An Analysis of the Full

Alpha-Beta Pruning Algorithm

GErard M. Baudet

Department of Computer ScLence CarnegLe-Metton Uni,versl.ty

Pi.ttsburgh, Pehnsylvanl, a 152[3

An anatysi,s of the alpha-beta pruning alBori,thm i.s presented which takes i.nto account both shaU.ow and deep cut-offs. A formula i,s fi.rst devetoped to measure the average number of termi,na[ nodes examined by the alEori,thm [n a uni.form tree of deRree n and depth d when ties are all,owed among the bottom posi.tions: specifi.caliºy, act bottom values are assumed to he {ndependent identi,catty distributed random variables drawn from a di,screte probabii,i.ty di.stri.butLon. A worst case analysi,s over all. possi.bte probabi.ti.i,y distri,bui,i,ons i,s then presented by ¢onslderin6 the tlmLti,ng case when the dL';crete probablU.ty dl.stri.buti.on tends to a conti.nuous probabitLty distri.butl.on. The branchi.ng factor of the alpha--beta pruni.nt~ atp, ori.thm [s shown to grow wi.th n as e(nAn -), therefore confi.rming a ctai.m by Knuth and Moore that deep cut-ells only have a second order effect on the behavi.or of the at6ori.thm.

1 - Introduction

Most so-catted i, ntei,t[gent programs use some form of

t ree searchi.ng; among them, most game playing programs are

buLtt around an effi.ci.ent tree searchi,ng algorithm known as

the (~Lpha-beta prtzni.ng aLgorittznz. ThLs paper i,nvest[gates

the effi,ci,ency of this a[gori.thm wi, th respect to a test

measure fi.rst introduced by Knuth and Moore i.n [3] and

El.yen i.n the foltowi.ng.

D e f i n i t i o n ! .1 :

Let Nn, d be the number of terminal posi.ti.ons exami.ned

by some algori.thm A bm searchi.ng a uni.form tree of degree

n and depth d. The quanti.ty

°

is catted the branchin E factor corrOspondi.ng to the search

ai,gorLthm A. •

ThLs research was partly supported by the Nati.onal Sci,ence Foundati.on under. Grant MCS 75-222-55 and the Offi,ce of Naval Research under Contract N00014-76-C-0370, NR 044-422 and partly by a Research Grant from the ]nsti.tut de Recherche d'lnformat~que et d'Automati,que (IRIA), Rocquencourt, France.

Si.mi.i,ar analyses have been attempted i,n two recent

papers by Fuli,er, Gaschni,g and Gi.ttogty [ t ] and by Knuth and

Moore [3]. Both papers address the problem of searchi,ng a

uni.form game tree of degree n. and depth d wi, th the

~-/~ pruni.ng algori,thm under the assumpUons that the n d

stati.c values assi.gned to the terminal nodes are i.ndependent

i,dent[cali,y clistributed random variables and that they are czll

dis[[net. We immedialety observe that, i.n order to evaluate

the branching factor, the tar, t as.-,umpti.on requires that the

n d di,stinct values assi,gned to the termi,naiº posi.ti,ons be

taken from arm i.nfi.ni.te range. For most practi.ca[ appti.cati.ons

this i.s, however, unreali sti,c.

Fuller, Gaschnig and Gitlogty developed i.n [1] a general

formula for time average number of terminal posi.ti.ons

e×amined by the ~,../~ procedure. Thei.r formula, however, i.s

computationalty intractable and leads to undesi.rabte

rounding errors for large trees (L e., for large n and d) siince

i.t i nvoi,ves, [n particular, a 2d-2 nested summati.on of terms

wLth atternati.ng si,gns and requi,res on the order of n d steps

for i.ts evaluai,i,on. Then they gave some empi,ri,cal resui,ts

based on a seri,es of simutati.ons, and compared the results

wi,th actual measurements obtai.ned by running a modLfi.ed

version of time Technology Chess Program [2].

In [3], Knui,h and Moore have analyzed, under the same

condi.t[ons, a si, mpler version of the full. w-./~pruni.ng

algori thm by not (:onsi,deri,ng the possi.bi.ti,ty of deep

cut-offs; they have shown, i.n parti.cular, that the branchi.ng

- 2 9 6 -

factor of the resuttiºng algori,tl~m is (9(n/tn n). Knuth and

Moore also considered other assumptions to account for

dependencies among the static val,ues assigned to the

term[nat positions and devei,oped anatytiic results under

those assumptions. Their paper gives, in addition, an

excel lent presentation and historical, account of the

=-/3 pruning algorithm.

Departing from the a.~sumptiions of the two papers we

just mentiioned, we first consider the effect of possible

equati,tLes between the values assigned to the terminal, nodes

of a uniform tree, assuming that these values are

independent i.dontical,l,y distributed random variables drawn

from any d~serete probabiitity distribution. |n Section 2, we

establish some notations and preiºiminary results, and in

Section 3, we derive a general formui,a for the number of

terminal nodes examined by the ~../t pruning algorithm when

we take into account both shallow and deep cut-offs. The

evaluation of this formula requires oni,y a finite summation

over the range of possible values assigned to the terminal.

nodes and i.s retatl,vety easy. We show, in partiicui,arj that,

when the termiinat nodes can only take on two diisti,nct

values, the branching /actor of the w-/3 pruning algorithm

can grow wdh n as O(n/l.n n) for some choice of the

probabiºti.ty di,stribution. }n Section 4, we show that, when

the discrete probabii,l,ty distribution tends to a continuous

probabi.t ity distribuUon, the summation derived in Section 3

can be replaced by an i,ntegral., which constitutes the worst

case over ai,t discrete probabitiºty distributions, tn

Section 5, an analysiis of this integral shows that the

branchi,ng factor of the ot../T pruning algorithm for a uniform

tree of degree tz grows with r~ as ®(n/in n), therefore

confiºrmi,ng a clairn by Knuth and Moore [3] that deep cut-offs

only have a second order effect on the average behavior of

the w-/~ pruning al.gorithm. Some concl,uding remarks and

open problems are given in the last section.

2 - Presentation and initial properties of the ~ - ~

pruning algorithm

There are two usual approaches for dealinR wi,th

searching a game tree. In [I], Futi,er, Gaschni.g and Giºttogty

adopted the Min-Mrzz approach, whiite, in [3], Knuth and

Moore chose the Nege-Mcz~c approach. We wiºl.t briefi,y

present, in Section 2.t, the two approaches and introduce

the w-/~ procedure in terms of the Nega-Max modet. Then, in

Section 2.2, we witi, reestabl,ish an initial, result of [1] which

was stated in terms of the Miin-Max approach.

2.1 - The 0~-/3 procedure

Let us consider a game (IHke chess, checkers, tic-tac-toe

or kalah) played by two players who take turns. It is

common to represent the evotutiion of the game by means of

a g,zme tree, where each positi,on of the game is represented

by a node. Lf the position is a dead-end, the node is

terminal, otherwise all possible moves from that posiºti,on are

represented as the successors of the node. The structure of

the tree is preserved by not generating moves leadiºng to

some posi,ti.ons already generated (thus, avoiding cyctes)l

thiis is the function of tile mo=,e Eetlerator. The eucb~Qtion

f~nct~on is another important function i.n game playing

programs ~ it assigns to each terminal position a static u,l~e

by esti.mati.ng various parameters such as piece counts,

occupation of the board, etc. The evaluati.on function

evatuates the terminal nodes from one player's viewpoint,

giv ing higher val,ues to positions more favorabiie to thiis

ptayer. It is convenient at this point to name the two

ptay.ers Max and Min. Hence, Max's strategy is to lead the

game towards positions wdh higher values, white Mi,n's

strategy is to lead the game towards posiitions with tower

values.

The .~irLLmc=z p,.oced~Lre is directl,y based on this

formutation and can be used by eiither Max or Min to decide

on his next move from a giyen position, assuming that his

opponent wit[ respond with his best move. Using a rather

- 297-

brute force approach, the minimax procedure assigns values

to al l nodes of a game tree. It f irst assigns to terminal

nodes the results of the evaluation function, then it backs-up

to internal nodes corresponding to a position from which It

i s Max's (Min's) turn to play the maximum (minimum) of the

values assigned to its successors.

Suppose i t is Max's turn to play from an init ial position

(corresponding to the root of the game tree), then it is his

turn to play from any positions at even depth and Mi.n's turn

to p lay from any positions at odd depth. Therefore, the

minimax procedure w i l l back-up values to the nodes of the

game tree through a succession of Minimazi.ng/Maximazing

operat ions. This corresponds to the Mi . , -M~ approach.

By observing that:

max{ mini ~cl, z2, ... }, mini :yj, :)'2 .... } .... } =

max{ -max{ -=1, "z2 .... }, -max{ -Yl, -:)'2 .... } .... } ,

the Min-Max approach can be directly reformul.ated into the

NeEa-M~z= approach. In the Nega-Max formulation, a

termina l node of a game tree should be assigned the result

of the evaluation function only if it is at an even depth

(assuming i t is in i t ia l ly Max's turn to play) and it should be

assigned the opposite of the result of the evaluation

funct ion if i t is at an odd depth, The Nega-Max approach

requi res the same operator at all levels of a game tree, and

the uni formi ty of the notation wi l l make it easier to carry

out an analysis. This approach wit[ be used throughout.

