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<Paper uid="P06-2047">
  <Title>Graph Branch Algorithm: An Optimum Tree Search Method for Scored Dependency Graph with Arc Co-occurrence Constraints</Title>
  <Section position="4" start_page="0" end_page="362" type="intro">
    <SectionTitle>
2 Optimum Tree Search in a Scored DG
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="361" type="sub_section">
      <SectionTitle>
2.1 Basic Framework
</SectionTitle>
      <Paragraph position="0"> Figure 1 shows the basic framework of the optimum 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 dependency relations between nodes. Each arc has  a preference score representing plausibility of the relation. The well-formed dependency tree constraint is a set of well-formed constraints which should be satisfied by all dependency trees representing 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.</Paragraph>
    </Section>
    <Section position="2" start_page="361" end_page="361" type="sub_section">
      <SectionTitle>
2.2 Dependency Graph
</SectionTitle>
      <Paragraph position="0"> 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.</Paragraph>
      <Paragraph position="1"> DGs are also classified into non-labeled and labeled DGs. There are some types of arc labels such as syntactic label (ex. &amp;quot;subject&amp;quot;,&amp;quot;object&amp;quot;) and semantic label (ex. &amp;quot;agent&amp;quot;,&amp;quot;target&amp;quot;). Various types of DGs are used in existing systems according to these classifications, such as</Paragraph>
    </Section>
    <Section position="3" start_page="361" end_page="361" type="sub_section">
      <SectionTitle>
2.3 Well-formedness Constraints and Graph
Search Algorithms
</SectionTitle>
      <Paragraph position="0"> There can be a variety of well-formedness constraints 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.</Paragraph>
      <Paragraph position="1"> (C1) Coverage constraint: Every input word has a corresponding node in the tree  No two arcs in a tree occupy the same valence of a predicate (C1) and (C2), collectively referred to as &amp;quot;covering constraint&amp;quot;, 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.</Paragraph>
      <Paragraph position="2"> 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) principle (Ozeki, 1994; Eisner, 1996) and the algorithm based on the B&amp;B method (Hirakawa, 2001), are used for the optimum tree search in scored DGs. The applicability of these algorithms is closely related 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 algorithms can satisfy (C1)-(C3) and run efficiently, but seems not to satisfy (C4) as shown in 2.4.</Paragraph>
      <Paragraph position="3"> (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.</Paragraph>
    </Section>
    <Section position="4" start_page="361" end_page="362" type="sub_section">
      <SectionTitle>
2.4 SVOC and DP
</SectionTitle>
      <Paragraph position="0"> (Ozeki and Zhang, 1999) proposed the minimum cost partitioning method (MCPM) which is a partitioning computation based on the recurrence equation where the cost of joining two partitions is the cost of these partitions plus the cost of combining these partitions. MCPM is a generalization 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 algo5Another condition for projectivity, i.e. &amp;quot;no arc covers top node&amp;quot; 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.</Paragraph>
    </Section>
  </Section>
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