cpsc 668set 18: wait-free simulations beyond registers1 cpsc 668 distributed algorithms and systems...

CPSC 668 Set 18: Wait-Free Simulations Beyond Registers 1

CPSC 668Distributed Algorithms and Systems

Fall 2006

Prof. Jennifer Welch


Data Types Beyond Registers

• Registers support the operations read and write

• We've seen wait-free simulations of one kind of register out of another kind– different numbers of values, readers, writers

• What about (wait-free) simulating a significantly different kind of data type out of registers?

• More generally, what about (wait-free) simulating an object of type X out of objects of type Y ?


Key Insight

• Ability of objects of type Y to be used to simulate an object of type X is related to the ability of those data types to solve consensus!

• We are focusing on systems that are– asynchronous– shared memory– wait-free


FIFO Queue Example

• Sequential specification of a FIFO queue:– operation with invocation enq(x) and

response ack– operation with invocation deq and

response return(x)– a sequence of operations is allowable iff

each deq returns the oldest enqueued value that has not yet been dequeued (returns if queue is empty)


Consensus Algorithm for n = 2 Using FIFO QueueInitially Q = [0] and Prefer[i] =

Prefer[i] := pi's inputval := deq(Q)if val = 0 then

decide on pi's inputelse

temp := Prefer[1 - i]decide temp

one shared FIFO queuetwo shared registers

write my input into my register

use shared queue to arbitrate between the 2 procs: first oneto dequeue the initial 0 wins,decision value is its input

loser obtains decisionvalue from other proc'sregister


Implications of Consensus Algorithm Using FIFO Queue• Suppose we want to wait-free simulate

a FIFO queue using read/write registers.

• Is this possible?• No! If it were possible, we could solve

consensus:– simulate a FIFO queue using registers– use simulated queue and previous

algorithm to solve consensus


Extend Algorithm to More Procs?

• Can we use FIFO queues to solve consensus with more than 2 procs?

• The ability to atomically dequeue a value was key to the 2-proc alg:– one proc. learns it is the winner– the other learns it is the loser, therefore the

id of the winner is obvious

• Not clear how to handle 3 procs.• Suppose we have a different data type:


Initially First =

val := compare&swap(First, , pi's input)if val = then

decide on pi's inputelse

decide val

one shared C&S object

simultaneously indicate the winner and the value to be decided by all the losers

Consensus Algorithm Using Compare-and-Swap


Impossibility of 3-Proc Consensus with FIFO Queue

Theorem (15.3): Wait-free consensus is impossible using FIFO queues and registers if n > 2.

Proof: Same structure as for registers.

Key difference is when considering situation when

• C is bivalent

• p0(C) is 0-valent and p1(C) is 1-valent.


Impossibility of 3-Proc Consensus with FIFO Queues

• p0 and p1 must be accessing the same FIFO queue.

Case 1: Both steps are deq's.

p0 deq's p1 deq's

C0/1

0 1

p1 deq's p0 deq's

0 1look sameto p2


Impossibility Proof

Case 2: p0 deq's and p1 enq's.

Case 2.1: The queue is not empty in C

p0 deq's p1 enq's

C0/1

0 1

p1 enq's p0 deq's

?


Impossibility Proof

Case 2: p0 deq's and p1 enq's.

Case 2.2: The queue is empty in C

p0 deq's p1 enq's

C0/1

0

1

look thesame to p2

p0 deq's

1

queue is empty again

queue is empty

queue is still empty


Impossibility ProofCase 3: Both p0 and p1 enq (on same queue).

p0 enq's A p1 enq's BC

0/1

0 1p1 enq's B p0 enq's A

: p0 takessteps untildeq'ing A

: p1 takessteps untildeq'ing B

: p0 takessteps untildeq'ing B

: p1 takessteps untildeq'ing A

0 1look the same to p2

why do and exist?


Impossibility ProofCase 3 cont'd: Suppose does not exist:

p0 enq's A p1 enq's BC

0/1

0 1p1 enq's B p0 enq's A

p0 takessteps untildecidingbut neverdeq's A;decides 0

p0 takes same numberof steps as on the left;never deq's B; alsodecides 0

0 1


Impossibility Proof

Case 3 cont'd: Prove existence of similarly.

Thus there is no wait-free algorithm for consensus with 3 procs using FIFO queues and registers.