Figure 2.1 - Searching a game tree

wi th the minimax procedure

Figure 2.1 shows the effect of the minimax procedure in

a uniform tree of degree 2 and depth 4. The values assigned

to the terminal nodes have been chosen arbitrari ly. The path

indicated by a darker line show~ the sequence of moves

selected by the procedure. The minimax procedure is

c lear ly a brute force search and, when exploring a node, i t

uses none of the information already available from the

nodes previously explored. Obviou.,,ty, by taking advantage

of the information previously acquired we (:an easily

improve on the brute force search. Figure 2.2 presents

some simple patterns in which the distribution of the

in format ion could lead to such improvements.

Ca) shallow cut-off ".

(b) deep cut-off

Figure 2.2 - Example~ of po,.;sible cut-offs

The circ led nodes have already been explored, and they are

labe led w i th their hacked-up vah.les; the values of the other

nodes are yet to be determined. We are interested in the

value u of the top level node in both patterns Ca) and (b).

Let us consider the pattern of Figure 2,2 Ca) first. From

the def in i t ion of the minimax procedure, the values u and x

sat isfy:

=, = m e × { . 7 , - z . } , ~. = m a x { - 2 . . . . } ,

which shows that ~ >. --2 or 2 ~. -z. Since 3~. 2;~ -~, i t

fo l lows that i.ndepenclen.t o f file e~ecl uall.~e o [ X, we wit [

have IJ .-- 3. This .~,llows that we need not explore further the

succes.',ors of the node labeled :~ if we are only interested in

the value of u. This leads to a f irst type of elLt-o,t[s known

as sha.ZLouJ clLt-oJ'fs.

The pattern of Figure 2.2. (b) il lustrates a deeper clLt-o[[.

As w i th the previous example, there are immediate relations

between the values of the nodes. In particular, we have

- 298 -

y > -z, which reads u'.; to consider two cases. Either y > -Z,

and thi.s means lhat the va[Ue y is determi,ned by i,ts r[ght

son(s) and certai,nty does not depend on the r ight son(s) of z.

Or y .-- -z, i,n which case, sLnce z. ~ -y and z > -2, we deduce

~ - 2 or -~ .~ 2; but since u ,, max{3, -z } i,t fotl.ows that

,~ 3, i ,ndependent of the exact value of x and, a forti,or[,

i ndependent of the exact value of z. This shows that i.n

e i t he r case the successors of the node tabeted z need not be

fu r the r exp to red since the final vatue of u woutd LLn no way

be af fected.

The two examptes presented in Figure 2.2 indicate that a

reduc t ion of the search can be achieved Ef a node passes

down to i,ts sons the current value backed-up so far (3 [n the

case of the two above examptes) as a bound for pruni,ng

branches 2, 4, 6, ... [evel.-~ betow; the bound can, of course, be

i ,mproved as the search progresses down the tree (!.eadi.ng to

more and more possible cut-.offs).

Using two bounds for even and odd revels of a tree,

these improvements are implemented in the foltowi.ng

p rocedure adapted from [3].

i.nteKe[ I~rocedure ALPHA~3ETA(posHior~ P, Lnte~.'r atpha, integer beta):

beKi, n !nt/9_g- e_r j , t, , ; determit~e the successor positions: Pl ..... Pr#) i,l n ~ o . t h e n

ALPHABETA := [(P) etse

fo~r j := I s_t_e:p_ I tjntE_At n do

t := -.ALPHAf]ITA(Pj,-beta,-a[pha)~ [( t > alpha then alpha := t; .if atpha ;~ beta then F, ot__~o done (2.J) .e n_ql;

done: AI.PttABETA := alpha enc___ I_

end

The atpha-beta procedure (from [3])

The functi.on denoted by [ i,s the evatuation function whi.ch

assigns s ta t i c ualue,¢ to termi,nat posiHons.

Knuth and Moore [3 ] have shown this procedure to be

co r rec t i,n the sense that the ca|t ALPHAf3ETA(P,-oo,+oo)

assigns to pos[ti,on P the value MINIMAX(P), which i,s the

va lue assi,gned by the minimax procedure. More genera[i,y,

t hey showed [3, p. 297] that:

AI.PHABETA(P,alpha,beta) ~ atpha,

i.f MINIMAX(P) ~ atpha, (2.2)

ALPHABETA(P,atpha,beta) = MINIMAX(P),

i.f alpha < MINIMAX(P) < beta, (2.3)

ALPHABETA(P,atpha,beta) ~ beta,

[f MINIMAX(P) ~ beta. (2.4)

The same tree used in Figure 2.1 to i t [us[rate the

mEn[max procedure i.s shown i.n Fi.gure 2.3 to i.ttustrate the

ef fects of the w-/~ pro¢edure.

"W

• : k , ,_

Fi.gure 2.3 - Searching a game'[tee with the c¢-/~ procedure

The branches pruned by ti le procedure are indi.cated wi th

dashed ti.nes, and the nodes marked with a ci.rcte have not

been comp[ete ty exptored. We observe that onty 8 out of

the 16 te rmina l posi,t(ons and 19 out of ati, t i le 31 nodes are

examined by the ~--/'3 pruni,ng algorithm i,n this exampte,

reduci,ng 8 rea t ty the cost of searchinF, the tree. As i.s seen

by compar[nt~ Fi,gures 2.! and 2.3, the vatues backed-up by

the ~ - f l procedure to some [nternat nodes are not

necessar i ty the same as the values backed-up by the

mini.max procedure, as ref lected by the i,ndetermi,nati,on i,n

equati .ons.(2.2) and (2.4). The t o p value, however, i.s not

a f fec ted by thi,s i,ndetermi,nation.

2.2 - Some properties of the o~-f3 pruning algorilhm

In thi.s secti,on, we wi t [ introduce some notations whLch

wi.tt be used throughout, and we welt reestabti.sh, i.n terms of

the Nega-Max approach, an i,nit~at resutt of [],] givi,n 8 a

- 299-

necessary and suffi.ci.ent conddi.on for any node of a game

t ree to be exami,ned by the ot-/3 pruni,ng atgori,thm.

2.2.1 - Notations

As [n [3], we wi l t use the Dewey decimal notati,on to

represent a node i,n a tree. Ivloro preci,sety, let c, the empty

sequence, denote the root of the game tree. The.n, i.f

denotes some i.nternat node of the tree wi,th n sons, ~. j wi.tt

denote the j - t h son of node ~, for j = 1 ..... n. In Fi.gure 2.4,

node 4.L3.4.3 i,s the node at depth .5 whose path from the

root i,s i.ndi.cated wi,th a darker ti,ne.

¢(~) ,, 2

~4.D - <o

C(4.1 .3 ) - 3

C ( 4 . L 3 . 4 ) - - $

c ( 4 . 1 . 3 . 4 . 3 ) - 0

c~(4.1J.4J) - max{ o(4.1J.4J), ~ 4 . 1 J ) , c (4) } - 3

13(4.1.3.4.3). - .msx{ ¢(4.JJ.4), c (4 . | ) } - 5

Fi.gure 2.4 - PortLon of a game tree showi.ng the path

to node <4.1.3.4.3>

The value assodated wi,th some node c7 of a game tree

by the minimax procedure tsee Secti,on 2.1) wi, tt be denoted

by u(J). Then, if ~7 i,s a termi,nat node, rioT) i,s the #tat/c

ua£~ce asi,gned to that terminal posi.tion0 and, i.f c7 i,s an

Lnternat node, o(J) i,s the value backed-up to node ~ by the

mLni.rnax procedure. In the tatter case, i,f node ~/ has r¢ sons,

u(cT) i.s gi.ven by:

u(J) = max{ -v (2 . j ) l ! < j < , } . (2.5)

In Fi,gure 2.4, the nodes on the path from the root to node

4.1.3.4.3 are evaluated through formula (2.5) whi,te the other

nodes (Lnctuding 4.1.3.4.3) are shown as terminal nodes and

are assLgned arbi t rary values. (Nodes are labeled wi.th thei.r

values.)

While the values u(J) deal wi.th the stati,c aspect of a

game tree, the quantit ies we wi,tt i.ntroduce next deat more

wi . th the dynamic aspect of the tree when bei,ng searched by

the o~-/3 procedure.

For any node J . j at depth d ;,_ 1, we define:

c(J . j ) = max{ -v(; / . i ) I I .~ i ~ j - I } .

(By convention, the maxi,mum over an empty set i.s defi,ned to

be -co; Ln parti,cular, e(~/. l)= -oo.) For the root of the tree

we also deft.no c(~) = -co, The quanti.ty c(~7) accounts for the

i.nformati,on provi.ded to node 3 by Lts e/tier brothers. These

values are i.ndi,cated to the right of the game tree shown Ln

Fi.gure 2.4 for al l nodes on the path to node 4.L3.4.3; only

the nodes i.ndk:ated wi,th .,;quares are used i.n computi.ng

these vatues.

