THE PROFILE MINIMIZATION PROBLEM IN TREES

THE PROFILE MINIMIZATION PROBLEM IN

TREES* DAVID KUO ANDGERARDJ.

CHANGt

Abstract. Theprofileminimizationproblemis to finda one-to-onefunction_ffrom thevertex setV (G)of a graph Gtothesetof all positive integers such that

_xeV(G)

{f(x) minyN[x]f(y)}isassmallaspossible, where N[x] {x}t3{y yisadjacenttox} isthe closed neighborhood ofx inG. Thispapergivesan O(n1"722)time

algorithm for theproblemina treeofnvertices.

Keywords,sparsematrix, profile,labeling,tree,leaf, centroid, basicpath,algorithm AMSsubject classifications. 05C78, 05C85, 68R10

1. Introduction. The profile minimizationproblemwas introducedby [5], [6]asa tech-nique for handlingsparsematrices. Forinstance,inthe finite element method[8],[9],we want tosolve a system oflinearequations

Ax

b where

A

is asparse symmetricn n matrix.

Suppose

for each rowi,aii 0 andPi isthe position of the first non-zero element in this row.

Wecall

UO _Pi min{j aij

0}

the width of rowi,and call

n

P(A)

Z

11)i

i=1

the profile ofmatrixA. Tostore

A,

weneedonlystoreL0 J- elements in eachrow i,which are from positionPi topositioni. The totalamountof storage for this schemeisthenP

(A)

+

n.

In orderto reduce theamount of storage, we needonly permute the rows and columns of

A

simultaneouslysuch that the resultingmatrixhasminimumprofile, i.e., weneedtofinda permutation matrix

Q

such that theprofile

P(QA

Qt)

isminimized.

Wecanreformulate thisproblemin termsofgraphs. Associatethe matrix

A

withagraph G such that

V(G)

{v,/)2

On}

and E(G) {(1)i,

l)j)

:/:

j andaij

0}. Note

that

P(A)-

wi

-(i-i=1 i=1

min

j)

vjEN[vi]

where N[vi] {1)i} I,.J

_{1)j

l) is adjacent to

_vj}

is the closed neighborhood of 1) _in G.

The row and column permutation

Q

corresponds to a one-to-one function

_f

from

V(G)

onto {1,2

n}

and

P(QAQ

t)

yxEv(6)

(f

(x)

minyENtxl f(Y)).

Thismotivatesthe definitionof the profile of agraphgiven below.

Fortechnicalreasons, however,weshall giveaslightlymoregeneraldefinitionthan that described inthe previousparagraph. A labelingofagraph Gisaone-to-onefunction

_f

from thevertex set V

(G)

tothesetof all positive integers.

A

labelingis simple ifitmaps

V (G)

onto

{1,

2

V(G)I}.

Foralabeling

f,

theprofile-widthof avertexx isdefined as

wf(x)-

f(x)-

min f(y).

yEN[x]

*Receivedbythe editors March 11, 1991"acceptedforpublication(in revisedform) July30, 1992. This work

wassupportedin part by theNational ScienceCouncil oftheRepublicofChinaunder grant NSC79-0208-M009-31. Institute ofAppliedMathematics, National ChiaoTungUniversity,Hsinchu30050, Taiwan, Republic of China

(gjchang@cc. nctu. edu.tw).

71

The profile

_of

Gwithrespectto

_f

is

Pu(G)

Z

wf(x)

xEV

andthe profile ofGis

P(G) min{

Pf(G)

f

isalabelingof

G}.

A

labeling

_f

isoptimal if

Pf(G)

P(G).

Thepurposeofthispaperis tostudytheprofileminimizationproblem, i.e., theproblem

ofdeterminingthe profile

P(G)

of agraph

G,

fromanalgorithmicpointofview. The profile minimization problem is analogous to the linear arrangementproblem, which is to find a labeling

f

ofa graph G such that

’{If(x)

f(Y)l

(x, y) is anedgein

G}

is minimized

(see [1],[3],[7]). Reference[5] provedthat theprofileminimizationproblemisequivalentto theproblemofintervalgraphcompletion,whichis knowntobeNP-completeevenwhenGis stipulatedtobe anedgegraph (see [4]). Themainresult of thispaperistogive an O(n

1"722)

timealgorithmfor theproblemwhen Gis a treeofnvertices.

Therestof thispaperisorganizedas follows.

In 2,

weestablish severalbasicproperties

that motivate the developmentof our algorithm.

