THIRD PART Algorithms for Concurrent Distributed Systems: The Mutual Exclusion problem

Concurrent Distributed Systems Changes to the model from the MPS: –MPS basically models a distributed system in which processors needs to coordinate to perform a widesystem goal (e.g., electing their leader) –Now, we turn our attention to concurrent systems, where the processors run in parallel but without necessarily cooperating (for instance, they might just be a set of laptops in a LAN)

Shared Memory Model Changes to the model from the MPS: –No cooperation  no communication channels between processors and no inbuf and outbuf state components –Processors communicate via a set of shared variables, instead of passing messages  no any communication graph! –Each shared variable has a type, defining a set of operations that can be performed atomically (i.e., instantaneously, without interferences)

Shared Memory processors

Types of Shared Variables 1.Read/Write 2.Read-Modify-Write 3.Test & Set 4.Fetch-and-add 5.Compare-and-swap. We will focus on the Read/Write type (the simplest one to be realized)

Read/Write Variables Read(v)Write(v,a) return(v); v = a; In one atomic step a processor can: – read the variable, or – write the variable … but not both!

write 10

10

write read

write read 10

write 10 write 20 Simultaneous writes

write 10 write 20 Possibility 1 10 Simultaneous writes are scheduled:

write 10 write 20 Possibility 2 20 Simultaneous writes are scheduled:

write In general: write xє{a 1,…,a k } Simultaneous writes are scheduled:

read Simultaneous Reads: no problem! aa a All read the same value

Computation Step in the Shared Memory Model When processor p i takes a step: –p i 's state in old configuration specifies which shared variable is to be accessed and with which operation –operation is done: shared variable's value in the new configuration changes according to the operation –p i 's state in new configuration changes according to its old state and the result of the operation

The mutual exclusion (mutex) problem –The main challenge in managing concurrent systems is coordinating access to resources that are shared among processes –Assumptions on the system: Non-anonymous Asynchronous Fault-free

Each processor's code is divided into four sections: –entry: synchronize with others to ensure mutually exclusive access to the … –critical: use some resource; when done, enter the… –exit: clean up; when done, enter the… –remainder: not interested in using the resource Mutex code sections entrycriticalexitremainder

Mutex Algorithms A mutual exclusion algorithm specifies code for entry and exit sections to ensure: –mutual exclusion: at most one processor is in its critical section at any time, and –some kind of liveness condition. There are three commonly considered ones…

Mutex Liveness Conditions no deadlock: if a processor is in its entry section at some time, then later some processor is in its critical section no lockout: if a processor is in its entry section at some time, then later the same processor is in its critical section bounded waiting: no lockout + while a processor is in its entry section, any other processor can enter the critical section no more than a bounded number of times. These conditions are increasingly strong.

Mutex Algorithms: assumptions The code for the entry and exit sections is allowed to assume that –no processor stays in its critical section forever –shared variables used in the entry and exit sections are not accessed during the critical and remainder sections

Complexity Measure for Mutex Main complexity measure of interest for shared memory mutex algorithms is amount of shared space needed. Space complexity is affected by: –how powerful is the type of the shared variables –how strong is the progress property to be satisfied (no deadlock vs. no lockout vs. bounded waiting)

Mutex Results Using R/W Liveness Condition upper boundlower bound no deadlockn no lockout3n booleans (tournament alg.) bounded waiting 2n unbounded (bakery alg.)