We fLnatl.y defi,ne for any node c7 = Jl . . . . . Jd at depth

d a X i.n a game tree two quanti.ti.es di.rectty associ,ated wi,th

node ~ by the ¢v-/~ procedure. For i = 0 ..... d-X, tet

o7i = J l . . . . . Jd-i" We deft.he:

¢e(~) = max{ c ( ~ i ) I i is e=/en, 0 ~ i ~; d - I } ,

/~(~/) = -max{ c (~ i ) l i i.s odd, O < i < d - I } .

I t i.s convenient to £1efi.ne these two quanti.ti.es for the root

of the game tree by ~(~)=-oa and /~(¢)= +oo (whi.ch i.s

consi,stent wi,th the defi.ni,ti,on). These u- and /~-vatues are

shown i.n Fi.gure 2.4 for the node 4.1.3.4.3 along wi.th thei.r

defi.nLti.ons.

2.2.2 - Necessary and sufficient condition for a node to be

explored by the ¢~-/J procedure

The fottowi,ng lemma justifi,es the notati,ons we just

in t roduced i.n the preceding secti,on.

Lemma 2.1 :

Assume that, i.nitia[ty, the root of a game tree i.s

exp lored by the ~-/3 procedure through the call

ALPHAE3E TA(root,-oo,÷oo) . (2.6)

Then, i,f node J is examined, i.t i.s through a cart of

procedure AI.PHA|3ETA i,n which the parameters alpha and

beta sati,sfy:

alpha = o f f ~ ) , and beta = /9(~) . (2.7)

- 3 0 0 -

Proof:

If ~7 = Jl . . . . . Jd denotes some node explored by the

procedure at depth d ~ 1, t,et, as before, ,-7 i ~ Jl ..... Jd4, for

0 < i ~d - l . Thus node //1 is the father of node ~, white, Lf

j~y> 2, node ~ l l . ( j d - l ) Ls the brother of ~ immediately

preceding ~/. (and explored just before ~/). Observe that, Lf

Jd = 1, c(~ O) = e(~]) = -¢o and therefore:

oKo~) = max{ c(J i ) } i Ls even, 0 ~ i ~ d - I }

= - [ -max{ e(Ji,+l) I i, is odd, 0 < i, < d-2 } ]

(similarly,/~(,7) = -(v(t71)). Observe also that, if Jd ~ 2:

or(2) = max{ oK(72), eL,7) }

= max{ ~ q 2 ) , ¢ [ ; t l . (?d -D ] , -u [21 . ( ja -D] }

and that i3(,7) = t~ [J l . ( j d - l ) ] .

By the cat,L of tLne (2.6), retat~ons (2.7) certainly hold for

the root of the game tree, since eL( t )=- (~ and 13(t)= +co.

Then the proof folt,ows by induction from EnspectEon of the

procedure ALPHA(ETA, and 'from the relations we derived

above. I

The for,Lowing theorem states a useful ret,ati.on that

characteri.zes the fact that a node of a tree is explored by

the o¢-/3 pruning algorithm. This relation was first

estabtLshed by Futter, Gaschnig and GiLtogty [1] wi.th

d i f ferent notations Ln terms of the MLn-Max model,

Theorem 2.1 •

Assume that, ini t ial ly, the root of a game tree Ls

explored by the 0~-/3 procedure through the call

ALPHABE TA(root,-oo,+oo) .

Then, an arbitrary node ~7 of the game tree wi l l be

subsequen t l y explored i.f and only if

~(o~) < /~(~). (2.8)

Proof:

Because of the presence of Line (2.1) Ln the procedure

ALPHABETA, the result for,tows directly from the result of

Lemma 2. l . i

Since i.t wi l t be more convenient in the following

sections, rather than ~(c7) and /~(~), we wi l l use the

quantit ies:

At2) = rn~x{ c(c7 i ) l i i seve" ,O~;L~;d- ] } ,

B(J) ,,- max{ c(Ji:) I/. is odd, 0 < ~ < d- ! } ,

where J~ i.s defi, ned as before. The definEU.ons of A(~) and

B(~.) are more symmetricaL,, and relation (2.8) can at,so be

rewri.tten Ln a more symmetric:at way:

A(~I) + B i l l ) < 0 .

3 - Number of nodes exp lored

p rocedure : d iscrete case

(2.9)

by the o~-/3

As Ln [1] and [3], we wLtt evat,uate in this and the

fotlowLng section the amount of work performed Ln searchi.ng

a r a n d o m uni form, game tree usi.ng the ¢v-/~ pruning

algorithm. The defLni.tLon and some properties of random

uniform game trees are given tn Section 3.1. The amount of

work performed by the at-/3 procedure is measured by the

number of terminal nodes examLned during the search and Ls

evaluated tn SectLon 3.2.

3.1 - Random uniform game trees

In order to perform an anat,ysis of the ot-/~ pruning

atgorLthm, we wi.tt remit ourselves and consider the for,towing

class of game trees,

Definition 3.1 •

A game tree in which

(a) art internal, nodes have exactly n sons, and

(b) al l termLnat, nodes (or bottonz posit~on~) are at

depth d

Ls cat,ted a u n i [ o r m lzame tree of degree n and depth d. A

uniform game tree which satisfies the additional condi.tLon

(c) the vat.ues assigned to at,t terminal, nodes (or

bot tom ualu.es) are independent i.dentLcaLLy

distr ibuted random variables

Ls called a tone{on= uni.J'orn= g~nze tree, or, for short, a rug

tree. i

- 301 -

UnLess otherwise specified, we wttt only consider

throughout a rug tree of degree n and depth d.

Since the value backed-up to a node by the minimax

procedure only depends on the backed-up values of its sons,

we immediately observe that, by condition (c), the backed-up

values of aLL nodes at the sam.e depth are also independent

identicaLLy dist r ibuted random variables. In the remainder of

the section, we wiLL assunie that the bottom values are

drawn from the f inite set { k / m l - m ~ k ~m }, for some

nt ~ O, and we wi l t denote by {p i ( k ) }_m~k~m or simply

{ P i ( k ) } the common probabi l i ty distribution for the

backed-up values of at[ nodes at depth d - i ([. e., p i ( k ) [s

the p robab i l i t y that the value, u(J) , backed-up by the

minimax procedure to some node c7 at depth d - i be k /m) . In

par t icu lar , {pO(k) } t.q the common probabi l i ty distribution for

al l bottom values, and {pd (k ) } is the probabil i ty distribution

for the value backed-up to the root of the rug tree.

The foLLowing temma states the relations between these

p robab i l i t y distribution.-,.

Lemma 3.1 :

For i = O, .., d - I , we have:

p i . l ( - n t ) + ;.. + P i + l ( k ) = [P i t - k ) + ... * p i ( m ) ] n . (3.1)

Proof=

Let ~/ be some internal node at depth d - i - l , then by

equat ion (2.5), v ( d ) ~ k if and only if -u(o?.j) < k, for

j = 1, ..., n. Equation (3.1) foLLows easily from the fact that

a l l var iables u( ; ] . j ) are independent. B

Since the quanti ty P i t - k ) * ... * P i (m) wil t occur again

tater on, we define for i = O, 1, ... and - m ~; k ~; m:

F i ( k ) = p i ( - k ) + ... * P i ( m ) .

For c o n v e n i e n c e , we also define p i ( - m - J ) = O. Note that

p i ( k ) ts a non-decreasi.ng function of k which satisfies

P i t - m - l ) = 0 and P i (m) = P i t - m ) * ... ÷ p i (m) = 1. By

rewr i t i ng equation (3.1), we see that Pi satisfies:

p i . l ( - k - l ) = 1 - [ p i ( k ) ] n for i = O, 1 .... , (3.2)

ancI, therefore:

p i , 2 ( k ) = 1 - { ! - [ p i . ( k ) ] n ] a f o r i = O, 1, . . . . ( 3 . 3 )

The foLLowing quantities wi l t also be useful in

Section 3.2. For i = O, I, ... and -m. - I .~ k ~ m, define:

p i ( k ) = I + [ p i ( k ) ] * ... + [P i ( k ) ] n ' l , (3.4)

and

o ' i ( k ) = I * [ p i t - k - l ) ] + . . . . [ p i ( - k - l ) ] n - I . (3.5)

Observe that P i t - m - l ) = o ' i (nt) . 1 and p i (m) = o ' i ( - n t - l ) = n.

Lemma 3.1 establishes the probabi l i ty distributions for

at[ the values in the nodes of a rug tree. The next [emma

establ ishes a similar result for the quantities c(rT) defined in

Section 2.

Lemma 3.2:

Let J . j denote any node at depth i, where i = 1, .... d. If

, j = 1, c ( J . j ) = -oo. If j ~. 2, then the probabi l i ty distr ibution

of c ( ~ . j ) , denoted by {qk(c ] . j ) }_nt~k~m, satisfies:

q _ m ( ~ . y ) + .:. + qk( [ ] .y) = [ p d _ i ( k ) ] J ' l . (3.6)

Proof:

When j = 1, ¢ ( J . j ) =-oo by definition. When j ~ 2 ,

equat ion (3.6) foLLows from the same argument given in the

proof of Lemma 3.1. |

In order to evaluate, through equat[on (2.9), the

p robab i l i t y that a terminal node is explored, we first need to

determine the probabi l i ty distributions for the two

quant i t ies A( [ / ) and B{,7). This is clone in the foLLowing.

