Proceedings of the COLING/ACL 2006 Main Conference Poster Sessions, pages 361–368,
Sydney, July 2006. c©2006 Association for Computational Linguistics
Graph Branch Algorithm: An Optimum Tree Search Method for Scored
Dependency Graph with Arc Co-occurrence Constraints
Hideki Hirakawa
Toshiba R&D Center
1 Komukai Toshiba-cho, Saiwai-ku,
Kawasaki 210, JAPAN
hideki.hirakawa@toshiba.co.jp
Abstract
Various kinds of scored dependency
graphs are proposed as packed shared data
structures in combination with optimum
dependency tree search algorithms. This
paper classifies the scored dependency
graphs and discusses the specific features
of the “Dependency Forest” (DF) which is
the packed shared data structure adopted
in the “Preference Dependency Grammar”
(PDG), and proposes the “Graph Branch
Algorithm” for computing the optimum
dependency tree from a DF. This paper
also reports the experiment showing the
computational amount and behavior of the
graph branch algorithm.
1 Introduction
The dependency graph (DG) is a packed shared
data structure which consists of the nodes corre-
sponding to the words in a sentence and the arcs
showing dependency relations between the nodes.
The scored DG has preference scores attached to
the arcs and is widely used as a basis of the opti-
mum tree search method. For example, the scored
DG is used in Japanese Kakari-uke analysis1
to represent all possible kakari-uke(dependency)
trees(Ozeki, 1994),(Hirakawa, 2001). (McDon-
ald et al., 2005) proposed a dependency analysis
method using a scored DG and some maximum
spanning tree search algorithms. In this method,
scores on arcs are computed from a set of features
obtained from the dependency trees based on the
1Kakari-uke relation, widely adopted in Japanese sen-
tence analysis, is projective dependency relation with a con-
straint such that the dependent word is located at the left-hand
side of its governor word.
optimum parameters for scoring dependency arcs
obtained by the discriminative learning method.
There are various kinds of dependency analy-
sis methods based on the scored DGs. This pa-
per classifies these methods based on the types
of the DGs and the basic well-formed constraints
and explains the features of the DF adopted in
PDG(Hirakawa, 2006). This paper proposes the
graph branch algorithm which searches the opti-
mum dependency tree from a DF based on the
branch and bound (B&B) method(Ibaraki, 1978)
and reports the experiment showing the computa-
tional amount and behavior of the graph branch
algorithm. As shown below, the combination of
the DF and the graph branch algorithm enables the
treatment of non-projective dependency analysis
and optimum solution search satisfying the single
valence occupation constraint, which are out of the
scope of most of the DP(dynamic programming)-
based parsing methods.
2 Optimum Tree Search in a Scored DG
2.1 Basic Framework
Figure 1 shows the basic framework of the opti-
mum dependency tree search in a scored DG. In
general, nodes in a DG correspond to words in
the sentence and the arcs show some kind of de-
pendency relations between nodes. Each arc has
a0a1a2
Scored Dependency 
Graph
Dependency
Tree
Set of Scored Well-
formed Dependency 
Trees
Well-formed 
dependency tree 
constraint
Optimum Tree
Search 
Algorithm
Well-formed Dependency 
Tree with the highest score
s1
s2
s3
s4 s5
(score=s1+s2+s3+s4+s5 )
Figure 1: The optimum tree search in a scored DG
361
a preference score representing plausibility of the
relation. The well-formed dependency tree con-
straint is a set of well-formed constraints which
should be satisfied by all dependency trees repre-
senting sentence interpretations. A DG and a well-
formed dependency tree constraint prescribe a set
of well-formed dependency trees. The score of a
dependency tree is the sum total of arc scores. The
optimum tree is a dependency tree with the highest
score in the set of dependency trees.
2.2 Dependency Graph
DGs are classified into some classes based on the
types of nodes and arcs. This paper assumes three
types of nodes, i.e. word-type, WPP-type2 and
concept-type3. The types of DGs are called a word
DG, a WPP DG and a concept DG, respectively.
