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<Paper uid="C92-4185">
  <Title>UNIFYING DISJUNCTIVE FEATURE STRUCTURES</Title>
  <Section position="10" start_page="0" end_page="0" type="evalu">
    <SectionTitle>
8 Complexity
</SectionTitle>
    <Paragraph position="0"> To analyze the complexity of this algorith m 1 will look at threc cases. If we assume that there are no disjunctions in the formulas the procedure can be implemented almost linearly. If we have local disjunction in the formulas, i.e. disjunctions which do not contain variables and which not are connected by switch nantes, the total complexity becomes exponential on the maximum depth of disjunctions occurring within each other. For the third case we have to add the complexity for the removal strategies when alternatives have failed. The complexity for this procedure is also exponential in the size of a&amp;quot;, where a is the total nnmbcr of alternatives OccutTing in the formulas. For a more complete discussion of the complexity see StrOmhack (1991, 1992) When considering complexity one must remember that the second case will only be pcrforn~ed when there are disjunctions in the formula and when these disjunctions are actually affecWA by the unification. Disjunctions in some subpart of the formula not affected by the unification never affect the complexity. It is also reasonable to assume that m most case.q when a disjunction really participates in the unification, some of its alternatives will be removed due to failure. The same thing holds for the last case; it will only be performed when some global alternative has failed. This means that this procedure can at most be performed once for each ordinary alternative in the initial formulas.</Paragraph>
    <Paragraph position="1"> Comparing this to the other proposed alternatives we can see that Kasper's (1987) algorithm has a better worst case complexity (2a/2). On the other hand this complexity holds for all disjunctions in the structure regardless of whetlmr they arc ',fffected by the unification or not. The algorithm by Eisele &amp; D0rre (1988) has a similar worst case complexity.</Paragraph>
    <Paragraph position="2"> The disadvantage here is that this 'algorithm is expensive even if the structures do not contain any disjunctions at all. The third algorithm (Eisele &amp; D~3rre 1990a, b) will also be NP-complete in rite worst case and will probably have a stinilar performartce compared to the algorithm descritxxl in this paper.</Paragraph>
  </Section>
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