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Computing the Span of two Normalized Spanning Intervals

The span of two intervals  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 , denoted tex2html_wrap_inline9792 , is the smallest interval whose extension contains the extensions of both  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 . For example, the span of (1,4) and [2,6] is (1,6]. Similarly, the span of [3,7) and (3,7] is [3,7]. More generally, the lower endpoint of tex2html_wrap_inline9792 is the minimum of the lower endpoints of  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 . The lower endpoint of tex2html_wrap_inline9792 is open or closed depending on whether the smaller of the lower endpoints of  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 is open or closed. Analogously, the upper endpoint of tex2html_wrap_inline9792 is the maximum of the upper endpoints of  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 . The upper endpoint of tex2html_wrap_inline9792 is open or closed depending on whether the larger of the upper endpoints of  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 is open or closed. More precisely, tex2html_wrap_inline9792 can be computed as follows:

displaymath1755

The notion of span will be used in Section 4.7.

Let us extend the notion of span to two sets of intervals by the following definition:

displaymath1773

We will want to compute the span of two sets of intervals  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 , when both  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 are represented as spanning intervals. Additionally, we will want the resulting span to be represented as a small set of spanning intervals.

Given two normalized spanning intervals  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 , their span tex2html_wrap_inline9792 is a set of normalized spanning intervals whose extension is the span of the extensions of  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 . One can compute tex2html_wrap_inline9792 as follows:

displaymath1786

An important property of normalized spanning intervals is that for any two normalized spanning intervals  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 , tex2html_wrap_inline9792 contains at most four normalized spanning intervals. In practice, however, fewer normalized spanning intervals are needed, often only one.

The intuition behind the above definition is as follows. Consider, first, the lower endpoint. Suppose that the lower endpoints  tex2html_wrap_inline9862 and  tex2html_wrap_inline9864 of  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 are in tex2html_wrap_inline9870 and tex2html_wrap_inline9872 respectively. That means that tex2html_wrap_inline9874 and tex2html_wrap_inline9876 . The lower endpoint of tex2html_wrap_inline9792 will be  tex2html_wrap_inline9862 , when tex2html_wrap_inline9882 , and  tex2html_wrap_inline9864 , when tex2html_wrap_inline9886 . Thus it will be  tex2html_wrap_inline9862 , for all tex2html_wrap_inline9890 , and will be  tex2html_wrap_inline9864 , for all tex2html_wrap_inline9894 , where tex2html_wrap_inline9896 , when tex2html_wrap_inline9898 , and tex2html_wrap_inline9900 , when tex2html_wrap_inline9902 . Thus there will be two potential ranges for the lower endpoint of tex2html_wrap_inline9792 : tex2html_wrap_inline9906 and tex2html_wrap_inline9908 . When the lower endpoint of tex2html_wrap_inline9792 is taken from the former, it will be open or closed depending on whether the lower endpoint of  tex2html_wrap_inline9216 is open or closed. When it is taken from the later, it will be open or closed depending on whether the lower endpoint of tex2html_wrap_inline9218 is open or closed. Thus the lower endpoint of tex2html_wrap_inline9792 can be either tex2html_wrap_inline9918 or tex2html_wrap_inline9920 . Analogous reasoning can be applied to the upper endpoints. If the upper endpoints of  tex2html_wrap_inline9216 and  tex2html_wrap_inline9218 are tex2html_wrap_inline9926 and tex2html_wrap_inline9928 respectively, then there are two possibilities for the upper endpoint of tex2html_wrap_inline9792 , namely tex2html_wrap_inline9932 and tex2html_wrap_inline9934 , where tex2html_wrap_inline9936 , when tex2html_wrap_inline9938 , and tex2html_wrap_inline9940 , when tex2html_wrap_inline9942 .


next up previous
Next: Computing the of a Up: An Efficient Representation and Previous: Computing the Complement of

Jeffrey Mark Siskind
Wed Aug 1 19:08:09 EDT 2001