DGs are also classified into non-labeled and la-
beled DGs. There are some types of arc labels
such as syntactic label (ex. “subject”,“object”)
and semantic label (ex. “agent”,“target”). Var-
ious types of DGs are used in existing sys-
tems according to these classifications, such as
non-label word DG(Lee and Choi, 1997; Eisner,
1996; McDonald et al., 2005)4, syntactic-label
word DG (Maruyama, 1990), semantic-label word
DG(Hirakawa, 2001), non-label WPP DG(Ozeki,
1994; Katoh and Ehara, 1989), syntactic-label
WPP DG(Wang and Harper, 2004), semantic-label
concept DG(Harada and Mizuno, 2001).
2.3 Well-formedness Constraints and Graph
Search Algorithms
There can be a variety of well-formedness con-
straints from very basic and language-independent
constraints to specific and language-dependent
constraints. This paper focuses on the following
four basic and language-independent constraints
which may be embedded in data structure and/or
the optimum tree search algorithm.
(C1) Coverage constraint: Every input word has
a corresponding node in the tree
(C2) Single role constraint(SRC): No two nodes
in a dependency tree occupy the same input
position
2WPP is a pair of a word and a part of speech (POS). The
word “time” has WPPs such as “time/n” and “time/v”.
3One WPP (ex. “time/n”) can be categorized into one or
more concepts semantically (ex. “time/n/period time” and
“time/n/clock time”).
4This does not mean that these algorithms can not handle
labeled DGs.
(C3) Projectivity constraint(PJC): No arc crosses
another arc5
(C4) Single valence occupation constraint(SVOC):
No two arcs in a tree occupy the same valence
of a predicate
(C1) and (C2), collectively referred to as “cover-
ing constraint”, are basic constraints adopted by
almost all dependency parsers. (C3) is adopted
by the majority of dependency parsers which are
called projective dependency parsers. A projective
dependency parser fails to analyze non-projective
sentences. (C4) is a basic constraint for valency
but is not adopted by the majority of dependency
parsers.
Graph search algorithms, such as the Chu-
Liu-Edmonds maximum spanning tree algorithm
(Chu and Liu, 1965; Edmonds, 1967), algorithms
based on the dynamic programming (DP) princi-
ple (Ozeki, 1994; Eisner, 1996) and the algorithm
based on the B&B method (Hirakawa, 2001), are
used for the optimum tree search in scored DGs.
The applicability of these algorithms is closely re-
lated to the types of DGs and/or well-formedness
constraints. The Chu-Liu-Edmonds algorithm is
very fast (C7B4D2BEB5 for sentence length D2), but it
works correctly only on word DGs. DP-based al-
gorithms can satisfy (C1)-(C3) and run efficiently,
but seems not to satisfy (C4) as shown in 2.4.
(C2)-(C4) can be described as a set of co-
occurrence constraints between two arcs in a DG.
As described in Section 2.6, the DF can represent
(C2)-(C4) and more precise constraints because it
can handle co-occurrence constraints between two
arbitrary arcs in a DG. The graph branch algorithm
described in Section 3 can find the optimum tree
from the DF.
2.4 SVOC and DP
(Ozeki and Zhang, 1999) proposed the minimum
cost partitioning method (MCPM) which is a parti-
tioning computation based on the recurrence equa-
tion where the cost of joining two partitions is
the cost of these partitions plus the cost of com-
bining these partitions. MCPM is a generaliza-
tion of (Ozeki, 1994) and (Katoh and Ehara, 1989)
which compute the optimum dependency tree in a
scored DG. MCPM is also a generalization of the
probabilistic CKY algorithm and the Viterbi algo-
5Another condition for projectivity, i.e. “no arc covers top
node” is equivalent to the crossing arc constraint if special
root node , which is a governor of top node, is introduced at
the top (or end) of a sentence.
362