Lemma 3.3:

Let ~ = J d - I . . . . . j l . J o denote any terminal node.

(1) If Ji = I for aLL e~zeft integers i in the range

0 < i ~ d - l , then A ( d ) ~ -co.

(2) Otherwise, the. probabi l i ty distribution for A(~) ,

denoted by {ak(~ t ) }_m<k<m, satisfies:

[9,i(k)] h-I a - m ( 2 ) * "'" * ak ( ; t ) -- ]-[e , (3.7)

where the product denoted by ]"[e is extended to aLL euen

in tegers in the range 0 ~ i ~ d - l .

- 302 -

SimiLar[y,

(1') If Ji = I for art odd i,ntegers i Ln the range

1 ~ i ~ d - t , then B(07) = -oo.

(2') Otherwi,se, the probab{l[ty d[stribut{on for B(ol),

denoted by { b k ( J ) } _ m < k ~ r n , satLsfies:

b 'm(~7 ) ÷ "'" ÷ b k ( 2 ) = ['[o [ ~ i ( k ) ] j i - I , (3.8)

where the product denoted by Fie i.s extended to art odd

i,ntegers i,n the range I .~ i ~; d . - l .

Proof:

We wi,tt only consider A(,')) since the proof retat{ve to

B(07) [s the same. Part (1) follows di,rectLy from the

def[ni.t{on. For par| (2), let o/~; denote the node Jd-J . . . . . JP

We note that A(/~) .~ k if and only {f e ( J i ) < k for all. even

i.ntegers i i,n the range 0 ~ i ~ d - I such that Ji > 2. S[nce the

var iables c(~7~) are i,ndependent, equal{on (3.7) foLLows from

equati.on (3.6) by observi.ng that, Ln the product T'[e, a factor

correspondEng to Ji = 1 amounts to 1. iii

The Last lemma [n thEs sect[on states the probab[t{ty of

exp lor ing a terminal node.

Lemma 3.4:

Let J = J d - I . . . . . j l . J o denote any termi.nat node. The

probab{t i . ty ;Y(07) that node ~/ i.s exam{ned by the

r~--/~ procedure {s gLven by ;Y(,}) = I if j~ = I for at{ euett

i,ntegers i: i,n the range 0 ~ i .~ d - l , or for at[ odd i,ntegers /,

[n the range 1 .,; i .~ d-- I , and by

~(~7) -- - m . ~ m - I ~k( '7) [ 6 - m ( 7 ) * . . . . b - k - l ( ~ t ) ] (3.9)

otherwLse.

Proof:

When j i = I for at[ even i'ntegers i [n the ranF.e

0 ~; ~ s; d - I t by Lemma 3.3 A(~)) ~ -co. Hence A(~/) + B(~/) = -oo

too, and by Theorem 2.1 node 0 7 [s certai,n[y exptored.

Si,.m[[arty when J i = 1 for al.L odd {ntegers [n the range

1 ~ i ~ d - l .

Otherw{se, both A ( J ) and H(,'~) are f[n{te. Let A(J ) ,= ~'k"

We observe that A(~7) + B(f'J) < 0 i,f and only [f -m ~; k ~; m- I

and -Z,n~ .~ B(, ')) .~ ~ - k - l " Hence, equati,on (3.9) foLLows from

Theorem 2.1 and the fact that A(,} ) and B(J ) are i,ndependent

var iables, m

Us[ng equati,ons (3.7) and (3.8), equati,on (3.9) can be

rewrLtten as:

~r(~) ,. -m,.~-.~n~-! ak(cT) Fie [Pi ( -k- l )] j i - I ,

~c~) = -m~.,-1 {Tie [W~)14-1 " Fie { W k ' l ) ] 4 - 1 )

~, T-[ o {(,,~-k-l)] j i - I (3.1o)

( recal l that p i ( - m - l ) = 0).

3.2 - Number of terminal nodes examined by the ~ - ~ pruning

algorithm: discrete ca.~e

We are now able to evaluate the amount of work

per formed by the w../;t procedure while searchi,ng a rug tree.

As i,n [1] and [3], we have chosen to measure the amount of

work by the number of termi,nat nodes exam[ned by the

procedure. (We wi.tt also consi,der briefly, at the end of the

secti,on, the total number of [nternat and fermi,nat nodes

exp lored by the procedure as a measure of performance.)

Theorem 3.1 :

The average number, Nn,d(nz) , of bottom posit[ons

exam{ned by the (~../;t procedure {n searchi,ng a rug tree of

degree n and depth d, for which the bottom values are

( f istr [buted accordi,ng to the di.screte probab[t[ty

di,str[buti,on {Po(k)} . .m.~k~m., i.s 6i.ven by:

N n , d ( n ' ) = t~[d/2J

* _,,L//~m [T[ e pi(k) - TT e pi(k-l)] TT o cri(k) , (3 .11)

where the quanti,t{es p£(k) and o' i (k) are def[ned by

equat{ons (3.4) and (3.5), and where the products denoted

by ]']'e and Fie are deft,ned i,n Lemma 3.3.

Proof:

By (:lefi,ni.ti,on of the probabi.t[ty ;Y(,}), the ave rage number

of bottom posi,tLons examined by the w-/3 procedure [s

Nn,d(n=) = ~ ~(fT) ,

where the sum I.s extended to air termi,nat nodes

- 303 -

~l = Jd - I . . . . . Jl'JO, and is actually a d-nested summation over

the range I .~ Jo -~ , , 1 .~ J l -~ n ..... 1 ~ Jd- I "~ n. The

summation can be rearranged as:

Nn,d (m) = ~ e ~([~) * ~o t((~/) + ~ " tC(~)

- ~t(l . . . . . 1 ) ,

where the three summations Ye, Z~o and z~' correspond to

the three expressions for ~(~) given Ln Lemma 3.4. The

four th term ~'(1 . . . . . f ) is subtracted from the sum si,nce i t Ls

counted by both z~ e and Yo" These two sums are easily

evaluated si,nce aLL the terms K(~) are 1. As ~(1 . . . . . 1) itself

Ls 1, we obtain:

Nn,d (m) - n [d/2] + n [d/2j - 1 "* Z ' tt(,7). (3.12)

It Ls to be noted that the first three terms correspond

exact ly to the number of term[nat nodes examined by the

u-/3 procedure under opti,mat ordering of the bottom' val,ues

(see [5, p. 201]).

We now evaluate the sum ~ ' . inside the sum the terms

~(~) can be eval,uated H1rough equation (3.10). We note that

aLL the summati,ons relat ive to Ji, for i = O, l , ..., d - l , can be

done i.ndependentLy, each one being the sum of a geometri.c

seri,es. Using the quanti,ties pi(k) and ¢ri(k) defined by

equations (3.4) and (3.5), we obtain:

Y ' tt([#) , - r n ~ , t - I []'Te pi(k) - H e I~i(k-l)] T'[° cri(k)

- ]"[e Pi ( m - l ) * 1 .

The theorem fol,tows from this Las t equation and

equatLon (3.12), using the facts that P i ( m ) - n and that

cr i (n t ) = 1. II

The formula of equati.on (3.11) can be easi.ty evaluated

and provides us wi th a measure of' performance for the

at-/3 prunLng al,gorithm. For some appti.cati.ons, however

(espocLatty when the cost of generati.ng moves i.s greater

than the cost of eval,uati,ng posi.ti.ons), i t i.s more convenient

to use the total number of nodes (internal. and termi,nat)

exp lo red by the procedure as a measure of performance. Let

Tr t ,d(m) denote the average of this number. The same way

we evaluated Nn,d(m) , we can eval,uate Tn,d(m) by summLng

the probabit i t i ,es ~'(J) over al,t nodes of the tree. We obtai,n:

° +

where Nin.,d(m) is the average number of nodes examined at

depth i, and is directl.y derived from the expression of

Nn ,d (m) i.n equation (3. t 1 ) by repl,acing d by /. and {PoCk)} by

{Pd_i(I¢)} (recal l that {PoCk)} i.s the probabiLity distri,bution

for the values assigned to the term[nat nodes and that

{Pd_ i (k ) } is the probabil,i.ty distributi,on for the values

backed-up to nodes at depth i).

3.3 - Bi -valued rug trees

Aiithough i t Ls retati,vety easy in most Kame pl,aying

programs to obtain (by inspecti,on of the eval,uation function)

an accurate bound for the range of distinct values assi,gned

to the various posi,ttons of the game, Lt is usuaLLy not so

easy to deri,ve a good estimate for the probabi l i ty

dLstri,buti,on of these val,ues. In the remainder of the section

we. w i l t study rug trees in which the terminal nodes can onty

take on two dist inct values, and we wit[ see, in parti.cutar,

that a change i,n the probabil,ity dL'~tributi.on of these val,ues

can Lead to very i.mportant differences in the growth rate of

Nn,d(n t ) .

