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Allocation of balls into bins is a well studied abstraction for loadbalancing problems.The literature hosts numerous results for sequential(singledimensional) allocation case when m balls are thrown into n bins. In this paperwe study the symmetric multiple choice process for both unweighted and weightedballs as well as for both multidimensional and scalar models.Additionally,wepresent the results on bounds on gap for (1+beta) choice process withmultidimensional balls and bins. We show that for the symmetric d choiceprocess and with m=O(n), the upper bound on the gap is O(lnln(n)) w.h.p.Thisupper bound on the gap is within D=f factor of the lower bound. This is thefirst such tight result.For the general case of m>>n the expected gap isbounded by O(lnln(n)).For variable f and non-uniform distribution of thepopulated dimensions,we obtain the upper bound on the expected gap asO(log(n)).  Further,for the multiple round parallel balls and bins,we show that the gapis also bounded by O(loglog(n)) for m=O(n).The same bound holds for theexpected gap when m>>n. Our analysis also has strong implications in thesequential scalar case.For the weighted balls and bins and general case m>>n,weshow that the upper bound on the expected gap is O(log(n)) which improves uponthe best prior bound of n^c.Moreover,we show that for the (1 + beta) choiceprocess and m=O(n) the upper bound(assuming uniform distribution of f populateddimensions over D total dimensions) on the gap is O(log(n)/beta),which iswithin D=f factor of the lower bound.For fixed f with non-uniform distributionand for random f with Binomial distribution the expected gap remainsO(log(n)/beta) independent of the total number of balls thrown. This is thefirst such tight result for (1 +beta) paradigm with multidimensional balls andbins.

Multidimensional Balanced Allocation for Multiple Choice & (1 + Beta) Processes