# 程序代写代做代考 algorithm distributed system concurrency cache Remember Integration with mutexes?

Remember Integration with mutexes?

What does Amdahl’s law say here?

Assume we have one processor per slice?

1

Some (simplifiying) assumptions.

• Assume

•

•

•

•Parallelisable codecomputesa,b,andarea(using`trapezoid`,whichuses`f`)

• •

• Total cost for a, b and area is 15.

• Serial Code is mutex lock/unlock and answer update: total cost is 4.

we can ignore the time spent creating threads

assume mutex lock and unlock each require one time-step

assume assignment and each arithmetic operation or function call require one time step.:

Cost of `f` is 3

Cost of `trapezoid` is 10

2

Using a thread to do several trapezoids

We get two formulas

Sequential Part: Seq(n) = 5n+1

Parallel Part: Par(w) = 17w+7

Time in terms of n, if W=1000:

T(n) = 5n + 8 + 17000/n

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T(n) = 5n + 8 + 17000/n

• A bit of calculus shows T(n) a minimum for n = 92, when it equals 653 steps

•

•

•

•

• So throwing 3400 processors at this problem is as slow as using one!

T(1) = 17013 T(92) = 653

T(1000) = 5025 T(3400) = 17013

4

Revisiting Matrix Multiplication

(keeping Amdahl’s law in mind)

fori=1 to2{ for k = 1 to 3 {

sum = 0.0; for j = 1 to 2

sum = sum + a[i,j] * b[j,k]; c[i,k] = sum;

} }

What is `p`, the proportion of this program that can be sped-up ?

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We need to consider a specific strategy…

e.g., parallelising just the `k` loop:

with k = 2; fori=1 to2{

for k = 1 to 3 { sum = 0.0; for j = 1 to 2

sum = sum + a[1,j] * b[j,2]; c[i,2] = sum;

} }

with k = 1; fori=1 to2{

for k = 1 to 3 { sum = 0.0; for j = 1 to 2

sum = sum + a[1,j] * b[j,1]; c[i,1] = sum;

} }

with k = 3; fori=1 to2{

for k = 1 to 3 { sum = 0.0; for j = 1 to 2

sum = sum + a[1,j] * b[j,3]; c[i,3] = sum;

} }

What is `p`, the proportion of this program that can be sped-up ?

How do we handle updating `sum`? Make it a critical resource? Local copies?

6

More specific…

parallelising just the `k` loop, with local sums

with k = 1; fori=1 to2{

for k = 1 to 3 { sum[1] = 0.0; for j = 1 to 2

sum[1] = sum[1] + a[1,j] * b[j,1]; c[i,1] = sum[1];

} }

with k = 2; fori=1 to2{

for k = 1 to 3 { sum[2] = 0.0; for j = 1 to 2

sum[2] = sum[2] + a[1,j] * b[j,2]; c[i,2] = sum[2];

} }

with k = 3; fori=1 to2{

for k = 1 to 3 { sum[3] = 0.0; for j = 1 to 2

sum[3] = sum[3] + a[1,j] * b[j,3]; c[i,3] = sum[3];

} }

What is `p`, the proportion of this program that can be sped-up ? Each `sum[k]` is only used locally!

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Try Six Execution Agents again

with k = 1, i=1; fori=1 to2{

for k = 1 to 3 { sum[1,1] = 0.0; for j = 1 to 2

sum[1,1] = sum[1,1] + a[1,j] * b[j,1]; c[1,1] = sum[1,1];

} }

with k = 3, i=1; fori=1 to2{

for k = 1 to 3 { sum[1,3] = 0.0; for j = 1 to 2

sum[1,3] = sum[1,3] + a[1,j] * b[j,3]; c[1,3] = sum[1,3];

} }

…

`sum` is redundant now, and we can simply use`c` itself to compute the result.

with k = 1, i=2; fori=1 to2{

for k = 1 to 3 { sum[2,1] = 0.0; for j = 1 to 2

sum[2,1] = sum[2,1] + a[2,j] * b[j,1]; c[2,1] = sum[2,1];

} }

with k = 3, i=2; fori=1 to2{

for k = 1 to 3 { sum[2,3] = 0.0; for j = 1 to 2

sum[2,3] = sum[2,3] + a[2,j] * b[j,3]; c[2,3] = sum[2,3];

