Alignment of Shared Forests for Bilingual Corpora 
Adam Meyers, Roman Yangarber, Ralph Grishman 
New York lJlfiv<;rsity 
715 Broadway, 7th 1+1oo, ', NY, NY 10003, USA 
meyers/roman/grishman@cs, nyu. edu 
Abstract 
Research in example-based machine 
translation (F,I~MT) has been hampered 
by the lack of efficient tree alignment al- 
gorithms for bilingual corpora. *ghis pa- 
per <lescribes an alignment algorithm for 
F,I~MT whose running time is quadratic 
in tile size of the input parse trees. 'Phe 
algorithm uses dynamic programming to 
score all possible matching nodes be- 
tween structure-sharing trees or forests. 
We describe the algorithm, various opti- 
mizations, and onr implementation. 
1 Introduction 
The development of a glachine translation (MT) 
system re<luires the lengthy manual preparation 
of bilingual lexicons an(t transfer rules. I{esearch 
over the past few years using parallel senten<:e- 
aligned bilingual corpora sugg{'.sts ways in which 
this manual effort <'an be partly replaced by 
corpus-based training. Some o1' l;his research 
has treated the sentenees as unstructured word 
sequences to be aligned; this work has primar- 
ily involved the acquisition of bilingual lexical 
correspondences (Chen, 1993), although there 
has also been a,n attempt to create a full MT 
system based on such trcat, ment (Brown et al., 
1993). I{ecently, several groul)S have been ex- 
ploring the possibility of aligning t)arMlel syn- 
tacticalhj analyzed sentences fr<)m the source and 
target languages (el. (Sate and Nagao, 1990), 
(Klawms and Tzoukermann, 1990), (Grishman 
and Kosaka, 1992), (Kaji et al., 1992), (Mat- 
sumoto et al., 11993) and (Grishman, 11994)). 
Tiffs offers the potential for acquiring not j ust lcx- 
ical but also structural correspondences between 
the two languages, q'he specific goal in aligning 
syntax trees is to identify tile (:orresponding tree 
fragments in the source and target trees. By pro- 
cessing a. substantial corpus, a large set of such 
corresponding fragments can be collected. These 
(:an then serve as the example base for a form of 
examph;-based MT (of. (Nagao, 198d), (Sate and 
Nagao, 1990), (IG\ii et al., 1992), (Matsumoto 
<% al., 19!)3) and (leuruse and lida, 1994)). This 
approach requires a fast tree alignment teehuiqu<~; 
research has I)een ham/)ered by the lack of efli<:icnt 
algorithms. This pa,per des<:ril/cs an efficient al- 
gorithm for bilingual tree alignment. 
2 Our Approach 
For each input sent;ellee our parser t)rodlmes a set 
of trees, corresponding to each possible syntactic 
analysis. O.r parse trees are transformed illl, O a 
"regularized" format, to represent the l)redicate - 
Argument structure. For each senten<:e, the out 
put of the parser is a stru(:tm:e-sharing tbrest. An 
example of strltetllre sharing I:)etween two ila, rse 
trees <ff l,}le same input senten('.e is shown in Fig 
are 1. We apply the parser to the source and target 
sentences, using a Spanish and an F, nglish grain 
mar, respeetbely. The resulting sets of structure- 
shin:lug parse trees form the input to the alignment 
procedure. 
Our alignment program employs dynamic 
programming I algorithms, which are described in 
detail in later sect;ions. The program begins a.t the 
roots of the source and target trees, and 1)roeeeds 
top down reeursively, filling a matrix of scorcs. 
(liven N nodes in the som:ee tree 7',+ -- "/'(V~, I5',) 2 
and M nodes in the target tree "/}. -- 7'(Vt, El), the 
score matrix is an N × M matrix. For each pair of 
nodes xi,i = I .... N 6 V~ and Yi,J - 1,...M C ~, 
the corresponding entry in the score matrix is a. 
measure of how well z/ illatc:hes >{. The score for 
each pair of nodes depends only on the closeness of 
the lexieal entries associated with the nodes and 
i()\[., e.g. (Cornlen (% al., \]990), pp.299-328 
>l'he expression T(<, 1'}.~) <lcnotes a tree as a pair 
of sets: V~ is the, set of vertices (nodes) in (,he tree, 
and l','.+ is the set; of edges (arcs). 