We wi,tt assume i,n the fol,towing that the val,ues assigned

to the termLnat nodes of a rug tree can only be ei.ther -1 or

*1 wi,th respec t i ve probabili,ties l - p and p, for some

p ~" [0, 1]. Under these conditions, the number, Tn,d(p), of

termi.nal, nodes exami_ned by the at--/'3 procedure can be

obtai.ned as a parti.cular case of equation (3.11) in whLch

m = 1 and {Po(k)}_m~k~nt is defi,nod by P o t - l ) - l - p ,

Po(O) = O, po ( l ) = p.

Theorem 3.2:

Let PO = p, and , f ° r i . . . . 1,2 , ,i ,et p; = ! " P t - l "

Trt ,d(p) = n [d/2] + n[ d /2 j - I + (Pe-J ) (Po- l ) , (3.13)

wi,th

P e = ]-re P i * l , P o =' I -re Pi* l 1 - P i I - Pi '

- 3 0 4 -

where the product~ Tie and ]7" 0 are defined as before.

P r o o f :

Choose n= = I and define the probabi l i ty distribution

{po(k)}_m<k<m by po( -J )= J-p, PolO)= 0 and p o ( l ) - p .

Hence po(-2) = O, pO(- l ) = PolO) = p =, PO and p0(/) =, J. By

equat ion (3.2) we obtain, for i = O, l , ...:

p i ( - 2 ) = O, p i ( - l ) = Pi(O) - Pi ' Pi (I) = I ,

Then equation (3.13) fol lows directly from Theorem 3.1 and

equations (3.4) and (3.5). I

Equation (3.13) can be evaluated very easily and, tn

partEcutar, we note that for 0 < p < 1:

Tn,d(P) > Tn,d(O) = Tn,d(J)

= nfd/21 + nld/2J _ I . (3.14)

This last equation shows that Tn,d(p) reaches its minimum

n [d/2] * n ld /2 j .. I for p = 0 and p = 1. This is in agreement

w i th the result of Slagte and Dixon [5, p. 201] since i t

corresponds to the case when all terminal nodes are

assigned the same value and therefore all possibte cut-offs

do occur. Equation (3.1q) also shows that Tn,d(p) admits a

maxi.mum for p C (0, I); at:hough the exact maximum cannot

be read i ly obtained, we wi l t derive a lower bound m the

fo l low ing . We f i rst establish a preliminary result.

Lemma 3.5:

The unique positive, root, ~n, of the equation

~tt + ~ - 1 : 0

tS In the ~nterva| (0, I). Asymptotically (for large n) ~t

sat isf ies:

1 - ~'n ~ / in n . ( 3 . 1 5 )

P r o o f :

As there is no ambiguity, we wi l l drop the index n from

~'n i.n the fol lowing.

Let g(~.)= x n * x. - 1, note that ~ ' (0)=-1< 0 and

R(I) = I > O. Since i~(x,) is continuous and strict ly increases

for ~ posi t ive, the equation R'(~) ~ 0 admi.ts a unique positive

root , [ , Whi.ch is in the interval (0, 1).

We observe that equation ~.n + ~. _ I - 0 can be rewri t ten

as

I - ~ " = I

f rom which we deduce that

I. ~ E > _ _ L _ _ . ( 3 , 1 6 ) r ~ + l

On the other hand, since ~.n = I - ~', we obtain

rt (.~ - 1) > n ln~" = in(I-T) ,

which shows, along wi th equation (3.16), that

1 - '~ < I l n ( n 4 1 ) .-- / L n . 0(.-2). (3.17) ÷

S~mi.tarl.y, taki.ng the logarithm of both sides- of

equati~on (3.17), and using the facts that I - ~" - .~rt and that

tn ~" > I - ~ , we, obtain:

1 : ~" < I + |n(nAn n+l ) '

hence:

I - ~" > / | n ( n A n n+l) + (9 [ ( / in n) 2]

- - I . . • In t . . ) .

Equation (3.15) fol lows direct ly from the previous equation

and equation (3.17). I

When p = ~'n we obtain immediately that, for i = O, 1, ...,

Pi = ~'n" Hence

Pe = [.~n/(l-.~n )] [d/2] and eo = [ .~n/( l - [n ) ] [d /2 j "

From equations (3.13) and (3.15) i t follows that, for large n:

T n , d ( [ n ) ~ [nAn n] d , (3.18)

wh i le equation (3.14)shows that

Tn,d(O ) = Tn,d( i ) ~ (3.19)

Equations (3,18) and (3.19) indicate that Tn,d(p) can be

la rge ly influenced by the variations of the probabiKty

d is t r ibu t ion for the static values. This result can be easily

genera l ized to Nn,d(m). In the next section, we wi l l derive

an approximat ion to Nn,d(.m) which corresponds to its worst

case behavior.

4 - Number of nodes exp lo red by t h e o~-/3

p r o c e d u r e : cont inuous case

In this section, we derEve an approximation to Nn,d(m)

by cons ide r ing , time l imit of the finite series of

- 305 -

equati.on (3.1 1) when m |ends to infi.ni.ty whi le the discrete

probabiAi. ty distri.buti,on {Po(k)}_m.~k~m tends to a conti.nuous

probabi.U.ty di.stributi.on. Thi,s corresponds to the case

studi,ed by Fui,ter, Gaschn[g and Gi.ttogty [1] and by Knuth

and Moore [3 ] when the terminal, nodes of a rug tree are at[

assi.gned di.sti.nct va!.ues. In parti.cutar, we wi,tt r~estabtiºsh

(wi,th a much si,mp[er formul.a) a resutt of [1].

4.1 - Notations end preliminary results

We fi.rst i.ntroduce the sequence of functiºons { f i } mapping

the i.ntervai, [0, / ] i.nto i,tsetf, and clef[ned recursi.vety by:

re(=.) = x ,

f i t= ) = I -- {z - [ , r~_l (~ . ) ]"}" f o r ~ = 1, 2 . . . . .

] [ [s read [ ty veri,f[ed by i,nduct[on on i that art functi,ons f i

a re s l t r [c t ty i,ncreasing on [0, I ] and sati,sfy [i(O)= 0 and

f i t 1 ) = 1, L e., 0 and I are two fi,xed poi,nts of the functi,ons

f i , fo r at[ n and i. The functi,on [ i wi,i,t be shown to be

re ta ted to the quanti,ti,es ~2i(k) defi,ned i,n Secti,on 3.1.

Si.mi.l.arty, i.n rel.ati, on to the quant[ti.es P2i.(k) and o~2~,j(k) ,

we defi.ne the foitowi.ng functiºons on [0, 1]: for /. = 1, 2, ...,

Let

l - [ f i _ / ( z ) ] n

r i (=) l_fi_l<X ) '

f /=> si(x) [ [ i_ l<X)]n

If we def ine r i ( l ) = n and si(O)= 1, we observe that at[

funct ions r i and s i are cbnti,nuous on [0, 1] (they are

actua[ ty poi,ynomi,als i,n x), and that r i [ s stri,ctty i,ncreasi,ng

whi.te s i i,s str i ,ctty decreasi,ng.

in retati.on to the two products ]-r e and ]7"o, we atso

I .ntroduce, for i = J, 2, .., the fottowi,ng functi.ons on [0, 1]:

Ri(x) = ,.l(X) ..... rli/21(x) ,

~i (~) = s l (x ) × ... × s[i/2J(Z) ,

w h e r e Sl (~ ) = 1. Observe here, too, that functi,ons R i and S i

are potynom[ats , and that, when ~ increases from 0 to 1,

Ri(x.) [ncreases from I to n [i'/21 whi,[e Si(~) decreases from

nL~12J t o 1.

Lastty, for k = 0, 1 ..... 2m.1, tet

c k = ~ O ( k - m - l ) .

Lemma 4.15

For i = 1, 2, ... and k = 0, ..., 2re+J, we have:

r i(t : k) = P2i_2(k-m.-J) , (4.1)

si(c k) = c~2b. l(k-m.-J). (4.2)

Proof:

The proof re[ tows tmmed[atety from the defi.ni,ti,ons by

[nducti.on on i. I

4.2 - Number of bottom posilions examined by the ~-/~

procedure: continuous case

Let us re turn to the defi,ni,ti.on of the sequence

T m = {Ck}O~k~2nz. 1. As was observed i,n Secti,on 3.1 wi,th

the sequence {piCk)}, the sequence T m i,s non-decreasLng end

de f ines a parti,ti,on of the [nterva| [0, J], i,. e.:

0 = c O ~ ~I ~''' ~ ~2m ~ ~2m+1 -- I .

The norm of the parti.ti.on T m iºs

I ITml l - m a x { c k - ok_ 1 I I ~ k ~ 2,n*~ }

= max{ PoCk) [ -m ~ k ~ m } .

In the remainder of the secti,on we requi,re the fottowi.ng.

