What is an example of a weakly universal hash function that is not pairwise independent?












3












$begingroup$


A family of hash functions $H_w$ is said to be weakly universal if for all $x ne y$ :



$$P_{h in H_w}(h(x) = h(y)) leq 1/m$$



Here the function $h:U rightarrow [m]$ is chosen uniformly from the family $H$ and we assume $|U| > m$.



A family of hash functions $H_s$ is said to be strongly universal if for all $x ne y$ and $k, ell in [m]$:



$$P_{h in H_s}(h(x) = k land h(y) = ell) = 1/m^2$$



What is a concrete example of a hash function family which is weakly universal but not strongly universal?






hash hash-tables probabilistic-algorithms







share|cite|improve this question











$endgroup$

















    3












    $begingroup$


    A family of hash functions $H_w$ is said to be weakly universal if for all $x ne y$ :



    $$P_{h in H_w}(h(x) = h(y)) leq 1/m$$



    Here the function $h:U rightarrow [m]$ is chosen uniformly from the family $H$ and we assume $|U| > m$.



    A family of hash functions $H_s$ is said to be strongly universal if for all $x ne y$ and $k, ell in [m]$:



    $$P_{h in H_s}(h(x) = k land h(y) = ell) = 1/m^2$$



    What is a concrete example of a hash function family which is weakly universal but not strongly universal?






    hash hash-tables probabilistic-algorithms







    share|cite|improve this question











    $endgroup$















      3












      3








      3





      $begingroup$


      A family of hash functions $H_w$ is said to be weakly universal if for all $x ne y$ :



      $$P_{h in H_w}(h(x) = h(y)) leq 1/m$$



      Here the function $h:U rightarrow [m]$ is chosen uniformly from the family $H$ and we assume $|U| > m$.



      A family of hash functions $H_s$ is said to be strongly universal if for all $x ne y$ and $k, ell in [m]$:



      $$P_{h in H_s}(h(x) = k land h(y) = ell) = 1/m^2$$



      What is a concrete example of a hash function family which is weakly universal but not strongly universal?






      hash hash-tables probabilistic-algorithms







      share|cite|improve this question











      $endgroup$




      A family of hash functions $H_w$ is said to be weakly universal if for all $x ne y$ :



      $$P_{h in H_w}(h(x) = h(y)) leq 1/m$$



      Here the function $h:U rightarrow [m]$ is chosen uniformly from the family $H$ and we assume $|U| > m$.



      A family of hash functions $H_s$ is said to be strongly universal if for all $x ne y$ and $k, ell in [m]$:



      $$P_{h in H_s}(h(x) = k land h(y) = ell) = 1/m^2$$



      What is a concrete example of a hash function family which is weakly universal but not strongly universal?






      hash hash-tables probabilistic-algorithms




      hash hash-tables probabilistic-algorithms






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Feb 14 at 17:07







      Anush

















      asked Feb 14 at 16:33









      AnushAnush

      1407




      1407






















          1 Answer
          1






          active

          oldest

          votes


















          4












          $begingroup$

          Let $U = [m]$, and let $h$ be the identity function.



          If you insist that $|U| > m$, then you can take $U = [m+1]$, and consider the functions $h_i$, for $i in [m]$, given by
          $$
          h_i(x) = begin{cases}
          x & text{if } x neq m+1, \
          i & text{if } x = m+1.
          end{cases}
          $$
          The same approach can be used for arbitrary $|U|$: fix the first $m$ coordinates, and make all other coordinates uniformly and independently random.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Oh sorry. I meant $|U|$ to be larger than $m$? Let me fix that.
            $endgroup$
            – Anush
            Feb 14 at 17:07






          • 2




            $begingroup$
            Is there anything else you forgot about the question? I don't like continuously changing my answer to fit an ever-changing question.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:12










          • $begingroup$
            No I don’t think so. Thank you for your very nice answer to the first version.
            $endgroup$
            – Anush
            Feb 14 at 17:13






          • 1




            $begingroup$
            Well, it makes a nice exercise.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:24






          • 1




            $begingroup$
            I don’t see any lookup tables in your question. Perhaps you need to spend more time formulating your question. When you have a concrete follow-up question, you can ask it separately.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 18:58











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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          4












          $begingroup$

          Let $U = [m]$, and let $h$ be the identity function.



          If you insist that $|U| > m$, then you can take $U = [m+1]$, and consider the functions $h_i$, for $i in [m]$, given by
          $$
          h_i(x) = begin{cases}
          x & text{if } x neq m+1, \
          i & text{if } x = m+1.
          end{cases}
          $$
          The same approach can be used for arbitrary $|U|$: fix the first $m$ coordinates, and make all other coordinates uniformly and independently random.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Oh sorry. I meant $|U|$ to be larger than $m$? Let me fix that.
            $endgroup$
            – Anush
            Feb 14 at 17:07






          • 2




            $begingroup$
            Is there anything else you forgot about the question? I don't like continuously changing my answer to fit an ever-changing question.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:12










          • $begingroup$
            No I don’t think so. Thank you for your very nice answer to the first version.
            $endgroup$
            – Anush
            Feb 14 at 17:13






          • 1




            $begingroup$
            Well, it makes a nice exercise.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:24






          • 1




            $begingroup$
            I don’t see any lookup tables in your question. Perhaps you need to spend more time formulating your question. When you have a concrete follow-up question, you can ask it separately.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 18:58
















          4












          $begingroup$

          Let $U = [m]$, and let $h$ be the identity function.