In

particular, we provethat for a tree

T

thereexistsabasicpath c(x, y)suchthatP(T) P(T -or(x, y))

₊

IE(T)I.

Sotheproblem

becomes that of finding apathc(x, y)such that

P

(T-or(x, y))is minimized. Forthepurposes

of recurrence,wealsointroducetheproblemof findingapathc(x, y)such thatP(T-ot(x, y))

is minimized,withtheboundaryconditionthatyisfixed.

In

ordertodeterminethe basicpath,

3

developstheoremsthatnarrowthepossibilitiesforthe basicpath. Forinstance, weprove

thatc(x, y)containscentroids of thetree. Thisalso means that the number ofverticesofeach component ofT ot(x, y)isnomorethanhalf the numberof vertices of

T.

This isimportant in determining thespeedofourrecursive algorithm. Section4 uses these resultstodesign an algorithm, and

₅

analyzesthe timecomplexityof thealgorithm.

2. Motivating properties. This sectionshows theexistenceofabasicpathc(x, y)such that P(T)

P(T

a(x, y))

₊

IE(T)I

andintroduces theproblemoffinding aminimum suchpathwiththeboundarycondition thatyis fixed. The followingpropertiesareobvious and theirproofsare omitted.

PROPOSITION 2.1.

An

optimal labeling

_of

aconnectedgraph GmapsV (G)ontoaset

_of

consecutiveintegers.

PROPOSITION2.2([5]).

_If

His asubgraph

_of

G,then

P(H)

<

P(G).

PROPOSITION2.3 ([5]).

_If

G has m componentsG1,

G2

Gm,

then

P(G)

_Eim=l

P(Gi).

We can in fact assume that an optimal labeling ofa graph is simple even ifit is not connected.

Suppose

T is atreeofn vertices. For anyleaf x andanyvertexyin

T,

consider the unique (x, y)-path c(x, y) (v0,vl vr),where v0 x and

vr

y.

Suppose

that

foreachi, < < r,T c(x, y)hasn components Til, T/2

_Tin

eachwith a vertex _vii adjacentto vi in T (seeFig.2.1).

Let

f

be anoptimalsimple labeling of

F/

tO<_j<_niTij.

We

consider asimple labeling

fxy

definedby

fxy(1))

Ly(Vi_I)

+

J(/))

Ly(Ui-1)

_-"

IV(F/)I

+

ifv v0,

ifv 6 V(Fi), ifv vi.

SeeFig. 2.2 foranexampleof

_fxy

withoe(x,y) (a, b,c,d). Notethat the numbersbeside thevertices are theirlabels. Then

and

w./..,.(vo) 0,

w,.;,.(v)

fx(V)

fxy(V_)

-IV(F,.)I

+

forl <i _<r,

forv V(F,-).

Consequently

(2.1)

Wecalloe(x,y)thebasicpath(with respecttothe labeling

fy).

Notethat

fxy(Vo)

<

Ly()l)

<"" <

Ly(Vr)-

n. In general,an optimal labeling ofa tree isofthistype.

FIG. 2.1. Tree T.

THEOREM2.4.

_If

_f

isanoptimal labeling

_of

a tree T

_of

n vertices, then

_f

_.Ly

where x

f-

(1)isa

_leaf

andy

_f-

(n)isadjacentto at mostone

_non-leaf

vertex.

Proof.

Let oe(z, u) (vo,Vl v) be a longest path containingboth x and y, say,

x

v

andy vt for 0_< s < _< r. Notethatsinceris the maximum,v0and v,.areleaves. Inthis case

_n

0 and

_PU:,,

(T)

PJi,,,

(T),where

u’

v,._.

Suppose

T and oe(z,u) are as shown inFig. 2.1. Let

_fj

_f[v(r/)

be the labeling

_f

restricted onV

_(Tij).

Then,bydefinition,

(2.2) P(T)

Pu(T)

>

_wf(vi)

₊

i=0 i=! ./’=1

2(

1 1

Tll

T2

Ta

1 7 8 13

T

FIG. 2.2.An example_offy.