Assumption;

CA1) ti,m max{ PoCk) I -m < k ~ m } = O. 1

This assumption ensures that the norm of the parti,ti,on

T m tends to 0 when m tends to infi,ni,ty: It also shows that,

as m tends to i,nfi,ni,ty, the probabi,ti,ty of two termi,nat nodes

bei,ng assigned the same value vani,shes. This corresponds

to the case studi,ed by Futter, Gaschni, 8 and Gi,ttogty [1], and

by Knuth and Moore [3].

WLth thi,s assumpti,on, we wi,tt now see that the fi,ni,te

ser ies of equat i ,on(3.11) can be reptaced by an i,ntegra[

when m-~oo. Thi,s i,s estabti,shed i,n the fottowi,ng.

Theorem 4.1,

Under assumpti.on CA1), we have:

Nn,d(m) = n[d/2J" . /01 R'd(t).Sd(t).dt, i,i,m (4.3)

- 306-

where R~(~.) is the f i rst derivative of Rd(X).

Proof:

Since there is no risks of confusion, we will. drop, Ln the

fottowi.n6, the l.ndex d from the functions R d and S d.

It fol lows direct ly from Lemma 4.1 that, for

k = O, .... 2m.+h

R(~k) = Tie Pi (k-"t-l) , S(~ k) = T~ o %(h--hi-l),

which shows that equation (3.1 1) can be simply rewrit ten as:

Nrt,d(m) = n[ d/2j * z~_ [R(~ k) - R(~k_l) ] S(~ k) l ~k<2nt+ l

Let A m denote the series defined in this Last equation.

Recall that R(~.) is a polynomial. By considering the

Taylor development of R(t:k_l), we obtain for k = 1, ..., 2nt*h

R(c k) - R(ck_ 1) = [Ck-Ck_l] R'(c k)

, L 2 [Ck-~'k-l]2 R"(tk) '

where ~k- I "~ tl¢ ~; ~k" t4ence:

An~ = ~ " [ t :k-~k_l] R'(~" k) S('z~k) l~k~2m*l

+ 1~k~2nz*!Z /2. [8"h-ok_l]2 R"(t k) S(~" h) . (4.4)

SLnce R and S are polynomials, the quantity IR"(x)S(~)/21 ts

bounded by some constant, say M, for any x and y in [0, 1].

In part icular, the second sum in equation (4.4) Ls bounded Ln

module by M.IITmll.[r.2,~,l-CO] =/4.117,.11 and therefore tends

to 0 when m -* co since, from assumption (A1), IIr,tll -* 0.

As fo r the f i rst sum in equation (4.4), we observe that i t

corresponds to a Riemann sum for the function R'(x.)S(;¢) over

the par t i t ion T m of [0, 1]. Therbfore since, in particular,

this functl.on is continuous and since UTmll tends to 0, the

sum tends to the integral of equation (4.3). This proves the

theorem. |

In the remainder of the section we wi l t reinterpret the

Limit of Nn,d(nz) established En Theorem 4.1.

Let C be the distr ibut ion function of some continuous

p robab i l i t y density function ~', and assume, to simpLi.fy the

discussion, that O(- l ) = 0 and C(l ) = I (therefore, C(~) ~ 0

for x s; -1 and C(x) ~ ! for x>. 1). We define a sequence of

functions C m for m = O, 1, . . as follows. For - m ~ k ~m, Let

~'1¢ = k /m. Functi.on C,~ Is defined as the following step

function;

0

C.,(x) = C(x. k)

!

i f x < ~ _ t n =0,

if x k < x. < Xk. l , for -m ~; k ~; m-X,

if 1 = x.nt ~ ~..

The sequence of functions {C,m} constitutes a sequence of

approxi,mations to the continuous function G. (]t should be

noted that the convergence of the sequence is uniform on

the interval [0, 1].) The function Cnt corresponds to the

cumulat ive distributLon of the discrete probabi,tl.ty

d{stributl.on Polk )= Cm(xk ÷) -Cm(X.k-) associated with the

poi,nts =k = k/m., for k = -m, ..., m.

Usl.ng the approximation {Po(k)}_nt~;k~m to the densdy

functl.on g, equation (3.1 1 ) provides us with an

approxLmati.on to the average number of bottom posi.tLons

examl.ned by the oz-fl procedure in a rug tree in which the

bottom values are drawn from the continuous probabffity

densi,ty functLon @'. When nt becomes Larger, the

approxLmatl.on becomes better, and (due to the uni,form

convergence of the sequence C,t) it can actually be shown

(i,n a rather technical way) that the Limit of Nn,d(m) when

m -~ co corresponds exactly to the average number of bottom

posi t ions examined by the ~../3 procedure Ln the continuous

case. As .a matter of fact, equation (4.3) could be derived

d i rec t l y by considering a continuous probabUity distribution

rather than a discrete one in very much the same way we

der ived equatl.on (3.1 1) in Section 3. This result is stated [n

the fof fowing.

Theorem 4.2:

Let tO(X) = x, and, for i = 1, 2 ..... define:

f~(x) = I - { I - [f~_f~)].}n,

l - [r i_l(~)]" r i ( = ; l - f ~ _ / ~ ) '

fix.) si/~') [fi.l(x)]a '

- 307-

R~(¢) = r1(~) ~ ... × r[d2](~.) ,

S;/z) = s1(~) .... ~ ,[i./2j(~).

The average number, Nn,d, of termi,nal, nodes examined by

the =-/3 pruni.ng al.gorithm in a rug tree of degree n and

depth d for whi,ch the bottom values are drawn from a

conti,nuous distri,buti.on is given by:

Nn,c t = ntdl2J , fo I R'd(t).Sd(t).dt. (4.5)

]t i.s to be noted that, unl,ike the case of a diºscrete

prebabi t i ty diºstributi.on, when the bottom values are drawn

from a ¢onti.nuous di.stri,buti.on, the number of termi,nal,

posLLtiºons examined by the 0,-/3 procedure does not depend

on the di.stri,bution function.

4 . 3 - D i s c r e t e case v e r s u s continuous case

SLnce equation (4.5) has been derived as the U.mi.t of

equatiºon (3.1 1), i t is reasonable to investigate the val,i.di,ty of

the approximation of Nn,d(m) by Nn, d. As was seen iºn

Sectiºon 3.3, Nrt,d(m) strongi,y depends on the probabil,i,ty

di.strtbtJtion {PO(k)}_nz.,;k<m and, l,herefore, we cannot expect

Nn, d to be a close approximation of Nn,d(m) i.n all cases. We

wi,tl, see beLLow, however, that Nn, d provi,des us with a good

i.nsi.ght i,nto the behavior of the 0`-/3 pruning al,gori,thm.

Name[[y, we wtl,LL see that it consti,tutes the worst case of

Nn,d(rn) over all. discrete probabil,i.ty distributions.

Since Nn, d was obtai.ned as the l,imit of Nn,d(m), i.t is

suffi,cient to show that, for all. probabi,lLty distri,butions

{Po(k)}_m<k~;nL, we have:

Nn, d ;~ Nn.d(m). (4.6)

In order to prove inequality (4.6), Lt is conveni,ent to give a

geometric Lnl,erpretation of both Nn, d and Nn,d(m).

Consider the curve (/;) defined by the Cartes[an

coordi,nates ix., :y) through the paramel,ric equations

(,C): [ ~ = R d ( t ) , ~, = S d ( t ) ] ,

where the parameter t varies in the interval. [0, 1]. The

integraL[ of equati.0n (ll.5) represents the area dei,imited by

the curve (,C), the =-axLs and the paral,lel,s to the 3,-axi,s at

the abscissas Rd(O) = I and Fed(l) = n[ d/21 (see Figure 4.1).

Siºnce Rd(O) = 1 and Sd(O)= n[ d/2j, the term r= [d/2j of

equati,on (4.5) can be accounted for by the area of the

rectangl,e del,imited by the ~-axls. tbe y-axis and the tines

x. = ! and :), = n[ d/2j (tl~e tatter lºi,ne extends the curve (/;) Ln

a continuous way). Figure 11.1 represents the curve (a~) end

its e~cter~i, on in the case n = 3, d = 6. The area beLLow the

unbroken LLLnes represents the quantity Nn, d.

S d(t)

27

24

2t

19

15

12

9

6

3

1 0

\ r

! - - i

L . . . . . .

I I I 1 6 9 112 115 118 211 214

1 27

Figure 4.1 - Geometric intrepretation of Nn, d and Nn,d(m)

The sum of equation (3.1 1) can also be represented along

wi th the curve (£,). It follows di,rectl,y from the relations of

equations (4.1) and (4.2) that the terms of the sum represent

the areas of the rectangLLes detiºmi,ted by the I.iºnes

~c = R(t:k_i) , ~ ~ R(~k) , y =.- 0 and y = R(~k) , for

k = 1, 2, ..:, 2nz-l. The quantity Nn,d(m) represents therefore

the area of Fi.gure 4.J shown beLLow the broken l,iºnes.

Inequal i ty (4.6), l,hen~ follows directly from the fact that,

when t iºncreases in [0, I], R(t) increases whLl,e Sit)

decrea.,;es.

