifies If this is not the case, we can replace “A(y)” Y. or IfB(y)” by ‘Iz := y ; B(z),t! by “Z := y ; A(z)t! as necessary.) If there is a unique base case X. the such that p(x) is true if and only if X=xo? initial assignment and while-loop may be replaced by the single assignment y :. Xo. if

9,,

present eliminating The methods we for redundancy in 9’=92 can easily be extended to In general we will require that yn, n=3, 4, . . . . the descent conditions hold for all of the descent Explict redundancy can functions simultaneously. the same way as for Pn in exactly be eliminated

a

Even eliminate follows:

to

to one such oblong couldbe stored in a il x i2 x .. . improvement applying an x in array, or, by similar to that presented in Section 4, by n arrays, each of n-1 dimensions and representing a The the n-dimensional oblong. cross-section of redundancy for P can solution for commutative We require that all the also be extended to ~n. descent functions be pairwise commutative on L%, and we use an (n-l)-dimensional array BACK whose ith length in the dimension is set to The procedure f2 takes on 2
we can always a counter, as

;

procedure

f2(x,y2,

. . ..yn)

p(x) then begin Set

BACK[i2,.

;

u p(y) h(y)

Q

@ ; n

:=

n+l

[email protected] [email protected]

from

1 until

i @

y

:.

a(c212. o
,.,in]

..Cnln(x)), . . ..O~i&n

to ;

d;

;

f2(cl(x),c2-depth(x), . to Set BACK[i2, ..., in] all such (i22c2-depth(x) QE . . . . @ (O~i2fi2 m

gnJ Although this increases from some value n to

the number of calls on h (n2-n)/2, it is a reasonable quiokly, so computed very approach if h can be that most of the time is still spent in the if there computation of A and B. In particular, is a way to compute the value of hi(x) in an amount of time independent of i, the body of the outer ~-loop can be replaced by the equivalent of !tB(hl(x)) .!! Even if none of these conditions holds, we can compute f2 with a fixed amount of storage in time proportional to nl+(l/k), where k is an arbitrary integer, using the construction of Chandra [1973]. The amount of space required is O(k).

‘e~~;;;;t;~~i~~~

. .,cn~depth(~)) ; a(cz=z.. .cn=n(x)) that . . J2C in~cn-depth(x)) ~ O~infin) ;

for

~JY,in],...,

BACK[i2, . . ..in_l.in+ll) for all such that ,. ..,O~ificdepth(x)x) 0~i2ic2-depth(x) in an order which guarantees that BACK[i2, . . ..+l.l, in],in] is always earlier than &ACK[i2,..., ij, in],in],

, set 2~jin

@@ four For each of the condition frontier an obvious way.

154

for

kinds ~2

of can

redundancy, be generalized

the in

functions to some value is less than nk for all Clearly, sufficiently high k. whenever this condition holds, there is going to be a duplication of ar’gument values on the kth level of the descent tree. In fact, a sufficient condition for redundancy is that the number of values obtainable by k gr fewer applications of the n descent functions is less than (nk+l-1)/(n-1) for sufficiently high k. This guarantees that there is a duplication of argument values somewhere in the first k levels of the descent tree.

We have not addressed the question of’ what we can do when a particular form of redundancy exists among certain subsets of the descent functions in pn, but not among all of them simultaneously. Additional problems arise when we consider the possibility of different subsets of the descent functions sharing different forms of redundancy, or the possibility that only a proper subset of the descent functions are in any way redwdant. Neither have we discussed the more ,general any n) where situation (not an instance of &’n for the number of recursive calls invoked by tlhe call f(x) depends on x. The dynamic programming examples of Aho, Hopcroft, and Unman [19’74] are of this form.

One example of redundancy commutativity is a recursive program the solution to the Tower of Hanoi string. The program is:

not due to which returns Problem as a

procedure ~ n.O then else

The program transformations presented here can be applied to actual progranrs. More importantly, they can be incorporated in automatic program transformation systems. One such system is described by Burstall and Darlington [1977]. That system uses heuristic methods to spply a series of simple but powerful transformatic>ns to recursively defined functions. One of the most effective transformations, however, the definition of new functions in terms of old ones, has not yet been automated. Among the examples of definition presented by Burstall and Darlington is the definition of a function which returns a tuple consisting of the nth & (n.l)th elements of the Fibonacci series, by calling itself recursively to obtain the values of the (n-l)th and (n-2)th elements of the series. But that is exactly the program that result by applyirlg our would transformation for common generator redundancy to the usual recursive Fibonacci program (provided that we used tuple-valued return values instead of global variables). The methods presented in this paper provide both a way to recognize a particular situation in which a definition would be useful, and a way to formulate the definition itself.

HANOI(n,from,to,using) return return

;

~ HANOI(n-l,from,using,to) move(from,to) II HANOI(n-l,using ,to,from)

!!