460 
S~I + S~I" 
(llliDItl°2 el / pos (Icl)(te 
deporte \[ qm,~'///f -- ---~ 
t'\])OS/\] "~+ ~ UlI P' } /:"-"'"2 
,':,.,r,,,, / 
nn favorilo / ~ "/ 
IlU~l() in2| ~iiiil ,, 11\]1~ 
cl rico 
I,'ip;uro \]: Two l}a,t'scs I'or: El I,+~+i.~ (~,~ +~ dcp<)'rLu 
.fi~vorilo (h!l 'm.'+u:/~a<:ho 'ri(:<). 
on l,h{ 2 s(:or<'.8 of {,h<+ I)esl, n\];tA+(:hiNg t)a, ir8 of {,hch' 
(l<'~+(:<'mlatH,,~. I)yna, tiii(: l)ro~ra.tiHui.~ a,'+,'+Hr<~H thal, 
ea(:h (;Ill,ry ill i,h<+ tna,t,rix, (i.<;. i,he +(:or(~ for l,lie (:or- 
r<;,'-;I)(>lMin<~ I)air o1+ uo(Io8) is (:(mil}Ul,<~(I {>.ly ou(:('. 
We lla,v(~ iult)h~l.(ml,(~(t Lhi,<+ al)l)ro.(:h I'()r 18D ,~ell= 
• ) i,(2\]IC()H \['rOlIl iA, vo SOil I+(;('~,<;: l':l (.<m~.< Ih,.I (a HI)ati 
ish I,<':x{,l>ool<: fronl wh i<:h w<' .sod 73 ~elll,(qlC{;S wit,h 
Limit l",.gli~;h tra tmlai,iolm), a, ti(l (/lu'io+,+,+, (;(:<rr:\]<~ or 
,/or:\]+' <:1 6"urio,+o (l,lm l';ngliMi +LII(t SI}a..i+h vet'-- 
8io\]m o1' 11, A. Flay + l)Ot)ula, r (:hil(h'<'ti>~ I)o()k, I'rom 
whi(:h w(" u~<-.I all Ill I:..gIish H<mi,(m(:('H atM 112 
Sl)anish Senl,('+tl.c<~,<+). ()\['i,h(~ I,oi,M 185 Hctlt,(;n<:<++H, 57 
l'ronI El (~;'(m+,ino R+'+(d attd ,35 |'P(}III (/?l?'J¢)ll,+ (',( ()1~(\](" 
l)rodu<:('(l a,t, leaHI, oil<~ l)a,l+~(~ t, re{~ iu bol/~ lanzHa~2\]es. 
Th('+ aligum<'+.l+ i>ro<>'+(hlre wa+ al)t)lie(1 o.ly i,(> {,he+e 
i)a.ir.<+ <>1" H(2III,CII(:(2H. :~ 
3 Data StPu('l;ures 
Fl,(%tflariz<;(l I>ar~<'s are shtdhu + t,o l,tio I:.,'-;t, rl.:t,lu'(~ 
o1' I,(;xi{:al I,'u.(.:l,iollal (,ranu la, r (I,1:(;)+ cx(:(;t)l, 
lJml+ a (Icl)<'JMen<:y t.yl)c ,.-;t.ru(:t, ttr<', iH a.,~minl(;(I. 4 ., ) p ) 
Ih'mc a r{;la,t,i{)n h, oh:( l IU, \] , A R,(:) is r('l)r<;~<;l\]l~e{I 
I)y an ar(:, t,h<~ l'lUql)o\[ , (., at iJic tail ot'i+l~<~ arc, 
and I,h('+ ALP.(; (or 1,he roh~ r('(:ii)i<m{,) al, I,h<! h(;a(l 
. 
:~t~y hlii)\['ovhll~, I.h,.: ~l+sylnl)(.(}l,h:: 8t){!{!(1 o1 ;tliznin{'lit 
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iil<t~ lilllch la, r<~,::r c{.wF,(}t';t Jri \['lllMr(', ",,v{}il,:. 
4(~;+~+{,o ~++tl{| N;~.p,~++()~ 199{}) ;+u\](I (M,q.l,sHltiot() ('t, ~d., 
If)!)',l) also ,:b~StUltC {l(G:,('lt{h!n(:y l,.yt),:' Sl.l'tl(:i,tit'{~ ht I,\]l{!\]r 
e x ,q. m t } h:- t );_is,.,. ( l woi'k. 