5 - On the branching factor of the ~ - / 3 pruning

a lgo r i t hm

We have del,iberatety chosen to introduce first the case

308 -

When the bottom values of a game tree are drawn from a

di,sc:rete probab[li.ty distribution since it is of most interest

i,n actual appi,Ecations. The case of a continuous distribution,

however, tends itself to an easier analysis, and, si.nce i,t

constitutes the worst case over a[i, discrete probabii,i.ty

distri.butions, we witºt, iºn this section, examine the i.ntegraiº of

equati,on (4.5) rather than the series of equation (3 . I l k

5.1 - Previous results

In Section 1, we i.ntroduced the branchin~ factor as a

cost measure for the work involved in searching a tree.

Rather than cons[[tiering the number, Nn,d, of termi,na[

positi,ons examined by a search algorithm, as a measure of

performance of the algorithm, we could have cons{dered the

total number, Tn,d, of nodes (terminal, and internal) explored

during the search. In the case of the ~-/~ pruning ai,gorithm,

si.nce Nn,d, gi,ven by equation (4.5), does not depend on the

distributi.on function of the bottom values, we deduce that

Tn, d sati,sfies:

Tn, d = 1 * Nn, ! *... + Nn, d .

St,nee i,t can be checked easli,y that 0 < Nn,i_ I ~ Nn,i, we

obtain Nn~,d =; TR, d ~; dNn,d, and therefore:

~<,> ( r , , ,d / Id : ~.,>,<,> (N,, ,d/ le : S~_i3(,).

Thus Defi.ni.tion 1.1 provides us with a measure of

performance useful to compare search algorithms. In the

fottowi,ng, we review some of the resui,ts which have already

been presented in the literat.ure.

Minimax search

The mi,nimax search examines systematically all. nodes of

a tree. It, therefore, examines Nn, d = n d terminal, nodes in a

uniform tree of degree n and depth d, [eading to a branchi.ng

factor

~ntittintct~(n) = n.

~- /3 procedure under optimal ordering

Stagte and Dixon [.5, p. 20J] have shown that, when al.t

possi.bte oz- and /3-(ut-offs occur, the u=/3 procedure

exami,nes

t , /n, d = n r d / 2 ] + nLd /2J _ S

termi.nat positions. In this case, the corresponding branching

factor [s

~opt(n) = e l~ 2 ,

o~-,d procedure (experimental results from El])

Based on a series of s[mulation results, Fuller, Gaschni,g

and Gi,i,i,ogty [1] have argLled that the formula

Nn,d = c(dj.nO.72d + 0.277

consti.tutes a reasonable approximation to the number of

bottom positions examined by the ~--/~ procedure for stoat{

values of n and d, and that 1 :; c(d)~; 2 (at [,east for the

range of vatues they considered). For comparison purposes,

tel us assume that their approximation can be extrapolated

for any n and d. Provided that c(d) l i d --~ I when d -~ oa, we

obtain

~oL..//n) ~ n 0.72 "

In v iew of the results of Section 3.3, we can questi,on the

accuracy of the approximation for i,arge n st,ace it foi,i,ows

from Theorem 3.2 that

t~oo [Tn,d([n)] l id = O<nAn n) . i

o~-/J procedure without deep cut-ells

Knuth and Moore [3] have anai,yzed a si.mpl.er verst,on of

the o~-/~ procedure by not considering the possi,.biiºi,ties of

deep cut-offs. Thi.s /~-procedttre is the same as the

~-/~ procedure except that no ~,-vatues are passed to the

~-/3 procedure; instead, the lower value o~ is always set to

-co before exploring the successors of a node. Knuth and

Moore have shown that the branching factor of this

procedure sati.sfies

~p(n) = e(nAn n).

Note that, since the /%procedure always explores more

nodes at any depth in a tree than the fui,t or-/3 procedure

does i,n the same tree, 7~/,3(n) provides us with an upper

bound for ~OZ./~(n).

- 309-

5.2 - Bounds on the branching factor of the o¢-fl procedure

In this section we w[t[t derive some tower and upper

bounds on the branching factor of time at-/.~ pruning algorithm.

In partLcutar, since the tower bound we derive Brows with n

as nan n, we wU_t be ab[te to conc[tude, using the resutt on

the branching factor of the ~zT/~ procedure uAthout deep

cut-offs estabt[shed by Knuth and Moore [n [3], that the

branching factor of the v¢-/3 procedure {s (9(n/tn n).

We introduced {n Section 4.1 the sequence of functions

,f'p /. = O, ! .... , from [0, 1] to itsetf, and we observed that aft

functi.ons .¢~ share {he two fixed points 0 and I (independent

of n). Another common fixed point, Whi.ch depends on np was

i.ntroduced [n Section 3.3.

Lemma 5.1:

For a given n, aft functions /i, for i = O, I,..., share the

common f ixed point [ n ~ (0, I), the unique positive root of

the equation

~rt + ~ _ 1 = 0.

Proof:

Since J'O(X) = x, .~" [s certainty a fixed point of JrO; then,

the proof fottows [mmedLatety by induction on i. J

Since ~r a is a fixed point common to a[tt functions f/,

/. = 0, 1, ..., i t is easy to eva[tu'ate at fhLs point the functions r i

a n d s~ d e f i n e d in S e c t i o n & l . For i = 1, 2, ..., w e d e d u c e

t h a t "

ri(~ n) = si(~ n) = ~ n / ( l - ~n ) . (5.1)

In partLcutar, [t lot:tows from Lemma 3.5 that, for targe n:

r~(~. n) = sif~ n) " nan I t . (5,2)

Equations (5.1) and (5.2) walt be useful to obtain the desired

bounds [n the remainder of the .,;ect[on.

The geometric representation of equation (4.5), given [n

Fi.gure 4.1, makes it easy to derive bounds on the quantity

Nn, d, They are stated [n the lot[Lowing.

Theorem 5.1 :

The branching factor of the w-/~ pruninK algorithm in the

search of a rug tree of degree n satisfies:

nan n ~-~n/(1-~n) < )~cz_fl(.)

< " ¢ n f n / ( l - f n ) ~ n/~Cn--n, (5,3)

f o r n = 2, 3, ....

Proof:

Si.nce, when t increases in [0, 1], Rd(t) increases white

Sd(t) decrea.,;es, i t for[tows directty that for any at in [0, 1]

we have the felt[towing inequalities:

Rd(~).Sd(~) < Nn, d

< Rd(~).Sd(O) * i R a ( I ) - Rd(W.)].Sd(~). (5.4)

If we choose at = J'n, we have Rd(~ :) = [ [n / ( l - . [n) ] [d /2] and

Sd(~) = [~n/( I -~n) ] [d /2J. SEnce Rd(l) = n [d/2] and

Sd(O) = n [d/2j, [nequatLty(5.3) felt[tows immediately from

inequat i ty (5.4) and the resutts of Lemma 3.5. |

As an immediate consequence, we obtain the fo[ttowing.

Theorem 5.2:

The branching factor of the ~-/~ pruning algorithm [n the

search of a rug tree of degree n satisfies, for targe n:

~ot. f l fn) = @(nAn n) .

Proof:

The resutt comes directty from the [tower bound nan n of

Theorem 5.1, and from the upper bound f~fl(n) obtained for

the ez.-/3 procedure ,,i.thout deep cut-offs, which Knuth and

Moore have shown to be e(./tn n). II

This resutts confirms, as was suggested by Knuth and

Moore [3, p. 310], that deep cut-offs have only a second

order effect on the behavior of the t~-/3 pruning a[tgorithm.

On the other hand, it shows that the formu[ta proposed by

Fu[tter, Gaschnig and Git[togty i n [ l ] and mentioned i.n

Section 5.1, if it constitutes a reasonab[te approximation for

sma[tt values of n and d (the range of values they considered

is n * d ~ 12), [s certa[n[ty not adequate for [targe vatues.

- 310 -

We note thai the bounds of Theorem 5.1 were obtained

w i t h o u t d i f f i cu t ty by ¢onvenientty choosing just one point,

~'n, on the curve (,C) since i t was easy to evacuate both

Rd(~n) and Sd(~n). ]n the next section, using a d i f ferent

approach, we wUt der ive a t ighter upper bound for Nn,d, and

hence for ~o~..f3(n).

5.3 - Improved upper bound

Since, for d = 1, 2, ..., Nn,d < Nn,d+ 1 < nNn,d, then, if

(Nn ,d ) l / d tends to some t imit when d tends to inf in i ty as an

euen i n teger , this quant i ty tends to the same t imit when d

tends to in f in i t y as an odd integer. Therefore, wi thout toss

of genera t i t y , we w i l t onty con.,;ider, in this section, the case

when d [ s an even integer. Let d = 2h.

For ~. in [0, 1] and for i = 1, 2,..., we define

pi (~) = ri(t~)si(~).

Lemma 5.2 =

At: funct ions Pi, for i = 1, 2 ..... have the same absotute

maximum, Mn, in the {ntervat [0, l ] .

Proof :

From the def in i t ions of r i (x) and si(=) we have for

i = I, 2,. . . :

r i (x ) = r l [ f i _ l ( x ) ] ,

and

s/x) = sl [ f i_/x) ] .