Bakery Algorithm Guaranteeing: –Mutual exclusion –Bounded waiting Using 2n shared read/write variables –booleans Choosing[i] : initially false, written by p i and read by others –integers Number[i] : initially 0, written by p i and read by others

Bakery Algorithm Code for entry section: Choosing[i] := true Number[i] := max{Number[0],..., Number[n-1]} + 1 Choosing[i] := false for j := 0 to n-1 (except i) do wait until Choosing[j] = false wait until Number[j] = 0 or (Number[j],j) > (Number[i],i) endfor Code for exit section: Number[i] := 0

BA Provides Mutual Exclusion Lemma 1: If p i is in the critical section, then Number [i] > 0. Proof: Trivial. Lemma 2: If p i is in the critical section and Number [k] ≠ 0 (k ≠ i), then ( Number [k],k) > ( Number [i],i). Proof: Observe that a chosen number changes only after exiting from the CS. Since p i is in the CS, it passed the second wait statement for j=k. There are two cases: p i in CS and Number [k] ≠ 0 p i 's most recent read of Number [k]; Case 1: returns 0 Case 2: returns ( Number [k],k) > ( Number [i],i)

Case 1 p i in CS and Number [k] ≠ 0 p i 's most recent read of Number [k], returns 0. So p k is in remainder or choosing number. p i 's most recent read of Choosing [k], returns false. So p k is not in the middle of choosing number. p i 's most recent write to Number [i] So p k chooses its number in this interval, sees p i 's number, and chooses a larger one (and so it will never enter in CS before p i, which means that its number cannot change before p i exits from the CS, and so the claim is true)

Case 2 p i in CS and Number [k] ≠ 0 p i 's most recent read of Number [k] returns (Number[k],k)>(Number[i],i). So p k has already taken its number. So p k chooses a number not less than that of p i in this interval. Observe that it maybe p k finished its choice before than p i, but since (Number[k],k)>(Number[i],i), it must be Number[i]≤Number[k]. Thus, p k will never enter in CS before p i. Indeed, either p k will be stopped by p i in the first wait statement (in case Number[i]=Number[k] but p i is still choosing its number), or in the second one (in case Number[i]<Number[k]), and so the claim is true END of PROOF

Mutual Exclusion for BA Mutual Exclusion: Suppose p i and p k are simultaneously in CS. –By Lemma 1, both have number > 0. –By Lemma 2, ( Number [k],k) > ( Number [i],i) and ( Number [i],i) > ( Number [k],k)

No Lockout for BA Assume in contradiction there is a starved processor. Starved processors are stuck at the wait statements, not while choosing a number. Starved processors can be stuck only at the second wait statement, clearly. Let p i be a starved processor with smallest ( Number [i],i). Any processor entering entry section after p i has chosen its number, chooses a larger number, and therefore cannot overtake p i Every processor with a smaller ticket eventually enters CS (not starved) and exits, setting its number to 0. Thus p i cannot be stuck at the second wait statement forever by another processor.

What about bounded waiting? YES: It’s easy to see that any processor in the entry section can be overtaken at most once by any other processor (and so in total it can be overtaken at most n-1 times).

Space Complexity of BA Number of shared variables is 2n · Choosing variables are booleans · Number variables are unbounded: as long as the CS is occupied and some processor enters the entry section, the ticket number increases Is it possible for an algorithm to use less shared space?

Bounded 2-Processor ME Algorithm with ND Start with a bounded algorithm for 2 processors with ND, then extend to NL, then extend to n processors. Uses 2 binary shared read/write variables: W[0] : initially 0, written by p 0 and read by p 1 W[1] : initially 0, written by p 1 and read by p 0 Asymmetric code: p 0 always has priority over p 1

Bounded 2-Processor ME Algorithm with ND Code for p 0 's entry section: W[0] := wait until W[1] = 0 Code for p 0 's exit section: 7. 8W[0] := 0

Bounded 2-Processor ME Algorithm with ND Code for p 1 's entry section: 1W[1] := 0 2wait until W[0] = 0 3W[1] := if (W[0] = 1) then goto Line 1 6. Code for p 1 's exit section: 7. 8W[1] := 0

Discussion of 2-Processor ND Algorithm Satisfies mutual exclusion: processors use W variables to make sure of this (a processor enters only when the other W variable is set to 0) Satisfies no deadlock But unfair w.r.t. p 1 (lockout) Fix by having the processors alternate in having the priority