This

is an with x = instance of 9’, , ❑ (x1=O), a(x) ❑ e, p(x) b(x,y,z) = ❑ c(x) y! lmove(x2,x3){lz, , and d(x) = . Observe that c and d do not commute. (If they periodic did, this would be an example of ❑ . redundancy, since c2(x) = d2(x) On each level of the descent tree n is constant, and there are only six possible permutations of from, to, and using, so there can be no more than six distinct values at each level of the descent tree. (In fact, there are at most three distinct three of values at each level of the tree, since the permutations can only result from an odd number of applications of c and d, and the other three only from an even number.) The descent tree typical and minimal compressed descent dag for a computation of HANOI are shown in Figure 6-1.

Effective procedures exist to find true !!blocksn of straightdescent conditions, given line code which compute descent functions. Let Ul,... ,uq and Vi,... ,vr be represent the block obtained by and then using its output valus block Lewis

b2; and let bn=bbn-l [1977] proves that

th~se values u~+l.. .uq+q VIJ1 . .. VrJr. A General

it

of it, . . ..iqand the computes

Definition

blocks; executing as input

for n>l, is possible

let blb2 block bl values to with to

bl=b. compute which jl, . . ..jr for same function as

5,0,b,c

4, c,b,a

4, o,c,b 3, b.c,o ..+

d

c

d c

of

d

c

3,c, o,b

3,0,b, c

*c-= l, b,c,a -- .

Each form of redundancy we have discussed has involved descent functions which commute. Redundant calls arise because arguments are just repeated applications of the same two (or n) descent functions, in differing but irrelevant orders. (In fact, each minimal compressed descent dag we have discussed is a compression of the minimal compressed descent dag for commutative redundancy.) This is not meant to imply that the descent functions must commute in order .For an interpretation or ~n to define a redundant program. To the contrary, redundancy exists whenever the maximum number of distinct values obtainable by k applications of the descent

s~

d

c

d

c 0,o, c,b

2, b,o,c

2,c,b,o

2,a, c,b

Redundancy

I,o, b,c

d

c

d

d

c O,C,b,a

. .*

d

c

d

c

d

I,c,a,b

~

+--*

d

C* O,b,a,c

tree minimal The descent and Figure 6-1. call for dag the descent compressed HANOI.(5,a,b,c). In the tree, left sons represent right represent sons and of c applications applications of d. In the dag, edges are labelled they descent functions the according to represent. Edges coming in from the right edge of the page are continuations of edges leaving the leftmost node of the previous level, and visa versa; the dag does not get any wider after the third level.

155

As this example demonstrates, our taxonomy of redunancy is far from complete. The characterization of other forms of redundancy, and the formulation of program transformations applicable to those forms of redundancy, are topics for future research. It may be possible to formulate a general theory of redundancy which encompasses all these forms. (The characterization of redundancy in terms of the number of distinct values obtainable by k or fewer applications of the descent functions would be a typical component of such a theory.)

APPENDIX. RELATIVELY

THE SET miN+n~. PRIME

Proof

1. The largest Sm,n 2. natural

is

natural

not

there are not included

exactly in .\$m,n.

of

the

First

included

Let

(m-l)(n-1)/2

which follows

of

Sm,n

Result

is

itself

a member

of

k .

(m-1)(n-1)-1

= inn-m-n.

We have

shown

that k is the highest integer not expressible in suppose i, 04iSk, is expressible the form am+bn. Then k-i cannot be, for if k-i=a’m+b’n, as am+bn. . (a+a’)m+ then k . i+(k-i) = (am+bn)+(a’m+b’n) suppose that i cannot be (b+b’)n. Conversely, BY “ the lemma, i can be am+bn. expressed as expressed either as am-bn, with aO, or as In the first case, k-i = bn-am, with bO. = (n-a-l )m+(b-l)n, where n-a-1~0 (mn-m-n)-(am-bn ) case, k-i = second the and b-l~O; in (mn-m-n)-(bn-am) = (a-l)m+(m-b-l)n) where a-l~O and m-b-120,

It “ well-known that the set {Om,lm, . . ..~.-l)m} s Sm,n contains exactly one representative of each congruence class mod n. In fact, each of the n numbers in that set is the smallest member of its congruence class which is in Sm n. Once a congruence class is “introduced” 9 n numbers, all larger into Sm,n by one of these numbers in that congruence class may be generated by adding appropriate multiples of n. Clearly, any number generated by adding a multiple of n to a member

second elegant makes

By Euclid’s algorithm, any integer can be Proof —. expressed in at least one of the four forms am+bn, By hypothesis, x cannot be -am+bn -am-bn, am-bn, Since x>O (x is in in the first form. expressed ~ and not zero, Since O=Om+On), x T~nO~ be is fourth form. the expressed in Assume am-bn or bn-am. expressible either as Then x . am-bn ❑ and let q.floor(a/n). x=am-bn (a-qn+qn)m-bn = (a-qn)m+qnm-bn . (a-qn)m-(b-qm)n. Since x cannot Since a-qn = (a mod n)! Oia-qnO. i.e., negative, lx! x=bn-am, the proof is analogous.

(Note that if m and n are relatively prime, all integers are expressible in the form am+bn, where a and b are integers. Sm,n is the subset of the integers obtained by restricting a and b to be non-negative.)

