1 Adaptive and Efficient Mutual Exclusion Presented by: By Hagit Attya and Vita Bortnikov Mian Huang

2 Talk Outline  Introduction  Related Work  Preliminaries  Algorithms  Future Work

3 Introduction  Mutual Exclusion Problem  Limitations of most of current solutions  Adaptive solutions  Contributions of this Paper - Adaptive algorithms using only read and write operations that achieve:  Remote step complexity O(k), where k is actual contention  System response time O(log k)  Space complexity  Long-lived renaming - O(nN) where N is the range of processes’ names  One-shot renaming – O(n 2 ) where n is total # of processes

4 Related Work  Dijkstra  Lamport A fast mutual exclusion algorithm  Alur and Taubenfeld Results about fast mutual exclusion  Step complexity could not be adaptive for any asynchronous Algo.  Choy and Singh Adaptive solutions to the mutual exclusion problem  Without assuming hardware-provided synchronization primitives  Performance to be governed solely by # of contending processes

5 Preliminaries  Adaptive algorithms  Point contention (strongest)  Interval contention  Renaming algorithms (getName/releaseName)  Long-lived  One-shot  Non-wait-free algorithms

6 Preliminaries (Cont.)  The Computation Model A standard asynchronous shared-memory model of computation.  n processes  Multi-writer multi-reader registers  shared primitives – atomic read/write registers

7 Preliminaries(Cont.)  The Complexity Measures  Remote step complexity - Max # of shared memory operations performed by a process, where a wait is counted as a single operation  Remote memory references - A stronger version of the above parameter - Assumption of local spin  System response time - The time interval between subsequent entries into the critical section, where a time unit is the min execution interval in which each active process performs at least one step

8 How the Basic Algorithm Works How the Basic Algorithm Works  Architecture  An adaptive long-lived non-wait-free k-renaming  An adaptive tournament tree for mutual exclusion  What it does  A process gets a name in a range of size O(k) using long-lived renaming  Uses this name to enter an adaptive tournament tree for mutual exclusion  The winner of the tournament tree enters the critical section

9 The Basic Algorithm Critical Section Long-Lived Renaming Figure 1. Algorithm Structure Adaptive Tournament Tree

10 Non-Wait-Free Renaming  The basic building block  Filter with an entry point and an exit point  success or fail  Modified filter improves time complexity  An array of n entries  Each entry contains a pointer to a chain of filters/a chain  The name that the process receives is the index of the chain it wins  A chain of filters  Concatenated one after the other  Process succeeds  Process fails - If failed in the r’th filter of the l’th chain, then it skips the next r-1 chains and tries to win in the (l+r)’th chain

Non-Wait-Free Renaming Figure 2. The next chain calculation 0l (r-1) l + r n r’th r-1 2 Filters Chains

12 The code of Basic Algorithm Private variables lastFilter : interger Procedure getName() //get a new name 1. l := 0 2. Index := Repeat 4. l := l + index +1 //calculate next chain to enter 5. := executeChain(l) 6. Until result = win 7. lastFilter := startFilter[l] + index 8. Return l+1 Procedure releaseName(name) // release a name 1. startFilter[name] := lastFilter + 1 Procedure executeChain(l) // access chain l 1. filters := Chains[l][startFilter[l]] 2. curr := 0 3. while (true) 4. if executeFilter(Filters[curr]) = success then 5. if curr > 0 and  Filters[curr-1].c then 6. return 7. else curr else return Figure 3. Adaptive long-lived non-wait-free k-renaming with unbounded memory

13 The code of Basic Algorithm(Cont.) Procedure executeFilter(filter) // access a specific filter 1. if filter.turn   then return fail 2. filter.turn :=id 3. if filter.d then return fail // there is a winner - fail 4. wait until  filter.b or filter.turn  id or filter.d 5. if filter.d then return fail 6. if filter.turn = id then filter.b := true // try to succeed 7. if filter.turn  id then // failing in the filter 8. filter.c := true 9. filter.b := false 10. return fail 11. else // succeeding in the filter 12. filter.d := true 13. return success Figure 3. Adaptive long-lived non-wait-free k-renaming with unbounded memory

14 The Procedures – Basic Algorithm  getName()  Obtains a new name  Invokes procedure executeChain ()  executeChain (l)  Returns a pair:  Executes a filter of a chain  executeFilter(filter)  Modification of the filter suggested by Choy and Singh  Shared variables  releaseName(name)

15 Result of the Algorithm - Filter Properties The following properties ensure that exactly one process eventually remains in the chain and wins it.  Safety If k processes enter the filter, then at most  k/2  processes succeed in it.  Progress If one or more processes enter the filter, then at least one process succeeds in it. Modified filter will also have the property:  Time complexity Some process succeeds in the filter O(1) time units after the first process enters it

16 An execution interval of a process p i includes one iteration of enter, critical section, and exit  If the point contention during p i ‘s execution interval is k, then p i wins in chain l <= k-1  A process wins some chain within O(k) steps Remote step complexity  If k processes enter the chain, then some process wins the chain in  log k  + 2 filters  Some process wins the chain within O(logk) time units after the chain became busy, where k is the point contention of the winner’s execution interval System response time Result of the Algorithm - Chain Properties

17 An Adaptive Tournament Tree  What is it  Is an unbalanced binary tree  Constructed from log N complete binary tree of exponentially growing sizes(1,2,2 2,… nodes) – N is size of name space  Connected by a single path of nodes  Embedded two-process mutual exclusion algorithm with O(1) remote steps- Yang and Anderson

18 An adaptive Tournament Tree A process with a name in the range 0, …, k-1 climbs at most 2logk +1 nodes Root The leaves are numbered from left to right

19 Complexity of the Basic Algorithm  Renaming Algorithm + Tournament Tree  Step complexity: O(k) + O(logk) O(k)  Time complexity: O(logk) + O(logk)O(logk)

20 Bounding The Number of Filters  Basic idea  The number of filters in a chain is bounded by recycling previously used filters  Solutions  A chain of only 2N filters are used cyclically  The process that exits from the critical section detects “slow” processes and promotes them to enter the critical section  In addition, the scan also initializes the recycled filters  Space Complexity of the Renaming Algorithm  Is dominated by the size of array Chains – O(nN)

21 Reducing The Space Complexity  Basic idea  one-shot f(n)-renaming  Complexity of the n-renaming Algorithm  System response time O(logk)  Remote step complexity O(k’), where k’ is the interval contention  Space complexity O(n log n)  Space Complexity of the Mutual Exclusion Algorithm  O(n 2 ) – n times the range of names

22 Summary of Result AlgorithmsRemote Step Complexity System Response Time Space Complexity Choy & Singh O(N)O(k)O(N) Adaptive Bakery Algorithm O(k 4 ) O(N 3 ) Afek et al. O(min(k 2,klogN)) O(k 2 )O(N2 2n ) Anderson & Kim O(k) O(N) Algorithm with long- lived renaming O(k)O(logk)O(nN) Algorithm with one- shot renaming O(k)O(logk)O(n 2 ) Talbe1. Comparison with previous adaptive mutual exclusion algorithms

23 Future Work  Improve the remote step complexity of the algorithm or show that it is optimal  Prove a lower bound on the system response time  Design an algorithm with O(k) remote memory references and O(logk) system response time