  Allocation of balls into bins is a well studied abstraction for load
balancing problems.The literature hosts numerous results for sequential(single
dimensional) allocation case when m balls are thrown into n bins. In this paper
we study the symmetric multiple choice process for both unweighted and weighted
balls as well as for both multidimensional and scalar models.Additionally,we
present the results on bounds on gap for (1+beta) choice process with
multidimensional balls and bins. We show that for the symmetric d choice
process and with m=O(n), the upper bound on the gap is O(lnln(n)) w.h.p.This
upper bound on the gap is within D=f factor of the lower bound. This is the
first such tight result.For the general case of m>>n the expected gap is
bounded by O(lnln(n)).For variable f and non-uniform distribution of the
populated dimensions,we obtain the upper bound on the expected gap as
O(log(n)).
  Further,for the multiple round parallel balls and bins,we show that the gap
is also bounded by O(loglog(n)) for m=O(n).The same bound holds for the
expected gap when m>>n. Our analysis also has strong implications in the
sequential scalar case.For the weighted balls and bins and general case m>>n,we
show that the upper bound on the expected gap is O(log(n)) which improves upon
the best prior bound of n^c.Moreover,we show that for the (1 + beta) choice
process and m=O(n) the upper bound(assuming uniform distribution of f populated
dimensions over D total dimensions) on the gap is O(log(n)/beta),which is
within D=f factor of the lower bound.For fixed f with non-uniform distribution
and for random f with Binomial distribution the expected gap remains
O(log(n)/beta) independent of the total number of balls thrown. This is the
first such tight result for (1 +beta) paradigm with multidimensional balls and
bins.
