Mutual Exclusion
Companion slides for
The Art of Multiprocessor Programming
by Maurice Herlihy & Nir Shavit
Modified by Pavol Černý,
Programming Paradigms for Concurrency,
Fall 2010, IST Austria
Review: Peterson’s Algorithm
Alice
flag[0] ←0
no access
flag[0]
←0
Bob
flag[1] ←0
no access
flag[0] ← 1
flag[1] ← 1
turn ← 1
turn ← 0
request
flag[1] = 0 or
turn = 0
access
flag[0]
request
flag[0] = 0 or
turn = 1
access
←0
Implementing a Counter
public class Counter {
private long value;
Use Peterson’s
public long getAndIncrement() {
temp = value;
value = temp + 1;
return temp;
Make these
}
}
steps
indivisible using
locks
Art of Multiprocessor
3
Programming
Locks (Mutual Exclusion)
public interface Lock {
public void lock();
public void unlock();
}
Art of Multiprocessor
Programming
4
Locks (Mutual Exclusion)
public interface Lock {
public void lock();
acquire lock
public void unlock();
}
Art of Multiprocessor
Programming
5
Locks (Mutual Exclusion)
public interface Lock {
public void lock();
acquire lock
public void unlock();
release lock
}
Art of Multiprocessor
Programming
6
Using Locks
public class Counter {
private long value;
private Lock lock;
public long getAndIncrement() {
lock.lock();
try {
int temp = value;
value = value + 1;
} finally {
lock.unlock();
}
return temp;
}}
Art of Multiprocessor
Programming
7
Using Locks
public class Counter {
private long value;
private Lock lock;
public long getAndIncrement() {
lock.lock();
acquire
try {
int temp = value;
value = value + 1;
} finally {
lock.unlock();
}
return temp;
}}
Art of Multiprocessor
Programming
Lock
8
Using Locks
public class Counter {
private long value;
private Lock lock;
public long getAndIncrement() {
lock.lock();
try {
int temp = value;
value = value + 1;
} finally {
Release lock
lock.unlock();
}
(no matter what)
return temp;
}}
Art of Multiprocessor
Programming
9
Using Locks
public class Counter {
private long value;
private Lock lock;
public long getAndIncrement() {
lock.lock();
try {
int temp = value;
value = value + 1;
} finally {
lock.unlock();
}
return temp;
}}
Art of Multiprocessor
Programming
Critical
section
10
Peterson’s in Java
public void lock() {
flag[i] = 1;
turn = 1-i;
while (!((flag[1-i] == 0) || (turn == i))) { }
}
flag[0]
←0
public void unlock() {
flag[i] = 0;
}
flag[0] ←0
no access
flag[0] ← 1
turn ← 1
request
flag[1] = 0 or
turn = 0
access
Review: Peterson’s Algorithm
Questions from last lecture:
1.Can we make do with two shared bits
(instead of three)?
2.How can one extend this idea to n
processes?
3.Does the algorithm work in Java?
Run it and see. What is the problem?
Where is the fault in our proof?
Question 1: Modify Peterson’s to use two bits
public void lock() {
if (i==0) {
flag[0] = 1;
while (!(flag[1] == 0)) { }
}
else {
while (!( flag[1] == 1)) {;
flag[1] = 1;
while (!(flag[0] == 0)) {
flag[1] = 0;
}
}
}
}
public void unlock() {
flag[i] = 0;
}
Question 2: Modify Peterson’s for n processes
Today’s class
Question 3: But does it work in Java?
public void lock() {
flag[i] = 1;
turn = 1-i;
while (!((flag[1-i] == 0) || (turn == i))) { }
}
public void unlock() {
flag[i]=0;
}
java Peterson … (results in deadlock)
java Peterson ... (mutual exclusion fails
(if a counter is protected
with these locks, we observe
a problem after enough
iterations))
Fact
• Most hardware architectures don’t
support sequential consistency
• Because they think it’s too strong
Art of Multiprocessor
Programming
16
The Flag Example
y.read(0)
x.write(1)
y.write(1)
x.read(0)
time
time
Art of Multiprocessor
Programming
17
The Flag Example
y.read(0)
x.write(1)
y.write(1)
x.read(0)
• Each thread’s view is sequentially
consistent
– It went first
time
Art of Multiprocessor
Programming
18
The Flag Example
y.read(0)
x.write(1)
y.write(1)
x.read(0)
• Entire history isn’t sequentially
consistent
– Can’t both
go first
time
Art of Multiprocessor
Programming
19
The Flag Example
y.read(0)
x.write(1)
y.write(1)
x.read(0)
• Is this behavior really so wrong?