Note

that

(2.3)

_Wf(Ui)

_Wf(Ui)

>

_{f(vi)

_f(vi-1)}

n

IE(T)I.

i=0 i=s+l i=s+l

Consequently, by(2.1), (2.4)

ni

P(T) >

IE(T)I

+

P(Tij)-

PL.(T)

> P(T).

i=1 j=l

Therefore, allinequalitiesin(2.2)to

(2.4)

areequalities. Thisimpliesthe following:

(1)each

fj

isanoptimallabeling for

T/j,

(2)

wf(vo)

wf(Vl)

tof(Vs)

O,

(3)

Wf(l)t+l)

//)f(l)t+2)

Wf(Ur)

O, (4)

f(vi-1)

minyeN[vi]

f(y)fors

+

< < t.

Statement (2) impliesthats 0, otherwise

_wf(Vs-1)

f(vs-1)

f(Vs)

> 0. That is, x z, whichisaleaf. Statement

(3)

impliesthatr < t,otherwise either

_f

(vt₊ >

_f

(vt

+2)

or

f(vt+l)

<

f(vt+2),

i.e., either

Wf(1)t+l)

> 0 or

tof(vt+2)

> 0. Soeithery uory

u’.

In

the former case,y uisaleaf.

In

the latter case,y

u’

is adjacentto at mostonenon-leaf vertex, otherwise we can choose alonger c(z, u). Inthiscase, since

_Pfz,,

(T)

_Pfz,

(T),we

replaceuby

u’

and assumey u.

Note

that in this case

_nr

>0.

So,

nowcg(x,y) or(z,u). Finally,statement(4) impliesthefollowing"

(5)

_f

(vo) <

_f

(v) <... <

_f

(Vr)"--n, (6)

_f

(vi-1) <

_f

(vij)

for < < rand < j <hi.

Onthe otherhand,statement(1)andProposition2.1 imply that each

f(V (Tij))

contains consecutive integers. Fromthis,togetherwith statements

(5)

and(6), weobtain

_f

fz,

fxy"

[-]

COROLLARY2.5. For anytree

T

thereisanoptimal labeling

fxy

inwhichboth x andy

areleaves.

From

now on, alloptimal labelingswe consider are as specifiedinCorollary 2.5. The

path a(x, y)iscalled a basicpathfor

P(T).

Theorem2.4 and

(2.1)

tell us that in ordertofindthe profile of atree

T

we needonlyfind a basicpathct(x, y) whose deletionresults in a forest with the smallestpossible profile.

For

technicalreasons, we now consider thefollowingrestrictedpathdeletionproblem.

Suppose

yis a fixedvertexintree

T;

find apathot(x, y) endingatysuch that

P (T

ct(x, y)) isminimum. Weuse

P’(T,

y)todenote this minimumvalue.

We

also calla(x, y)the basic

pathfor

P’

(T, y).

Suppose

fxy

isanoptimal labelingof

T

andthetree

T

is asshown in Fig. 2.1.

Denote

byk

T

(respectively

T

k)

the subtree of

T

that containsv0,vl vk,

F1

Fk

(respectively vk vr,

Fk

Fr).

From

Theorem 2.4 and(2.1),we obtainthefollowing corollary.

COROLLARY2.6. Forabasicpath (vo,Vl v)

for

P(T),thefollowinghold: (1) (vo, Vl Vk) is a basicpath

for

P’(kT,

vk) and

P’(kT,

vk)

Eki__l

P(Fi)

Ei=,

Y.=,

P(Tij)

for

1 <k < r.

(2) (vk, _vk+l

v)

is a basicpath

_for

P’(T

,

vk) and

P’(T

k,

vk)

_.i=k

P(Fi)

Ei=

ET-_,

P(T,.)

for

1 <_ k <_r.

t-1

(3)

P(T) --IE(T)I

+

P’(ST, vs)

₊

P’(T

vt)

₊

_i=s+l

P(Fi)

for

<s < <_ r.

PROPOSITION2.7.

P(T)

<

P’(T,

y)

₊

IE(T)lfor

anyvertexyin

T.

3. Maintheorems. This sectiondevelopstheoremsthat restrict thepossibilitiesof the basicpathsforP(T)and

P’(T,

y).

In

particular, the basicpath or(x, y)forP(T)containsthe centroidsof

T. We

alsoprovethat the basicpathfor

P’(T, u)

iseitherct(x,

u)

oror(y,u),and the deletion of the basicpathfor

P’

(T,u)from

T

results inaforest each of whose components hasat most

_21V

(T)

I/3

vertices. These results arethekeystoneof our algorithm for theprofile

maximizationproblem.

A

centroid of atreeof n vertices is avertexwhose deletion results in a forest each of whose components hasat most

_/

vertices.