Bounded 2-Processor ME Algorithm with NL Uses 3 binary shared read/write variables: W[0] : initially 0, written by p 0 and read by p 1 W[1] : initially 0, written by p 1 and read by p 0 Priority : initially 0, written and read by both

Bounded 2-Processor ME Algorithm with NL Code for p i ’s entry section: 1W[i] := 0 2wait until W[1-i] = 0 or Priority = i 3W[i] := 1 4if (Priority = 1-i) then 5 if (W[1-i] = 1) then goto Line 1 6else wait until (W[1-i] = 0) Code for p i ’s exit section: 7Priority := 1-i 8W[i] := 0

Analysis: ME Mutual Exclusion: Suppose in contradiction p 0 and p 1 are simultaneously in CS. W.l.o.g., assume p 1 last write of W before entering CS happens not later than p 0 last write of W before entering CS W[0]=W[1]=1, both procs in CS p 1 's most recent write of 1 to W[1] (Line 3) p 0 's most recent write of 1 to W[0] (Line 3) p 0 's most recent read of W[1] before entering CS (Line 5 or 6): must return 1, not 0!

Analysis: No-Deadlock Useful for showing no-lockout. If one processor ever enters remainder forever, other one cannot be starved. –Ex: If p 1 enters remainder forever, then p 0 will keep seeing W[1] = 0. So any deadlock would starve both processors in the entry section

Analysis: No-Deadlock Suppose in contradiction there is deadlock. W.l.o.g., suppose Priority gets stuck at 0 after both processors are stuck in their entry sections. p 0 and p 1 stuck in entry Priority =0 p 0 not stuck in Line 2, skips Line 5, stuck in Line 6 with W[0]=1 waiting for W[1] to be 0 p 0 sees W[1]=0, enters CS p 1 sent back in Line 5, stuck at Line 2 with W[1]=0, waiting for W[0] to be 0

Analysis: No-Lockout Suppose in contradiction p 0 is starved. Since there is no deadlock, p 1 enters CS infinitely often. The first time p 1 executes Line 7 in exit section after p 0 is stuck in entry, Priority gets stuck at 0. p 1 at Line 7; Priority =0 forever after p 0 stuck in entry p 0 stuck at Line 6 with W[0]=1, waiting for W[1] to be 0 p 1 enters entry, gets stuck at Line 2, waiting for W[0] to be 0: it cannot reenter in CS! DEADLOCK!

Bounded Waiting? NO: A processor, even if having priority, might be overtaken repeatedly (in principle, an unbounded number of times) when it is in between Line 2 and 3.

Bounded n-Processor ME Alg. Can we get a bounded space NL mutex algorithm for n>2 processors? Yes! Based on the notion of a tournament tree: complete binary tree with n-1 nodes –tree is conceptual only! does not represent message passing channels A copy of the 2-proc. algorithm is associated with each tree node –includes separate copies of the 3 shared variables

Tournament Tree p 0, p 1 p 2, p 3 p 4, p 5 p 6, p 7

Tournament Tree ME Algorithm Each proc. begins entry section at a specific leaf (two procs per leaf) A proc. proceeds to next level in tree by winning the 2-proc. competition for current tree node: –on left side, play role of p 0 –on right side, play role of p 1 When a proc. wins the 2-proc. algorithm associated with the tree root, it enters CS.

The code

More on Tournament Tree Alg. Code is recursive p i begins at tree node 2 k +  i/2 , playing role of p i mod 2, where k =  log n  -1. After winning node v, "critical section" for node v is –entry code for all nodes on path from  v/2  to root –real critical section –exit code for all nodes on path from root to  v/2 

Tournament Tree Analysis Correctness: based on correctness of 2- processor algorithm and tournament structure: –ME for tournament alg. follows from ME for 2- proc. alg. at tree root. –NL for tournament alg. follows from NL for the 2- proc. algs. at all nodes of tree Space Complexity: 3n boolean read/write shared variables. Bounded Waiting? No, as for the 2- processor algorithm.