How many repetitions does the loop












1












$begingroup$


Here is the following algorithm:



for(k=2; k<n; k=k^k)


I understand that I need to check when
$n=k^k$.



But I'm stuck on $k=log_k(n)$



How many repetitions are performed by the loop?










share|cite|improve this question











$endgroup$












  • $begingroup$
    Have you computed what $k$ is the first few passes through the loop? It grows very rapidly. I doubt there is a nice formula, but for any reasonable $n$ it will be no more than $3$
    $endgroup$
    – Ross Millikan
    Jan 8 at 19:44










  • $begingroup$
    I know it grow very rapidly, but it's depends on "n", i guess that's will be log(log(n)) but i still don't know how to show it.
    $endgroup$
    – shay
    Jan 8 at 20:35
















1












$begingroup$


Here is the following algorithm:



for(k=2; k<n; k=k^k)


I understand that I need to check when
$n=k^k$.



But I'm stuck on $k=log_k(n)$



How many repetitions are performed by the loop?










share|cite|improve this question











$endgroup$












  • $begingroup$
    Have you computed what $k$ is the first few passes through the loop? It grows very rapidly. I doubt there is a nice formula, but for any reasonable $n$ it will be no more than $3$
    $endgroup$
    – Ross Millikan
    Jan 8 at 19:44










  • $begingroup$
    I know it grow very rapidly, but it's depends on "n", i guess that's will be log(log(n)) but i still don't know how to show it.
    $endgroup$
    – shay
    Jan 8 at 20:35














1












1








1





$begingroup$


Here is the following algorithm:



for(k=2; k<n; k=k^k)


I understand that I need to check when
$n=k^k$.



But I'm stuck on $k=log_k(n)$



How many repetitions are performed by the loop?










share|cite|improve this question











$endgroup$




Here is the following algorithm:



for(k=2; k<n; k=k^k)


I understand that I need to check when
$n=k^k$.



But I'm stuck on $k=log_k(n)$



How many repetitions are performed by the loop?







calculus computational-complexity






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 8 at 21:04









the_candyman

8,80122045




8,80122045










asked Jan 8 at 19:25









shayshay

103




103












  • $begingroup$
    Have you computed what $k$ is the first few passes through the loop? It grows very rapidly. I doubt there is a nice formula, but for any reasonable $n$ it will be no more than $3$
    $endgroup$
    – Ross Millikan
    Jan 8 at 19:44










  • $begingroup$
    I know it grow very rapidly, but it's depends on "n", i guess that's will be log(log(n)) but i still don't know how to show it.
    $endgroup$
    – shay
    Jan 8 at 20:35


















  • $begingroup$
    Have you computed what $k$ is the first few passes through the loop? It grows very rapidly. I doubt there is a nice formula, but for any reasonable $n$ it will be no more than $3$
    $endgroup$
    – Ross Millikan
    Jan 8 at 19:44










  • $begingroup$
    I know it grow very rapidly, but it's depends on "n", i guess that's will be log(log(n)) but i still don't know how to show it.
    $endgroup$
    – shay
    Jan 8 at 20:35
















$begingroup$
Have you computed what $k$ is the first few passes through the loop? It grows very rapidly. I doubt there is a nice formula, but for any reasonable $n$ it will be no more than $3$
$endgroup$
– Ross Millikan
Jan 8 at 19:44




$begingroup$
Have you computed what $k$ is the first few passes through the loop? It grows very rapidly. I doubt there is a nice formula, but for any reasonable $n$ it will be no more than $3$
$endgroup$
– Ross Millikan
Jan 8 at 19:44












$begingroup$
I know it grow very rapidly, but it's depends on "n", i guess that's will be log(log(n)) but i still don't know how to show it.
$endgroup$
– shay
Jan 8 at 20:35




$begingroup$
I know it grow very rapidly, but it's depends on "n", i guess that's will be log(log(n)) but i still don't know how to show it.
$endgroup$
– shay
Jan 8 at 20:35










4 Answers
4






active

oldest

votes


















1












$begingroup$

After the first pass we have $k=2^2=4$ If $n$ is $3$ or $4$ there will be only one pass.

After the second pass we have $k=4^4=256$. If $4 lt n le 256$ there will be two passes.

After the third we have $k=256^{256}$, which is enormous-about $600$ digits. Only if $n$ is larger than this will there be a fourth pass.



