Hashing and Natural Parallism Companion slides for The Art of Multiprocessor Programming by Maurice Herlihy & Nir Shavit.

Hashing and Natural Parallism

Companion slides forThe Art of Multiprocessor

Programmingby Maurice Herlihy & Nir Shavit

Code

• Please review the code for the presented data structures from Chapter 13 of the textbook

Art of Multiprocessor Programming© Herlihy-Shavit

2007 2


2007 3

Linked Lists

• We looked at a number of ways to make highly-concurrent list-based Sets:– Fine-grained locks– Optimistic synchronization– Lazy synchronization– Lock-free synchronization

• What’s missing?


2007 4

Linear-Time Set Methods

• Problem is– add(), remove(), contains()– Take time linear in set size

• We want– Constant-time methods– (at least, on average)


2007 5

Hashing

• Hash function– h: items integers

• Uniformly distributed– Different item most likely have

different hash values

• Java hashCode() method


2007

6

Sequential Hash Map

0

1

2

3

16

9

h(k) = k mod 4h(k) = k mod 4

2 Items2 Items

buckets


2007

7

Add an Item

0

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16

9

7


3 Items3 Items


2007

8

Add Another: Collision

0

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16 4

9

7


4 Items4 Items


2007

9

More Collisions

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16 4

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7 15


5 Items5 Items


2007

10

More Collisions

0

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16 4

9

7 15


5 Items5 ItemsProblem:

buckets getting too long


2007

11

Resizing

0

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16 4

9

7 15

4

5

6

7 Grow the array

5 Items5 Items



2007

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5 Items5 Items

Resizing

0

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16 4

9

7 15

4

5

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7


Adjust hash function


2007

13

Resizing

0

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16

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7 15

h(4) = 0 mod 8

4

5

6

7

4



2007

14

Resizing

0

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7 15

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7


h(4) = 4 mod 8


2007

15

Resizing

0

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7 15

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7


h(15) = 7 mod 8


2007

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Resizing

0

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4

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7


h(15) = 7 mod 8

157


2007 17

Hash Sets

• Implement a Set object– Collection of items, no duplicates– add(), remove(), contains()

methods– You know the drill …


2007 26

No Brainer?

• We just saw a– Simple– Lock-free– Concurrent hash-based set

implementation

• What’s not to like?


2007 27

No Brainer?

• We just saw a– Simple– Lock-free– Concurrent hash-based set

implementation

• What’s not to like?• We don’t know how to resize …


2007 28

Is Resizing Necessary?

• Constant-time method calls require– Constant-length buckets– Table size proportional to set size– As set grows, must be able to resize


2007 29

Set Method Mix

• Typical load– 90% contains()– 9% add ()– 1% remove()

• Growing is important• Shrinking not so much


2007 30

When to Resize?

• Many reasonable policies. Here’s one.

• Pick a threshold on num of items in a bucket

• Global threshold– When ≥ ¼ buckets exceed this value

• Bucket threshold– When any bucket exceeds this value


2007 31

Coarse-Grained Locking

• Good parts– Simple– Hard to mess up

• Bad parts– Sequential bottleneck


2007

32

Fine-grained Locking

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4 8

9

7 11

17

Each lock associated with one bucket


2007

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Resize ThisMake sure table reference didn’t change between resize decision and

lock acquisition

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4 8

9

7 11

17

Acquire locks in ascending order


2007

34

Resize This

0

1

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4 8

9

7 11

170

1

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6

7

Allocate new super-sized

table


2007

35

Resize This

0

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2

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9

7

4 8

170

1

2

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4

5

6

7

8

4

9 17

7

1111


2007

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Resize This

0

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3

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3

4

5

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7

8

4

9 17

7

11

Striped Locks: each lock now associated with two buckets


2007 38

Observations

• We grow the table, but not locks– Resizing lock array is tricky …

• We use sequential lists– Not LockFreeList lists– If we’re locking anyway, why pay?


2007 53

Fine-Grained Locks

• We can resize the table• But not the locks• Debatable whether method calls

are constant-time in presence of contention …


2007 54

Insight

• The contains() method– Does not modify any fields– Why should concurrent contains()

calls conflict?


2007

55

Read/Write Locks

public interface ReadWriteLock { Lock readLock(); Lock writeLock();}


2007

56

Read/Write Locks


Returns associated read

lock


2007

57

Read/Write Locks


Returns associated read

lockReturns

associated write lock


2007 58

Lock Safety Properties

• No thread may acquire the write lock– while any thread holds the write lock– or the read lock.

• No thread may acquire the read lock– while any thread holds the write lock.

• Concurrent read locks OK


2007 59

Read/Write Lock

• Satisfies safety properties– If readers > 0 then writer == false– If writer = true then readers == 0

• Liveness?– Lots of readers …– Writers locked out?


2007 60

FIFO R/W Lock

• As soon as a writer requests a lock• No more readers accepted• Current readers “drain” from lock• Writer gets in


2007 61

The Story So Far

• Resizing the hash table is the hard part

• Fine-grained locks– Striped locks cover a range (not

resized)

• Read/Write locks– FIFO property tricky


2007 62

Optimistic Synchronization

• If the contains() method– Scans without locking

• If it finds the key– OK to return true– Actually requires a proof ….

