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CPSC 668 DISTRIBUTED ALGORITHMS AND SYSTEMS

CPSC 668 DISTRIBUTED ALGORITHMS AND SYSTEMS. Fall 2011 Prof. Jennifer Welch. Shared Memory Model. Processors communicate via a set of shared variables, instead of passing messages. Each shared variable has a type , defining a set of operations that can be performed atomically.

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CPSC 668 DISTRIBUTED ALGORITHMS AND SYSTEMS

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  1. Set 6: Mutual Exclusion in Shared Memory CPSC 668DISTRIBUTED ALGORITHMS AND SYSTEMS Fall 2011 Prof. Jennifer Welch

  2. Shared Memory Model • Processors communicate via a set of shared variables, instead of passing messages. • Each shared variable has a type, defining a set of operations that can be performed atomically. Set 6: Mutual Exclusion in Shared Memory

  3. Shared Memory Model Example p0 p1 p2 read write read write X Y Set 6: Mutual Exclusion in Shared Memory

  4. Shared Memory Model • Changes to the model from the message-passing case: • no inbuf and outbuf state components • configuration includes a value for each shared variable • only event type is a computation step by a processor • An execution is admissible if every processor takes an infinite number of steps Set 6: Mutual Exclusion in Shared Memory

  5. Computation Step in Shared Memory Model • When processor pi takes a step: • pi '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's semantics • pi 's state in new configuration changes according to its old state and the result of the operation Set 6: Mutual Exclusion in Shared Memory

  6. Observations on SM Model • Accesses to the shared variables are modeled as occurring instantaneously (atomically) during a computation step, one access per step • Definition of admissible execution implies • asynchronous • no failures Set 6: Mutual Exclusion in Shared Memory

  7. entry critical exit remainder Mutual Exclusion (Mutex) Problem • 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 Set 6: Mutual Exclusion in Shared Memory

  8. Mutual Exclusion 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" or "progress" condition. There are three commonly considered ones… Set 6: Mutual Exclusion in Shared Memory

  9. Mutex Progress 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, other processors enter the critical section no more than a certain number of times. • These conditions are increasingly strong. Set 6: Mutual Exclusion in Shared Memory

  10. Mutual Exclusion Algorithms • 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 Set 6: Mutual Exclusion in Shared Memory

  11. Complexity Measure for Mutex • An important complexity measure 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) Set 6: Mutual Exclusion in Shared Memory

  12. Mutex Results Using RMW • When using powerful shared variables of "read-modify-write" type Set 6: Mutual Exclusion in Shared Memory

  13. Mutex Results Using Read/Write • When using read/write shared variables Set 6: Mutual Exclusion in Shared Memory

  14. Test-and-Set Shared Variable • A test-and-set variable V holds two values, 0 or 1, and supports two (atomic) operations: • test&set(V): temp := V V := 1 return temp • reset(V): V := 0 Set 6: Mutual Exclusion in Shared Memory

  15. Mutex Algorithm Using Test&Set • code for entry section: repeat t := test&set(V) until (t = 0) An alternative construction is: wait until test&set(V) = 0 • code for exit section: reset(V) Set 6: Mutual Exclusion in Shared Memory

  16. Mutual Exclusion is Ensured • Suppose not. Consider first violation, when some pi enters CS but another pj is already in CS pi enters CS: sees V = 0, sets V to 1 pj enters CS: sees V = 0, sets V to 1 impossible! no node leaves CS so V stays 1 Set 6: Mutual Exclusion in Shared Memory

  17. No Deadlock • Claim:V = 0 iff no processor is in CS. • Proof is by induction on events in execution, and relies on fact that mutual exclusion holds. • Suppose there is a time after which a processor is in its entry section but no processor ever enters CS. no processor is in CS V always equals 0, next t&s returns 0 proc enters CS, contradiction! no processor enters CS Set 6: Mutual Exclusion in Shared Memory

  18. What About No Lockout? • One processor could always grab V (i.e., win the test&set competition) and starve the others. • No Lockout does not hold. • Thus Bounded Waiting does not hold. Set 6: Mutual Exclusion in Shared Memory

  19. Read-Modify-Write Shared Variable • The state of this kind of variable can be anything and of any size. • Variable V supports the (atomic) operation • rmw(V,f ), where f is any function temp := V V := f(V) return temp • This variable type is so strong there is no point in having multiple variables (from a theoretical perspective). Set 6: Mutual Exclusion in Shared Memory

  20. Mutex Algorithm Using RMW • Conceptually, the list of waiting processors is stored in a circular queue of length n • Each waiting processor remembers in its local state its location in the queue (instead of keeping this info in the shared variable) • Shared RMW variable V keeps track of active part of the queue with first and last pointers, which are indices into the queue (between 0 and n-1) • so V has two components, first and last Set 6: Mutual Exclusion in Shared Memory

  21. Conceptual Data Structure The RMW shared object just contains these two "pointers" Set 6: Mutual Exclusion in Shared Memory

  22. Mutex Algorithm Using RMW • Code for entry section: // increment last to enqueue self position := rmw(V,(V.first,V.last+1) // wait until first equals this value repeat queue := rmw(V,V) until (queue.first = position.last) • Code for exit section: // dequeue self rmw(V,(V.first+1,V.last)) Set 6: Mutual Exclusion in Shared Memory