– We can argue either way …
time
Art of Multiprocessor
Programming
20
Opinion1: It’s Wrong
• This pattern
– Write mine, read yours
• Is exactly the flag principle
– Beloved of Alice and Bob
– Heart of mutual exclusion
• Peterson
• Bakery, etc.
• It’s non-negotiable!
Art of Multiprocessor
Programming
21
Opinion2: But It Feels So Right
…
• Many hardware architects think that
sequential consistency is too strong
• Too expensive to implement in modern
hardware
• OK if flag principle
– violated by default
– Honored by explicit request
Art of Multiprocessor
Programming
22
Who knew you wanted to
synchronize?
• Writing to memory = mailing a letter
• Vast majority of reads & writes
– Not for synchronization
– No need to idle waiting for post office
• If you want to synchronize
– Announce it explicitly
– Pay for it only when you need it
Art of Multiprocessor
Programming
23
Explicit Synchronization
• Memory barrier instruction
– Flush unwritten caches
– Bring caches up to date
• Expensive
Art of Multiprocessor
Programming
24
Volatile
• In Java, can ask compiler to keep a
variable up-to-date with volatile
keyword
• Also inhibits reordering & other
“optimizations”
• Example:
public static volatile int[] flag;
public static volatile int z = 0;
Art of Multiprocessor
Programming
25
Mutual Exclusion
•
•
•
•
•
Problem definitions
Solutions for 2 threads
Solutions for n threads
Fair solutions
Inherent costs
Art of Multiprocessor
Programming
26
Warning
• You will never use these protocols
• Understanding these is necessary
– The same issues show up everywhere
– Except hidden and more complex
Art of Multiprocessor
Programming
27
Formalizing the Problem
• Two types of formal properties in
asynchronous computation:
• Safety Properties
– Nothing bad happens ever
– E.g. Mutual exclusion, No overtaking
• Liveness Properties
– Something good happens eventually
– E.g. Deadlock freedom, Starvation freedom
Art of Multiprocessor Programming
28
Events
• An event a0 of thread A is
– Instantaneous
– No simultaneous events (break ties)
a0
time
Art of Multiprocessor
Programming
29
Threads
• A thread A is (formally) a sequence
a0, a1, ... of events
– “Trace” model
– Notation: a0  a1 indicates order
a0
a1
a2
…
time
Art of Multiprocessor
Programming
30
Example Thread Events
•
•
•
•
•
Assign to shared variable
Assign to local variable
Invoke method
Return from method
Lots of other things …
Art of Multiprocessor
Programming
31
Threads are State Machines
Events are
transitions
a3
a2
Art of Multiprocessor
Programming
a0
a1
32
States
• Thread State
– Program counter
– Local variables
• System state
– Object fields (shared variables)
– Union of thread states
Art of Multiprocessor
Programming
33
Concurrency
• Thread A
time
Art of Multiprocessor
Programming
34
Concurrency
• Thread A
time
• Thread B
time
Art of Multiprocessor
Programming
35
Interleavings
• Events of two or more threads
– Interleaved
– Not necessarily independent (why?)
time
Art of Multiprocessor
Programming
36
Intervals
• An interval A0 =(a0,a1) is
– Time between events a0 and a1
a0
A0
a1
time
Art of Multiprocessor
Programming
37
Intervals may Overlap
b0
a0
A0
B0
b1
a1
time
Art of Multiprocessor
Programming
38
Intervals may be Disjoint
b0
a0
A0
B0
b1
a1
time
Art of Multiprocessor
Programming
39
Precedence
Interval A0 precedes interval B0
b0
a0
A0
B0
b1
a1
time
Art of Multiprocessor
Programming
40
Precedence
• Notation: A0  B0
• Formally,
– End event of A0 before start event of B0
– Also called “happens before” or
“precedes”
Art of Multiprocessor
Programming
41
Precedence Ordering
•
•
•
•
Never true that A  A
If A B then not true that B A
If A B & B C then A C
Funny thing: A B & B A might both
be false!