• What if it doesn’t find the key?


2007 63

Optimistic Synchronization

• If it doesn’t find the key– May be victim of resizing

• Must try again– Getting a read lock this time

• Makes sense if– Keys are present– Resizes are rare


2007 64

Stop The World Resizing

• The resizing we have seen up till now stops all concurrent operations

• Can we design a resize operation that will be incremental

• Need to avoid locking the table• A lock-free table with incremental

resizing


2007

65

Lock-Free Resizing Problem

0

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4 8

9

7 15


2007

66


0

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4 8

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7 15

12

Need to extend table


2007

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4 12


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8

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7 15

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4 12

Need to extend table


2007

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0

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84

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7 15

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200769


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12

to remove and then add even a single item single location CAS not enough

4

84 12

We need a new idea…


2007

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Don’t move the items

Move the buckets instead Keep all items in a single lock-free

list Buckets become “shortcut pointers”

into the list16 4 9 7 15

0123


2007

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Recursive Split Ordering

0

0 4 2 6 1 5 3 7


2007

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0

1/2

1

0 4 2 6 1 5 3 7


2007

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0

1/2

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1/4 3/4

2

3

0 4 2 6 1 5 3 7


2007

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0

1/2

1

1/4 3/4

2

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0 4 2 6 1 5 3 7

List entries sorted in order that allows recursive splitting. How?


2007

75


0

0 4 2 6 1 5 3 7


200776


0

1

0 4 2 6 1 5 3 7

LSB 0 LSB 1

LSB = Least significant Bit


2007

77


0

12

3

0 4 2 6 1 5 3 7

LSB 00 LSB 10 LSB 01 LSB 11


2007 78

Split-Order

• If the table size is 2i,– Bucket b contains keys k

• k = b (mod 2i)

– bucket index consists of key's i LSBs


2007 79

When Table Splits

• Some keys stay– b = k mod(2i+1)

• Some move– b+2i = k mod(2i+1)

• Determined by (i+1)st bit– Counting backwards

• Key must be accessible from both– Keys that will move must come later


2007

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A Bit of Magic

0 4 2 6 1 5 3 7

Real keys:


2007

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A Bit of Magic

0 4 2 6 1 5 3 7

Real keys:

0 1 2 3 4 5 6 7

Split-order:

Real key 1 is in the 4th location


2007

82

A Bit of Magic

0 4 2 6 1 5 3 7

Real keys:

0 1 2 3 4 5 6 7

Split-order:

000 100 010 110 001 101 011 111

000 001 010 011 100 101 110 111

Real key 1 is in 4th location


2007

83

A Bit of MagicReal keys:

Split-order:

000 100 010 110 001 101 011 111

000 001 010 011 100 101 110 111


2007

84

A Bit of MagicReal keys:

Split-order:

000 100 010 110 001 101 011 111

000 001 010 011 100 101 110 111

Just reverse the order of the key bits


2007

85

Split Ordered Hashing

0

12

3

0 4 2 6 1 5 3 7

000 001 010 011 100 101 110 111

Order according to reversed bits


2007

86

Parent Always Provides a Short Cut

0

12

3

0 4 2 6 1 5 3 7

search


2007 87

Sentinel Nodes

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16 4 9 7 15

Problem: how to remove a node pointed by 2 sources using CAS


2007 88

Sentinel Nodes

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16 4 9 7 153

Solution: use a Sentinel node for each bucket

0 1


2007 89

Sentinel vs Regular Keys

• Want sentinel key for i ordered – before all keys that hash to bucket i– after all keys that hash to bucket (i-1)


2007 90

Splitting a Bucket

• We can now split a bucket • In a lock-free manner• Using two CAS() calls ...


2007 91

Initialization of Buckets

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16 4 9 7 150 1


2007 92

Initialization of Buckets

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16 4 9 7 150 1

3

Need to initialize bucket 3 to split bucket 1

3 in list but not connected to bucket yet

Now 3 points to sentinel – bucket has been split


2007 93

Adding 10

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16 4 9 3 70 1

2

10 = 2 mod 4

2

Must initialize bucket 2Then can add 10


2007 94

Recursive Initialization

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8 120

7 = 3 mod 4To add 7 to the list

3

Must initialize bucket 3


= 1 mod 2 1

Could be log n depthBut EXPECTED depth is constant


2007

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Lock-Free List

int makeRegularKey(int key) { return reverse(key | 0x80000000);}int makeSentinelKey(int key) { return reverse(key);}


2007

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Lock-Free List


Regular key: set high-order bit to 1 and reverse


2007

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Lock-Free List


Sentinel key: simply reverse (high-order bit is

0)


2007 98

Main List

• Lock-Free List from earlier class• With some minor variations


2007

99

Lock-Free List

public class LockFreeList { public boolean add(Object object, int key) {...} public boolean remove(int k) {...} public boolean contains(int k) {...} public LockFreeList(LockFreeList parent, int key) {...};}