Implications

• Suppose we want to wait-free simulate a compare&swap object using FIFO queues (and registers).

• Is this possible?• Not if n > 2! If it were possible, we could

solve consensus using FIFO queues (and registers):– simulate a compare&swap object using FIFO

queues (and registers)– use simulated compare&swap object and c&s

algorithm to solve consensus


Generalize these Arguments

• Previous results concerning FIFO queues and compare&swap suggest a criterion for determining if wait-free simulations exist:

• based on ability of the data types to solve consensus for a certain number of procs.


Consensus NumberData type X has consensus number n if n is the largest

number of procs. for which consensus can be solved using only objects of type X and read/write registers.

data type consensus

number

read/write register 1

FIFO queue 2

compare&swap ∞


Using Consensus NumbersTheorem (15.5): If data type X has consensus

number m and data type Y has consensus number n with n > m, then there is no wait-free simulation of an object of type Y using objects of type X and read/write registers in a system with more than m procs.

X X X …

reg reg reg …Y


Using Consensus Numbers

Proof: Suppose in contradiction there is a wait-free simulation S of Y using X and registers in a system with k procs, where m < k ≤ n.

• Construct consensus algorithm for k > m procs using objects of type X (and registers):– Use S to simulate some objects of type Y using

objects of type X (and registers)– Use the (simulated) type Y objects (and registers)

in the k-proc consensus algorithm that exists since CN(Y) = n.


Corollaries

• There is no wait-free simulation of any object with consensus number > 1 using just read/write registers.

• There is no wait-free simulation of any object with consensus number > 2 using just FIFO queues and read/write registers.


Universality

• Let's now consider positive results relating to consensus number.

• A data type is universal if objects of that type (together with read/write registers) can wait-free simulate any data type.

• Theorem: If data type X has consensus number n, then it is universal in a system with at most n procs.


Proving Universality Result

1. Describe an algorithm that simulates any data type

– uses compare&swap (instead of any object with consensus number n)

– simulation is only non-blocking, weaker than wait-free

2. Modify to use any object with consensus number n

3. Modify to be wait-free 4. Modify to bound shared memory used


Non-Blocking

• Non-blocking vs. wait-free is analogous to no-deadlock vs. no-lockout for mutual exclusion.

• Non-blocking simulation: at any point in an execution, if at least one operation is pending (response is not yet ready to be done), then there is a finite sequence of steps by a single proc that completes one of the pending operations.

• Does not ensure that every pending operation is eventually completed.


Universal Construction

• Keep history of operations that have been applied to the simulated object as a shared linked list.

• To apply an operation on the simulated object, the invoking proc. must insert an appropriate "node" into the linked list:– it is convenient to put the newest node at the head

of the list

• A compare&swap object is used to keep track of the head of the list


Details on Linked ListEach linked list node has• operation invocation• new state of the simulated object• operation response• pointer to previous node (previous op)

invocation

state

response

before

invocation

state

response

before

initial state

anchor

Head


SimulationInitially Head points to anchor node

– represents initial state of simulated object

When inv is invoked:allocate a new linked list node in shared memory,

pointed to by local var pointpoint.inv := invrepeat

h := Head // h is a local varpoint.state, point.response := apply(inv,h.state)point.before := h

until compare&swap(Head,h,point) = hdo the output indicated by point.response

depends on simulated data type

if Head still points tosame node h pointsto, then make Headpoint to new node.


Strengthenings of Algorithm

• To use replace compare&swap object with any object with consensus number n (the number of procs):– define a consensus object (data type

version of consensus problem)– get around the difficulty that a consensus

object can only be used once by adding a consensus object to each linked list node that points to next node in the list


Strengthenings of Algorithm

• To get a wait-free implementation, use idea of helping: procs help each other to finish pending operations (not just their own)

• To reduce the size of the linked list (so it doesn't grow without bound), need to keep track of which list nodes can be recycled.


Effect of Randomization

• Suppose we relax the liveness condition for linearizable shared memory:– operations must terminate with high

probability

• Now a randomized consensus algorithm can be used to simulate any data type out of any other data type, including read/write registers

• I.e., hierarchy collapses.

cpsc 668set 18: wait-free simulations beyond registers1 cpsc 668 distributed algorithms and systems...

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