Art of Multiprocessor
Programming
42
Partial Orders
(review)
• Irreflexive:
– Never true that A  A
• Antisymmetric:
– If A  B then not true that B  A
• Transitive:
– If A  B & B  C then A  C
Art of Multiprocessor
Programming
43
Total Orders
(review)
• Also
– Irreflexive
– Antisymmetric
– Transitive
• Except that for every distinct A, B,
– Either A  B or B  A
Art of Multiprocessor
Programming
44
Repeated Events
while (mumble) {
a0; a1;
}
a0
k
A0k
k-th occurrence
of event a0
k-th occurrence of
interval A0 =(a0,a1)
Art of Multiprocessor
Programming
45
Mutual Exclusion
• Let CSik
be thread i’s k-th critical
section execution
Art of Multiprocessor
Programming
46
Mutual Exclusion
• Let CSik
be thread i’s k-th critical
section execution
• And CSjm
be thread j’s m-th critical
section execution
Art of Multiprocessor
Programming
47
Mutual Exclusion
• Let CSik
be thread i’s k-th critical
section execution
• And CSjm
be j’s m-th execution
• Then either
–
or
Art of Multiprocessor
Programming
48
Mutual Exclusion
• Let CSik
be thread i’s k-th critical
section execution
• And CSjm
be j’s m-th execution
• Then either
–
or
CSik  CSjm
Art of Multiprocessor
Programming
49
Mutual Exclusion
• Let CSik
be thread i’s k-th critical
section execution
• And CSjm
be j’s m-th execution
• Then either
–
or
CSik  CSjm
Art of Multiprocessor
Programming
CSjm  CSik
50
Deadlock-Free
• If some thread calls lock()
– And never returns
– Then other threads must complete lock()
and unlock() calls infinitely often
– This is true for all fair schedules
– Schedule is fair if all threads in CS or
concurrency parts proceed
• System as a whole makes progress
– Even if individuals starve
Art of Multiprocessor
Programming
51
Starvation-Free
• If some thread calls lock()
– It will eventually return
• Individual threads make progress
Art of Multiprocessor
Programming
52
Example – proving mutual
exclusion
class LockOne implements Lock {
private boolean[] flag = new boolean[2];
public void lock() {
flag[i] = true;
while (flag[j]) {}
}
Art of Multiprocessor
Programming
53
LockOne
class LockOne implements Lock {
private boolean[] flag =
new boolean[2];
public void lock() {
flag[i] = true;
while (flag[j]) {}
Set my flag
}
Art of Multiprocessor
Programming
54
LockOne
class LockOne implements Lock {
private boolean[] flag =
new boolean[2];
public void lock() {
flag[i] = true;
while (flag[j]) {}
Set my flag
}
Wait for other
flag to go false
Art of Multiprocessor
Programming
55
LockOne Satisfies Mutual
Exclusion
• Assume CSAj overlaps CSBk
• Consider each thread's last (j-th
and k-th) read and write in the
lock() method before entering
• Derive a contradiction
Art of Multiprocessor
Programming
56
From the Code
• writeA(flag[A]=true) 
readA(flag[B]==false) CSA
• writeB(flag[B]=true) 
readB(flag[A]==false)  CSB
class LockOne implements Lock {
…
public void lock() {
flag[i] = true;
while (flag[j]) {}
}
Art of Multiprocessor
Programming
57
From the Assumption
• readA(flag[B]==false) 
writeB(flag[B]=true)
• readB(flag[A]==false) 
writeA(flag[A]=true)
Art of Multiprocessor
Programming
58
Combining
• Assumptions:
– readA(flag[B]==false)  writeB(flag[B]=true)
– readB(flag[A]==false)  writeA(flag[A]=true)
• From the code
– writeA(flag[A]=true)  readA(flag[B]==false)
– writeB(flag[B]=true)  readB(flag[A]==false)
Art of Multiprocessor
Programming
59
Combining
• Assumptions:
– readA(flag[B]==false)  writeB(flag[B]=true)
– readB(flag[A]==false)  writeA(flag[A]=true)
• From the code
– writeA(flag[A]=true)  readA(flag[B]==false)
– writeB(flag[B]=true)  readB(flag[A]==false)
Art of Multiprocessor
Programming
60
Combining
• Assumptions:
– readA(flag[B]==false)  writeB(flag[B]=true)
– readB(flag[A]==false)  writeA(flag[A]=true)
• From the code
– writeA(flag[A]=true)  readA(flag[B]==false)
– writeB(flag[B]=true)  readB(flag[A]==false)
Art of Multiprocessor
Programming
61
Combining
