robust network supercomputing with malicious processes (reliably executing tasks upon estimating the...
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Robust Network Supercomputing with Malicious Processes
(Reliably Executing Tasks Upon Estimating the Number of Malicious
Processes)Kishori M. Konwar*
Sanguthevar RajasekaranAlexander A. Shvartsman
*Computer Science & Engineering DepartmentUniversity of Connecticut
Storrs, CT
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Motivation Internet supercomputing is increasingly becoming a powerful tool for harnessing massive amounts of computational resources
availability of high bandwidth Internet connections there is an enormous number of processes around the world comes at a cost substantially lower than acquiring a supercomputer or building a cluster of powerful machines
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Master
Worker
Workers
Worker
Worker
Worker
Worker
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Master
Worker
Workers
Worker Worker
Worker
Worker
TASKS
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Master
Worker
Workers
Worker Worker
Worker
Worker
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PrimeNet Server PrimeNet Server is a distributed, massively parallel scientific
computing Internet Supercomputer
Supported by Entropia.com and ranks among the most powerful computers in the world
A project comprised of about 30,000 PCs and laptops
Currently sustains a 22,296 billion floating point operations per second (gigaflops) (operations that involve fractional
numbers )
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SETI@home
SETI@home project a massive distributed cooperative computer
Used for analysis of gigabytes of data for Search for Extraterrestrial Intelligence (SETI)
Comprises of millions of voluntary machines around
SETI@home project reported its speed to be more than 57,290 billion floating point operations per second
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Reliability Issues The master and perhaps certain workers are reliable
they will correctly execute the tasks assigned by the server
However, workers are commonly unreliable they may return to the master incorrect results due to
unintended failures caused, e.g., by over-clocked processors
may deceivingly claim to have performed assigned work so as to obtain incentive such as getting higher rank
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Master
Worker
Workers
Worker Worker
Worker
Worker
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Some Previous Studies [FGLS05] Assumed the worker processes
might act maliciously and hence deliberately return wrong results. goal is to design algorithm that enable the
master to accept correct results with high probability at a lower cost
they provided a randomized algorithm unfortunately the cost complexity results
depend on several parameters and hard to interpret
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Some Previous Studies (cont’d)
[GM05] considered the problem of maximizing the expected number of correct result the tasks are dependent any worker computes correctly with probability p < 1
any incorrectly computed task corrupts all dependent tasks
the goal is to compute a schedule that maximizes expected number of correct results under a given time constraint
they showed the optimization problem to be NP-hard provided some solutions on a restricted DAG
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Overview
Models of Computation Stopping Rule Algorithm based solution Detection of Faulty Processors Performing Tasks with Faulty Workers Conclusions
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Overview
Models of Computation Stopping Rule Algorithm based solution Detection of Faulty Processors Performing Tasks with Faulty Workers Conclusions
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Models of Computation Processes takes steps in lock steps, i.e., in synchrony
Processes communicate by exchanging messages
The tasks are independent and idempotent
Processes are subject to failures and can return incorrect results maliciously
Workers, P = {1,2, . . ., n} and a master M
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Work Complexities
[CDS01] defined as work complexity or available processor steps
All steps taken by processes during execution of the algorithm are counted including the steps of the idling and waiting non-faulty processes
work [DHW92] define work as the number of performed tasks
counting multiplicities Approach does not charge for idling and waiting this is
called task oriented work
work
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Few Comments
work
We say that an even E occurs with high probability (w.h.p.) to mean that Pr[E] = 1 – O(n -) for some constant > 0.
work
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Modeling Failures
Failure model Fa
f-fraction, 0 < f < ½ of the n workers may fail
Each possibly faulty worker independently exhibits faulty behavior with probability
0 < p < ½. The master has no a priori knowledge of f and p.
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Modeling Failures (cont’d)
Failure model Fb
There is a fixed bound on the f-fraction, 0 < f < ½ of the n workers that can be faulty
Any worker from the remaining (1-f)-fraction of the workers fails with probability 0 < p <1/2 independently of other workers
The master knows the values of f and p.