  23. Correctness Sketch • Mutual Exclusion: • Only the processor at the head of the queue (V.first) can enter the CS, and only one processor is at the head at any time. • n-Bounded Waiting: • FIFO order of enqueueing, and fact that no processor stays in CS forever, give this result. Set 6: Mutual Exclusion in Shared Memory

  24. Space Complexity • The shared RMW variable V has two components in its state, first and last. • Both are integers that take on values from 0 to n-1, n different values. • The total number of different states of V thus is n2. • And thus the required size of V in bits is 2*log2n . Set 6: Mutual Exclusion in Shared Memory

  25. Spinning • A drawback of the RMW queue algorithm is that processors in entry section repeatedly access the same shared variable • called spinning • Having multiple processors spinning on the same shared variable can be very time-inefficient in certain multiprocessor architectures • Alter the queue algorithm so that each waiting processor spins on a different shared variable Set 6: Mutual Exclusion in Shared Memory

  26. RMW Mutex Algorithm With Separate Spinning • Shared RMW variables: • Last : corresponds to last "pointer" from previous algorithm • cycles through 0 to n–1 • keeps track of index to be given to the next processor that starts waiting • initially 0 Set 6: Mutual Exclusion in Shared Memory

  27. RMW Mutex Algorithm With Separate Spinning • Shared RMW variables: • Flags[0..n-1] : array of binary variables • these are the variables that processors spin on • make sure no two processors spin on the same variable at the same time • initially Flags[0] = 1 (proc "has lock") and Flags[i] = 0 (proc "must wait") for i > 0 Set 6: Mutual Exclusion in Shared Memory

  28. Overview of Algorithm • entry section: • get next index from Last and store in a local variable myPlace • increment Last (with wrap-around) • spin on Flags[myPlace] until it equals 1 (means proc "has lock" and can enter CS) • set Flags[myPlace] to 0 ("doesn't have lock") • exit section: • set Flags[myPlace+1] to 1 (i.e., give the priority to the next proc) • use modular arithmetic to wrap around Set 6: Mutual Exclusion in Shared Memory

  29. Question • Do the shared variables Last and Flags have to be RMW variables? • Answer: The RMW semantics (atomically reading and updating a variable) are needed for Last, to make sure two processors don't get the same index at overlapping times. Set 6: Mutual Exclusion in Shared Memory

  30. Invariants of the Algorithm • At most one element of Flags has value 1 ("has lock") • If no element of Flags has value 1, then some processor is in the CS. • If Flags[k] = 1, then exactly (Last - k) mod n processors are in the entry section, spinning on Flags[i], for i = k, (k+1) mod n, …, (Last-1) mod n. Set 6: Mutual Exclusion in Shared Memory

  31. 0 0 1 0 0 0 0 0 5 Example of Invariant 0 1 2 3 4 5 6 7 Flags Last k = 2 and Last = 5. So 5 - 2 = 3 procs are in entry, spinning on Flags[2], Flags[3], Flags[4] Set 6: Mutual Exclusion in Shared Memory

  32. Correctness • Those three invariants can be used to prove: • Mutual exclusion is satisfied • n-Bounded Waiting is satisfied. Set 6: Mutual Exclusion in Shared Memory

  33. Lower Bound on Number of Memory States Theorem (4.4): Any mutex algorithm with k-bounded waiting (and no-deadlock) uses at least n states of shared memory. Proof: Assume in contradiction there is an algorithm using less than n states of shared memory. Set 6: Mutual Exclusion in Shared Memory

  34. Lower Bound on Number of Memory States • Consider this execution of the algorithm: • There exist i and j such that Ciand Cj have the same state of shared memory. pn-1 p0 p0 p0 … p1 p2 …… C C0 C1 C2 Cn-1 p2 in entry sec. pn-1 in entry sec. p1 in entry sec. p0 in CS by ND Why? Set 6: Mutual Exclusion in Shared Memory

  35. Lower Bound on Number of Memory States pi+1, pi+2, …, pj Ci Cj p0 in CS, p1-pi in entry, rest in rem. p0in CS, p1-pjin entry, rest in rem.   = sched. in which p0-pi take steps in round robin by ND, some ph has entered CS k+1 times ph enters CS k+1 times while pi+1 is in entry contradiction! Set 6: Mutual Exclusion in Shared Memory

  36. Lower Bound on Number of Memory States • But why does ph do the same thing when executing the sequence of steps in  when starting from Cj as when starting from Ci? • All the processes p0,…,pi do the same thing because: • they are in same states in the two configs • shared memory state is same in the two configs • only differences between Ci and Cj are (potentially) the states of pi+1,…,pj and they don't take any steps in  Set 6: Mutual Exclusion in Shared Memory

  37. Discussion of Lower Bound • The lower bound of n just shown on number of memory states only holds for algorithms that must provide bounded waiting in every execution. • Suppose we weaken the liveness condition to just no-lockout in every execution: then the bound becomes n/2 distinct shared memory states. • And if liveness is weakened to just no-deadlock in every execution, then the bound is just 2. Set 6: Mutual Exclusion in Shared Memory

  38. "Beating" the Lower Bound with Randomization • An alternative way to weaken the requirement is to give up on requiring liveness in every execution • Consider Probabilistic No-Lockout: every processor has non-zero probability of succeeding each time it is in its entry section. • Now there is an algorithm using O(1) states of shared memory. Set 6: Mutual Exclusion in Shared Memory

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