• Assumptions:
– readA(flag[B]==false)  writeB(flag[B]=true)
– readB(flag[A]==false)  writeA(flag[A]=true)
• From the code
– writeA(flag[A]=true)  readA(flag[B]==false)
– writeB(flag[B]=true)  readB(flag[A]==false)
Art of Multiprocessor
Programming
62
Combining
• Assumptions:
– readA(flag[B]==false)  writeB(flag[B]=true)
– readB(flag[A]==false)  writeA(flag[A]=true)
• From the code
– writeA(flag[A]=true)  readA(flag[B]==false)
– writeB(flag[B]=true)  readB(flag[A]==false)
Art of Multiprocessor
Programming
63
Combining
• Assumptions:
– readA(flag[B]==false)  writeB(flag[B]=true)
– readB(flag[A]==false)  writeA(flag[A]=true)
• From the code
– writeA(flag[A]=true)  readA(flag[B]==false)
– writeB(flag[B]=true)  readB(flag[A]==false)
Art of Multiprocessor
Programming
64
Cycle!
Art of Multiprocessor
Programming
65
Deadlock Freedom
• LockOne Fails deadlock-freedom
– Concurrent execution can deadlock
flag[i] = true;
while (flag[j]){}
flag[j] = true;
while (flag[i]){}
– Sequential executions OK
Art of Multiprocessor
Programming
66
The Filter Algorithm for n
Threads
There are n-1 “waiting rooms” called
levels
ncs
• At each level
– At least one enters level
– At least one blocked if
many try
cs
• Only one thread makes it through
Art of Multiprocessor
Programming
67
Filter
class Filter implements Lock {
int[] level; // level[i] for thread i
int[] victim; // victim[L] for level L
2
public Filter(int n) {
n-1
0
level = new int[n];
level 0 0 4 0 0 0 0 0
victim = new int[n];
1
for (int i = 1; i < n; i++) {
level[i] = 0;
}}
2 4
…
}
Thread 2 at level 4
Art of Multiprocessor
Programming
n-1
victim
68
Filter
class Filter implements Lock {
…
public void lock(){
for (int L = 1; L < n; L++) {
level[i] = L;
victim[L] = i;
while (($ k != i level[k] >= L) &&
victim[L] == i );
}}
public void unlock() {
level[i] = 0;
}}
Art of Multiprocessor
Programming
69
Filter
class Filter implements Lock {
…
public void lock() {
for (int L = 1; L < n; L++) {
level[i] = L;
victim[L] = i;
while (($ k != i) level[k] >= L) &&
victim[L] == i);
}}
public void release(int i) {
level[i] = 0;
}}
One level at a time
Art of Multiprocessor
Programming
70
Filter
class Filter implements Lock {
…
public void lock() {
for (int L = 1; L < n; L++) {
level[i] = L;
victim[L] = i;
while (($ k != i) level[k] >= L) &&
victim[L] == i); // busy wait
}}
public void release(int i) {
level[i] = 0;
}}
Art of Multiprocessor
Programming
Announce
intention to
enter level L
71
Filter
class Filter implements Lock {
int level[n];
int victim[n];
public void lock() {
for (int L = 1; L < n; L++) {
level[i] = L;
victim[L] = i;
while (($ k != i) level[k] >= L) &&
victim[L] == i);
}}
public void release(int i) {
level[i] = 0;
}}
Art of Multiprocessor
Programming
Give priority to
anyone but me
72
Filter
class Filter implements Lock {
Wait
as long as someone else is at
int level[n];
int
victim[n];
higher
level, and I’m designated
public void lock() {
for (int L = 1; L < n; L++) {
level[i] = L;
victim[L] = i;
same or
victim
while (($ k != i) level[k] >= L) &&
victim[L] == i);
}}
public void release(int i) {
level[i] = 0;
}}
Art of Multiprocessor
Programming
73
Filter
class Filter implements Lock {
int level[n];
int victim[n];
public void lock() {
for (int L = 1; L < n; L++) {
level[i] = L;
victim[L] = i;
while (($ k != i) level[k] >= L) &&
victim[L] == i);
}}
public void release(int i) {
Thread
enters level L when
level[i] = 0;
the loop
}}
Art of Multiprocessor
Programming
it completes
74
Claim
• Start at level L=0
• At most n-L threads enter level L
• Mutual exclusion at level L=n-1
ncs
L=0
L=1
L=n-2
cs L=n-1
Art of Multiprocessor
Programming
75
Induction Hypothesis
• No more than n-L+1 at level L-1 or above
• Induction step: by contradiction
• Assume all at level
L-1 enter level L
• A last to write
victim[L]
• B is any other
thread at level L
ncs
assume
L-1 has n-L+1
L has n-L
cs
