Question

Topic: Discrete Mathematics and its Applications" Chapter 9:Equivalence Relations and Partial Orderings.

9. hash function H : {0,1)" → {0,1)" maps a bit string of length n to a bit string of length k. Hash functions are used to give a short label to a long string. The set of all "collisions" with a given string s defines an equivalence class for a given hash function H, that is: (a) What is the average cardinality of the equivalence classes [s]H in terms of n and k, where n > k? (b) What is the probability for any two random bit strings from s,te { 0, 1 }n that

Answer 1

Answer to (a):

The number different binary string of length n = 2ⁿ and the number different binary string of length k = 2^k

So we need to map 2ⁿ strings to 2^k strings.

When n > k, then each of 2^k strings will be mapped on average by 2ⁿ/2^k strings out of 2ⁿ strings.

Therefore, the average cardinality of the equivalence class [s]_H = 2ⁿ/2^k = 2^n-k

Answer to (b):

The number of equivalence class = 2^k

So the probability of a random string $t \epsilon \{0,1\}^n$ to be a member of a particular equivalence class = 1/2^k = 2^-k

Therefore, the probability for any two random strings $s,t \epsilon \{0,1\}^n$ such that [s]_H = [t]_H= 2^-k

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