$(30$ points$)$ Given two $2-$dimensional points $A=($:$x_1,y_1$:$)$ and $B=($:$x_2,y_2$:$),$ we say that $S=\{{}_1,{}_2,$cdots,${}_n\}{S}^{\text{'}}subeSCD{S}^{\text{'}}{S}^{\text{'}}$SDinS$-S^{\text{'}}C{S}^{\text{'}}D.CorD.D.$ Give a divide and conquer algorithm $to$ solve this problem. Write

the recurrence relation describing the runtime $of$ the algorithm and give the solution $to$ the

recurrence. You $do$ not need $to$ show your work for deriving the solution $to$ the recurrence.

Your will receive zero credit $if$ you $do$ not use divide$-$and$-$conquer.

Note that a subset $S^{\text{'}}ofS$ maximal $if$ the following holds: For every DinS$-S^{\text{'}},$ there exists

$Cin{S}^{\text{'}}$ such that either $CorD.C$ nor $D.$ Give a divide and conquer algorithm $to$ solve this problem. Write

the recurrence relation describing the runtime $of$ the algorithm and give the solution $to$ the

recurrence. You $do$ not need $to$ show your work for deriving the solution $to$ the recurrence.

Your will receive zero credit $if$ you $do$ not use divide$-$and$-$conquer.

Note that a subset $S^{\text{'}}ofS$ maximal $if$ the following holds: For every DinS$-S^{\text{'}},$ there exists

$Cin{S}^{\text{'}}$ such that either $CorD.y_1.$ Given a set $ofS=\{{}_1,{}_2,$cdots,${}_n\}of2-$dimensional points, the goal

$isto$ output a maximal set $S^{\text{'}}$subeS such that for every pair $of$ points $C$ and $Din{S}^{\text{'}},$ neither

$C$ nor $D.$ Give a divide and conquer algorithm $to$ solve this problem. Write

the recurrence relation describing the runtime $of$ the algorithm and give the solution $to$ the

recurrence. You $do$ not need $to$ show your work for deriving the solution $to$ the recurrence.

Your will receive zero credit $if$ you $do$ not use divide$-$and$-$conquer.

Note that a subset $S^{\text{'}}ofS$ maximal $if$ the following holds: For every DinS$-S^{\text{'}},$ there exists

$Cin{S}^{\text{'}}$ such that either $CorD.x_1$ and $y_1.$ Given a set $ofS=\{{}_1,{}_2,$cdots,${}_n\}of2-$dimensional points, the goal

$isto$ output a maximal set $S^{\text{'}}$subeS such that for every pair $of$ points $C$ and $Din{S}^{\text{'}},$ neither

$C$ nor $D.$ Give a divide and conquer algorithm $to$ solve this problem. Write

the recurrence relation describing the runtime $of$ the algorithm and give the solution $to$ the

recurrence. You $do$ not need $to$ show your work for deriving the solution $to$ the recurrence.

Your will receive zero credit $if$ you $do$ not use divide$-$and$-$conquer.

Note that a subset $S^{\text{'}}ofS$ maximal $if$ the following holds: For every DinS$-S^{\text{'}},$ there exists

$Cin{S}^{\text{'}}$ such that either $CorD.A$

$if{x}_1$ and $y_1.$ Given a set $ofS=\{{}_1,{}_2,$cdots,${}_n\}of2-$dimensional points, the goal

$isto$ output a maximal set $S^{\text{'}}$subeS such that for every pair $of$ points $C$ and $Din{S}^{\text{'}},$ neither

$C$ nor $D.$ Give a divide and conquer algorithm $to$ solve this problem. Write

the recurrence relation describing the runtime $of$ the algorithm and give the solution $to$ the

recurrence. You $do$ not need $to$ show your work for deriving the solution $to$ the recurrence.

Your will receive zero credit $if$ you $do$ not use divide$-$and$-$conquer.

Note that a subset $S^{\text{'}}ofS$ maximal $if$ the following holds: For every DinS$-S^{\text{'}},$ there exists

$Cin{S}^{\text{'}}$ such that either $CorD.$