Proof

Result

a>O.

(m-1)(n-11-1.

In all, numbers

Second

Let m and n be relatively prime positive Lemma —. Any x in ~ not expressible in the form integers. am+bn, ae~, belhl, is expressible either as am-bn, with ()~aO, or as bn-am, With O
m AND n

number

the

In place of our original proof of the result, we present a much shorter and more proof by Christos Papadimitriou. The proof use of the following lemma:

We will prove the following results about the set Sm,n ❑ {am+bn ! m and n are relatvely prime positive integers, aclt4, belN}:

in

of

am+bn o~k-i~k

Thus if

for OSiSk, i is expressible in the form Furthermore, if k-i is not. and only establishes a This whenever OLiLk.

one-to-one correspondence between those members are in Sm n and those which {0,1,. ... k} which It follows that exactiy half, or (k+l)/2, not. Since k is these integers are in Sm,n. in Sm n, there exist in highest integer not (k+l)/2 . (m-l)(n-1)/2 eleme~ts of ~ which not in Sm,n.

Sm,n.

(n-l)m is the last number in Sm n a new congruence class. it numbers greater than (n-l)m are Furthermore, the n-1 integers preceding (n-l)m belong to congruence

of are of the all are

The number introduces that all

in Sm,n. immediately classes introduced into Sm,n, and therefore belong to Sm,n themselves. The number preceding these n-l integers, however, belongs to the mod n congruence class which is not introduced until (n-l)m, so it cannot be a member of Sm n. This number, (n-1)m-(n-1)-1 = (m-l)(n-1)-1, ‘is therefore the highest number not in Sm n, establishing the first result. (A similar prhof of this result appears in a different context in Rangel [1974].)

is grateful to The author Acknowledgements. Thomas Cheatham, Harry Lewis, and Professors Christos Papadimitriou for their careful reading of the manuscript and many helpful suggestions.

156

REFERENCES

Aho ,

Hopcroft, J.E., and [:i;ii. The Design and Analvsis Addison-Wesley, Algorithms, Massachusetts

Bird,

R.S. elimination, 434-439

Bird,

[1977al. ~

Notes &

R.S. [1977b]. Improving recursion, introduction of (November), pp. 856-863

Burstall, R.M., transformation recursive (January), Chandra, A.K. linear Record ~ Switchin~&

Darlington, system programs, ~ PP. 44-67

Unman, J.D. of Computer Reading,

recursion (June), pp.

on 6

No.

programs by the CACM 20, No. 11

and

J. for ~,

[19771. developing No.

A 1

[1973]. Efficient compilation Of recursive programs, Conference 14th Annual SvM Dosium g .— Automata Theorv, pp. 16-25

Darlington, J., and Burstall, R.M. [1976]. A system which automatically improves programs, [email protected] Informatica ~, No. 1, pp. 41-60 Lewis,

ERRATUM

H.R. [1977], A new decidable problem, with applications, Proceedings, IEEE 18th .A—— SvmQosium Foundations QQ ~ ComDuter Science, pp. 62-73

the

In the sequence

solution

Paterson, M.S,, Comparative ~ Conference parallel

and Hewitt, C.E. [19701. schematology, Record of Pro.iect ~ Concurrent Svstems ~ Commutation, pp. llg-127

H.R. Translating [19711. equations into flow charts, (June), PP. 254-285

commutative

Mi&l

ARGUMENT[i] x := d(x)

:= x

;

[email protected];

&

Rangel, J.L. [1974]. The equivalence problem for regular expressions over one letter is elementary, Conference Record. ~ti Annual SvmDosium ~ Switching and Aut~mata Theorv, pp. 24-27 Strong,

to

should

i from BACKd[i]

be replaced ~i

m until n @ :. a(ARGUMENT[i]) by:

from

O until

max(m,n)

[email protected]

&Ya!l

recursion JCSS ~, INo. 3

ARGUMENT[i] := x ; x := d(x) [email protected]; -i ~o~m until max(m,n) @ BACKd[i] := a(ARGUMENT[i 1)

,

Without this correction, a problem arises for some x: In d-depth(c(x) ).d-depth(x) with arguments c(x) invocation of f2 be set to n+l, and the d-depth(x) - 1, m will of the second ~-loop will not be executed Consequently, the first time that the once. of the loop ~

is

i ~m, m-1 * BACKd[i] := b(ARGUMENT[i ], BACKd[i], BACKd[i+l])

executed,

undefined. one element, base rule in

157

BACKd[i+l

The correction BACKd[m], is each invocation

-1

until

[email protected]

].BACKd[m]

assures initialized of f2.

when the and body even body

will

that

at using

be

least the

## Characterization and Elimination of Redundancy in Recursive Programs

common generator redundancy, lprovided that the frontier condition holds for the set of functions. {gi!Omax(m,n)}. The. Fibonacci program discussed in. Section. 1 provides an example of common generator redundancy. Recall that under that interpretation c(x) . x-2 and d(x) . x-1. If we define g(x) . x-1, then c(x)=gm(x) and.

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