          If you insist that $|U| > m$, then you can take $U = [m+1]$, and consider the functions $h_i$, for $i in [m]$, given by
          $$
          h_i(x) = begin{cases}
          x & text{if } x neq m+1, \
          i & text{if } x = m+1.
          end{cases}
          $$
          The same approach can be used for arbitrary $|U|$: fix the first $m$ coordinates, and make all other coordinates uniformly and independently random.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Oh sorry. I meant $|U|$ to be larger than $m$? Let me fix that.
            $endgroup$
            – Anush
            Feb 14 at 17:07






          • 2




            $begingroup$
            Is there anything else you forgot about the question? I don't like continuously changing my answer to fit an ever-changing question.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:12










          • $begingroup$
            No I don’t think so. Thank you for your very nice answer to the first version.
            $endgroup$
            – Anush
            Feb 14 at 17:13






          • 1




            $begingroup$
            Well, it makes a nice exercise.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:24






          • 1




            $begingroup$
            I don’t see any lookup tables in your question. Perhaps you need to spend more time formulating your question. When you have a concrete follow-up question, you can ask it separately.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 18:58














          4












          4








          4





          $begingroup$

          Let $U = [m]$, and let $h$ be the identity function.



          If you insist that $|U| > m$, then you can take $U = [m+1]$, and consider the functions $h_i$, for $i in [m]$, given by
          $$
          h_i(x) = begin{cases}
          x & text{if } x neq m+1, \
          i & text{if } x = m+1.
          end{cases}
          $$
          The same approach can be used for arbitrary $|U|$: fix the first $m$ coordinates, and make all other coordinates uniformly and independently random.






          share|cite|improve this answer











          $endgroup$



          Let $U = [m]$, and let $h$ be the identity function.



          If you insist that $|U| > m$, then you can take $U = [m+1]$, and consider the functions $h_i$, for $i in [m]$, given by
          $$
          h_i(x) = begin{cases}
          x & text{if } x neq m+1, \
          i & text{if } x = m+1.
          end{cases}
          $$
          The same approach can be used for arbitrary $|U|$: fix the first $m$ coordinates, and make all other coordinates uniformly and independently random.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Feb 14 at 17:19

























          answered Feb 14 at 16:59









          Yuval FilmusYuval Filmus

          193k14181345




          193k14181345












          • $begingroup$
            Oh sorry. I meant $|U|$ to be larger than $m$? Let me fix that.
            $endgroup$
            – Anush
            Feb 14 at 17:07






          • 2




            $begingroup$
            Is there anything else you forgot about the question? I don't like continuously changing my answer to fit an ever-changing question.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:12










          • $begingroup$
            No I don’t think so. Thank you for your very nice answer to the first version.
            $endgroup$
            – Anush
            Feb 14 at 17:13






          • 1




            $begingroup$
            Well, it makes a nice exercise.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:24






          • 1




            $begingroup$
            I don’t see any lookup tables in your question. Perhaps you need to spend more time formulating your question. When you have a concrete follow-up question, you can ask it separately.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 18:58


















          • $begingroup$
            Oh sorry. I meant $|U|$ to be larger than $m$? Let me fix that.
            $endgroup$
            – Anush
            Feb 14 at 17:07






          • 2




            $begingroup$
            Is there anything else you forgot about the question? I don't like continuously changing my answer to fit an ever-changing question.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:12










          • $begingroup$
            No I don’t think so. Thank you for your very nice answer to the first version.
            $endgroup$
            – Anush
            Feb 14 at 17:13






          • 1




            $begingroup$
            Well, it makes a nice exercise.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 17:24






          • 1




            $begingroup$
            I don’t see any lookup tables in your question. Perhaps you need to spend more time formulating your question. When you have a concrete follow-up question, you can ask it separately.
            $endgroup$
            – Yuval Filmus
            Feb 14 at 18:58
















          $begingroup$
          Oh sorry. I meant $|U|$ to be larger than $m$? Let me fix that.
          $endgroup$
          – Anush
          Feb 14 at 17:07




          $begingroup$
          Oh sorry. I meant $|U|$ to be larger than $m$? Let me fix that.
          $endgroup$
          – Anush
          Feb 14 at 17:07




          2




          2




          $begingroup$
          Is there anything else you forgot about the question? I don't like continuously changing my answer to fit an ever-changing question.
          $endgroup$
          – Yuval Filmus
          Feb 14 at 17:12




          $begingroup$
          Is there anything else you forgot about the question? I don't like continuously changing my answer to fit an ever-changing question.
          $endgroup$
          – Yuval Filmus
          Feb 14 at 17:12












          $begingroup$
          No I don’t think so. Thank you for your very nice answer to the first version.
          $endgroup$
          – Anush
          Feb 14 at 17:13




          $begingroup$
          No I don’t think so. Thank you for your very nice answer to the first version.
          $endgroup$
          – Anush
          Feb 14 at 17:13




          1




          1




          $begingroup$
          Well, it makes a nice exercise.
          $endgroup$
          – Yuval Filmus
          Feb 14 at 17:24




          $begingroup$
          Well, it makes a nice exercise.
          $endgroup$
          – Yuval Filmus
          Feb 14 at 17:24




          1




          1




          $begingroup$
          I don’t see any lookup tables in your question. Perhaps you need to spend more time formulating your question. When you have a concrete follow-up question, you can ask it separately.
          $endgroup$
          – Yuval Filmus
          Feb 14 at 18:58




          $begingroup$
          I don’t see any lookup tables in your question. Perhaps you need to spend more time formulating your question. When you have a concrete follow-up question, you can ask it separately.
          $endgroup$
          – Yuval Filmus
          Feb 14 at 18:58


















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