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Ctrie Data Structure

Aleksandar Prokopec, profile picture
Aleksandar Prokopec

The document presents a discussion on concurrent tries with a focus on efficient non-blocking snapshots, particularly utilizing hash array mapped tries (HAMT). It includes detailed algorithms for operations like insertions and size calculations, as well as mechanisms for maintaining thread safety in their implementation. Various methods for creating snapshots, both using locks and immutability, are also explored with implications for performance and data integrity.

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Related topics:
Data Structures
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Concurrent Tries with Efficient Non-blocking Snapshots Aleksandar Prokopec Phil Bagwell Martin Odersky École Polytechnique Fédérale de Lausanne Nathan Bronson Stanford
Motivation val numbers = getNumbers() // compute square roots numbers foreach { entry => x = entry.root n = entry.number entry.root = 0.5 * (x + n / x) if (abs(entry.root - x) < eps) numbers.remove(entry) }
Hash Array Mapped Tries (HAMT)
Hash Array Mapped Tries (HAMT) 0 = 0000002
Hash Array Mapped Tries (HAMT) 0
Hash Array Mapped Tries (HAMT) 0 16 = 0100002
Hash Array Mapped Tries (HAMT) 0 16
Hash Array Mapped Tries (HAMT) 0 16 4 = 0001002
Hash Array Mapped Tries (HAMT) 16 0 4 = 0001002
Hash Array Mapped Tries (HAMT) 16 0 4
Hash Array Mapped Tries (HAMT) 16 0 4 12 = 0011002
Hash Array Mapped Tries (HAMT) 16 0 4 12 = 0011002
Hash Array Mapped Tries (HAMT) 16 0 4 12
Hash Array Mapped Tries (HAMT) 16 33 0 4 12
Hash Array Mapped Tries (HAMT) 16 33 0 4 12 48
Hash Array Mapped Tries (HAMT) 16 0 4 12 48 33 37
Hash Array Mapped Tries (HAMT) 16 4 12 48 33 37 0 3
Hash Array Mapped Tries (HAMT) 4 12 16 20 25 33 37 0 1 8 9 3 48 57
Immutable HAMT •used as immutable maps in functional languages 4 12 16 20 25 33 37 0 1 8 9 3
Immutable HAMT •updates rewrite path from root to leaf 4 12 16 20 25 33 37 0 1 8 9 3 4 12 8 9 11 insert(11)
Immutable HAMT •updates rewrite path from root to leaf 4 12 16 20 25 33 37 0 1 8 9 3 4 12 8 9 11 insert(11) efficient updates - logk(n)
Node compression 48 57 48 57 1 0 1 0 48 57 1 0 1 0 48 57 10 BITPOP(((1 << ((hc >> lev) & 1F)) – 1) & BMP)
Node compression 48 57 48 57 1 0 1 0 48 57 1 0 1 0 48 57 10 48 57
Ctrie Can mutable HAMT be modified to be thread-safe?
Ctrie insert 4 9 12 16 20 25 33 37 0 1 3 48 57 17 = 0100012
Ctrie insert 4 9 12 16 20 25 33 37 0 1 3 48 57 17 = 0100012 16 17 1) allocate
Ctrie insert 4 9 12 20 25 33 37 0 1 3 48 57 17 = 0100012 16 17 2) CAS
Ctrie insert 4 9 12 20 25 33 37 0 1 3 48 57 17 = 0100012 16 17
Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 16 17 20 25
Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 16 17 20 25 1) allocate 16 17 18
Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 20 25 2) CAS 16 17 18
Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 20 25 2) CAS 16 17 18 Unless…
Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 16 17 20 25 T1-1) allocate 16 17 18 Unless… 28 = 0111002 T1 T2
Ctrie insert 4 9 12 0 1 3 18 = 0100102 16 17 20 25 T1-1) allocate 16 17 18 Unless… 28 = 0111002 T1 T2 20 25 28 T2-1) allocate
Ctrie insert 4 9 12 0 1 3 18 = 0100102 16 17 20 25 T1-1) allocate 16 17 18 28 = 0111002 T1 T2 20 25 28 T2-2) CAS
Ctrie insert 4 9 12 0 1 3 18 = 0100102 16 17 20 25 T1-2) CAS 16 17 18 28 = 0111002 T1 T2 20 25 28 T2-2) CAS
Ctrie insert 4 9 12 0 1 3 18 = 0100102 16 17 20 25 16 17 18 28 = 0111002 T1 T2 20 25 28 Lost insert!
Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 20 25 Solution: I-nodes
Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 20 25 18 = 0100102 28 = 0111002 T1 T2
Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 T1 T2 20 25 18 = 0100102 28 = 0111002 16 17 18 20 25 28 T2-1) allocate T1-1) allocate
Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 T1 T2 20 25 16 17 18 20 25 28 T2-2) CAS T1-2) CAS
Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 18 20 25 28
Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 18 20 25 28 Idea: once added to the Ctrie, I-nodes remain present.
Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 18 20 25 28 Remove operation supported as well - details in the paper.
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 0
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 0
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 0
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 0
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 1
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 2
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 3
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 5
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 5 actual size = 12
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 5 0 1 actual size = 12
Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 5 0 1 CAS actual size = 11
Ctrie size 4 9 12 16 17 18 20 25 28 size = 5 0 1 actual size = 11
Ctrie size 4 9 12 16 17 18 20 25 28 size = 6 0 1 actual size = 11
Ctrie size 4 9 12 16 17 18 20 25 28 size = 6 0 1 actual size = 11 19
Ctrie size 4 9 12 16 17 18 20 25 28 size = 6 0 1 actual size = 11 16 17 18 19
Ctrie size 4 9 12 16 17 18 20 25 28 size = 6 0 1 actual size = 12 16 17 18 19 CAS
Ctrie size 4 9 12 20 25 28 size = 6 0 1 actual size = 12 16 17 18 19
Ctrie size 4 9 12 20 25 28 size = 6 0 1 actual size = 12 16 17 18 19
Ctrie size 4 9 12 20 25 28 size = 7 0 1 actual size = 9 16 17 18 19
Ctrie size 4 9 12 20 25 28 size = 8 0 1 actual size = 12 16 17 18 19
Ctrie size 4 9 12 20 25 28 size = 9 0 1 actual size = 12 16 17 18 19
Ctrie size 4 9 12 20 25 28 size = 10 0 1 actual size = 12 16 17 18 19
Ctrie size 4 9 12 20 25 28 size = 11 0 1 actual size = 12 16 17 18 19
Ctrie size 4 9 12 20 25 28 size = 12 0 1 actual size = 12 16 17 18 19
Ctrie size 4 9 12 20 25 28 size = 13 0 1 actual size = 12 16 17 18 19
Ctrie size 4 9 12 20 25 28 size = 13 0 1 actual size = 12 16 17 18 19 But the size was never 13!
Global state information 4 9 12 20 25 28 0 1 16 17 18 19 •size •find •filter •iterator
Global state information 4 9 12 20 25 28 0 1 16 17 18 19 •size •find •filter •iterator  snapshot
Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19
Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19 •copy expensive
Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19 •copy expensive •not lock-free
Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19 •copy expensive •not lock-free •can insert or remove remain lock-free? 0 1 2 CAS
Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19 •copy expensive •not lock-free •can insert or remove remain lock-free? 0 1 2 CAS
Snapshot using logs 4 9 12 20 25 28 0 1 16 17 18 19 •keep a linked list of previous values in each I-node
Snapshot using logs 4 9 12 20 25 28 0 1 16 17 18 19 0 1 2 •keep a linked list of previous values in each I-node
Snapshot using logs 4 9 12 20 25 28 0 1 16 17 18 19 •keep a linked list of previous values in each I-node •when is it safe to delete old entries? 0 1 2
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 root
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 root
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 snapshot! root
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 snapshot! #2 root 1) create new I-node at #2
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 snapshot! #2 root 2) set snapshot snapshot #1
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 snapshot! #2 root 3) CAS root to new I-node snapshot #1
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 generation #2 - ok!
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 generation #1 not ok, too old!
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root 1) create updated node at #2 snapshot #1 2 #2 #2
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root 2) CAS to the updated node snapshot #1 2 #2 #2
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 #2 #2 #1 too old!
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 #2 #2 4 9 12 #2 1) create updated node at #2
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 #2 #2 4 9 12 #2 2) CAS
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 finally, create a new leaf and CAS
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 another insert #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 another insert #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 But... this won't really work... why? #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 CAS
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 CAS How to fail this last CAS?
