$($a$)$ We wrote $$Open that polling place j$.$ You might legitimately worry that the algorithm is underspecified

if there are multiple polling places j$,$ j$$

that each reach n$/$k voters at exactly the same time t$.$ Explain why

this is not a concern here.

$($b$)$ As we said above, the description about continuously increasing t and looking what happens is not $$programming friendly$.$ Here, you are to describe how to implement this algorithm to run in time O$($n

$2$

log n$).$

$($See below for the case when you find that difficult.$)$ In doing so$,$ you will almost certainly want to draw on

multiple algorithms and data structures you learned in CSCI $104,$ as well as some mathematical knowledge

about logs and other functions.

You should not give actual code here $$ rather, give pseudo$-$code which a competent CSCI $104$ student

could easily turn into actual Python$/$C$++$ code. Also, if you use algorithms or data structures from CSCI

$104($or CSCI $170),$ you do not need to actually give the code $$ simply say which algorithm or operation

you are calling $($such as $$Run BFS from node v and store the distances in an array x$.).$

$($c$)$ Explain informally why your detailed algorithm returns the same result as our informal continuous$-$time

algorithm.

$($d$)$ Carefully analyze the running time and show that it is O$($n

$2$

log n$).($Note that we are only asking for the

upper bound of O$($n

$2$

log n$)$ here, not a lower bound of $($n

$2$

log n$)$ though the latter would be easy to

prove.$)$ For any algorithms or data structures you learned in CSCI $104,$ you can simply state the running

times that you were taught and use in your analysis $$ no need to reprove them yourself.

If there are some parts that you cannot get to run in O$($n

$2$

log n$),$ explain which those parts are, what obstacles

you were facing, and what you would need to make it work. The more parts you can get in O$($n

$2$

log n$),$ the

closer to full credit you will be$.$ If your algorithm runs in O$($n

$3$

$),$ you will get partial credit, depending on how

many key parts are faster.