surprising property of the tree-based partition data structurewhen implemented using the union-by-size and path-compression heuristicsis that performing series of nunion and find operations takes (nlog*ntimewhere log* is the log-star functionwhich is the inverse of the tower-of-twos function intuitivelylog* is the number of times that one can iteratively take the logarithm (base of number before getting number smaller than table shows few sample values table some values of logn and critical values for its inverse as is demonstrated in table for all practical purposeslogn < it is an amazingly slow-growing function (but one that is growing nonethelessin factthe running time of series of partition operations implemented as above can actually be shown to be (na( ))where (nis the inverse of the ackermann function,awhich grows asymptotically even slower than logn although we will not prove this factlet us define the ackermann function hereso as to appreciate just how quickly it growshencehow slowly its inverse grows we first define an indexed ackermann functionaias followsa ( ) for >=