CPSC 668 Distributed Algorithms and Systems

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CPSC 668Distributed Algorithms and Systems

• Fall 2006
• Prof. Jennifer Welch

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Distributed Shared Memory

• A model for inter-process communication
• Provides illusion of shared variables on top of
  message passing
• Shared memory is often considered a more
  convenient programming platform than message
  passing
• Formally, give a simulation of the shared memory
  model on top of the message passing model
• We'll consider the special case of
• no failures
• only read/write variables to be simulated

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Shared Memory Issues

• A process will invoke a shared memory operation
  at some time
• The simulation algorithm running on the same node
  will execute some code, possibly involving
  exchanges of messages
• Eventually the simulation algorithm will inform
  the process of the result of the shared memory
  operation.
• So shared memory operations are not
  instantaneous!
• Operations (invoked by different processes) can
  overlap
• What should be returned by operations that
  overlap other operations?
• defined by a memory consistency condition

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Sequential Specifications

• Each shared object has a sequential
  specification specifies behavior of object in
  the absence of concurrency.
• Object supports operations
• invocations
• matching responses
• Set of sequences of operations that are legal

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Sequential Spec for R/W Registers

• Operations are reads and writes
• Invocations are readi(X) and writei(X,v)
• Responses are returni(X,v) and acki(X)
• A sequence of operations is legal iff each read
  returns the value of the latest preceding write.

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Memory Consistency Conditions

• Consistency conditions tie together the
  sequential specification with what happens in the
  presence of concurrency.
• We will study two well-known conditions
• linearizability
• sequential consistency
• We will only consider read/write registers, in
  the absence of failures.

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Definition of Linearizability

• Suppose ? is a sequence of invocations and
  responses.
• an invocation is not necessarily immediately
  followed by its matching response
• ? is linearizable if there exists a permutation ?
  of all the operations in ? (now each invocation
  is immediately followed by its matching response)
  s.t.
• ?X is legal (satisfies sequential spec) for all
  X, and
• if response of operation O1 occurs in ? before
  invocation of operation O2, then O1 occurs in ?
  before O2 (? respects real-time order of
  non-concurrent operations in ?).

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Linearizability Examples
Suppose there are two shared variables, X and Y,
both initially 0
p0
1
3
0
p1
2
4
Is this sequence linearizable?
Yes - green triangles.
What if p1's read returns 0?
No - see arrow.
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Definition of Sequential Consistency

• Suppose ? is a sequence of invocations and
  responses.
• ? is sequentially consistent if there exists a
  permutation ? of all the operations in ? s.t.
• ?X is legal (satisfies sequential spec) for all
  X, and
• if response of operation O1 occurs in ? before
  invocation of operation O2 at the same process,
  then O1 occurs in ? before O2 (? respects
  real-time order of operations by the same process
  in ?).

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Sequential Consistency Examples
Suppose there are two shared variables, X and Y,
both initially 0
0
4
3
p0
1
2
p1
Is this sequence sequentially consistent?
Yes - green numbers.
What if p0's read returns 0?
No - see arrows.
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Specification of Linearizable Shared Memory Comm.
System

• Inputs are invocations on the shared objects
• Outputs are responses from the shared objects
• A sequence ? is in the allowable set iff
• Correct Interaction each proc. alternates
  invocations and matching responses
• Liveness each invocation has a matching
  response
• Linearizability ? is linearizable

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Specification of Sequentially Consistent Shared
Memory

• Inputs are invocations on the shared objects
• Outputs are responses from the shared objects
• A sequence ? is in the allowable set iff
• Correct Interaction each proc. alternates
  invocations and matching responses
• Liveness each invocation has a matching
  response
• Sequential Consistency ? is sequentially
  consistent

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Algorithm to Implement Linearizable Shared Memory

• Uses totally ordered broadcast as the underlying
  communication system.
• Each proc keeps a replica for each shared
  variable
• When read request arrives
• send bcast msg containing request
• when own bcast msg arrives, return value in local
  replica
• When write request arrives
• send bcast msg containing request
• upon receipt, each proc updates its replica's
  value
• when own bcast msg arrives, respond with ack

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The Simulation
user of read/write shared memory
read/write
return/ack
read/write
return/ack

Shared Memory
alg0
algn-1
to-bc-send
to-bc-recv
to-bc-send
to-bc-recv
Totally Ordered Broadcast
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Correctness of Linearizability Algorithm

• Consider any admissible execution ? of the
  algorithm
• underlying totally ordered broadcast behaves
  properly
• users interact properly
• Show that ?, the restriction of ? to the events
  of the top interface, satisfies Liveness, and
  Linearizability.