Art of Multiprocessor
Programming
prove
76
Proof Structure
ncs
A
B
Assumed to enter L-1
n-L+1 = 4
Last to
write
victim[L]
n-L+1 = 4
cs
By way of contradiction
all enter L
A must have seen
B in level[L] and since victim[L] == A
could not have entered
Art of Multiprocessor
Programming
77
No Starvation
• Filter Lock satisfies properties:
– Just like Peterson Alg at any level
– So no one starves
• But what about fairness?
– Threads can be overtaken by others
Art of Multiprocessor
Programming
78
Bounded Waiting
• Want stronger fairness guarantees
• Thread not “overtaken” too much
• Need to adjust definitions ….
Art of Multiprocessor
Programming
79
Bounded Waiting
• Divide lock() method into 2 parts:
– Doorway interval:
• Written DA
• always finishes in finite steps
– Waiting interval:
• Written WA
• may take unbounded steps
Art of Multiprocessor
Programming
80
First-Come First-Served
• For threads A and B:
– If DAk  DB j
• A’s k-th doorway precedes B’s j-th doorway
– Then CSAk  CSBj
• A’s k-th critical section precedes B’s j-th
critical section
• B cannot overtake A
Art of Multiprocessor
Programming
81
Break!
Art of Multiprocessor
Programming
82
Bakery Algorithm
• Provides First-Come-First-Served
• How?
– Take a “number”
– Wait until lower numbers have been
served
• Lexicographic order
– (a,i) > (b,j)
• If a > b, or a = b and i > j
Art of Multiprocessor
Programming
83
Bakery Algorithm
class Bakery implements Lock {
boolean[] flag;
Label[] label;
public Bakery (int n) {
flag = new boolean[n];
label = new Label[n];
for (int i = 0; i < n; i++) {
flag[i] = false; label[i] = 0;
}
}
…
Art of Multiprocessor
Programming
84
Bakery Algorithm
class Bakery implements Lock {
boolean[] flag;
6
Label[] label;
2
0
n-1
public Bakery (int n) {
flag = new boolean[n]; f f t f f t f f
label = new Label[n];
0 0 4 0 0 5 0 0
for (int i = 0; i < n; i++) {
flag[i] = false; label[i] = 0;
}
}
…
CS
Art of Multiprocessor
Programming
85
Bakery Algorithm
class Bakery
…
public void
flag[i] =
label[i] =
implements Lock {
lock() {
true;
max(label[0], …,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) > (label[k],k));
}
Art of Multiprocessor
Programming
86
Bakery Algorithm
class Bakery
…
public void
flag[i] =
label[i] =
implements Lock {
Doorway
lock() {
true;
max(label[0], …,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) > (label[k],k));
}
Art of Multiprocessor
Programming
87
Bakery Algorithm
class Bakery
…
public void
flag[i] =
label[i] =
implements Lock {
I’m interested
lock() {
true;
max(label[0], …,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) > (label[k],k));
}
Art of Multiprocessor
Programming
88
Bakery Algorithm
class Bakery
…
public void
flag[i] =
label[i] =
implements Lock {
Take increasing
label (read labels
in some arbitrary
order)
lock() {
true;
max(label[0], …,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) > (label[k],k));
}
Art of Multiprocessor
Programming
89
Bakery Algorithm
class Bakery
…
public void
flag[i] =
label[i] =
implements Lock {
Someone is
interested
lock() {
true;
max(label[0], …,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) > (label[k],k));
}
Art of Multiprocessor
Programming
90
Bakery Algorithm
class Bakery implements Lock {
boolean flag[n];
Someone
int label[n];
is
interested
public void lock() {
flag[i] = true;
label[i] = max(label[0], …,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) > (label[k],k));
}
With lower (label,i)
in lexicographic order
Art of Multiprocessor
Programming
91
Bakery Algorithm
class Bakery implements Lock {
…
public void unlock() {
flag[i] = false;
}
}
Art of Multiprocessor
Programming
92
Bakery Algorithm
class Bakery implements Lock {
…
No longer
interested
public void unlock() {
flag[i] = false;
}
}
labels are always increasing
Art of Multiprocessor
Programming
93
No Deadlock
• There is always one thread with
earliest label
• Ties are impossible (why?)
