figure sequence-based implementation of partition consisting of three groupsa { } { }and { sizeand inserting them in the sequence with larger size each time we take position from the smaller group and insert it into the larger group bwe update the group reference for that position to now point to hencethe operation union(pqtakes time (min( nq ))where (resp nq is the cardinality of the group containing position (resp qclearlythis time is (nif there are elements in the partition universe howeverwe next present an amortized analysis that shows this implementation to be much better than appears from this worst-case analysis proposition when using the above sequence-based partition implementationperforming series of make groupunionand find operations on an initially empty partition involving at most elements takes ( log ntime justificationwe use the accounting method and assume that one cyber-dollar can pay for the time to perform find operationa make group operationor the movement of position object from one sequence to another in union operation in the case of find or make group operationwe charge the operation itself cyber-dollar in the case of union operationwe assume that cyber-dollar pays for the constant-time work in comparing the sizes of the two sequencesand that we charge cyber-dollar to each position that we move from the smaller group to the larger group clearlythe cyber-dollar charged for each find and make group operationtogether with the first cyber-dollar collected for each union operationaccounts for total of cyber-dollars considerthenthe number of charges made to positions on behalf of union operations the important observation is that each time we move position from one group to anotherthe size of that position' group at least doubles thuseach position is moved from one group to another at most logn timeshenceeach position can be charged at most (log ntimes since we assume that the partition is initially emptythere are (ndifferent elements referenced in the given series of operationswhich implies that the total time for moving elements during the union operations is ( log