How many balls will be left at the end of this process?












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$begingroup$


Consider having $N$ colored balls. Each color has at least $N/2k$ and at most $N/k$ balls in the beginning, for some parameter $kll N$.



At each iteration, we remove $k$ balls with different colors, choosing from the colors that have most balls (breaking ties arbitrarily).



The process ends when there are less than $k$ colors still present.



For example, consider $k=3$ and having $4$ balls from color $1$, $2$ balls from colors $2$, $3$ and $4$ and a single ball from color $5$.
Then the process may be (where each tuple is the frequency of each color):
$(4,2,2,2,1)to(3,2,1,1,1)to(2,1,1,1,0)to(1,1,0,0,0)$.



That is, we are left with just one ball in this example.




Can we guarantee that the process terminates with at most $k-1$ balls?




Clearly, this cannot hold without the bound on the number of balls from each color. E.g., if we had $6$ balls (which is more than $13/3=N/k$) from the first color instead of $3$ we could get:
$(6,2,2,2,1)to(5,2,1,1,1)to(4,1,1,1,0)to(3,1,0,0,0)$.



It seems that, for all examples I tested, we are always left with at most $k-1$ balls from different colors. Is that true for any distribution that satisfies the size constraints?










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$endgroup$

















    0












    $begingroup$


    Consider having $N$ colored balls. Each color has at least $N/2k$ and at most $N/k$ balls in the beginning, for some parameter $kll N$.



    At each iteration, we remove $k$ balls with different colors, choosing from the colors that have most balls (breaking ties arbitrarily).



    The process ends when there are less than $k$ colors still present.



    For example, consider $k=3$ and having $4$ balls from color $1$, $2$ balls from colors $2$, $3$ and $4$ and a single ball from color $5$.
    Then the process may be (where each tuple is the frequency of each color):
    $(4,2,2,2,1)to(3,2,1,1,1)to(2,1,1,1,0)to(1,1,0,0,0)$.



    That is, we are left with just one ball in this example.




    Can we guarantee that the process terminates with at most $k-1$ balls?




    Clearly, this cannot hold without the bound on the number of balls from each color. E.g., if we had $6$ balls (which is more than $13/3=N/k$) from the first color instead of $3$ we could get:
    $(6,2,2,2,1)to(5,2,1,1,1)to(4,1,1,1,0)to(3,1,0,0,0)$.



    It seems that, for all examples I tested, we are always left with at most $k-1$ balls from different colors. Is that true for any distribution that satisfies the size constraints?










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Consider having $N$ colored balls. Each color has at least $N/2k$ and at most $N/k$ balls in the beginning, for some parameter $kll N$.



      At each iteration, we remove $k$ balls with different colors, choosing from the colors that have most balls (breaking ties arbitrarily).



      The process ends when there are less than $k$ colors still present.



      For example, consider $k=3$ and having $4$ balls from color $1$, $2$ balls from colors $2$, $3$ and $4$ and a single ball from color $5$.
      Then the process may be (where each tuple is the frequency of each color):
      $(4,2,2,2,1)to(3,2,1,1,1)to(2,1,1,1,0)to(1,1,0,0,0)$.



      That is, we are left with just one ball in this example.




      Can we guarantee that the process terminates with at most $k-1$ balls?




      Clearly, this cannot hold without the bound on the number of balls from each color. E.g., if we had $6$ balls (which is more than $13/3=N/k$) from the first color instead of $3$ we could get:
      $(6,2,2,2,1)to(5,2,1,1,1)to(4,1,1,1,0)to(3,1,0,0,0)$.



      It seems that, for all examples I tested, we are always left with at most $k-1$ balls from different colors. Is that true for any distribution that satisfies the size constraints?










      share|cite|improve this question









      $endgroup$




      Consider having $N$ colored balls. Each color has at least $N/2k$ and at most $N/k$ balls in the beginning, for some parameter $kll N$.



      At each iteration, we remove $k$ balls with different colors, choosing from the colors that have most balls (breaking ties arbitrarily).



      The process ends when there are less than $k$ colors still present.



      For example, consider $k=3$ and having $4$ balls from color $1$, $2$ balls from colors $2$, $3$ and $4$ and a single ball from color $5$.
      Then the process may be (where each tuple is the frequency of each color):
      $(4,2,2,2,1)to(3,2,1,1,1)to(2,1,1,1,0)to(1,1,0,0,0)$.



      That is, we are left with just one ball in this example.




      Can we guarantee that the process terminates with at most $k-1$ balls?




      Clearly, this cannot hold without the bound on the number of balls from each color. E.g., if we had $6$ balls (which is more than $13/3=N/k$) from the first color instead of $3$ we could get:
      $(6,2,2,2,1)to(5,2,1,1,1)to(4,1,1,1,0)to(3,1,0,0,0)$.



      It seems that, for all examples I tested, we are always left with at most $k-1$ balls from different colors. Is that true for any distribution that satisfies the size constraints?







      algorithms recursion integers recursive-algorithms






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      asked Dec 6 '18 at 14:46









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