amortized analysis total time total potential figure sequence of inserts that works is found neverthelessthe discussion above suggests few ruleswhich tell us the properties that good potential functions have the potential function should always assume its minimum at the start of the sequence popular method of choosing potential functions is to ensure that the potential function is initially and always nonnegative all the examples that we will encounter use this strategy cancel term in the actual time in our caseif the actual cost was cthen the potential change was when these are addedan amortized cost of is obtained this is shown in figure we can now perform complete analysis of binomial queue operations theorem the amortized running times of insertdeleteminand merge are ( ) (log )and (log )respectivelyfor binomial queues proof the potential function is the number of trees the initial potential is and the potential is always nonnegativeso the amortized time is an upper bound on the actual time the analysis for insert follows from the argument above for mergeassume the two queues have and nodes with and treesrespectively let + the actual time to perform the merge is (log( log( ) (log nafter the mergethere can be at most log treesso the potential can increase by at most (log nthis gives an amortized bound of (log nthe deletemin bound follows in similar manner