On Vertical Visibility in Arrangements of Segments And the Queue Size in the Ben
On Vertical Visibility in Arrangements of Segments And the Queue Size in the Ben
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Proof: The upper bound follows immediately from Furedi's analysis. For the lower bound we use the following recursive construction. We construct collections {Sr}rs^\ so that S^ has T segments (all having [0, 1] as their x-projection), with Kr^r-2'~^ pairs of non-intersecting vertically visible segments. Sx is just a pjiir of parallel segments (with the same x-projection [0, 1]), so Kx = \ as required. Suppose Sr has already been constructed. To obtain 5^ + ] we construct two copies of Sr- One o...f them, S], is exactly S, . The second copy S} is obtained by first rigidly translating Sr slightly upwards, and then by "shearing" it further upwards by leav- ing the left endpoints undisturbed and by moving each right endpoint upwards by the same very large distance c. C is chosen sufficiently large so that all intersections between segments of S] and segments of S} occur to the left of the leftmost intersection of any pair of segments in Sr- We take Sr+\ to be 5jU5?. It is easily checked that for any x€[0, l] and any pair of segments e], e) in S\, if at x the segment e] lies above e] (respectively lies below e].
What to read after On Vertical Visibility in Arrangements of Segments And the Queue Size in the Ben?
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To quickly assess the difficulty of the text, read a short excerpt:
Proof: The upper bound follows immediately from Furedi's analysis. For the lower bound we use the following recursive construction. We construct collections {Sr}rs^\ so that S^ has T segments (all having [0, 1] as their x-projection), with Kr^r-2'~^ pairs of non-intersecting vertically visible segments. Sx is just a pjiir of parallel segments (with the same x-projection [0, 1]), so Kx = \ as required. Suppose Sr has already been constructed. To obtain 5^ + ] we construct two copies of Sr- One o...f them, S], is exactly S, . The second copy S} is obtained by first rigidly translating Sr slightly upwards, and then by "shearing" it further upwards by leav- ing the left endpoints undisturbed and by moving each right endpoint upwards by the same very large distance c. C is chosen sufficiently large so that all intersections between segments of S] and segments of S} occur to the left of the leftmost intersection of any pair of segments in Sr- We take Sr+\ to be 5jU5?. It is easily checked that for any x€[0, l] and any pair of segments e], e) in S\, if at x the segment e] lies above e] (respectively lies below e].
What to read after On Vertical Visibility in Arrangements of Segments And the Queue Size in the Ben?
You can find similar books in the "Read Also" column, or choose other free books by Janos Pach to read onlineMoreLess
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