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Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md

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+### Aravind’s algorithm
+Problem with Lamport's "Bakery" algorithm: registers must be unbounded (every invocation of lock potentially increases the counter by 1 -> domain of the registers is all natural numbers!)
+
+For all processes, we have a FLAG and a STAGE (both binary MRSW) and a DATE (MRMW) register that ranges from 1 to 2n.
+
+```
+For all i, initialize
+	FLAG[i] to down
+	STAGE[i] to 0
+	DATE[i] to i
+
+lock(i) :=
+	FLAG[i] <- up
+	repeat
+		STAGE[i] <- 0
+		wait (foreach j != i, FLAG[j] = down OR DATE[i] < DATE[j])
+		STAGE[i] <- 1
+	until foreach j != i, STAGE[j] = 0
+
+unlock(i) :=
+	tmp <- max_j{DATE[j]}+1
+	if tmp >= 2n
+		then foreach j, DATE[j] <- j
+		else DATE[i] <- tmp
+	STAGE[i] <- 0
+	FLAG[i] <- down
+```
+
+#### MUTEX proof
+**Theorem:** if $p_i$ is in the CS, then $p_j$ cannot simultaneously be in the CS.
+*Proof:* by contradiction.
+
+Let's consider the execution of $p_i$ leading to its CS:
+![[Pasted image 20250310172134.png]]
+
+**Corollary** (of the MUTEX proof)**:** DATE is never written concurrently.
+
+#### Bounded bypass proof
+**Lemma 1:** exactly after n CSs there is a reset of DATE.
+*Proof:*
+- the first CS leads $max_j{DATE[j]}$ to n+1
+- the seconds CS leads ... to n+2
+- ...
+- the n-th read leads ... to n+n = 2n -> RESET
+
+**Lemma 2:** there can be at most one reset of DATE during an invocation of a lock
+*Proof:*
+- let $p_i$ invoke lock, if no reset occurs, ok
+- otherwise, let us consider the moment in which a reset occurs
+	- if pi is the next process that enters the CS, ok
+	- Otherwise let $p_j$ be the process that enters; its next date is $n+1 > DATE[i]$
+		- $p_{j}$ cannot surpass $p_i$ again (before a RESET)
+	- The worst case is then all processes perform lock together and $i = n$ (i am process n)
+		- all $p_{1}\dots p_{n}$ surpass $p_{n}$
+		- then $p_n$ enters and it resets the DATE in its unlock
+			- only 1 reset and it is the worst case!
+
+**Theorem:** the algorithm satisfies bounded bypass with bound $2n-2$.
+*Proof:*
+![[Pasted image 20250310103703.png]]
+so by this, the very worst possible case is that my lock experiences that.
+
+It looks like I can experience at most $2n-1$ other critical sections, but it is even better, let's see:
+- $p_n$ invokes lock alone, completes its CS (the first after the reset) and its new DATE is n+1
+- all processes invoke lock simultaneously
+- $p_{n}$ has to wait all other processes to complete their CSs
+- when $p_{n-1}$ completes its CS, its new DATE will be $n+(n-1)+1=2n$ -> RESET
+- now all $p_{1}\dots p_{n-1}$ invoke lock again and complete their CSs (after that $p_i$ completes its CS, now it has `DATE[i] <- n+i`, because as everyone invoked lock after the RESET, max date was `n`)
+- so $p_n$ has to wait n-1 CSs for the reset, and another n-1 CSs before entering again.  **Literally the worst case is when the process is the first of the first round, and the last of the last round.**
+
+#### Improvement of Aravind’s algorithm
+```
+unlock(i) :=
+	∀j≠i.if DATE[j] > DATE[i] then
+		DATE[j] <- DATE[j]-1
+		DATE[i] <- n
+		STAGE[i] <- 0
+		FLAG[i] <- down
+```
+
+Since the LOCK is like before, the revised protocol satisfies MUTEX. Furthermore, you can prove that it satisfies bounded bypass with bound n-1 -> EXERCISE!
+