} }

…

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Try Six Execution Agents again

with k = 1, i=1; fori=1 to2{

for k = 1 to 3 { c[1,1] = 0.0; for j = 1 to 2

c[1,1] = c[1,1] + a[1,j] * b[j,1]; }

}

with k = 3, i=1; fori=1 to2{

for k = 1 to 3 { c[1,3] = 0.0; for j = 1 to 2

c[1,3] = c[1,3] + a[1,j] * b[j,3]; }

}

…

with k = 1, i=2; fori=1 to2{

for k = 1 to 3 { c[2,1] = 0.0; for j = 1 to 2

c[2,1] = c[2,1] + a[2,j] * b[j,1]; }

}

with k = 3, i=2; fori=1 to2{

for k = 1 to 3 { c[2,3] = 0.0; for j = 1 to 2

c[2,3] = c[2,3] + a[2,j] * b[j,3]; }

}

…

Here we see we have full output independence between each component of `c`.

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Matrix Multiply is easy (?)

• Matrix multiply turns out to be highly (“embarrasingly”, “massively”) parallelisable!

• In principle we should be able to get `p` very close to 1.

•

• However, for a M x M (square) matrix, the equivalent of the “Six Execution Agents” solution requires M2 processors!

• Also, there are always hidden “non-p” costs in practice

• •

•

Indeed many supercomputing facilities use massive parallelism to do matrix calculations

Getting fast access by N cores to main memory gets harder to achieve as N grows large

While each location in matrix C is only written by one core, each location in both A and B are read by many. Again, this can lead to “bus contention” as processors wait to read shared memory.

(Clever understanding of cache behaviour can lead to algorithms that minimise such contention)

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Dining Philosophers Problem

fork 3

phil3

phil4

phil2

Spaghetti

fork 2

Philosophers want to repeatedly

think and then eat.

Each needs two forks. Infinite supply of pasta (ugh!). How to ensure they don’t starve.

phil1

•

fork 4

fork 1

A model problem for thinking about problems in Concurrent Programming: Deadlock

Livelock Starvation/Fairness

Data Corruption

phil0

fork 0

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Dining Philosophers

•Features:

•A philosopher eats only if they have two forks.

•No two philosophers may hold the same fork simultaneously

•Characteristics of Desired Solution: •Freedom from deadlock

•Freedom from starvation

•Efficient behaviour generally.

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Modelling the Dining Philosphers

• We imagine the philosophers participate in the following observable events:

Event Shorthand

Description

think.p

Philosopher p is thinking.

eat.p

Philosopher p is eating

pick.p.f

Philosopher p has picked up fork f.

drop.p.f

Philosopher p has dropped fork f.

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What a philosopher does:

• A philosopher wants to: think ; eat ; think ; eat; think ; eat ; ….

• In fact each philosopher needs to do: think ; pick forks ; eat ; drop forks ; … • We can describe the behaviour of the ith philosopher as:

Phil(i) = think.i ; pick.i.i ; pick.i.i+ ; eat.i ; drop.i.i ; drop.i.i+ ; Phil(i)

•

Phil(0) || Phil(1) || Phil(2) || Phil(3) || Phil(4)

Here i+ is shorthand for (i+1) mod 5.

• What we have are five philosophers running in parallel (||):

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What can (possibly) go wrong ?

• Consider the following (possible, but maybe unlikely) sequence of events, assuming that, just before this point, all philosophers are think.p-ing…

pick.0.0 ; pick.1.1 ; pick.2.2 ; pick.3.3 ; pick 4.4 ;

• At this point, every philosopher has picked up their left fork.

• •

•

•

• DEADLOCK !

Now each of them wants to pick up its right one.

But its right fork is its righthand neighbours left-hand fork!

Every philosopher wants to pick.i.i+ , but can’t, because it has already been pick.i+.i+-ed! Everyone is stuck and no further progress can be made

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“Implementing” pick and drop

• In effect pick.p.f attempts to lock a mutex protecting fork f.

• So each philosopher is trying to lock two mutexes for two forks before they can eat.p. • The drop.p.f simply unlocks the mutex protecting f.

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You can’t always rely on the scheduler…

• A possible sequence we might observe, starting from when philosophers 1 and 2 are thinking, could be:

pick.1.1 ; pick.2.2 ; pick.2.3 ; eat.2 ; drop.2.2

• •

•

• STARVATION (and its close friend UN-FAIRNESS)

now, philosopher 1 has picked fork 1 but is waiting for it to be dropped by philosopher 2.