2007

100

Lock-Free List


Change: add takes key argument


2007

101

Lock-Free List


Inserts sentinel with key if not already present …


2007

102

Lock-Free List


… returns new list starting with sentinel (shares with

parent)


2007

103

Split-Ordered Set: Fields

public class SOSet { protected LockFreeList[] table; protected AtomicInteger tableSize; protected AtomicInteger setSize;

public SOSet(int capacity) { table = new LockFreeList[capacity]; table[0] = new LockFreeList(); tableSize = new AtomicInteger(2); setSize = new AtomicInteger(0); }


2007

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Fields



For simplicity treat table as big array …


2007

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Fields



In practice, want something that grows

dynamically


2007

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Fields



How much of table array are we actually using?


2007

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Fields



Track set size so we know when to resize


2007

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Fields



Initially use 1 bucket and size is zero


2007

109

Add() Method

public boolean add(Object object) { int hash = object.hashCode(); int bucket = hash % tableSize.get(); int key = makeRegularKey(hash); LockFreeList list = getBucketList(bucket); if (!list.add(object, key)) return false; resizeCheck(); return true;}


2007

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Add() Method

public boolean add(Object object) { int hash = object.hashCode(); int bucket = hash % tableSize.get(); int key = makeRegularKey(hash); LockFreeList list = getBucketList(bucket); if (!list.add(object, key)) return false; resizeCheck(); return true;} Pick a bucket


2007

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Add() Method


Non-Sentinel split-ordered

key


2007

112

Add() Method


Get pointer to bucket’s sentinel, initializing if

necessary


2007

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Add() Method


Call bucket’s add() method with reversed key


2007

114

Add() Method


No change? We’re done.


2007

115

Add() Method


Time to resize?


2007 116

Resize

• Divide set size by total number of buckets

• If quotient exceeds threshold– Double tableSize field– Up to fixed limit


2007 117

Initialize Buckets

• Buckets originally null• If you find one, initialize it• Go to bucket’s parent

– Earlier nearby bucket– Recursively initialize if necessary

• Constant expected work


2007 118

Recall: Recursive Initialization

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8 120

7 = 3 mod 4To add 7 to the list

3



= 1 mod 2 1

Could be log n depthBut EXPECTED depth is constant


2007

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Initialize Bucket

void initializeBucket(int bucket) { int parent = getParent(bucket); if (table[parent] == null) initializeBucket(parent); int key = makeSentinelKey(bucket); LockFreeList list = new LockFreeList(table[parent], key); }


2007

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Initialize Bucket

void initializeBucket(int bucket) { int parent = getParent(bucket); if (table[parent] == null) initializeBucket(parent); int key = makeSentinelKey(bucket); LockFreeList list = new LockFreeList(table[parent], key); } Find parent, recursively

initialize if needed


2007

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Initialize Bucket

void initializeBucket(int bucket) { int parent = getParent(bucket); if (table[parent] == null) initializeBucket(parent); int key = makeSentinelKey(bucket); LockFreeList list = new LockFreeList(table[parent], key); } Prepare key for new

sentinel


2007

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Initialize Bucket

void initializeBucket(int bucket) { int parent = getParent(bucket); if (table[parent] == null) initializeBucket(parent); int key = makeSentinelKey(bucket); LockFreeList list = new LockFreeList(table[parent], key); }

Insert sentinel if not present, and get back reference to rest

of list


2007 123

Correctness• Linearizable concurrent set

implementation• Theorem: O(1) expected time

– No more than O(1) items expected between two dummy nodes on average

– Lazy initialization causes at most O(1) expected recursion depth in initializeBucket()


2007 124

Empirical Evaluation• On a 30-processor Sun Enterprise

3000• Lock-Free vs. fine-grained (Lea)

optimistic • In a non-multiprogrammed

environment• 106 operations: 88% contains(),

10% add(), 2% remove()


2007 125

Work = 0ops/

tim

e

threads

(2)

locking

lock-free


2007 126

Work = 500ops/

tim

e

threads

(2)

locking

lock-free


2007 127

Expected Bucket Lengthops/

tim

e

Load factor

(2)

locking

lock-free

64 threads


2007 128

Varying The Mixops/

tim

e

(2)

locking

lock-free

64 threads

More reads More updates


2007 129

Summary

• Concurrent resizing is tricky• Lock-based

– Fine-grained– Read/write locks– Optimistic

• Lock-free– Builds on lock-free list


2007 130

Additional Performance

• The effects of the choice of locking granularity

• The effects of bucket size


2007

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Number of Fine-Grain Locks


2007

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Lock-free vs Locks


2007

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Hash Table Load Factor

(load factor = nodes per bucket)


2007

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Varying Operations


2007

135

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Hashing and Natural Parallism Companion slides for The Art of Multiprocessor Programming by Maurice Herlihy & Nir Shavit.

Documents

long slide

average slide

drill slide

items hk

hash function slide

free lists slide

items problem

items integers