It

iswell known that atreehaseitherexactlyone centroidorexactlytwoadjacent centroids(see

[2]).

A"from leavestocenter"method can be

employedtoderivethe centroids of atree. Thismethod requires linear time.

THEOREM3.1.

Any

basicpatht(x, y)

_for

P (T)

containsall centroids

_of

T.

Proof.

Suppose

thereisa centroid of

T

notin the basicpath

or(x, y) (x l)1,u,

v

y).

Then

T

isof the form shown inFig. 3.1,with

_IV

(T’)I

>_

n/2

where n

V (T)1.

By

Corollary

2.6(3),wehave

(3.1)

k

P(T)

IE(T)I-t-

P’(T1,

1)1)

_-’

P’(T2,

v2)

₊

E

P(T/)

₊

P(T’).

i=3

Up

toasymmetric argument,wemayassumethat

IV(T1)I

IV(T2)I. Let

a(z, v)be a basic

pathfor

P(T’).

Corollary2.6

(3)

andProposition2.2give

(3.2)

P(T’)

>

IE(T’)I

+

P’(Ta,

a)

₊

P’(Tb,

b)

₊

Z

P()"

j--1

We

also assumethat

_IV

(Ta)l

_-<

_IV

(T6)I.

Now

considerthelabeling

fry

for

T.

By

(2.1)and

Corollary2.6, we have

(3.3)

k

Pfoy(T)

--IE(T)I

₊

P’(Tb,

b)

2r-

P’(T2,

/)2)

-

P(Ta) 2t-

y

P(Fj)

+

P(T1)

₊

P(Ti).

j=l i=3

Equations (3.1)to(3.3)togetherleadtothat

E T’)

<

P

_Ta

P’

Ta

a

+

P

_TI

P’

T

v Then

IE(T’)I

<

IE(Ta)I

q-

IE(T)[

byProposition 2.7. Thus

IE(T)I

>

IE(T’)I-

IE(Ta)I

>

IE(T’)[/2

(since

IE(Ta)I

<_ IE(Tb)I and

T’-

(Ta t3Tb)

5

0)

> (n

--IE(T’)I)/2

(since

IE(T’)[

>_

n/2)

>_

IE(T)I

(since

IE(T1)I

_<

IE(T2)]),

which is a contradiction.

FIG. 3.1.

Similararguments leadtothefollowingtheorem.

THEOREM3.2.

Suppose

yisa

_fixed

vertex

_of

atree T

_of

n vertices. Forany basicpath

ot(xy)

of

P’

(T, y),every component

of

T ot(x, y)hasat most2n

/

3vertices.

Proof.

The proofofthis theorem isexactly the same as that for Theorem 3.1, except now we assume

IE(T’)I

>

2n/3

and

IE(Ta)I

<

IE(Tb)[,

and there isno assumptionthat

IE(T1)I

<

IE(T2)I. However,

westillhave

IE(T’)I

<

IE(Ta)I

₊

IE(T)I.

Then

IE(T1)I

>_

IE(T’)I-

IE(Ta)I

>_

IE(T’)I/2

(since

IE(T)I

_<

IE(T)I)

> n-

IE(T’)I

(since

IE(T’)I

>

2n/3)

>_ IE(T)I,

whichis acontradiction.

Figure3.2 givesanexampleinwhich abasicpathcg(x,y) for

P’(T,

y)doesnotcontain thecentroid zofT.

FIG. 3.2.

THEOREM3.3.

_If

ot(x, y)isabasicpath

_for

P(T)andu is a

_fixed

vertexinT,theneither

or(x, u)oror(y,u) is abasicpath

_for

U(T,

u).

Proof.

Suppose

or(x,y) (x vl, ul, v2 y) and (u,u2 Ur u) is the uniquepathfromc(x, y)tou, as shown inFig. 3.3. Let or(z, u)beabasicpathfor

P’(T,

u).

FIG.3.3.

Case 1. z V(Tx).

In

thiscase,

us

u,v3 v, and

Tz

Tx.

By

Corollary2.6 (1), or(x,v)isabasicpathfor

P’(Tx,

Vl), andsoP(Tx or(x, vl)) < P(Tz c(z, v3)). Then

P’(T, u)

P(T or(z,

u))

P(Tz -or(z, v3)) q-

P(Ty)

q- P(Fi)

i=1

> P(Tx -or(x, v))

+

P(Ty)

+

P(Fi)

i=1

P(T o(x,

u)).

Hence c(x, u)isalso a basicpathfor

P’(T,

u).

