A frequently used hash family is the matrix multiplication hash family that we have introduced in class. Suppose we have n buckets {1, 2, ..., n}, we will use a binary string of length b = log₂ n to index each bucket. (For example, if we have 4 buckets, they will be indexed as 00, 01, 10, 11.) Now, we would like to hash a u-bit binary string x into the hash table (For example, x could be an 8-bit string 10100110). To hash this u-bit string x into a bucket, we use the hash function hд(x) = (Ax)mod 2, where A is a b x u dimensional binary matrix, and x is a u x 1 dimensional column vector. As a result, h(x) will be a b × 1 dimensional vector which shows the bucket index that x will be hashed to. (For example, x is an 8-bit string [10100110]™, and A is a [1 0 1 1 0 0 0 then h₁(x) ) = (Ax)mod 2 = 2 x 8 dimensional matrix [ 0 1 1 0 1 0 1 [1, 0]T, which means that x will be hashed to bucket 2.) Prove that the above hash function h₁(x) = (Ax)mod 2 is a universal hash family. [Hint: Consider hashing two arbitrary keys x and y into the hash table, both x and y are u-bit binary strings.]

A frequently used hash family is the matrix multiplication hash family that we have introduced in class. Suppose we have n buckets {1, 2, ..., n}, we will use a binary string of length b = log₂ n to index each bucket. (For example, if we have 4 buckets, they will be indexed as 00, 01, 10, 11.) Now, we would like to hash a u-bit binary string x into the hash table (For example, x could be an 8-bit string 10100110). To hash this u-bit string x into a bucket, we use the hash function hд(x) = (Ax)mod 2, where A is a b x u dimensional binary matrix, and x is a u x 1 dimensional column vector. As a result, h(x) will be a b × 1 dimensional vector which shows the bucket index that x will be hashed to. (For example, x is an 8-bit string [10100110]™, and A is a [1 0 1 1 0 0 0 then h₁(x) ) = (Ax)mod 2 = 2 x 8 dimensional matrix [ 0 1 1 0 1 0 1 [1, 0]T, which means that x will be hashed to bucket 2.) Prove that the above hash function h₁(x) = (Ax)mod 2 is a universal hash family. [Hint: Consider hashing two arbitrary keys x and y into the hash table, both x and y are u-bit binary strings.]

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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