I don't know of a nice function that can take $n$ and give back the number of passes.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    Notice that



    $$2^2=4,4^4=256,256^{256}approx3.2317cdot10^{616}.$$



    Needless to say, the next term is astronomical.



    So you take little risk by saying zero to three iterations. (If $n$ is a 32 bits integer uniformly drawn, say three and you'll be right with probability $0.999999940$)






    share|cite|improve this answer











    $endgroup$





















      0












      $begingroup$

      We start with $k=2$. Since it is the first, let's call it $k_0$. We observe that:



      $$k_0 = 2 = 2^1 = 2^{2^0}.$$



      The next element of the sequence is:



      $$k_1 = k_0^{k_0} = 2^2 = 2^{2^1}.$$



      By continuing...



      $$k_2 = k_3^{k_3} = (2^2)^{2^2} = 2^{2^3}.$$
      $$k_3 = k_4^{k_4} = (2^{2^3})^{2^{2^3}} = 2^{2^{11}}.$$
      $$k_4 = k_5^{k_5} = (2^{2^{11}})^{2^{2^{11}}} = 2^{2^{2059}}.$$



      Let's introduce
      $s_i = log_2 log_2 k_i$ (i.e. $k_i = 2^{2^{s_i}})$. In our case, $s_0 = 0$, $s_1 = 1$, $s_2 = 3$, $s_3=11$, $s_4 = 2059$, ...



      After a deep analysis of the showed calculation, it turns out that the $s$ sequence satisfies the following:



      $$s_{i+1} = s_i + 2^{s_i},$$



      with $s_0 = 0.$






      share|cite|improve this answer











      $endgroup$





















        0












        $begingroup$

        I looked for the number of repetitions, here is the answer I thought of:
        $$
        2^(2^k) = n
        $$

        $$
        log(2^(2^k)) = log(n)
        $$

        $$
        2^klog(2)=log(n)
        $$

        $$
        2^k=log(n)
        $$



        $$
        k=log(log(n))
        $$






        share|cite|improve this answer









        $endgroup$













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          4 Answers
          4






          active

          oldest

          votes








          4 Answers
          4






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          After the first pass we have $k=2^2=4$ If $n$ is $3$ or $4$ there will be only one pass.

          After the second pass we have $k=4^4=256$. If $4 lt n le 256$ there will be two passes.

          After the third we have $k=256^{256}$, which is enormous-about $600$ digits. Only if $n$ is larger than this will there be a fourth pass.



          I don't know of a nice function that can take $n$ and give back the number of passes.






          share|cite|improve this answer









          $endgroup$


















            1












            $begingroup$

            After the first pass we have $k=2^2=4$ If $n$ is $3$ or $4$ there will be only one pass.

            After the second pass we have $k=4^4=256$. If $4 lt n le 256$ there will be two passes.

            After the third we have $k=256^{256}$, which is enormous-about $600$ digits. Only if $n$ is larger than this will there be a fourth pass.



            I don't know of a nice function that can take $n$ and give back the number of passes.






            share|cite|improve this answer









            $endgroup$
















              1












              1








              1





              $begingroup$

              After the first pass we have $k=2^2=4$ If $n$ is $3$ or $4$ there will be only one pass.

              After the second pass we have $k=4^4=256$. If $4 lt n le 256$ there will be two passes.

              After the third we have $k=256^{256}$, which is enormous-about $600$ digits. Only if $n$ is larger than this will there be a fourth pass.



              I don't know of a nice function that can take $n$ and give back the number of passes.






              share|cite|improve this answer









              $endgroup$



              After the first pass we have $k=2^2=4$ If $n$ is $3$ or $4$ there will be only one pass.

              After the second pass we have $k=4^4=256$. If $4 lt n le 256$ there will be two passes.

              After the third we have $k=256^{256}$, which is enormous-about $600$ digits. Only if $n$ is larger than this will there be a fourth pass.



              I don't know of a nice function that can take $n$ and give back the number of passes.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Jan 8 at 21:02









              Ross MillikanRoss Millikan

              293k23197371




              293k23197371























                  1












                  $begingroup$

                  Notice that



                  $$2^2=4,4^4=256,256^{256}approx3.2317cdot10^{616}.$$



                  Needless to say, the next term is astronomical.