,::>\[' th,:+ a r,::. l;'c,r ,:+xa.tJll:.l('~ , ,':l tu;'ri,+' +s l,h,:+ sul)j<+ct, 
or i,hc l)redicatc ,+;('?' iu l<'izure I. It+ a, rezu\]ar- 
izcd l)arse, corral. (:/()+~,+(I 8y.ta<:tic cla,sse,% +u(:h 
+m prc'l>OSitkm+ and +.l)oMhmt,e cottjunct, hcms, are 
rel)re+e.l,ed a,~ a r<: lab(q,+ dc.oI,+.~ role+, (e.Z. , tim 
l)l'(+l)OHitiO/l (\[(: ill \["i,'~llro l) r+1{Jv':r i,h+111 +ts ilO(I(~s 
itl I:- 81,t'Hct,tH'e. 
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l'<>r<++<, all(>w,~ u+~ i,o red.c(+ the ntHnl)(':r o\[' c,:)ti+l)Ut,{x\[ 
,'+(:orc~s, We (:oltil)Ht,(': |,\]t(~ s(:()r(; \[or a, ,Zivcli \[)air 
of mll)t,t'ees only otlc,:~, "e~a,'( les~ of Lhe mmfl)er 
or i+re+cs which share the,~c +ul>t, rees l)<+<'ause the 
s(mor<> <)\[+ a l)ah" <)|' no(les <l<q><mcLs o.ly on l,he +('ore,~ 
(>l' their ,.le,+<:<m(huH,+ (a.(l .oi+ ol' t, heh' a.t.:e,~l::>r,~). 
('.tirreiHJy ,:)tit' Im, rser re<:{>rd+ M, rtl<:l, Hr(+ mha.ritl Z 
..rely h('.t,w{ml+ NI}+'+. l'\]×I)(q'ittl(mls ill whi<:h .II uon~ 
tH(m MrH,::l,ur<'~ i,'+, ,'-;h,+~re,:l, a,'+ iH I"i~t.'<~ I, :-+;.~Z<~sI+ 
Lhal, ('~xi,<m<li.~ ,'.+l+ru(:t.re ,'4mt'ill~; Lo od.~r Lyl>e,~ of 
H<>(h!s ',,,.,,()tllcl \['tH'l,lmr iHiF, r{}v<+ I)erl'(.'nmn<:( +. 'l'\]d~ 
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w(M( in <>l>l,iJllizinz I+'<mt+Hr<~ Slmu(:l,Hr<+ I)a,scd pars: 
iu~+ (S<'{+ For (+xamlJ('. (ICu'i,i, utt(m, 1!)85) aim 
(I)<~t'(d,'a, 1985)). 
4 The LCA-Preservhlg Algorithm8 
\,Vc lit+i, di+(:u.+.<-; I,h<+ I'oruia, I a:,+F.,{'.<:I+H o1' LIt(; a lizulll<ml, 
t)t'ol)l(mi a,u(t i lit, rodu(:(' l,<;rtili,oh>gy. 
4.1 The Ma×imal Tr(;e Ali~titnl.('+nt 
Prol)l{:ni 
'l'h<'. ol).i<><:i,iv(~ i+ t;o lind . tnaxiumHi s{:or<> <:+)rr<'- 
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a lt<:l !\]), <:orl+(+Hl)()u(Is rlo.+dy to thal, \['()tm<l in ( M at,-- 
sUHiOl,O (;t, al., t9(,7'3). Ottr algorii;liitm are I)aH(~(I 
o. I, hose I>r<>~<'.t;<',-I iu (Steel all{I War.ow, 1!)9:1), 
(l:ara<:h ct+ al.> 1!)95I>) +'rl' ( \[ ( 1995~j. 
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<:ollittloti a.tw(~t,or (h:a,) (>l' Lv.,,<} H()(I(~,'+ (~ aH(l l, is i,h(~ 
m}(h'. :+:0 : : lc(~(<,,, b), ~tl(:ti l,hat, 
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;+:{} iH a (l<>'-;C<;ll(/atH+ o\[' x. 
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a. {}t>t,ill\]i~e<l v;~,i'i;).Iil, (:,I' LIic a, lg(}ril,linl, ;vhi(:\]i aligH+ 
l)air+ ,.}1' sl,iticLtlrc:,'-;ha+rhl Z t'(}lesl,H. 
4:61 
SOURCE 
e 
a d /i 
b e 
TARGET 
d' 
a' b' e' 
Figure 2: There is no lea-preserving alignment be- 
tween a, b and c, and a', b' and e'. 
If nodes a E V and b ¢ V map into nodes 
a' = f(a) E V' and b'= f(b) E V', 
then f(lca(a, b)) = lca(f(a), f(b)) = 
lca( a+, b+) 
To illustrate, in Figure 2 there is no lca-preserving 
alignment of the two trees which maps all three of 
the leaf nodes a, b and c into the nodes a', b' and 
c'. Lea-preserving alignments are possible which 
map any two of the leaves. 