Case2. z V

(Ty).

By

asimilarargument, or(y,u) isalso a basicpathfor

P’

(T, u).

78 DAVID KUO AND GERARD J. CHANG

Case3. z V(Tx) andz

T(Ty).

Let

T’,

T",

and

T’"

be subtrees,asshowninFig. 3.3.

Note

thatinthe caseofs 1,

T’

_Tx

tO

Ty

is notatree. Now

(3.4)

P(T -a(z, u))

P’(Tz,

v3)

₊

P(T’)

₊

_

P(Fi).

i--s

Note

that

P’

(Tz, v3) P (Tz a(z,v3)).

By

Proposition 2.2,wehave

(3.5)

s-1

P(T’)

>_ P(T)

+

P(Ty)

+

_

P(Fi).

i--I

Sincect(x, y)is a basicpathfor P(T),wehave

_Pfxz(T)

>

_efxy(T).

By

(2.1)andCorollary 2.6 (3)wehave

(3.6)

IE(T)I

+

P(Tx -or(x, Vl))

₊

P(Ty)

+

L

P(Fi)

+

P(T")

+

P(Tz -or(z,v3))

i=1

>

IE(T)I

₊

P’(Tx,

v)

₊

P’(Ty,

v2)

+

P(F)

₊

P(T"’).

Notethat

P’(Tx,

vl) P(Tx or(x, vl)). Again, by Proposition 2.2,

(3.7)

P(T’)

>

L

P(F)

+

P(Tz)

+

P(T").

i=2

Equations

(3.4)

to

(3.7)

togetherleadto

P(T -a(z,

u))

>

_P’(Ty,

/32) q-P(Tx)

₊

P(Fi)

₊

P(T) P(T -or(y, u)).

i--1

Hence

ct(y, u)isa basicpathfor

P’(T,

u). ]

4. Thealgorithm.

We

can usethe theorems in

₃

todesignanefficientalgorithmfor the

profileminimizationproblemin atree

T.

By

Theorem3.1, the basic idea of ouralgorithmis tofind a centroidzfirst in linear time.

Suppose T

z tO<i<mTi, whereu;istheonlyvertex of

T,.

that isadjacenttozin

T

(see Fig.

4.1). To

useCorollary 2.6(3), weneedto findall

FIG. 4.1.

profilesP(T/) and

P’(T/,

ui)recursively. Inthe following,Algorithm PROFILEfindsP(T)

and AlgorithmPROFILE1 finds

P’(T,

u).

Note

that, in ordertomake use of Theorem 3.3, AlgorithmPROFILEnotonlyhastooutputthe valueP(T)but alsoabasicpath.

ALGORITHMPROFILE

Input: A

treeTofnvertices.

Output: A

basicpath c(x, y) (v0, v, ,1)r) for P(T) and thevalues P(T) and

P(Tij)

for 1 < < r- and <_ j <_hi. Method:

1. findacentroidzofT.

2. letT z

U

<k<m

Tk

andzbe adjacentto uk V (Tk)for < k < m.

3. foreach 1 < k < m,recursively callPROFILEfor

_Tk

togetabasicpath ot(xk,

Yk)

and values P(Tk)and

P(Tkij),

where

_Tki

j arethecomponentsof

Tk

ot(Xk,

Yk).

4. for each < k < m, recursively call PROFILE1 for (Tk, Uk) to get a basicpath c(zk, uk) andvalues

P’(Tk,

uk)and

_P(Tij),

where

_Ti

_jarethe components of

Tk

(z,u).

5. let

P(T)

n

+

minl<_p<q<_m{P’(Tp, Up)

+

P’(Tq,

Uq)

-+-

Zip,q(Ti)},

wherep* and

q*attainthe above minimum.

6. letc(x, y)

ot(Zp,,

Up,)

+

z

+

ot(Uq,, Zq,).

7. combineprofiles

P(Tp,

ij),

P(T,)fork

:

p*, q*,

and

_P(Tq,ij)

togetprofiles

P(Tij).

To find

P’(T,

u),wenotethatbyTheorem3.3,eitheror(x,u) orc(y,u) is abasicpath

for

P’

(T, u). Sowe considertheconfigurationinFig. 3.3 with

Tz

omitted.

ALGORITHM PROFILE

Input:

Tree T ofn vertices with a basicpathc(x, y) (v0, Vl Vr) for P(T) and the

values

P(Tij)

for 1 < < r and < j < ni. uis afixedvertexin T.