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 DCAS How to fail this last CAS? DCAS
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 How to fail this last CAS? DCAS - software based DCAS
Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 How to fail this last CAS? DCAS - software based ...creates intermediate objects DCAS
GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 T2: remove 19 16 17 18 prev 1) set prev field
GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 T2: remove 19 16 17 18 prev 2) CAS
GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 T2: remove 19 16 17 18 prev 3) read root generation
GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 prev 4) if root generation changed CAS prev to FailedNode(prev) FN
GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 prev 4) if root generation changed CAS prev to FailedNode(prev) FN
GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 prev 5) CAS to previous value FN
GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 prev 4) if root generation unchanged CAS prev to null
GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 4) if root generation unchanged CAS prev to null
GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 1) Replace all CAS with GCAS 2) Replace all READ with GCAS_READ (which checks if prev field is null)
Snapshot-based iterator def iterator = if (isSnapshot) new Iterator(root) else snapshot().iterator()
Snapshot-based size def size = { val sz = 0 val it = iterator while (it.hasNext) sz += 1 sz }
Snapshot-based size def size = { val sz = 0 val it = iterator while (it.hasNext) sz += 1 sz } Above is O(n). But, by caching size in nodes - amortized O(logkn)! (see source code)
Snapshot-based atomic clear def clear() = { val or = READ(root) val nr = new INode(new Gen) if (!CAS(root, or, nr)) clear() } (roughly)
Evaluation - quad core i7
Evaluation – UltraSPARC T2
Evaluation – 4x 8-core i7
Evaluation – snapshot
Conclusion •snapshots are linearizable and lock-free •snapshots take constant time •snapshots are horizontally scalable •snapshots add a non-significant overhead to the algorithm if they aren't used •the approach may be applicable to tree-based lock-free data-structures in general (intuition)
Thank you!

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Ctrie Data Structure

  • 1. Concurrent Tries with Efficient Non-blocking Snapshots Aleksandar Prokopec Phil Bagwell Martin Odersky École Polytechnique Fédérale de Lausanne Nathan Bronson Stanford
  • 2. Motivation val numbers = getNumbers() // compute square roots numbers foreach { entry => x = entry.root n = entry.number entry.root = 0.5 * (x + n / x) if (abs(entry.root - x) < eps) numbers.remove(entry) }
  • 3. Hash Array Mapped Tries (HAMT)
  • 4. Hash Array Mapped Tries (HAMT) 0 = 0000002
  • 5. Hash Array Mapped Tries (HAMT) 0
  • 6. Hash Array Mapped Tries (HAMT) 0 16 = 0100002
  • 7. Hash Array Mapped Tries (HAMT) 0 16
  • 8. Hash Array Mapped Tries (HAMT) 0 16 4 = 0001002
  • 9. Hash Array Mapped Tries (HAMT) 16 0 4 = 0001002
  • 10. Hash Array Mapped Tries (HAMT) 16 0 4
  • 11. Hash Array Mapped Tries (HAMT) 16 0 4 12 = 0011002
  • 12. Hash Array Mapped Tries (HAMT) 16 0 4 12 = 0011002
  • 13. Hash Array Mapped Tries (HAMT) 16 0 4 12
  • 14. Hash Array Mapped Tries (HAMT) 16 33 0 4 12
  • 15. Hash Array Mapped Tries (HAMT) 16 33 0 4 12 48
  • 16. Hash Array Mapped Tries (HAMT) 16 0 4 12 48 33 37
  • 17. Hash Array Mapped Tries (HAMT) 16 4 12 48 33 37 0 3
  • 18. Hash Array Mapped Tries (HAMT) 4 12 16 20 25 33 37 0 1 8 9 3 48 57
  • 19. Immutable HAMT •used as immutable maps in functional languages 4 12 16 20 25 33 37 0 1 8 9 3
  • 20. Immutable HAMT •updates rewrite path from root to leaf 4 12 16 20 25 33 37 0 1 8 9 3 4 12 8 9 11 insert(11)
  • 21. Immutable HAMT •updates rewrite path from root to leaf 4 12 16 20 25 33 37 0 1 8 9 3 4 12 8 9 11 insert(11) efficient updates - logk(n)
  • 22. Node compression 48 57 48 57 1 0 1 0 48 57 1 0 1 0 48 57 10 BITPOP(((1 << ((hc >> lev) & 1F)) – 1) & BMP)
  • 23. Node compression 48 57 48 57 1 0 1 0 48 57 1 0 1 0 48 57 10 48 57
  • 24. Ctrie Can mutable HAMT be modified to be thread-safe?
