Range Queries in Non-blocking

k-ary Search Trees

Trevor BrownHillel Avni

Problem Statement

● Want to store keys in a dynamic data structure supporting insertion, deletion, and:

● RangeQuery(a, b): returns all keys of the data structure in range [a, b]

Previous Solutions● Software transactional memory ● Locks● Persistence

a d

f

e

b

root pointer

e

b

d

c

Insert(c)

k-ary Search Tree (k-ST)

● Add or remove keys by replacing node(s)● Related to persistent data structures

[Brown, Helga]

The Range Query AlgorithmRangeQuery(a, b):

– Traverse the tree, skipping sub-trees which cannot contain a key in [a, b]

– During this traversal, save a pointer to each leaf that contains a key in [a, b]

– […]

– Problem: how to efficiently tell if a key was added or removed during this traversal?

Extending the Data Structure● Add a dirty bit to each leaf● Each leaf has its dirty bit set just before it is

replaced– Consequence: If a leaf's dirty bit is not set, then

it has not been replaced

The Range Query AlgorithmRangeQuery(a, b):

– Traverse the tree, skipping sub-trees which cannot contain a key in [a, b]

– During this traversal, save a pointer to each leaf that contains a key in [a, b]

– After this traversal, check the dirty bits of these leaves, one by one

– If no dirty bit is set, then return “the result”– Otherwise, retry

● Reading dirty bit is far faster than re-traversing

Example: RangeQuery(3, 14)

8, 13, 25

2, 3, 5 8, 9, 12

1 4 5, 7

14, 19, 21

1315, 16,

1823, 24

29, 35

Saved pointers3, 4 Insert(3)

RangeQuery sees 4 is dirty… Retry!

Retrying RangeQuery(3, 14)

8, 13, 25

2, 3, 5 8, 9, 12

1 4 5, 7

14, 19, 21

1315, 16,

1823, 24

29, 35

Saved pointers3, 4

Success! Return the result…

Ctrie● Taking a “snapshot:” atomically replace root

● Old tree no longer changes● Future searches and updates copy nodes from

the old tree

[Prokopec, Bronson, Bagwell, Odersky]

| | |

… … … …

root pointer

| | |00 01 10 1101 10 11

When our Algorithm is Good● When workloads contain range queries over

small ranges (i.e., where snapshots are bad)– Example: database applications such as airline

database of flights

When it might not be● Very large ranges increase the chance that a

range query will have to retry– Our experiments explore how much this matters– In extreme cases Ctrie or Snap might be better

Experiment: compare performance of

● k-ST: k=16, 32, 64● Snap● Ctrie● Java’s Concurrent Skip List (SL)

– NOT LINEARIZABLE!

Experiment● Throughput vs. number of concurrent threads● Each thread repeatedly chooses a random

operation (Search, Insert, Delete, RangeQuery) with arguments chosen uniformly randomly in [0, 10^6)

● Each experiment ran with a fixed amount of memory, for a fixed, sufficiently long amount of time

Hardware● Intel 4-chip, 40-core, 80-thread● Sun 2-chip, 16-core, 128-thread

Many queries with small ranges

Snap

Ctrie

16-ST32-ST

SL

64-ST

Number of threads

Throughput

(millions)

50% search, 5% insert, 5% delete, 40% range query size 100

Many queries with bigger ranges

Snap

Ctrie

16-ST

32-ST

SL

64-ST

Number of threads

Throughput(hundred

thousands)

50% search, 5% insert, 5% delete, 40% range query size 10,000

Few queries (with small ranges)

Snap

Ctrie

16-ST

32-ST

SL

64-ST

Number of threads

Throughput(ten

millions)

59% search, 20% insert, 20% delete, 1% range query size 100

Throughput versus P(range query)

Snap

Ctrie

KST16KST32

SL

KST64

Probability of range query

Throughput(ten

millions) 1:10000 operations is RQ

Throughput versus arity

5i-5d-40r-size10000

20i-20d-1r-size10000

5i-5d-40r-size100

20i-20d-1r-size100

Degree of tree

Throughput(ten

millions)

Conclusion● Provably correct algorithm● Searches can ignore concurrent updates

● Although dirty bits invalidate range queries,they do not invalidate searches

● Range queries are invisible● No CAS, don’t change data structure

● Avoids excessive duplication of nodes● Appears to be practical when workloads contain

queries over small ranges

Future work● Adding balance

– (a, b)-tree, chromatic tree, relaxed AVL tree

● Wait-freedom?