Output: A

basic path or(z, u)

(Vo,

_v’

V’r,)

for

P’(T,

u) and the values

P’(T,

u) and

P(Ti.

)forl <i _{<r’andl <j<ni.} Method:

1. identifythepath (u,u2 Ur)asinFig.3.3.

2. recursively use PROFILE to solve P(Tx),

P(Ty),

P(Fi) (in fact

P(Tij)

for each

componentin Fi)for < < r.

3. a

P’(Tx,

v)

₊

P(Ty)

+

Yi=

P(Fi),

b

P’(Ty,

l)2)

₊

P(Tx)

₊

Yi=,

P(Fi),

where

P’(Tx,

v)and

P’(Ty,

1)2)canbecomputedfrom the input values

P(Tij).

4.

P’(T,

u) min{a,

b}.

ifa <b thenz x elsez y.

5. combine part of theprofiles

P (Tij),

P

_Tx

),orP

(Ty),

andP

Fi

toget profilesP

_(T,.}).

5. Time complexity. This section shows that the timecomplexities of the above two algorithms are

O(n1"722).

Let

f(n)

(respectively, g(n))be thetimecomplexityforAlgorithm PROFILE (respectively, PROFILE1).

In

Algorithm PROFILE,

Step

3 (respectively, 4)needs

_{Yim__}

f(ni)

(respectively,

Yim__

g(ni)) time, wheren V

(T

i)[for1 < < m. All other steps need O (n)time. Notethat for

Step

5 weonlyhave to find the smallest and the second smallest values ofP’(Ti, wi)

P

(Ti).

Therefore (5.) m

f

(n)

Z{f

(ni)

+

g(ni)}

+

cn, i=1 where

m

E

ni n and ni <

n/2

for

i=1

l<i<m.

Similarly, inAlgorithm PROFILE1,

Step

2 needs

_{zim___l}

_f(ni)

timeand all other steps need

O(n)

time. Thus,byTheorem3.2,

m

(5.2)

g(n)

_f

(ni)

₊

c2n, i=1 where m

Z

ni n and ni

2n/3

for i=1 l<i<m.

To

solve

(5.1)

and(5.2),we firstchoose a numbercr > 1, whichisverycloseto 1,say,

cr 1.001. Then choose <) _< 2 such that

e=

+

<1

and

6= (1

₊

o-e)2

()

x

Note

that a simple computerprogram givesthat)v 1.722forr 1.001.

THEOREM5.1. Thereexists a constantcsuch that

_f

(n) < cn andg(n) < ctren

_for

all n, i.e.,

_f

(n) O(n

)

andg(n)

O(nX).

Proof.

Theproofisbyinductionon n. Assumethat there exists a constant csuch that

f(n’)

< cn

’

and

g(n’)

< cren’z for all

n’

< n.

We

also assume thatc >

Cl/(1

3) and

>

C2/(0"

1).

For0 < a _< b, consider the function

h(x)

(b

₊

x)

+

(a x) b a where 0 < x < a.

Note

that

h’(x)

,k(b

+

x)

-

,k(a x)

-

>O. So his anincreasingfunction and thenh (x) > h (0) 0 for 0 <x < a. Thus

b+aZ<(b+x)+(a-x)

for O<x<a<b.

By

(5.1)and the inductionhypothesis,wehave

f(n)

< c(1

₊

re)

_’im=l

n}

+

clnwhere

Yi=I

n

_<

i=

ni n 1 and ni <

n/2

for 1 < < rn Repeatedlyapply

(5.3)

toget

n

);k n

(7)

+

(7

1)

z.

Therefore

f(n)

<_ c(1

+

r)2

₍

₊

cn

c3n

₊

c,n.

By

the choice ofc,

cn

< c(1 -6)n < c(1

-6)n

z.

Then

f(n)

< cn

.

m m

By (5.2)

and the inductionhypothesis,wehaveg(n) < c

_-,i=

nczn

where

_i=1

_ni < n 1 and_ni <

2n/3

for 1 < < rn. Repeatedly apply

(5.3)

toget

_Y.i=

n

<

_(2n/3)

z

₊

n Also,bythe choice of c,

czn

<_ c(

1)n

< c(r 1)n

z.

Theng(n) < cen

+

c(cr 1)en caen

.

Acknowledgments.

We

thank thetwoanonymousreferees formanyusefulsuggestions

onthe revision of thispaper.

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