Art of Multiprocessor
Programming
94
First-Come-First-Served
• If DA  DBthen A’s
label is smaller
• And:
– writeA(label[A]) 
readB(label[A]) 
writeB(label[B]) 
readB(flag[A])
• So B is locked out
while flag[A] is
true
class Bakery implements Lock {
public void lock() {
flag[i] = true;
label[i] = max(label[0],
…,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) >
(label[k],k));
}
Art of Multiprocessor
Programming
95
Mutual Exclusion
• Suppose A and B in
CS together
• Suppose A has
earlier label
• When B entered, it
must have seen
class Bakery implements Lock {
public void lock() {
flag[i] = true;
label[i] = max(label[0],
…,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) >
(label[k],k));
}
– flag[A] is false, or
– label[A] > label[B]
Art of Multiprocessor
Programming
96
Mutual Exclusion
• Labels are strictly increasing so
• B must have seen flag[A] == false
Art of Multiprocessor
Programming
97
Mutual Exclusion
• Labels are strictly increasing so
• B must have seen flag[A] == false
• LabelingB  readB(flag[A]) 
writeA(flag[A])  LabelingA
Art of Multiprocessor
Programming
98
Mutual Exclusion
• Labels are strictly increasing so
• B must have seen flag[A] == false
• LabelingB  readB(flag[A]) 
writeA(flag[A])  LabelingA
• Which contradicts the assumption
that A has an earlier label
Art of Multiprocessor
Programming
99
Bakery Y232K Bug
class Bakery
…
public void
flag[i] =
label[i] =
implements Lock {
lock() {
true;
max(label[0], …,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) > (label[k],k));
}
Art of Multiprocessor
Programming
100
Bakery Y232K Bug
class Bakery
…
public void
flag[i] =
label[i] =
Mutex breaks if
label[i] overflows
implements Lock {
lock() {
true;
max(label[0], …,label[n-1])+1;
while ($k flag[k]
&& (label[i],i) > (label[k],k));
}
Art of Multiprocessor
Programming
101
Does Overflow Actually
Matter?
• Yes
– Y2K
– 18 January 2038 (Unix time_t rollover)
– 16-bit counters
• No
– 64-bit counters
• Maybe
– 32-bit counters
Art of Multiprocessor
Programming
102
Is this Optimal Solution?
• The Bakery Algorithm is
– Succinct,
– Elegant, and
– Fair.
• Q: So why isn’t it practical?
• A: Well, you have to read N distinct
variables
Art of Multiprocessor
Programming
103
Shared Memory
• Shared read/write memory locations
called Registers (historical reasons)
• Come in different flavors
– Multi-Reader-Single-Writer (Flag[])
– Multi-Reader-Multi-Writer (Victim[])
– Not that interesting: SRMW and SRSW
Art of Multiprocessor
Programming
104
Theorem
At least N MRSW (multireader/single-writer) registers are
needed to solve deadlock-free
mutual exclusion.
N registers like Flag[]…
Art of Multiprocessor
Programming
105
Proving Algorithmic
Impossibility
•To show no algorithm exists:
• assume by way of contradiction
C
write
one does,
• show a bad execution that
CS
violates properties:
• in our case assume an alg for deadlock
free mutual exclusion using < N registers
Art of Multiprocessor
Programming
106
Proof: Need N-MRSW Registers
Each thread must write to some register
A
CS
B
C
write
write
CS
CS
…can’t tell whether A is in critical
section
Art of Multiprocessor
Programming
107
Upper Bound
• Bakery algorithm
– Uses 2N MRSW registers
• So the bound is (pretty) tight
• But what if we use MRMW registers?