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Correctness of Linearizability Algorithm

• Liveness (every invocation has a response) By
  Liveness property of the underlying totally
  ordered broadcast.
• Linearizability Define the permutation ? of the
  operations to be the order in which the
  corresponding broadcasts are received.
• ? is legal because all the operations are
  consistently ordered by the TO bcast.
• ? respects real-time order of operations if O1
  finishes before O2 begins, O1's bcast is ordered
  before O2's bcast.

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Why is Read Bcast Needed?

• The bcast done for a read causes no changes to
  any replicas, just delays the response to the
  read.
• Why is it needed?
• Let's see what happens if we remove it.

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Why Read Bcast is Needed
read return(1)
p0
write(1)
p1
to-bc-send
p2
read return(0)
Not linearizable!
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Algorithm for Sequential Consistency

• The linearizability algorithm, without doing a
  bcast for reads
• Uses totally ordered broadcast as the underlying
  communication system.
• Each proc keeps a replica for each shared
  variable
• When read request arrives
• immediately return the value stored in the local
  replica
• When write request arrives
• send bcast msg containing request
• upon receipt, each proc updates its replica's
  value
• when own bcast msg arrives, respond with ack

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Correctness of SC Algorithm

• Lemma (9.3) The local copies at each proc. take
  on all the values appearing in write operations,
  in the same order, which preserves the per-proc.
  order of writes.
• Lemma (9.4) If pi writes Y and later reads X,
  then pi's update of its local copy of Y (on
  behalf of that write) precedes its read of its
  local copy of X (on behalf of that read).

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Correctness of the SC Algorithm

• (Theorem 9.5) Why does SC hold?
• Given any admissible execution ?, must come up
  with a permutation ? of the shared memory
  operations that is
• legal and
• respects per-proc. ordering of operations

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The Permutation ?

• Insert all writes into ? in their to-bcast order.
• Consider each read R in ? in the order of
  invocation
• suppose R is a read by pi of X
• place R in ? immediately after the later of
• the operation by pi that immediately precedes R
  in ?, and
• the write that R "read from" (caused the latest
  update of pi's local copy of X preceding the
  response for R)

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Permutation Example
4
read return(2)
p0
write(2)
3
ack
p1
to-bc-send
to-bc-send
p2
read return(1)
write(1)
ack
1
2
permutation is given by green numbers
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Permutation ? Respects Per Proc. Ordering

• For a specific proc
• Relative ordering of two writes is preserved by
  Lemma 9.3
• Relative ordering of two reads is preserved by
  the construction of ?
• If write W precedes read R in exec. ?, then W
  precedes R in ? by construction
• Suppose read R precedes write W in ?. Show same
  is true in ?.

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Permutation ? Respects Ordering

• Suppose R and W are swapped in ?
• There is a read R' by pi that equals or precedes
  R in
• There is a write W' that equals W or follows W in
  the to-bcast order
• And R' "reads from" W'.

R
W
R'
?pi
W W' R' R
?

• But
• R' finishes before W starts in ? and
• updates are done to local replicas in to-bcast
  order (Lemma 9.3) so update for W' does not
  precede update for W
• so R' cannot read from W'.

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Permutation ? is Legal

• Consider some read R by pi and some write W s.t.
  R reads from W in ?.
• Suppose in contradiction, some other write W'
  falls between W and R in ?
• Why does R follow W' in ??

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Permutation ? is Legal

• Case 1 R follows W' in ? because W' is also by
  pi and R follows W' in ?.
• Update for W at pi precedes update for W' at pi
  in ? (Lemma 9.3).
• Thus R does not read from W, contradiction.

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Permutation ? is Legal

• Case 2 R follows W' in ? due to some operation
  O by pi s.t.
• O precedes R in ?, and
• O is placed between W' and R in ?

• Case 2.1 O is a write.
• update for W' at pi precedes update for O at pi
  in ? (Lemma 9.3)
• update for O at pi precedes pi's local read for R
  in ? (Lemma 9.4)
• So R does not read from W, contradiction.

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Permutation ? is Legal
W W' O' O R
?

• Case 2.2 O is a read.
• A recursive argument shows that there exists a
  read O' by pi (which might equal O) that
• reads from W' in ? and
• appears in ? between W' and O
• Update for W at pi precedes update for W' at pi
  in ? (Lemma 9.3).
• Update for W' at pi precedes local read for O' at
  pi in ? (otherwise O' would not read from W').
• Recall that O' equals or precedes O (from above)
  and O precedes R (by assumption for Case 2) in ?
• Thus R cannot read from W, contradiction.

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Performance of SC Algorithm

• Read operations are implemented "locally",
  without requiring any inter-process
  communication.
• Thus reads can be viewed as "fast" time between
  invocation and response is that needed for some
  local computation.
• Time for writes is time for delivery of one
  totally ordered broadcast (depends on how
  to-bcast is implemented).

