Dynamic Concurrent Van Emde Boas Array

Transcription

1 Dynamic Concurrent Van Emde Boas Array Data structure for high performance computing Konrad Kułakowski AGH University of Science and Technology HiPEAC Workshop 17 June 2016

2 Outline Instead of introduction: van Emde Boas Tree (veb Tree) - How does it work? From veb Tree to veb Array Why veb Array? Dynamic Concurrent veb Array Running time, operations scheme, synchronization Experimental results Test setup Benchmark tests Summary & further reading!2

3 van Emde Boas Tree Instead of introduction!3

4 van Emde Boas Tree Operations The structure that provides all the operations of the dynamic set structure: insert(int: key) delete(int: key) get(int: key) int minimum() int maximum() int successor(int: key) int predecessor(int: key)!4

5 van Emde Boas Tree Operations, performance Running time: O(log log n) - insert(int key) O(log log n) - delete(int key) O(log log n) - get(int key) O(1) - int minimum() O(1) - int maximum() O(log log n) - int successor(int key) O(log log n) - int predecessor(int key) where the tree is capable to hold keys from the range [1, n]!5

6 van Emde Boas Tree Operations What is really unique? Provided by hash maps: insert(int: key), delete(int: key), get(int: key) Easy to implement: int minimum(), int maximum() Unique: int successor(int: key), int predecessor(int: key)!6

7 van Emde Boas Tree Idea How to implement fast Successor()?!7

8 Array approach Idea Successor, Predecessor - alternatives Array O(n) X n-7 n-6 n-5 n-4 n-3 n-2 n-1 n X!8

9 Binary tree approach Idea Successor, Predecessor - alternatives Binary tree O(log n)..... h = log n X n-7 n-6 n-5 n-4 n-3 n-2 n-1 n X!9

10 Towards veb Tree Idea Successor, Predecessor - alternatives SQRT tree - the number of descendants decreases along the square root of ancestors (does not apply to the root) O(log log n)... h = log log n X n-7 n-6 n-5 n-4 n-3 n-2 n-1 n X!10

11 Towards veb Tree Idea The height of the SQRT tree How many times we can compute square root of n and be no smaller than 2? n 1 2 h = h logn = 1 log 1 2 h logn = log1 = 0 log 1 2 h + loglogn = 0 loglogn = h!11

12 Towards veb Tree From SQRT tree to veb Tree What the veb tree node includes: Array 1,, α of pointers to children Variables min and max denoting minimal and maximal values in its subtree Summary structure that provides answers to the question whether x-th subtree x 1,, α contains at least one element (x-th subtree is or not empty)!12

13 veb Tree Idea What does the trick? Summary itself is veb tree - hence it works in O(log log n) Although it is at first glance not obvious the values min, max and the response from Summary are sufficient to compute the position of successor and/or predecessor Hence, the given method either go down into a veb tree as such or go down into the Summary structure As a result every operation can be performed during the single passage from the top to the bottom of the tree.!13

14 veb Tree Idea veb Tree, 256 nodes!14

15 From veb Tree to dveb Array Where is the problem? Sequential perspective Trees containing a small number of elements with the relatively large keys may require disproportionately large amount of memory Allocating n of continuous memory when n is large might be difficult, especially in case of embedded systems. E.g. n = 2 require n > Concurrent perspective Since all the methods operate on the top-down principle, concurrent calls can lead to congestions in the upper levels of the three!15

16 Structure Dynamic Concurrent van Emde Boas Array (dcveb Array)!16

17 Structure Sequential perspective Memory allocation improvement Every node has descendants Concurrent perspective Synchronization improvement Summary structure is given as bit machine word, so all the operations on the summary can be implemented using CAS (Compare and Swap) mechanism Blocking (lock based) and non blocking (CAS) synchronization mechanisms are used Congestion preventing Depending on the method either top-down or bottom-up paradigm is adopted!17

18 Structure Concurrent perspective To meet concurrent objects consistency conditions: Quiescent consistency Sequential consistency Linearizability!18

19 Structure X n-7 n-6 n-5 n-4 n-3 n-2 n-1 n X!19

20 Performance Running time: O(log n) O(log n) O(log n) O(1) O(1) O(log n) O(log n) - insert(int key) - delete(int key) - get(int key) - int minimum() - int maximum() - int successor(int key) - int predecessor(int key) - expected concurrent running time!20

21 Operation scheme Operation scheme top-down - insert(int key) both - delete(int key) top-down - get(int key) top-down - int minimum() top-down - int maximum() both - int successor(int key) both - int predecessor(int key)!21

22 Operation scheme Synchronization method CAS & Lock based - insert(int key) CAS & Lock based - delete(int key) CAS - get(int key) CAS - int minimum() CAS - int maximum() CAS - int successor(int key) CAS - int predecessor(int key)!22

23 Insert vs. Delete insert() starts locking from the top, whilst delete() starts from the bottom lock. lock.. delete() insert() lock.. X lock n-7 n-6 n-5 n-4 n-3 n-2 n-1 n X!23

24 Insert vs. Insert insert() vs insert() do not interfere each other due to the (many)read-(single)write lock readwritelock insert() readwritelock. readwritelock X n-7 n-6 n-5 n-4 n-3 n-2 n-1 n X!24

25 Successor successor() traverses the tree up and down looking for the successor of the given element. successor() X concurrent delete n-7 n-6 n-5 n-4 n-3 n-2 n-1 n X!25

26 Successor Correctness property x, y! + x < y x For every such that and is stored in dcveb Array during the whole execution of Successor() (i.e. between call and return ) holds: r = Successor( ) where r y x x r y X X X y-1 X y y+1 y x x+1 x+2 x+3 r r+1 r+2 r+3!26

27 Results Experimental results!27

28 Results Experimental setup Language: Java Structures: dcveb Array concurrent skiplist map (JVM Library) snap tree k-ary search tree tree map (JVM Library, sequential) CPU: 6 core, 12 threads Intel i7-3930k at 3.8 GHz!28

29 Results Experiment 1 Four methods g - get, i - insert, r - delete, s - successor Methods, threads 25 - concurrent runs 4 threads, g = i = r = s = 1 8 threads, g = i = r = s = threads, g = i = r = s = 25 Every thread performs concurrent calls!29

30 Results Performance chart: ms $ tree map skiplist map dcveb array snap tree k-ary search tree!30

31 Results Performance chart: miliseconds % 925% % 937% inserter getter deleter successor searcher skiplist map dcveb array snap tree tree map k-ary search tree % 1162% % 1187%!31

32 Results Experiment 2 Four methods g - get, i - insert, r - delete, s - successor Methods, threads 8 threads, g = i = r = s = 2 variable key range, n=10,100,, Every thread performs concurrent calls per range!32

33 Results Performance chart: miliseconds $ tree map skiplist map dcveb array snap tree k-ary search tree!33

34 Literature cveb Array (1st version) Kułakowski, K. (2014), A concurrent van Emde Boas array as a fast and simple concurrent dynamic set alternative. Concurrency Computat.: Pract. Exper., 26: doi: /cpe dcveb Array (2nd version) Kułakowski K., Dynamic concurrent van Emde Boas array, CoRR,

35 Questions!35