There fo re , for i -- 1, 2, ..., we also have, from the def in i t ion

of pi(x) :

p~(~) _-- pl[[~_/~>].

The temma fot tows by observing that, for i = 1, 2, .., [ i -1 is a

o n e - t o - o n e funct ion from [0, 1] to i tself. •

Lemma 5.2 show~ that, in order to study the maximum of

pi(x) , when x (" [0, 1], i t is suff icient to study the maximum

of the po[ynomiat

pi(x ) = 1_1-__ xn-~ - J - (lx-nXn)n ' for xC [ 0 , 1 ] .

Observe that M n ~. p l (~n ) = [ f n / ( l - f n ) ] 2, in part icular,

s ince i t can be checked easi ly that, for n = 2, 3; ...,

~,t > J-E/(l+J-E), i t fot tows that

M n > n for n = ,2 ,3 , . . . . (5.5)

Theorem 5.3

The branching factor of the ~-/~ pruning argot[thin for a

rug t ree of degree n satisfkes:

/~_p(n) ~ ,/-M-~n, (5.6)

where M n is def ined [n Lemma 5.2.

Proof=

From the def in i t ion of R2h(t), we obtain for h = 2, 3, ...:

R'2h(t) = R'2h_2(t).rh(t) + R2h_2(t).r'h(t).

By mutt[pt[cation by S2h(t) i t fo l lows that

R'2h(t).S2h(t) = R'2h_2(t).S2h_2(t).Ph(t)

+ R2h_2(t).S2h_2(t).r'h(t).Sh(t).

Si.nce, for t 6 [0, I ] , atl. factors in this equation are

non-negat i .ve, we deduce, using the i'esuits of Lemma 5.2 and

the fact that Sh(t) ~; n when t C [0, 1], that:

R'2h(t)S2h(t) ~ MnR'2h_2(t)S2h.2(t) + n Mhh-I r'h(t).

Since , in addi t ion ,

R'2(t) S2(t) = r } ( t ) s l i t ) ~; n r ' l a ) ,

i t fo t tows that for t C [0, I ] and h = I, 2, ...:

R'2h(t) S2h(t) ~ n Mn h ' l [ r i ( t ) + ... + r 'h(t)]. (5.7)

Let In, d be the integral, def ined in equation (4.5). By

i .n tegrat ing ineqt la l i ty (5.7) over [0, I ] we see that In, d

sat is f ies :

In ,2h ~; n Mnh-I [h(n..l)] = n(n- l ) h Mnh ' l

s i n c e ri(O) = 1 and r i ( l ) ~ n for i = 1,2,. . . . This shows that

Nn,2h ~; n h + n(n.-l) h. Mnh"J .

Equat ion (5.6) now fottows d i r e d | y from inequati ty (5.5). II

5 . 4 - Numerical results

Tabte 5.1 summarizes the results of this section. ]t

p resents time variou.,, lower and upper bounds we have

d e r i v e d for the branching factor of the w-/3 pruning

a tgor i thm from equations (5.3) and (5.6). Atthough we have

not been abte to give an estimate for the asymptotic growth

of vr-~n, we can ea.,dty deri.ve an upper bound for this

- 3 1 1 -

quant i ty by studying rug trees of depth 2 since:

M n < Nn, 2 .~ 2n l ' n / ( l -~n ) - [~n/(l-~n)] 2 ~ 2n2An n ,

whi.ch shows that ~ ~ O(n/.CYff~). The humeri.cat results

of Table 5.1 l.ndical,e that VM n is a much better upper bound

for 1~0"./3(n)than .~n~n/(l-~n) for the range of values we

have consi.dered.

n

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

l,ower bnd.

~n/(1-~n )

1.618 2.148 2.630 3.080 3.506 3.915 4.309 4.692 5.064 5.427 5.782 6.130 6.473 6.809 7.141 7.468 7.791 8.110 8.425 8.736 9.045 9.350 9.653 9.952

10.250 10.545 10.838 11.128 11.416 11.703 11.987

1.622 2.168 2.678 3.166 3.638 4.098 4.549 4.993 5.430 5.862 6.290 6.713 7.133 7.549 7.963 8.373 8.782 9.188 9.591 9.993

10.393 10.791 11.188 11.583 11.976 12.369 12.759 13.149 13.537 13.924 14.310

upper bounds

Vn~nl(l-~n;

1199 2.538 3.243 3.924 4.587 5.235 5.872 6.498 7.116 7.726 8.330 8.927 9.519

10.107 10.689 11.268 11.842 12.413 12.980 13.545 14.106 14.665 15.221 15.774 16.325 16.873 17.420 17.964 18.507 19.047 19.586

from [3]

1284 2.666 3.397 4.095 4.767 5.421 6.059 6.684 7.298 7.902 8.498 9.086 9.668

10.243 10.813 11.378 11.938 12.494 13.045 13.593 14.137 14.678 15.215 15.748 16.265 16.778 17.288 17.796 18.300 18.802

Table 5.1 - Bounds on the brsnchi.ng factor of the.

0'-/3 pruning algorithm

6 - C o n c l u s i o n s and open problems

We have presented an an.atysi.s of the performance of the

o~-/3 prunl.ng algorithm for searchi.ng a uniform tree of

degree n and depth d when the values assi.gned to the

terrninat nodes are independent identically di.stributed

random variables. The analysis takes i.nto account both

shal low and deep cut-offs and we have also consi.dered the

effect of equatl, ti.es between the values assigned to the

terml.nat nodes.

A sl,mpl,e formula was derived, i.n Section 3, to measure

the number of termi.nal, nodes examined by the

or-/3 procedure when the bottom values are drawn from a

flni.te range accordi.ng to an arbitrary discrete probabi.l,ity

dl,stri.bution. Although the formula can be easi.l,y computed

numerical ly, a direct analysis is made difficult by the

presence of the probabfli.ty distr~butl.on. When only two

di.stinct values can be assigned to the terminal, nodes, i.t i.s

shown that the number of termi.nal, nodes examined by the

0..-/3 procedure (:an be at tea.~d el(n/In n)d]; and, i.n ti.ght of

the results of Section 5, l,hi.s corresponds to the worst case

behavi.or of the algorithm (over all. discrete probabi.l.i.ty

di.stri.buti.ons).

A formul,a was then presented i.n the form of an integral

to measure the number of termi.nal, nodes explored by the

e¢-/'3 procedure when the bottom values are all. di.st[nct. An

analysis of the i.ntegrat shows that the branching factor of

the ce-/3 pruning algorithm is E)(n//n n). a result whi.ch

confi.rms a claim by Knuth and Moore [3] that deep cut-offs

only have a second order effect on the behavi.or of the

¢e--/3 pruning algorithm.

Although the assumpti.on used in Secti.ons 4 and 5 when

the bottom values are all, distinct is not reati.stic for most

practi.cat appti.cati.ons, the results we have alert.red from l.t

gl,ve us some i.nsi.ght into the worst case behavior of the

¢e-/3 pruning algorithm when equati.ti.es between bqttom

values are possible, and they are a useful, complement to the

formula of Secti.on 3. Si.mil,arty, the branchi.ng factor

anatyzed in Section 5 provi.des us only wi.th an asymptoti.c

measure of performance for tile ez-/3 pruni.ng atgori.thm (i.. e.,

for trees of large depth). As i.ndicated by the results of

Sectl,on 3.3, however, the branchi.ng factor can at,so be used

as a real.isti.c measure of the worst case even for smart trees.

We have measured the offi.ci.ency of the u-/3 pruni.ng

atgori.thm by the average number of terminal, nodes explored

- 312 -

by the aLgorithm, it wouLd be interesting to also obtain an

estimate for the standard deviation of this number.

The scheme we have considered for assigning values to

terminal nodes of a uniform tree tent itself easily to

analysis; i t is, however, very simplistic. Different schemes

tot assigning static values have been proposed in [ t ] , [3]

and [4]. Analyses of those schemes would be helpful for

various applications~ a step in this direction was presented

in [4] for game trees of depth 2 and 3.

Acknowledgements

I wish to thank H. T. Kung and B. W. Weide for reading

and cornmenting on earlier (Irafls of this paper.

Re fe rences

[ I ] Fuller, S. Ft., GaschnEg, J. G., and Giltogty, J. J., Analysis of the alpha-beta prl|nin 8 algorithm, Carnegie-Mellon University, Computer Science Department Report, July 1973.

[2] Gitlol~ly, .1. J., The TechnolOgy chess proBram , Artificial InteUiEettce , Vet. 3, No. 3, 1972, pp. 145-163.

[3] Knuth, D. E., and Moore, R. W., An analysis of alpha-beta pruni~ng, A,.ti[tci~zl lnlelliRence, Vet. 6, No. 4, 1975, pp. 293-326.

[4] Newborn, M. M., The efficiency of the alpha-beta search on trees with branch--dependent terminal node scores, Art i f ic ia l Intelligence, Vet. 8, No. 2, 1977, pp. 137-153.

[5] Slagte, J. R., and Dixon, J. K., Experiments with some programs that search game trees, Journal of the ACM, Vet. 16, 1969, pp. 189-207.

- 313-