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In this problem, I was given the follow mapping

 U = {0, 1, 2, 3, 4, 5, 6, 7} to {0, 1}

From this, there is an explicit universal hashing function that must be derived, with the hint that this can be done with a set of 4 functions. Unfortunately, despite searching through articles on how to do this, I am still confused. Any help in understanding how to find this hashing function and moving towards the right direction is greatly appreciated!

EDIT:

After some deliberation, this is what I came up with; would this be correct?

     0  1  2  3  4  5  6  7
---------------------------
h1 | 1  1  0  0  0  0  0  0 
h2 | 0  0  1  1  0  0  0  0
h3 | 0  0  0  0  1  1  0  0
h4 | 0  0  0  0  0  0  1  1
Joe Cur
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1 Answers1

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Using the definition from Wikipedia:

A family of functions H = {h: U → [m]} is called a universal family if, ∀ x,y ∈ U, x ≠ y : Prh∈H[h(x) = h(y)] ≤ 1/m.

In your case, this means that for any two values x and y in the set {0, 1, 2, 3, 4, 5, 6, 7}, at most two of your four hash functions can map them to the same bit.

Your suggestion:

     0  1  2  3  4  5  6  7
---------------------------
h1 | 1  1  0  0  0  0  0  0 
h2 | 0  0  1  1  0  0  0  0
h3 | 0  0  0  0  1  1  0  0
h4 | 0  0  0  0  0  0  1  1

does not work, because there are four pairs (x, y) — namely (0,1), (2,3), (4,5), and (6,7) — where all four hash functions map them to the same bit.

Instead, here are some options that do work:

     0  1  2  3  4  5  6  7
---------------------------
h1 | 0  0  0  0  1  1  1  1
h2 | 0  0  1  1  0  0  1  1
h3 | 0  1  0  1  0  1  0  1
h4 | 0  1  1  0  1  0  0  1

     0  1  2  3  4  5  6  7
---------------------------
h1 | 0  0  0  1  0  1  1  1
h2 | 0  0  1  0  1  0  1  1
h3 | 0  1  0  0  1  1  0  1
h4 | 1  0  0  0  1  1  1  0
ruakh
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  • Thank you for your help! If I may ask a follow up question: how are the pairs (x, y) decided and how do you determine which space is considered a bit? – Joe Cur Oct 16 '18 at 05:14
  • @JoeCur: Re: "how are the pairs (x, y) decided?": All possible pairs. There are 28 such pairs: (0,1) through (0,7), (1,2) through (1,7), (2,3) through (2,7), etc. Re: "how do you determine which space is considered a bit?": I mean "bit" in the computing sense: a binary digit, either 0 or 1. (Usually the output of a hash function is more than just a single bit, but in your case it's not.) – ruakh Oct 16 '18 at 05:37