$[$Hashing : $20\%]$ Answer the following questions related to hashing.

$($a$)(7\%)$ Demonstrate what happens when we insert the keys $17,16,10,9,8,1,2,3,15,14,13,12$ into a hash table of size $13$ with collision resolved by double hashing. The hash functions are defined by $h_1(k)=$kmod$13$ and $h_2(k)=1+(k$ mod $7).$ You do not need to finish all but up to the moment when you just finish the insertion with the second collision.

$\backslash$table$[[0,],[1,],[2,],[3,],[4,],[5,],[6,],[7,],[8,],[9,],[10,],[11,],[12,]]$

$($b$)(7\%)$ If the function $h_1(k)$ is substituted by $h_1(k)=k$ mod $14,$ what will happen?

Try to discuss as many problems as possible.

$($c$)(6\%)$ Describe any defect for the quadratic probing designed by $h(k,i)=(h^{\text{'}}(k)+c_{1}i+c_2{i}^2)mod24,$ where $c_1=3$ and $c_2=6.$

$($d$)(5\%)$ Is $O(n^2)+(n^{3}lnn)=(n^{3}lnn)$ correct? Explain your answer.