The algorithm assumes that least common an- 
cestors are preserved in the alignment. We assign 
a score to each alignment based on the labels of 
the corresponding nodes and the arcs from these 
nodes, as described below. The algorithm seeks 
an alignment with maximal score. 
4.2 The Algorithm 
Let T~ and Tt be the source and the 
target trees. The algorithm uses dynamic pro- 
gramming to build up, in a bottom:up fashion, the 
scores for matching each node in T~ against each 
node of 2gl. There are O(n 2) such scores, where 
n = max(I T~ 1, I 7~ 1) is the number of nodes in 
the trees. Let d(v) be the degree of a node v. We 
denote children of v by vi, i = 1,..., d(v), and the 
arc (v, vi) by #i. 
Procedure SCORELcA: The dynamic pro- 
gramming builds up a score function S(v, v') for 
all v E T, and v' E Tt, which is stored in a 
I T, \] x I Tt I matrix S. The value S(v, v') is the 
score assigned to the best match between the two 
subtrees rooted at v in T., and at v' in Tt. Initially, 
S is filled with undefined values. When a value for 
S(v, v') is required, and the corresponding entry in 
the matrix is undefined, it is recursively computed 
by the following formula: 
{ MATCII)~(v, v') 
S(v, v') = max maxi:L...,a(~ ) S(vi, ~/) - P(vi) 
maxj=, ..... d(~') S(v,v}) - P(#}) 
(t) 
The function MATCHI~a(V, v') is a measure of 
how well the nodes v and v' align, and is computed 
as follows: 
MATCH,~(v, v') = Lex,o&(V, v')+ 
+ max 
pEP(v,v') " ' 3 @ 
(i,j)Ep 
where: 
(2) 
Lex~o&(V, v') >_ 0 is a measure of how closely 
the label on source node v corresponds to the 
label on target node v ~ in the bilingual dictio- 
nary. Lex~,.~ (#, ~') is the corresponding mea- 
sure for arcs. 
P(v, v') is the set of all possible pairings of 
the children of v against the children of v'. 
There are O(d!) such pairings, where d is the 
smaller of the degrees of v and v'. 
P(#i) _> 0 is the penalty for collapsing the 
edge vi, which may depend on the label of 
that edge. 
The summation in (2) ranges over all pairs, de- 
noted by (i,j), which appear in a given palnng 
p E P(v, v'). The summation is evaluated for all 
O(d!) possible pairings. The pairing with the max- 
imum score is then selected. 
The total running time for computing the scores 
of all of the O(n 2) nod(: pairs v and v', is O(d!n2), 
where d is the lesser of the degrees of the source 
and target trees. Comparing the max term in (2) 
can be mapped into tile Maxbnum-Weight Clique 
problem (which is NP-eomplete), of. (Farach et 
al., 1995b). However, in the NLP domain, the 
running time is contained becmlse d < 6 for most 
trees encountered in practice. Next we describe a 
heuristic ~which achieves a time bound quadratic 
in the size of the tree. 
5 A Greedy Heuristic 
We can reduce tile computation time of the max 
term in (2), if we do not consider all of the O(d!) 
pairings of the children of v and v'. Instead we 
462 
use a greedy approach and choose the d highest- 
scoring, mutually disjoint pairs from among the 
d 2 possible pairs of children of v and v/. The jus- 
tification for this heuristic is that we expect that 
the high-scoring pairs will dominate, and will be a 
priori mutually disjoint. 
The following is an alternative, greedy proce- 
dure for computing ,5'(v, v'): 
Procedure GflEEDYLcA: 
1. Vi,j s.t. I < i < d(v),\] <_ j _< d(,/) corn- 
pute the corresponding entry in a d(v) × d(v') 
matrix M: 
M<j = Lexa,.~(~;i, ~.) + H(vi, vj) 
The entry Mij of M = M(v,v') is the score 
of matching the ith child of v with jth child 
of v'.6 
2. l,et TOP +- {} be the set of highest scoring 
pairs. 
3. Find the largest entry Miojo ill the matrix, 
such that neither its row nor its cohmm is 
already occupied by some pair in TOI': 
7'OP ~-- 7'OP tO {(i0,j0)) 
where the coordinates (i0, jo) are such that 
Miojo z Ina.x{ Mij 
I V(i',f) C 7'OP, i 7 k i',j # j'} (3) 
4. Repeat the above step d times, where d = 
rnin(d(v), d(v')). 