                  So you take little risk by saying zero to three iterations. (If $n$ is a 32 bits integer uniformly drawn, say three and you'll be right with probability $0.999999940$)






                  share|cite|improve this answer











                  $endgroup$


















                    1












                    $begingroup$

                    Notice that



                    $$2^2=4,4^4=256,256^{256}approx3.2317cdot10^{616}.$$



                    Needless to say, the next term is astronomical.



                    So you take little risk by saying zero to three iterations. (If $n$ is a 32 bits integer uniformly drawn, say three and you'll be right with probability $0.999999940$)






                    share|cite|improve this answer











                    $endgroup$
















                      1












                      1








                      1





                      $begingroup$

                      Notice that



                      $$2^2=4,4^4=256,256^{256}approx3.2317cdot10^{616}.$$



                      Needless to say, the next term is astronomical.



                      So you take little risk by saying zero to three iterations. (If $n$ is a 32 bits integer uniformly drawn, say three and you'll be right with probability $0.999999940$)






                      share|cite|improve this answer











                      $endgroup$



                      Notice that



                      $$2^2=4,4^4=256,256^{256}approx3.2317cdot10^{616}.$$



                      Needless to say, the next term is astronomical.



                      So you take little risk by saying zero to three iterations. (If $n$ is a 32 bits integer uniformly drawn, say three and you'll be right with probability $0.999999940$)







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited Jan 8 at 22:02

























                      answered Jan 8 at 21:57









                      Yves DaoustYves Daoust

                      125k671222




                      125k671222























                          0












                          $begingroup$

                          We start with $k=2$. Since it is the first, let's call it $k_0$. We observe that:



                          $$k_0 = 2 = 2^1 = 2^{2^0}.$$



                          The next element of the sequence is:



                          $$k_1 = k_0^{k_0} = 2^2 = 2^{2^1}.$$



                          By continuing...



                          $$k_2 = k_3^{k_3} = (2^2)^{2^2} = 2^{2^3}.$$
                          $$k_3 = k_4^{k_4} = (2^{2^3})^{2^{2^3}} = 2^{2^{11}}.$$
                          $$k_4 = k_5^{k_5} = (2^{2^{11}})^{2^{2^{11}}} = 2^{2^{2059}}.$$



                          Let's introduce
                          $s_i = log_2 log_2 k_i$ (i.e. $k_i = 2^{2^{s_i}})$. In our case, $s_0 = 0$, $s_1 = 1$, $s_2 = 3$, $s_3=11$, $s_4 = 2059$, ...



                          After a deep analysis of the showed calculation, it turns out that the $s$ sequence satisfies the following:



                          $$s_{i+1} = s_i + 2^{s_i},$$



                          with $s_0 = 0.$






                          share|cite|improve this answer











                          $endgroup$


















                            0












                            $begingroup$

                            We start with $k=2$. Since it is the first, let's call it $k_0$. We observe that:



                            $$k_0 = 2 = 2^1 = 2^{2^0}.$$



                            The next element of the sequence is:



                            $$k_1 = k_0^{k_0} = 2^2 = 2^{2^1}.$$



                            By continuing...



                            $$k_2 = k_3^{k_3} = (2^2)^{2^2} = 2^{2^3}.$$
                            $$k_3 = k_4^{k_4} = (2^{2^3})^{2^{2^3}} = 2^{2^{11}}.$$
                            $$k_4 = k_5^{k_5} = (2^{2^{11}})^{2^{2^{11}}} = 2^{2^{2059}}.$$



                            Let's introduce
                            $s_i = log_2 log_2 k_i$ (i.e. $k_i = 2^{2^{s_i}})$. In our case, $s_0 = 0$, $s_1 = 1$, $s_2 = 3$, $s_3=11$, $s_4 = 2059$, ...



                            After a deep analysis of the showed calculation, it turns out that the $s$ sequence satisfies the following:



                            $$s_{i+1} = s_i + 2^{s_i},$$



                            with $s_0 = 0.$






                            share|cite|improve this answer











                            $endgroup$
















                              0












                              0








                              0





                              $begingroup$

                              We start with $k=2$. Since it is the first, let's call it $k_0$. We observe that:



                              $$k_0 = 2 = 2^1 = 2^{2^0}.$$



                              The next element of the sequence is:



                              $$k_1 = k_0^{k_0} = 2^2 = 2^{2^1}.$$



                              By continuing...