But philosopher 2 is still running, and so drops the other fork, has a quick think, and then gets quickly back to eating once more:

pick.1.1 ; pick.2.2 ; pick.2.3 ; eat.2 ; drop.2.2 ; drop.2.3 ; think.2 ; pick.2.2 ; …

Philosopher 1 could get really unlucky and never be scheduled to get the lock for fork 2. It is queuing on the mutex for fork 2, but when philosopher 2 unlocks it, somehow the lock, and control is not handed to philosopher 1.

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Communication

•A concurrent or parallel program consists of two or more separate threads of execution, that run independently except when they have to interact

•To interact, they must communicate

•Communication is typically implemented by • sharing memory

• One thread writes data to memory; the other reads it •passing messages

• One thread sends a message; the other gets it

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The Challenge of Concurrency

•Conventional testing and debugging is not generally useful.

•We assume that the sequential parts of concurrent programs are correctly developed.

•Beyond normal sequential-type bugs, there is a whole range of problems caused by errors in communication and synchronisation.They can not easily be reproduced and corrected.

•So, we are going to use notions, algorithms and formalisms to help us design concurrent programs that are correct by design.

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Sequential Process

•A sequential process is the execution of a sequence of atomic statements. •E.g. Process P could consist of the execution of P1,P2,P3,….

•Process Q could be Q1,Q2,Q3,….

•We think of each sequential process as being a distinct entity that has its own separate program counter (PC).

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Concurrent Execution

•A concurrent system is modelled as a collection of sequential processes, where the atomic statements of the sequential processes can be arbitrarily interleaved, but respecting the sequentiality of the atomic statements within their sequential processes.

•E.g. say P is P1,P2 and Q is Q1,Q2.

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Scenarios for P and Q.

p1→q1→p2→q2

p1→q1→q2→p2

p1→p2→q1→q2

q1→p1→q2→p2

q1→p1→p2→q2

q1→q2→p1→p2

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Notation

Trivial concurrent program: processes p and q each do one action that updates global n with the value of their local variable.

Trivial Concurrent Program (Title)

integer n ←0 (Globals)

p (Process Name)

q

integer k1 ← 1 (Locals)

p1: n ← k1 (Atomic Statements)

integer k2 ← 2 q1: n ← k2

All global and local variables are initialised when declared.

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Simple Sequential Program

Trivial Sequential Program

integer n ←0

integer k1 ← 1

integer k2 ← 2

p1: n ← k1

p2: n ← k2

p1:n ← k1 k1 = 1, k2 = 2 n=0

p2: n ← k2 (end)

k1 = 1, k2 = 2 k1 = 1, k2 = 2

n=1 n=2

First line gives next atomic action to be executed

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State

Transition Diagrams

p1:n ← k1 k1 = 1, k2 = 2 n=0

p2: n ← k2 k1 = 1, k2 = 2

(end)

k1 = 1, k2 = 2

n=1 n=2

k1 = 1, k2 = 2 n=0

p1 k1 = 1, k2 = 2 p2 (k1 = 1, k2 = 2 n=1 n=2

Transition

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Simple Concurrent Program (1)

Trivial concurrent program: processes p and q each do one action that updates global n with the value of their local variable.

Trivial Concurrent Program

integer n ←0 (Globals)

p

q

integer k1 ← 1 (Locals)

p1: n ← k1 (Atomic Statements)

integer k2 ← 2 q1: n ← k2

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Simple Concurrent Program (2)

p1:n ← k1

q1:n ← k2 k1 = 1, k2 = 2 n=0

(end) q1: n ← k2

p1: n ← k1 (end)

k1 = 1, k2 = 2 n=2

(end)

(end)

k1 = 1, k2 = 2 n=1

q1, p1

k1 = 1, k2 n=1

(end)

(end) k1 = 1, k2 n=2

= 2

p1, q1

= 2

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State Diagrams and Scenarios

•We could describe all possible ways a program can execute with a state diagram. •There is a transition between s1 and s2 (“s1:s2”) if executing a statement in s1 changes the state to s2. •A state diagram is generated inductively from the starting state.

• If ∃ s1 and a transition s1:s2, then ∃ s2 and a directed arc from s1:s2

•Two states are identical if they have the same variable values and the same directed arcs

leaving them.

•A scenario is one path through the state diagram.

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Example — Jumping Frogs

•A frog can move to an adjacent stone if it’s vacant.

•A frog can hop over an adjacent stone to the next one if that one is vacant. •No other moves are possible.

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Can we interchange the positions of the frogs?

to

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If the frogs can only move “forwards”, can we:

move from above to below?

