$3.$ A common pattern in parallel scientific programs is to have a set of threads do a computation in a sequence of phases. In each phase $\backslash ($ i $\backslash ),$ all threads must finish phase $\backslash ($ i $\backslash )$ before any thread starts computing phase $\backslash ($ i$+1\backslash ).$ One way to accomplish this is with barrier synchronization. At the end of each phase, each thread executes Barrier::Done$(\backslash ($ n $\backslash )),$ where $\backslash ($ n $\backslash )$ is the number of threads in the computation. A call to Barrier $\backslash ($ :: $\backslash )$ Done blocks until all of the $\backslash ($ n $\backslash )$ threads have called Barrier::Done. Then, all threads proceed. You may assume that the process allocates a new Barrier for each iteration, and that all threads of the program will call Done with the same value.

a$.$ Use pseudocode to write a monitor that implements Barrier.

$```$

monitor Barrier $\{$

$...$private variables...

void Done $($int n$)\{$

$...$

$\}$

$...$

$\}$

$```$

b$.$ Use pseudocode to implement Barrier using an explicit lock and condition variable. The lock and condition variable have the semantics described at the end of the "Semaphore and Monitor" lecture in the ping$\_$pong example, and as implemented by you in Project $1.$

$```$

class Barrier $\{$

$...$private variables...

void Done $($int n$)\{$

$...$

$\}$

$...$

$```$