  • 25. Ctrie insert 4 9 12 16 20 25 33 37 0 1 3 48 57 17 = 0100012
  • 26. Ctrie insert 4 9 12 16 20 25 33 37 0 1 3 48 57 17 = 0100012 16 17 1) allocate
  • 27. Ctrie insert 4 9 12 20 25 33 37 0 1 3 48 57 17 = 0100012 16 17 2) CAS
  • 28. Ctrie insert 4 9 12 20 25 33 37 0 1 3 48 57 17 = 0100012 16 17
  • 29. Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 16 17 20 25
  • 30. Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 16 17 20 25 1) allocate 16 17 18
  • 31. Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 20 25 2) CAS 16 17 18
  • 32. Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 20 25 2) CAS 16 17 18 Unless…
  • 33. Ctrie insert 4 9 12 33 37 0 1 3 48 57 18 = 0100102 16 17 20 25 T1-1) allocate 16 17 18 Unless… 28 = 0111002 T1 T2
  • 34. Ctrie insert 4 9 12 0 1 3 18 = 0100102 16 17 20 25 T1-1) allocate 16 17 18 Unless… 28 = 0111002 T1 T2 20 25 28 T2-1) allocate
  • 35. Ctrie insert 4 9 12 0 1 3 18 = 0100102 16 17 20 25 T1-1) allocate 16 17 18 28 = 0111002 T1 T2 20 25 28 T2-2) CAS
  • 36. Ctrie insert 4 9 12 0 1 3 18 = 0100102 16 17 20 25 T1-2) CAS 16 17 18 28 = 0111002 T1 T2 20 25 28 T2-2) CAS
  • 37. Ctrie insert 4 9 12 0 1 3 18 = 0100102 16 17 20 25 16 17 18 28 = 0111002 T1 T2 20 25 28 Lost insert!
  • 38. Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 20 25 Solution: I-nodes
  • 39. Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 20 25 18 = 0100102 28 = 0111002 T1 T2
  • 40. Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 T1 T2 20 25 18 = 0100102 28 = 0111002 16 17 18 20 25 28 T2-1) allocate T1-1) allocate
  • 41. Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 T1 T2 20 25 16 17 18 20 25 28 T2-2) CAS T1-2) CAS
  • 42. Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 18 20 25 28
  • 43. Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 18 20 25 28 Idea: once added to the Ctrie, I-nodes remain present.
  • 44. Ctrie insert – 2nd attempt 4 9 12 0 1 3 16 17 18 20 25 28 Remove operation supported as well - details in the paper.
  • 45. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28
  • 46. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 0
  • 47. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 0
  • 48. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 0
  • 49. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 0
  • 50. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 1
  • 51. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 2
  • 52. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 3
  • 53. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 5
  • 54. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 5 actual size = 12
  • 55. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 5 0 1 actual size = 12
  • 56. Ctrie size 4 9 12 0 1 3 16 17 18 20 25 28 size = 5 0 1 CAS actual size = 11
  • 57. Ctrie size 4 9 12 16 17 18 20 25 28 size = 5 0 1 actual size = 11
  • 58. Ctrie size 4 9 12 16 17 18 20 25 28 size = 6 0 1 actual size = 11
  • 59. Ctrie size 4 9 12 16 17 18 20 25 28 size = 6 0 1 actual size = 11 19
  • 60. Ctrie size 4 9 12 16 17 18 20 25 28 size = 6 0 1 actual size = 11 16 17 18 19
  • 61. Ctrie size 4 9 12 16 17 18 20 25 28 size = 6 0 1 actual size = 12 16 17 18 19 CAS
  • 62. Ctrie size 4 9 12 20 25 28 size = 6 0 1 actual size = 12 16 17 18 19
  • 63. Ctrie size 4 9 12 20 25 28 size = 6 0 1 actual size = 12 16 17 18 19
  • 64. Ctrie size 4 9 12 20 25 28 size = 7 0 1 actual size = 9 16 17 18 19
  • 65. Ctrie size 4 9 12 20 25 28 size = 8 0 1 actual size = 12 16 17 18 19
  • 66. Ctrie size 4 9 12 20 25 28 size = 9 0 1 actual size = 12 16 17 18 19
  • 67. Ctrie size 4 9 12 20 25 28 size = 10 0 1 actual size = 12 16 17 18 19
  • 68. Ctrie size 4 9 12 20 25 28 size = 11 0 1 actual size = 12 16 17 18 19
  • 69. Ctrie size 4 9 12 20 25 28 size = 12 0 1 actual size = 12 16 17 18 19
  • 70. Ctrie size 4 9 12 20 25 28 size = 13 0 1 actual size = 12 16 17 18 19
  • 71. Ctrie size 4 9 12 20 25 28 size = 13 0 1 actual size = 12 16 17 18 19 But the size was never 13!
