the sequence implementation above is simplebut it is also efficientas the following theorem shows proposition performing series of nmakesetunionandfind operationsusing the sequence-based implementation abovestarting from an initially empty partition takes (nlogntime justificationwe use the accounting method and assume that one cyberdollar can pay for the time to perform find operationa makeset operationor the movement of position object from one sequence to another in union operation in the case of find or makeset operationwe charge the operation itself cyber-dollar in the case of union operationhoweverwe charge cyber-dollar to each position that we move from one set to another note that we charge nothing to the union operations themselves clearlythe total charges to find and makeset operations sum to be (nconsiderthenthe number of charges made to positions on behalf of union operations the important observation is that each time we move position from one set to anotherthe size of the new set at least doubles thuseach position is moved from one set to another at most logn timeshenceeach position can be charged at most (logntimes 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 all the union operations is (nlognthe amortized running time of an operation in series of makesetunionand find operationsis the total time taken for the series divided by the number of operations we conclude from the proposition above thatfor partition implemented using sequencesthe amortized running time of each operation is