– Like victim[] ?
Art of Multiprocessor
Programming
108
Bad News Theorem
At least N MRMW multireader/multi-writer registers are
needed to solve deadlock-free
mutual exclusion.
(So multiple writers don’t help)
Art of Multiprocessor
Programming
109
Theorem (For 2Threads)
Theorem: Deadlock-free mutual
exclusion for 2 threads requires at
least 2 multi-reader multi-writer
registers
Proof: assume one register suffices
and derive a contradiction
Art of Multiprocessor
Programming
110
Two Thread Execution
A
B
R
Write(R)
CS
CS
• Threads run, reading and writing R
• Deadlock free so at least one gets in
Art of Multiprocessor
Programming
111
Covering State
B
Write(R)
A covering state for a Lock object is one in which
there is at least one thread about to write to
each shared location, but the object’s locations
appear as if no thread is in the critical section or
trying to enter the critical section.
Art of Multiprocessor
Programming
112
Covering State for One
Register Always Exists
B
Write(R)
In any protocol B has to write to
the register before entering CS,
so stop it just before
Art of Multiprocessor
Programming
113
Proof: Assume Cover of 1
A
B
Write(R)
CS
A runs, possibly writes to the
register, enters CS
Art of Multiprocessor
Programming
114
Proof: Assume Cover of 1
A
B
Write(R)
CS
CS
Art of Multiprocessor
Programming
B Runs, first
obliterating
any trace of A,
then also enters
the critical
section
115
Theorem
Deadlock-free mutual exclusion for 3
threads requires at least 3 multireader multi-writer registers
Art of Multiprocessor
Programming
116
Proof: Assume Cover of 2
A
B
C
Write(RB)
Write(RC)
Only 2 registers
Art of Multiprocessor
Programming
117
Run A Solo
A
B
C
Write(RB)
Write(RC)
Writes to one or both
CS registers, enters CS
Art of Multiprocessor
Programming
118
Obliterate Traces of A
A
CS
B
C
Write(RB)
Write(RC)
Other threads obliterate
evidence that A entered CS
Art of Multiprocessor
Programming
119
Mutual Exclusion Fails
A
CS
B
C
Write(RB)
Write(RC)
CS
CS looks empty, so
another thread
gets in
Art of Multiprocessor
Programming
120
Proof Strategy
• Proved: a contradiction starting from
a covering state for 2 registers
• Claim: a covering state for 2
registers is reachable from any state
where CS is empty
Art of Multiprocessor
Programming
121
Covering State for Two
A
B
Write(RA)
Write(RB)
• If we run B through CS 3 times, B must
return twice to cover some register, say RB
Art of Multiprocessor
Programming
122
Covering State for Two
A
B
Write(RA)
Write(RB)
• Start with B covering register RB for the 1st time
• Run A until it is about to write to uncovered RA
• Are we done?
Art of Multiprocessor
Programming
123
Covering State for Two
A
B
Write(RA)
Write(RB)
• NO! A could have written to RB
• So CS no longer looks empty
Art of Multiprocessor
Programming
124
Covering State for Two
A
B
Write(RA)
Write(RB)
• Run B obliterating traces of A in RB
• Run B again until it is about to write to RB
• Now we are done
Art of Multiprocessor
Programming
125
Inductively We Can Show
• There is a covering state
– Where k threads not in CS cover k distinct
registers
– Proof follows when k = N-1
• Reference:
J. Burns, N. Lynch : Bounds on shared
memory for mutual exclusion. Inf. And
Comp. 1993
Art of Multiprocessor
Programming
126
Summary of Lecture
• In the 1960’s many incorrect
solutions to starvation-free mutual
exclusion using RW-registers were
published…
• Today we know how to solve FIFO N
thread mutual exclusion using 2N
RW-Registers
Art of Multiprocessor
Programming
127
Summary of Lecture
• N RW-Registers inefficient
– Because writes “cover” older writes
• Need stronger hardware operations
– that do not have the “covering problem”
Art of Multiprocessor
Programming
128