5. Compute the result: 
M AT'C Ht~. (v, v') = 
= l;ex,~o~l~ (v, v') + (4) 
(<~)e~rc)v 
With sorting, this can be done in O(d log d+ d 2) 
time. 
The validity of this heuristic can be tested by 
comparing the performance of the procedures us- 
ing the computation it, (2) and in (4). 
6Note: if we disregard the arc labels for simplicity, 
and set Lex,,.~(., .) = O, then we do not need to build 
M, and may simply use Mij : S(vi, v5). 
6 Strict Lexical Matching Heuristic 
(Grishman, 11.994) employed an optimization 
heuristic which favored lexical matches. For each 
source node v with label L(v), the procedure using 
this heuristic would first attempt to find a target 
node v' with \]a, bel L(v') such that L(v) translated 
as L(v') in the bilingual dictionary (a perfcct lexi- 
cal match). If such a lexical match was found, the 
procedure did not attempt to match v with any 
other target node. 
A similar heuristic (Lex-Match) was incorpo- 
rated into our program as the following prepro- 
cessing steps: 
1 For each source node v, all possible lexical 
matches are identified in the target tree\] If. 
v has at least one possible lexical match, all 
of those positions in the score matrix S which 
do not correspond to a lexical match of v are 
set m zero. 
2 For each target node v' which has at least one 
lexical match, all of those positions in the 
score matrix which do nol, correspond to a 
lexicM match o\[ v' are set to zero. 
By setting to zero those positions in the score ma- 
trix which represent unlikely matches, this heuris- 
tic prevents these scores from ever being cah:u- 
lated, substantially reducing the rmming time. 
Lex-Match, unlike the (Grishman, 1994) heuris- 
tic, allows one source node to match lexically with 
more than one node in the target tree. 
7 Implementation 
We have implemented the greedy LCA-preserviug 
algorithm with the following features: 
Penalties: The penalties for collapsing edges 
were set to 0. s 
Scores: A LeX,~od,; score of 100 and a Lexa,.,, 
score of 21 was awarded for each match us- 
ing our bilingual dictionary. These fimctions 
have the value 0 if there is no lexica\] match. 
rA match M(v, v') is also a lexical match if either 
M(v, w') o,' M(w, v') is a lexical match, where w and 
w' are children of v and v j, respectively. 
aWhen penalties are set, to zero and an empty t)ilin- 
gum dictionary is used, the alignment algorithm \[ills 
tim scoring matrix with zeros. When we introduce 
no\[l-zero penalties, the alignment procedure prefers 
matches between nodes dominating similar structures, 
since nodes dominating dissimilar structures receive 
negative scores. Wc expect that non-zero penalties will 
improve precision with a nonempty bilingual dictio- 
nary, because they will favor similar structm'es. In pre- 
liminary testing, penalty values of 20 and 30 yielded 
iinprovenmnts in precision. 
463 
Text Baseline Struc-Share Lcx-Match Struc-ShaTv and Lex-Mateh 
El Camino Real \]\] .5 sec 11.3 see 8.3 sec 7.7 sec 
Curious George 98.0 sec 48.8 sec 87.4 see 44.7 set: 
Total 109.5 sec 60.1 set: 95.7 sec 52.4 sec 
Table 1: Time Improvements Due to Optimizations 
Text 
F,I Camino Real 
Lex-Match Off Lex-Match On 
47 out of 57 (82%) 47.5 out of 57 (83%) 
Curious George 44.6 out of 55 (81%) /14.(5 out of 55 (81%) 
q'otal 9:1.6 out of 1112 (82%) 92.1 out of 112 (82%) 
Table 2: Changes in Accuracy due to l,ex-Match Ileuristie 
Optimization Variables: We experimented with 
variants of thc. procedures which included 
Structure Sharing (Struc-Share) and the Lex- 
ieal Match Optimization (lmx-Match), as well 
as with those that did not. 
Table I shows the time consumed by our pro- 
gram to align sentences under different conditions. 
The baseline refers to our program without any 
optimiza.tions (which is at least 6 times faster 
than before using this algorithm.) The optimiza- 
tion w~riables have different effec.ts on the differ- 
ent texts. We believe that structure sharing has 
a much stronger el\[bet on Uurious Geo~ve than 
on El Camino Real because the tbrmer has longer 
sentences which produced more parses. The l,ex- 
M~tch optimization has a greater effect on l'SI 
(7amino Real than on Curious Ceor.q<: because all 
of the words contained in El Camino Real are in- 
eluded in our bilingual dictionary, but only a small 
portion of the words in Curious Geovje are in- 
cluded. We expect that as the size of our dictio- 
nary increases, the Lex-Match optimization will 
have a greater effect. 
