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( 2 0 points ) A d - dimensional box with dimensions ( x 1 , x 2 , . . . , xd )

(20 points) A d-dimensional box with dimensions (x1, x2,..., xd) nests
within another box with dimensions (y1, y2,..., yd) if there exists a permu-
tation \pi on {1,2,..., d} such that x\pi (1)< y1, x\pi (2)< y2,..., x\pi (d)< yd.
(a)(5 points) Show that the nesting relation is transitive and acyclic.
(b)(5 points) Let B1 and B2 be two given boxes. Describe an efficient
method to determine whether B1 nests within B2. Explain the cor-
rectness of your method and its running time.
(c)(10 points) Consider n d-dimensional boxes B1, B2,..., Bn. Describe
an efficient algorithm to find the longest sequence (Bi1, Bi2,..., Bik )
of boxes such that Bij nests within Bij+1 for j =1,2,..., k 1.
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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