Improving the Performance Competitive Ratios of Transactional Memory Contention Managers

1. Improving the Performance Competitive Ratios of Transactional Memory Contention Managers Gokarna Sharma Costas Busch Louisiana State University, USA WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

2. STM Systems • Progressis ensured through contention management (CM) policy • Performance is generally evaluated by competitive ratio • Makespan primarily depends on the TM workload • arrival times, execution time durations, release times, read/write sets • Challenge • How to schedule transactions such that it reduces the makespan? WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

3. Related Work • Mostly empirical evaluation • Theoretical Analysis • [Guerraoui et al., PODC’05] • GreedyContention Manager, Competitive Ratio = O(s2) (s is the number of shared resources) • [Attiya et al., PODC’06] • Improved to O(s) • [Schneider & Wattenhofer, ISAAC’09] • RandomizedRounds Contention Manager, Competitive Ratio = O(C logn) (C is the maximum number of conflicting transactions and n is the number of transactions) • [Attiya & Milani, OPODIS’09] • Bimodal scheduler, Competitive Ratio = O(s) (for bimodal workload with equi-length transactions) WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

4. Our Contributions • Balanced TM workloads • Two polynomial time contention management algorithms that achieve competitive ratio very close to O(√s) in balanced workloads • Clairvoyant – Competitive ratio = O(√s) • Non-Clairvoyant – Competitive ratio = O(√s¢logn) w.h.p. • Lower bound for transaction scheduling problem WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

5. Roadmap • Balanced workloads • CM algorithms and proof intuitions • Lower bound and proof intuition WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

6. Balanced Workloads • A transaction is balanced if: • ≤ , ≤ ≤ 1 is some constant called balancing ratio • and are number of writes and reads to shared resources by , respectively • For read-onlytransaction, = 0, and for write-onlytransaction, = 0 • A workload is balanced if: • It contains only balanced transactions WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

7. Algorithms • Clairvoyant • Clairvoyant in the sense it requiresknowledge of dynamic conflict graph to resolve conflicts • Competitive ratio = O(√s) • Non-Clairvoyant • Non-Clairvoyant in the sense it doesnot require knowledge of conflict graph to resolve conflicts • Competitive ratio = O(√s¢logn) with high probability • Competitive ratio O(logn) factor worse than Clairvoyant WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

8. Proof Intuition • Knowing ahead the execution times (t) and total number of shared resource accesses (|R(Ti)|)of transactions • Intuition • Divide transactions into l+1groups according to execution time, where l = d log()e • Again divide each group into · +1subgroups according to shared resource accesses needed, where ·= dlogse • Assign a total order among the groups and subgroups WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

9. Proof Intuition (Contd...) • Analysis • For a subgroup Aij, competitive ratio = O(min(¸j, )), where ¸j= 2j+1 -1 • For a group Ai, competitive ratio = O(√()) after combining competitive ratios of all the subgroups • After combining competitive ratios of all groups, competitive ratio = O(l ¢√()) • For l = O(1) and β = O(1), competitive ratio = O(√s) • O(logn) factorin Non-Clairvoyant due to the use of random priorities to resolve conflicts WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

10. Lower Bound • We prove the following theorem • Unless NP µ ZPP, we cannot obtain a polynomial time transaction scheduling algorithm such that for every input instance with β = 1 and l = 1 of the TRANSACTION SCHEDULING problem the algorithm achieves competitive ratio smaller than O((√s)1- ε) for any constant ε > 0. • Proof Intuition • Reduce the NP-Complete graph coloring problem, VERTEX COLORING, to the transaction scheduling problem, TRANSACTION SCHEDULING • Use following result [Feige & Kilian, CCC’96] • No better than O(n(1- ε)) approximation exists for VERTEX COLORING, for any constant ε > 0, unless NP µ ZPP WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

11. Lower Bound (Contd…) • Reduction: Considerinput graph G = (V, E) of VERTEX COLORING, where |V| = n and |E|= s • Constructa set of transactions Tsuch that • For each v 2V, there is a respective transaction Tv2T • For each e 2E, there is a respective resource Re2R • Let G’ be the conflict graph for the set of transactions T • G’ is isomorphic to G, for ¯ =1, tmin=tmax = 1, and l = 1 • Valid k-coloring in G implies makespan of step k in G’ for T • Algorithm Clairvoyant is tight for ¯= O(1) and l = O(1) WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

12. Conclusions • Balanced TM workloads • Two new randomized CM algorithms that exhibit competitive ratio very close to O(√s)in balanced workloads • Lower bound of O(√s)for transaction scheduling problem WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory

13. [Full paper to appear in OPODIS 2010] arXiv version: http://arxiv.org/abs/1009.0056v1.pdf Thank You!!! WTTM 2010 - 2nd Workshop on the Theory of Transactional Memory