                              $$k_2 = k_3^{k_3} = (2^2)^{2^2} = 2^{2^3}.$$
                              $$k_3 = k_4^{k_4} = (2^{2^3})^{2^{2^3}} = 2^{2^{11}}.$$
                              $$k_4 = k_5^{k_5} = (2^{2^{11}})^{2^{2^{11}}} = 2^{2^{2059}}.$$



                              Let's introduce
                              $s_i = log_2 log_2 k_i$ (i.e. $k_i = 2^{2^{s_i}})$. In our case, $s_0 = 0$, $s_1 = 1$, $s_2 = 3$, $s_3=11$, $s_4 = 2059$, ...



                              After a deep analysis of the showed calculation, it turns out that the $s$ sequence satisfies the following:



                              $$s_{i+1} = s_i + 2^{s_i},$$



                              with $s_0 = 0.$






                              share|cite|improve this answer











                              $endgroup$



                              We start with $k=2$. Since it is the first, let's call it $k_0$. We observe that:



                              $$k_0 = 2 = 2^1 = 2^{2^0}.$$



                              The next element of the sequence is:



                              $$k_1 = k_0^{k_0} = 2^2 = 2^{2^1}.$$



                              By continuing...



                              $$k_2 = k_3^{k_3} = (2^2)^{2^2} = 2^{2^3}.$$
                              $$k_3 = k_4^{k_4} = (2^{2^3})^{2^{2^3}} = 2^{2^{11}}.$$
                              $$k_4 = k_5^{k_5} = (2^{2^{11}})^{2^{2^{11}}} = 2^{2^{2059}}.$$



                              Let's introduce
                              $s_i = log_2 log_2 k_i$ (i.e. $k_i = 2^{2^{s_i}})$. In our case, $s_0 = 0$, $s_1 = 1$, $s_2 = 3$, $s_3=11$, $s_4 = 2059$, ...



                              After a deep analysis of the showed calculation, it turns out that the $s$ sequence satisfies the following:



                              $$s_{i+1} = s_i + 2^{s_i},$$



                              with $s_0 = 0.$







                              share|cite|improve this answer














                              share|cite|improve this answer



                              share|cite|improve this answer








                              edited Jan 8 at 21:47

























                              answered Jan 8 at 21:31









                              the_candymanthe_candyman

                              8,80122045




                              8,80122045























                                  0












                                  $begingroup$

                                  I looked for the number of repetitions, here is the answer I thought of:
                                  $$
                                  2^(2^k) = n
                                  $$

                                  $$
                                  log(2^(2^k)) = log(n)
                                  $$

                                  $$
                                  2^klog(2)=log(n)
                                  $$

                                  $$
                                  2^k=log(n)
                                  $$



                                  $$
                                  k=log(log(n))
                                  $$






                                  share|cite|improve this answer









                                  $endgroup$


















                                    0












                                    $begingroup$

                                    I looked for the number of repetitions, here is the answer I thought of:
                                    $$
                                    2^(2^k) = n
                                    $$

                                    $$
                                    log(2^(2^k)) = log(n)
                                    $$

                                    $$
                                    2^klog(2)=log(n)
                                    $$

                                    $$
                                    2^k=log(n)
                                    $$



                                    $$
                                    k=log(log(n))
                                    $$






                                    share|cite|improve this answer









                                    $endgroup$
















                                      0












                                      0








                                      0





                                      $begingroup$

                                      I looked for the number of repetitions, here is the answer I thought of:
                                      $$
                                      2^(2^k) = n
                                      $$

                                      $$
                                      log(2^(2^k)) = log(n)
                                      $$

                                      $$
                                      2^klog(2)=log(n)
                                      $$

                                      $$
                                      2^k=log(n)
                                      $$



                                      $$
                                      k=log(log(n))
                                      $$






                                      share|cite|improve this answer









                                      $endgroup$



                                      I looked for the number of repetitions, here is the answer I thought of:
                                      $$
                                      2^(2^k) = n
                                      $$

                                      $$
                                      log(2^(2^k)) = log(n)
                                      $$

                                      $$
                                      2^klog(2)=log(n)
                                      $$

                                      $$
                                      2^k=log(n)
                                      $$



                                      $$
                                      k=log(log(n))
                                      $$







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Jan 13 at 16:11









                                      shayshay

                                      103




                                      103






























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