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( 2 0 points ) A d - dimensional box with dimensions ( x 1 , x 2 , . . . , xd )
points A ddimensional box with dimensions x x xd nests
within another box with dimensions y y yd if there exists a permu
tation pi on d such that xpi y xpi y xpi d yd
a points Show that the nesting relation is transitive and acyclic.
b points Let B and B be two given boxes. Describe an efficient
method to determine whether B nests within B Explain the cor
rectness of your method and its running time.
c points Consider n ddimensional boxes B B Bn Describe
an efficient algorithm to find the longest sequence Bi Bi Bik
of boxes such that Bij nests within Bij for j k
Derive the running time of your algorithm and express it in terms of
n and d Marks will be deducted if your algorithm does not work in
a reasonably efficient manner.
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