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Solution Graph

•So, we have a finite state-transition diagram of a finite state machine (FSM) as a complete description of the behaviour of the four frogs, operating concurrently, no matter what they do according to the rules.

•By examining the FSM, we can state properties as definitely holding, i.e. we can prove properties of the system being modelled.

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Solution Graph

•The solution graph makes it clear that this concurrent system—of four frogs that share certain resources—can experience deadlock.

•Deadlock occurs when the system arrives in a state from which it can not make any transitions (and which is not a desired end-state.)

•Livelock (not possible in this system) is when the system can traverse a sequence of states indefinitely without making progress towards a desired end state.

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Proof

•We can prove interesting properties by trying to construct a state diagram describing •each possible state of the system and

•each possible move from one state to another

•We might use a state diagram to show that •A property always holds

•A property never holds in any reachable state •A property sometimes holds

•A property always holds eventually

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State Diagrams for Processes

•A state is defined by a tuple of

•for each process, the label of the statement available for execution. •for each variable, its value.

•Q:What is the maximum number of possible states in such a state diagram?

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Abstraction of Concurrent Programming

•A concurrent program is a finite set of [sequential] processes.

•A process is written using a finite set of atomic statements.

•Concurrent program execution is modelled as proceeding by executing a sequence of the atomic statements obtained by arbitrarily interleaving the atomic statements of the processes.

•A computation [a.k.a. a scenario] is one particular execution sequence.

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Atomicity

•We assume that if two operations s1 and s2 really happen at the same time, it’s the same as if the two operations happened in either order.

•E.g. simultaneous writes to the same memory locations:

Sample

integer g ← 0;

p

q

p1:g ← 2;

q1: g ← 1

•

We assume that the result will be that g will be 2 or 1 after this program, not, for example, 3.

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Interleaving

•We model a scenario as an arbitrary interleaving of atomic statements, which is somewhat unrealistic.

•For a concurrent system, that’s OK, it happens anyway.

•For a parallel shared memory system, it’s OK so long as the previous idea of atomicity

holds at the lowest level.

•For a distributed system, it’s OK if you look at it from an individual node’s POV, because either it is executing one of its own statements, or it is sending or receiving a message.

•Thus any interleaving can be used, so long as a message is sent before it is received.

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Level of Atomicity

•The level of atomicity can affect the correctness of a program.

Example: Atomic Assignment Statements

integer n ← 0;

p

q

p1: n ← n+1;

q1: n ← n+1;

Value before atomic statement on same row

process p

process q

n

p1: n ← n+1;

q1: n ← n+1;

0

(end)

q1: n ← n+1;

1

(end)

(end)

2

process p

process q

n

p1: n ← n+1;

q1: n ← n+1;

0

p1: n ← n+1;

(end)

1

(end)

(end)

2

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Different Level of Atomicity

Example: Assignment Statements with one Global Reference

integer n ← 0;

p

q

integer temp

integer temp

p1: temp ← n

q1: temp ← n

p2: n ← temp + 1

q2: n ← temp + 1

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Alternative Scenarios

process p

process q

n

p.temp

q.temp

p1: temp ← n

q1: temp ← n

0

?

?

p2: n ← temp+1

q1: temp ← n

0

0

?

(end)

q1: temp ← n

1

?

(end)

q2: n ← temp+1

1

1

(end)

(end)

2

process p

process q

n

p.temp

q.temp

p1: temp ← n

q1: temp ← n

0

?

?

p2: n ← temp+1

q1: temp ← n

0

0

?

p2: n ← temp+1

q2: n ← temp+1

0

0

0

(end)

q2: n ← temp+1

1

0

(end)

(end)

1

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Atomicity & Correctness

•Thus, the level of atomicity specified affects the correctness of a program •We will typically assume that:

•assignment statements and •condition statement evaluations

•are atomic

•Note:this is not true for programs written in C using concurrency libraries likepthreadsor similar.

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Concurrent Counting Algorithm

Example: Concurrent Counting Algorithm

integer n ← 0;

p

q

integer temp

integer temp

p1: do 10 times

q1: do 10 times

p2:

temp ← n

q2:

temp ← n

p3:

n ← temp + 1

q3:

n ← temp + 1

•Increments a global variable n 20 times, thus n should be 20 after execution. •But, the program is faulty.

•Proof: construct a scenario where n is 2 afterwards. •Wouldn’t it be nice to get a program to do this analysis?

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