Problem $1$: Find Crossover Indices. We have two cross over points. Your algorithm may output either $i=0$ or $i=3.$

$($A$)$ Design an algorithm to find an index iin$\{0,1,$dots,$n-1\}$ such that $x_i{y}_i$ but $x,yxxyyxy(n)0x_{i+1}.$

Describe your algorithm using python code for a function findCrossoverIndexHelper$(x,y,$ left, right$)$

$xis$ a list $ofx$ values sorted $in$ increasing order.

$yis$ a list $ofy$ values sorted $in$ increasing order.

$x$ and $y$ are lists $of$ same size $(n).$

left and right are indices that represent the current search region $in$ the list such that $0$ left $$ right $n$

Your solution must use recursion.

Hint: Modify the binary search algorithm $we$ presented $in$ class. In $[]$: #First write a "helper" function with two extra parameters

# left, right that ddedscribes the search region as shown below

def findCrossoverIndexHelper$($x$,$ y$,$ left, right$)$:

# Note: Output index i such that

# left $i$ right

# $,x[i]y[i]$

# First, Write down our invariants as assertions here

assert$($len $(x)==$len$(y)$

assert$($left $>=0)$

assert$($left right$-1)$

assert$($right $$ len$($x$))$

# Here is the key property we would like to maintain.

assert$($x$[$left$]>$ y$[$left$])$

assert right

You are given data that consists of points $(x_0,y_0),$dots,$(x_n,y_n),$ wherein $y_n>x_{n}in-1y_i{x}_i{y}_{i+1}>x_{i+1}i=3i=0i=3$iin$\{0,1,$dots,

Describe your algorithm using python code for a function findCrossoverIndexHelper$(x,y,$ left, right$)$

$xis$ a list $ofx$ values sorted $in$ increasing order.

$yis$ a list $ofy$ values sorted $in$ increasing order.

$x$ and $y$ are lists $of$ same size $(n).$

left and right are indices that represent the current search reaion $in$ the list such that $0$ left $$ right $n{y}_0$ and $y_n>x_n.$

Find a "cross$-$over" index i between $0$ and $n-1$ such that $y_i{x}_i$ and $y_{i+1}>x_{i+1}.$

Note that such $an$ index must always exist $(convince$ yourself $of$ this fact before $we$ proceed$).$

Example

Your algorithm must find the index $i=3as$ the crossover point.

$On$ the other hand, consider the data

$We$ have two cross over points. Your algorithm may output either $i=0ori=3.$

$(A)$ Design $an$ algorithm $to$ find $an$ index iin$\{0,1,$dots,$n-1\}$ such that $x_i{y}_i$ but $x_{i+1}.$

Describe your algorithm using python code for a function findCrossoverIndexHelper$(x,y,$ left, right$)$

$xis$ a list $ofx$ values sorted $in$ increasing order.

$yis$ a list $ofy$ values sorted $in$ increasing order.

$x$ and $y$ are lists $of$ same size $(n).$

left and right are indices that represent the current search reaion $in$ the list such that $0$ left $$ right $n{y}_{0}as$ well.

Furthermore, $itis$ given that $y_0$ and $y_n>x_n.$

Find a "cross$-$over" index i between $0$ and $n-1$ such that $y_i{x}_i$ and $y_{i+1}>x_{i+1}.$

Note that such $an$ index must always exist $(convince$ yourself $of$ this fact before $we$ proceed$).$

Example

Your algorithm must find the index $i=3as$ the crossover point.

$On$ the other hand, consider the data

$We$ have two cross over points. Your algorithm may output either $i=0ori=3.$

$(A)$ Design $an$ algorithm $to$ find $an$ index iin$\{0,1,$dots,$n-1\}$ such that $x_i{y}_i$ but $x_{i+1}.$

Describe your algorithm using python code for a function findCrossoverIndexHelper$(x,y,$ left, right$)$

$xis$ a list $ofx$ values sorted $in$ increasing order.

$yis$ a list $ofy$ values sorted $in$ increasing order.

$x$ and $y$ are lists $of$ same size $(n).$

left and right are indices that represent the current search reaion $in$ the list such that $0$ left $$ right $n{x}_0,$ and $y_{0}as$ well.

Furthermore, $itis$ given that $y_0$ and $y_n>x_n.$

Find a "cross$-$over" index i between $0$ and $n-1$ such that $y_i{x}_i$ and $y_{i+1}>x_{i+1}.$

Note that such $an$ index must always exist $(convince$ yourself $of$ this fact before $we$ proceed$).$

Example

Your algorithm must find the index $i=3as$ the crossover point.

$On$ the other hand, consider the data

$We$ have two cross over points. Your algorithm may output either $i=0ori=3.$

$(A)$ Design $an$ algorithm $to$ find $an$ index iin$\{0,1,$dots,$n-1\}$ such that $x_i{y}_i$ but $x_{i+1}.$

Describe your algorithm using python code for a function findCrossoverIndexHelper$(x,y,$ left, right$)$

$xis$ a list $ofx$ values sorted $in$ increasing order.

$yis$ a list $ofy$ values sorted $in$ increasing order.

$x$ and $y$ are lists $of$ same size $(n).$

left and right are indices that represent the current search reaion $in$ the list such that $0$ left $$ right $n$