Chernoff Bounds Flashcards
What are Chernoff bounds?
Chernoff bounds are probabilistic tools that provide bounds on the probability of large deviations of a random variable from its expectation.
What is the Chernoff bound for X deviating above its expectation?
For ( \mu \geq E[X] )
What is the Chernoff bound for X deviating below its expectation?
For ( \mu \leq E[X] )
What is the intuition behind Chernoff bounds?
Chernoff bounds leverage independence to show that the fluctuations of sums of independent random variables are likely to “cancel out.”
What is the probability of exceeding a load in random allocation?
Using Chernoff bounds
What happens to job loads with ( n ) processors and ( n ) jobs?
With high probability
How do loads balance when jobs are ( 16n \ln n )?
With high probability, all processors have loads between half and twice the average.
What is the relationship between Chernoff bounds and hashing?
Chernoff bounds help analyze hash table collisions by showing that the load on any hash bucket is unlikely to deviate significantly from the average.
What does ( X = X_1 + X_2 + … + X_n ) represent in Chernoff bounds?
It represents the sum of independent 0-1 random variables
How does independence affect Chernoff bounds?
Independence ensures that the deviations of individual random variables are uncorrelated
What is the importance of ( \mu ) in Chernoff bounds?
( \mu ) is the expectation of the random variable ( X )
How are Chernoff bounds used in load balancing?
They estimate the probability that a processor’s load significantly deviates from the average in a distributed system.
Why are Chernoff bounds asymmetric for upper and lower bounds?
For the upper bound
What is the role of logarithmic growth in load balancing?
Logarithmic growth
What is the Union Bound and how is it used with Chernoff bounds?
The Union Bound allows combining individual bounds across multiple processors, ensuring high-probability results for all processors collectively.
What happens if ( m ) (jobs) is much larger than ( n ) (processors)?
As ( m ) increases, processor loads become more evenly distributed, with deviations from the mean bounded by constants.
What does ( \Pr[X > (1+\delta)\mu] ) signify in Chernoff bounds?
It represents the probability that the random variable ( X ) exceeds ( (1+\delta) ) times its expectation.
What are the applications of Chernoff bounds in computer science?
Chernoff bounds are used in distributed systems, hashing, randomized algorithms, and load balancing to analyze probabilistic behavior.
What is the load deviation bound for ( O(n \log n) ) jobs?
With high probability, all processors have loads between ( 0.5\mu ) and ( 2\mu ), where ( \mu ) is the average load.
How does the exponential function relate to Chernoff bounds?
The exponential function ( e^x ) is used to derive tight probability bounds for deviations using moment generating functions.