  • 72. Global state information 4 9 12 20 25 28 0 1 16 17 18 19 •size •find •filter •iterator
  • 73. Global state information 4 9 12 20 25 28 0 1 16 17 18 19 •size •find •filter •iterator  snapshot
  • 74. Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19
  • 75. Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19 •copy expensive
  • 76. Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19 •copy expensive •not lock-free
  • 77. Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19 •copy expensive •not lock-free •can insert or remove remain lock-free? 0 1 2 CAS
  • 78. Snapshot using locks 4 9 12 20 25 28 0 1 16 17 18 19 •copy expensive •not lock-free •can insert or remove remain lock-free? 0 1 2 CAS
  • 79. Snapshot using logs 4 9 12 20 25 28 0 1 16 17 18 19 •keep a linked list of previous values in each I-node
  • 80. Snapshot using logs 4 9 12 20 25 28 0 1 16 17 18 19 0 1 2 •keep a linked list of previous values in each I-node
  • 81. Snapshot using logs 4 9 12 20 25 28 0 1 16 17 18 19 •keep a linked list of previous values in each I-node •when is it safe to delete old entries? 0 1 2
  • 82. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 root
  • 83. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 root
  • 84. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 snapshot! root
  • 85. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 snapshot! #2 root 1) create new I-node at #2
  • 86. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 snapshot! #2 root 2) set snapshot snapshot #1
  • 87. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 snapshot! #2 root 3) CAS root to new I-node snapshot #1
  • 88. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2
  • 89. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 generation #2 - ok!
  • 90. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 generation #1 not ok, too old!
  • 91. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root 1) create updated node at #2 snapshot #1 2 #2 #2
  • 92. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root 2) CAS to the updated node snapshot #1 2 #2 #2
  • 93. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 #2 #2 #1 too old!
  • 94. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 #2 #2 4 9 12 #2 1) create updated node at #2
  • 95. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 2 #2 #2 4 9 12 #2 2) CAS
  • 96. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 subsequent insert #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 finally, create a new leaf and CAS
  • 97. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 another insert #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3
  • 98. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 another insert #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3
  • 99. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 But... this won't really work... why? #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3
  • 100. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18
  • 101. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 CAS
  • 102. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 CAS How to fail this last CAS?
  • 103. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 DCAS How to fail this last CAS? DCAS
  • 104. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 How to fail this last CAS? DCAS - software based DCAS
  • 105. Snapshot using immutability 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 0 1 2 3 T2: remove 19 16 17 18 How to fail this last CAS? DCAS - software based ...creates intermediate objects DCAS
  • 106. GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 T2: remove 19 16 17 18 prev 1) set prev field
  • 107. GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 T2: remove 19 16 17 18 prev 2) CAS
  • 108. GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 T2: remove 19 16 17 18 prev 3) read root generation
  • 109. GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 prev 4) if root generation changed CAS prev to FailedNode(prev) FN
  • 110. GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 prev 4) if root generation changed CAS prev to FailedNode(prev) FN
  • 111. GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 prev 5) CAS to previous value FN
  • 112. GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 prev 4) if root generation unchanged CAS prev to null
  • 113. GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 16 17 18 4) if root generation unchanged CAS prev to null
  • 114. GCAS - generation-compare-and-swap 4 9 12 20 25 28 0 1 16 17 18 19 #1 #1 #1 #1 #1 #2 root snapshot #1 #2 #2 4 9 12 #2 0 1 2 3 1) Replace all CAS with GCAS 2) Replace all READ with GCAS_READ (which checks if prev field is null)
  • 115. Snapshot-based iterator def iterator = if (isSnapshot) new Iterator(root) else snapshot().iterator()
  • 116. Snapshot-based size def size = { val sz = 0 val it = iterator while (it.hasNext) sz += 1 sz }
  • 117. Snapshot-based size def size = { val sz = 0 val it = iterator while (it.hasNext) sz += 1 sz } Above is O(n). But, by caching size in nodes - amortized O(logkn)! (see source code)
  • 118. Snapshot-based atomic clear def clear() = { val or = READ(root) val nr = new INode(new Gen) if (!CAS(root, or, nr)) clear() } (roughly)
  • 119. Evaluation - quad core i7
  • 121. Evaluation – 4x 8-core i7
  • 123. Conclusion •snapshots are linearizable and lock-free •snapshots take constant time •snapshots are horizontally scalable •snapshots add a non-significant overhead to the algorithm if they aren't used •the approach may be applicable to tree-based lock-free data-structures in general (intuition)