$(20$ points$)$ A $d-$dimensional box with dimensions $(x_1,x_2,$dots,$x_d)$ nests within another box with dimensions $(y_1,y_2,$dots,$y_d)$ if there exists a permutation $$ on $\{1,2,$dots,$d\}$ such that $B_1{B}_2{B}_1{B}_{2}nd{B}_1,B_2,$dots,$B_n(B_{i_1},B_{i_2},$dots,$B_{i_k})B_{i_j}{B}_{i_{j+1}}j=1,2,$dots,$k-1nd{x}_{(1)}.$

$(a)(5$ points$)$ Show that the nesting relation $is$ transitive and acyclic.

$(b)(5$ points$)$ Let $B_1$ and $B_{2}be$ two given boxes. Describe $an$ efficient method $to$ determine whether $B_1$ nests within $B_2.$ Explain the correctness $of$ your method and its running time.

$(c)(10$ points$)$ Consider $nd-$dimensional boxes $B_1,B_2,$dots,$B_n.$ Describe $an$ efficient algorithm $to$ find the longest sequence $(B_{i_1},B_{i_2},$dots,$B_{i_k})of$ boxes such that $B_{i_j}$ nests within $B_{i_{j+1}}$ for $j=1,2,$dots,$k-1.$ Derive the running time $of$ your algorithm and express $itin$ terms $ofn$ and $d.$ Marks will $be$ deducted $if$ your algorithm does not work $in$ a reasonably efficient manner.