Prob $1.$ Given the input list S$=[79,28,49,17,68],$ assume we want to use a hash table with the length of $8.$ Please answer the following questions:

a$)$ Please list the potential benefits of hash table.

b$)$ Assuming we need to use the hashing function modular division: h$($k$)=$ k mod N$,$ where N is the length of the hash table, do we have the collision? How can we be able to handle the collison $($list $3$ possible methods$)?$

c$)$ Assuming we consider modular division: h$($k$)=$ k mod N$,$ if you are required to use the separate chain table to handle the collision, please draw the steps that we need to insert the elements in S$=[79,28,49,17,68]$ into an empty hash table.

d$)$ Assuming we consider modular division: h$($k$)=$ k mod N$,$ if you are required to use the linear probing to handle the collision, please draw the steps that we need to insert the elements in S$=[79,28,49,17,68]$ into an empty hash table.

e$)$ Assuming we consider modular division: h$($k$)=$ k mod N$,$ if you are required to use the quadratic probing to handle the collision, please draw the steps to insert the elements in S$=[79,28,49,17,68]$ into an empty hash table.

e$)$ Assuming we consider modular division: h$($k$)=$ k mod N and h$($k$)=3-($k mod $3),$ if you are required to use the double hashing to handle the collision, please draw the steps to insert the elements in S$=[79,28,49,17,68]$ into an empty hash table.

Prob $2.$ Assuming you work for a marketing company and you have $10$ customers living in a small community. Based on the posted information, you have the following information:

Emma and Liam are friends. Liam and Olivia are classmates. Liam and Isabella know each other. Olivia and Jackson are dating. Liam and Lucas had a road trip together. Locas and Sophia were close friend. Liam and Noah are doing the senior design together. Noah and Ava are in a small study group. Aiden and Isabella did field study together. Oliva and Ava go to the same church and belongs to the same group. Ava and Aiden are siblings.

a$)$ Your manager requests you to develop an algorithm to group the existing customers into different social networks. What kind of algorithm you can use?

b$)$ Please use the UFConnectedComponents function to classify customers into disjoint social networks. List your steps.

c$)$ Please use the tree$-$based implementation and consider the path compression at the same time. List your steps.