Intervals Distribution
An intervals distribution is an n-tuple of natural numbers where the index represents the interval length and the value is a count of its appearances in the interval chain.
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Let there be an interval chain.
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classDef c4 fill:#98df8a,color:#000;
classDef c4a fill:#98df8a8a,color:#000;
classDef c5 fill:#d62728,color:#fff;
classDef c6 fill:#ff9896,color:#000;
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classDef c8 fill:#c5b0d5,color:#000;
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classDef c12 fill:#f7b6d2,color:#000;
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class t1,t2,t5,e0,e1 position
s1 --> i1
s2 --> i2
s3 --> i3
s4 --> i2
s5 --> i4
s6 --> i6
Intervals distribution used as an input data in calculating characteristics. While characteristics could be calculated based on the itervals chain intervals distribution highlights that intervals themselves are enough to measure the order of a sequence and intervals connectivity in intervals chain does not affect measure values. Whether it is possible in general to reconstruct distinctly an interval chain by the given interval distribution is an open question.
Interval distribution is useful in comparing intervals produced from the same sequence with different Binding. In the interest of studying how intervals depend on Binding direction for Bounded Binding FOA introduce two operations on distributions:
- Lossy - takes two intervals distribution and produce new one only with intervals exists in both distributions.
- Redundant - extends intervals distribution
Awith intervals that appears only in intervals distrubutionB.
Mathematical Definition
Let \(IC\) is Interval Chain length of \(n\) described as function \(IC : \{1,...,n\} \longrightarrow \{1,...,n\}\)
Define