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Alternative SC Algorithm

• It is possible to have an algorithm that
  implements sequentially consistent shared memory
  on top of totally ordered broadcast that has
  reverse performance
• writes are local/fast (even though bcasts are
  sent, don't wait for them to be received)
• reads can require waiting for some bcasts to be
  received
• Like the previous SC algorithm, this one does not
  implement linearizable shared memory.

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Time Complexity for DSM Algorithms

• One complexity measure of interest for DSM
  algorithms is how long it takes for operations to
  complete.
• The linearizability algorithm required D time for
  both reads and writes, where D is the maximum
  time for a totally-ordered broadcast message to
  be received.
• The sequential consistency algorithm required D
  time for writes and C time for reads, where C is
  the time for doing some local computation.
• Can we do better? To answer this question, we
  need some kind of timing model.

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Timing Model

• Assume the underlying communication system is the
  point-to-point message passing system (not
  totally ordered broadcast).
• Assume that every message has delay in the range
  d-u,d.
• Claim Totally ordered broadcast can be
  implemented in this model so that D, the maximum
  time for delivery, is O(d).

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Time and Clocks in Layered Model

• Timed execution associate an occurrence time
  with each node input event.
• Times of other events are "inherited" from time
  of triggering node input
• recall assumption that local processing time is
  negligible.
• Model hardware clocks as before run at same
  rate as real time, but not synchronized
• Notions of view, timed view, shifting are same
• Shifting Lemma still holds (relates h/w clocks
  and msg delays between original and shifted execs)

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Lower Bound for SC

• Let Tread worst-case time for a read to
  complete
• Let Twrite worst-case time for a write to
  complete
• Theorem (9.7) In any simulation of sequentially
  consistent shared memory on top of point-to-point
  message passing, Tread Twrite ? d.

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SC Lower Bound Proof

• Consider any SC simulation with Tread Twrite lt
  d.
• Let X and Y be two shared variables, both
  initially 0.
• Let ?0 be admissible execution whose top layer
  behavior is
• write0(X,1) ack0(X) read0(Y) return0(Y,0)
• write begins at time 0, read ends before time d
• every msg has delay d
• Why does ?0 exist?
• The alg. must respond correctly to any sequence
  of invocations.
• Suppose user at p0 wants to do a write,
  immediately followed by a read.
• By SC, read must return 0.
• By assumption, total elapsed time is less than d.

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SC Lower Bound Proof

• Similarly, let ?1 be admissible execution whose
  top layer behavior is
• write1(Y,1) ack1(Y) read1(X) return1(X,0)
• write begins at time 0, read ends before time d
• every msg has delay d
• ?1 exists for similar reason.
• Now merge p0's timed view in ?0 with p1's timed
  view in ?1 to create admissible execution ?'.
• But ?' is not SC, contradiction!

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SC Lower Bound Proof
time
0
d
not SC - contradiction!
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Linearizability Write Lower Bound

• Theorem (9.8) In any simulation of linearizable
  shared memory on top of point-to-point message
  passing, Twrite u/2.
• Proof Consider any linearizable simulation with
  Twrite lt u/2.
• Let be an admissible exec. whose top layer
  behavior is
• p1 writes 1 to X, p2 writes 2 to X, p0 reads 2
  from X
• Shift to create admissible exec. in which p1 and
  p2's writes are swapped, causing p0's read to
  violate linearizability.

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Linearizability Write Lower Bound
linearizable
admissible
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Linearizability Write Lower Bound
u
time
0
u/2
read 2
p0
not linearizable
write 1
shift p1 by u/2
p1
shift p2 by -u/2
write 2
p2
contradiction!
admissible
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Linearizability Read Lower Bound

• Approach is similar to the write lower bound.
• Assume in contradiction there is an algorithm
  with Tread lt u/4.
• Identify a particular execution
• fix a pattern of read and write invocations,
  occurring at particular times
• fix the pattern of message delays
• Shift this execution to get one that is
• still admissible
• but not linearizable

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Linearizability Read Lower Bound

• Original execution
• p1 reads X and gets 0 (old value).
• Then p0 starts writing 1 to X.
• When write is done, p0 reads X and gets 1 (new
  value).
• Also, during the write, p0 and p1 alternate
  reading X.
• At some point, the reads stop getting the old
  value (0) and start getting the new value (1)

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Linearizability Read Lower Bound

• Set all delays in this execution to be d - u/2.
• Now shift p2 earlier by u/2.
• Verify that result is still admissible (every
  delay either stays the same or becomes d or d -
  u).
• But in shifted execution, sequence of values read
  is
• 0, 0, , 0, 1, 0, 1, 1, , 1

not linearizable!
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Linearizability Read Lower Bound