The precision for each aligned pair of sentences 
is computed according to the formula: 
\[ltc.s'ultSct N Answcr Kc?/\[ 
{ t~esuliSet I 
where Re.suit,fret is the set o1' source parses to 
which the alignment procedure assigned the high- 
cst score, and AnswcrKey is the sc't of best source 
parses as judged by one of the exDerJtnenteFs. 9 
This precision measure was previously used in 
(Matsumoto et al., 1993) a.nd ((h:ishman, 71994). 
Table 2 compares the precision of the alignment 
procedure with and without the Lex-Match heuris- 
tic (structure sharing had no eider on the scores.) 
The slight increase in precision observed with the 
91f there was no correct parse, the parscs wil,h the 
fewest errors were used for purposes of nligmnent. 
l,ex-Match optimization, may be an itldication that 
we should raise the score for lexical matches of 
node labels. 
8 Results and Future Directions 
The. era:rent implement~tion aligns trees 63 times 
faster than our previous program (Grishman, 
199d), with a 2.3% inq)rovement in pre(:ision. 1° 
We expect line-tuning el' the pa.rameters in our 
procedures to improve our performance. We ex- 
pect to gain greater efficiency if all COi\[HIHOH IlOdeS 
between forests are s\]lare(l, rather than .just the 
NPs. Another efficiency inlprovcment will be 
aehieved by factoring all ambiguity into the parse 
tree, as in (Matsumoto eta\]., 1993). In our cur 
rent approach, disjunctions are r(:prcse.ntetl only 
at; the root level. 
In order to inq)rovc the precision of alig,tment, 
we plan to ext)erimenl, with w~rying the values 
of the Lex fmletions mid penali, ies in our scor.- 
ing algorithm and e×l)anding our bilh,gua\] dictio 
nnry. We will also experiment with the non--gr('.edy 
algorithm discussed al)ove and a th)nthmn(:(.'. 
preserving algorithm (~ less consl.rained version of 
the algorithni which we have omitted due to .space 
limitations). In the dominance-preserving algo- 
rithm we relax the requirement of h:a-preserw~.tion, 
and require the l)rcserw~,tion of the (\]omi\]~ancc rc 
lationship between lm(les: 
If', for two nodes a C- 7' a~d b C 7', a 
dominates b ((telloted ~ts a ~ b), thell 
for f(a) ~ 7" and f(b) C '1", f(a) j@). 
The kh:a which makes it possible to align s(m~ 
tenets quickly is that we place restrictions on 
the ways in which we align the parse trees. W('. 
t°'I'ihe dynamic l)rogreumniug algorithm accounts for 
ml approximately 600% iucrc~tsc ill speed of align\]nent 
a rough estimate sin(:e much of Chc I)rogrmn has been 
r(>implemcntcd. 
464 
disallow ~,lignnmnts which viobtte the I,(:A con- 
stra.itd; or I,h(~ (\[o~nin~m(:(~ r('xluiren,enl. , and per- 
\[nil oltly one to ()ira a.lignm(ml.s I)(~l.w(~en no(los. 
Some cases wh(:rc on(: ~\[~ight. posit a (:orr(~sl)On- 
den(:e I)el, wcen a single node :~: ~m(I ~ group of 
nodes (; = {yl...y,~.}, ('.~m l)e interpr('A;('~(\] ~s ;~n 
~dignmcnt I)etwcen a: and ,~ti, for some. j, \] < j _< n, 
where 95 dominates the rem~dning no(les in (l. We 
(Io not consider other types of (me-to-many ~:dign- 
lll(':lltS. 
Acknowledgements 
We wish to thm~k Antonio Moreno San(lowd m,d 
(;rist;in~ ()htmd~ More.no for t)r(qyara.tion of the 
Spm~_ish a.nalyscs for ,\]or(,lC c/ CTurioso. We a.lso 
thank C, a therin(~ Ma(:h~o(l For l)rep~rati(m o1' the 
English l)a.rscs. 
'l'his rese~trch was SUl)l)Ort(~'d I)y t,hc Na.t,iouM 
S(:ieJt(:e Foun(lal.ion undc'r (\] I'.:L\[I{. 11{1-930:I013. 
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465 
