ta ta ta tb tb tb after the initial run construction phasethe number of passes required using -way merging is logk ( / )because the runs get times as large in each pass for the example abovethe formula is verifiedsince log ( / ) if we have tapesthen and our large example from the previous section would require log passes polyphase merge the -way merging strategy developed in the last section requires the use of tapes this could be prohibitive for some applications it is possible to get by with only tapes as an examplewe will show how to perform two-way merging using only three tapes suppose we have three tapest and and an input file on that will produce runs one option is to put runs on each of and we could then merge this result onto obtaining one tape with runs the problem is that since all the runs are on one tapewe must now put some of these runs on to perform another merge the logical way to do this is to copy the first eight runs from onto and then perform the merge this has the effect of adding an extra half pass for every pass we do an alternative method is to split the original runs unevenly suppose we put runs on and runs on we would then merge runs onto before was empty at this pointwe could rewind and and merge with runsand which has runsonto we could then merge runs until was emptywhich would leave runs left on and runs on we could then merge and and so on the following table shows the number of runs on each tape after each passt run const after after after after after after after the original distribution of runs makes great deal of difference for instanceif runs are placed on with on then after the first mergewe obtain runs on and runs on after another mergethere are runs on and runs on at this point the going gets slowbecause we can only merge two sets of runs before is exhausted then has runs and has runs againwe can only merge two sets of runsobtaining with runs and with runs after three more passest has two