precept4 - Computer Science 340 Reasoning about Computation...

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Computer Science 340 Reasoning about Computation Precept 4 Problem 1 Consider the family of hash functions discussed in class h a,b ( x ) = ( ax + b mod p )( mod n ) where a, b ∈ { 0 , . . . p - 1 } and a = 0. Consider a hash function h a,b drawn uni- formly and at random from this family. For any x 1 , x 2 ∈ { 0 , . . . , p - 1 } , x 1 = x 2 , show that Pr[ h a,b ( x 1 ) = h a,b ( x 2 )] 1 /n. Problem 2 Consider a variant of the family of hash functions discussed in class h a ( x ) = ( ax mod p )( mod n ) where a ∈ { 1 , . . . p - 1 } . Consider a hash function h a drawn uniformly and at random from this family. For any x 1 , x 2 ∈ { 0 , . . . , p - 1 } , x 1 = x 2 , show that Pr[ h a ( x 1 ) = h a ( x 2 )] 2 /n. Problem 3 Consider the algorithm outlined in class to estimate the number of distinct queries in a query log using a random hash function h : U [0 , 1]. Here U is the universe of all queries. Assume that we can use a truly random hash function. Recall that the estimator used by the algorithm applies the hash function to every query in the query log and reports the minimum value. Let Y be the minimum of h ( x
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