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Algorithmic Template procedure for master process M, task T
Choose a set S P
Send task T to each processor p S
Wait for the results from the processes in S
Decide on the result value v from the responses
procedure for worker w P
Wait to receive a task from master M
Upon receiving a task from M
Execute the task
Send the result to M
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Overview Models of Computation Stopping Rule Algorithm based solution Detection of Faulty Processors Performing Tasks with Faulty Workers Conclusions
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(, )-approximation algorithm
Z is a random variable distributed in the interval [0,1] with mean Z
Z1, Z2, Z3 .... are independently and identically distributed according to the random variable Z
An (, )-approximation algorithm, with 0 < < 1,
> 0 for estimating Z satisfies
Pr[Z (1- ) Z (1+ ) ] > 1 -
where is the estimated value of Z
~
Z~Z~
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Stopping Rule Algorithm
[Dagum, Karp, Luby, and Ross 1995]
Input Parameters (, ) with 0 < < 1, > 0
Let 1 = 1 + (1+ ) // = 0.72 & = 4 log(2/ )/2
Initialize N 0 , S 0
While S < 1 do: N N+1, S S + ZN
Output: Z 1 /N~
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Stopping Rule TheoremTheorem (Stopping Rule Theorem) [Dagum, Karp, Luby, and Ross]
Let Z be a random variable in [0,1] with Z = E[Z] > 0. Let
be the estimate produced and let NZ be the number of experiments that SRA runs with respect to Z on input and . Then,
(i) Pr[Z (1- ) Z (1+ ) ] > 1 -
(ii) E[NZ ] 1 /Z and
(iii) Pr[NZ >(1+ ) 1 /Z ] /2
Z~
Z~
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Algorithm Af,p to estimate f and p
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Work Complexity of Af,p
Theorem: Algorithm Af,p is an (, )-approximation algorithm,
0 < < 1, > 0, for the estimation of f and p with work
complexity O(log2n), complexity O(n log n), message
complexity O(log2 n) and time complexity O(log n), with high
probability.
work
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Overview
Models of Computation Stopping Rule Algorithm based solution Detection of Faulty Processors Performing Tasks with Faulty Workers Conclusions
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Detection of Faulty Processors
Lemma: It is not possible to perform all the n
tasks correctly, in the failure model Fa with linear
complexity (i.e., O(n)) with high probability.work
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Detection of Faulty Processors procedure for master process M
Initially, F For t = 0, …. k log n, k > 0
Choose a set S P \ F
Send each process p S “test” task
Wait for the results from the processes in S
If the response is faulty
F F {p: p is a faulty process}
End If
End For
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Detection of Faulty Processors Lemma: The algorithm detects all faulty processes among
the n workers in O(log n) time with O(n) work with high
probability
Theorem[Karp 04]: Suppose that a(x) is a non-decreasing,
continuous function that is strictly increasing on {x | a(x) >0},
and m(x) is a continuous function. Then for every positive real x
and every positive integer t,
Pr[T(x) > u(x) + ta(x)] (m(x)/x)t
where u(x) is the solution to the equation u(x)=a(x) + u(m(x))
with m0(x) :=0 and mi+1(x):= m(mi(x)).
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Overview
Models of Computation Stopping Rule Algorithm based solution Detection of Faulty Processors Performing Tasks with Faulty Workers Conclusions
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Performing Tasks under Faprocedure for master process M:
Initially, C , J set of n tasks Randomly choose a set, possibly with repetition, SP, |S|=kn/log n
workers k>0 is a constant For i = 1, …, k' log n, k' > 0 Send to each worker pS a “test” task Collect the responses from all the workers. End For If all the responses from a worker pS are correct then C C {p} End if For i=1, …, n/|C| Send |C| jobs from J, not sent in previous iteration, one to each
worker in C. Collect the responses from the C workers End For
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Work and Time Complexities
Theorem: The algorithm performs all n tasks correctly in
O(log n) time and has O(n) work and complexities,
with high probability.
work
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Overview
Models of Computation Stopping Rule Algorithm based solution Detection of Faulty Processors Performing Tasks with Faulty Workers Conclusions
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Performing Tasks under Fbprocedure for master process M,
For t = 0, …. k log n, k > 0
Choose a random permutation R Sn
Foreach j [n] Send task to processor (j) End For Collect the responses from all the workers End For Foreach j [n] Choose the majority of the results of computation for task as the result End For
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Work and Time Complexities
Theorem: The algorithm performs all n tasks correctly in
O(log n) time and has and work complexities O(n log n),
for 0 < p, f < ½ and (1- f)(1- p) > ½ with high probability
work
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Overview
Models of Computation Stopping Rule Algorithm based solution Detection of Faulty Processors Performing Tasks with Faulty Workers Conclusions
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Conclusions
Perform tasks under above models where the tasks are dependent The dependency graph can be DAG Quantify work and time complexities on some characteristics of the DAG