insertion is the same with duplicates allowed as when they're nota single step inserts the new item but rememberif duplicates are not allowedand there' possibility the user will attempt to input the same key twiceyou may need to check every existing item before doing an insertion deletion with duplicates deletion may be more complicated when duplicates are alloweddepending on exactly how "deletionis defined if it means to delete only the first item with specified valuethenon the averageonly / comparisons and / moves are necessary this is the same as when no duplicates are allowed but if deletion means to delete every item with specified key valuethen the same operation may require multiple deletions this will require checking cells and (probablymoving more than / cells the average depends on how the duplicates are distributed throughout the array the applet assumes this second meaning and deletes multiple items with the same key this is complicatedbecause each time an item is deletedsubsequent items must be shifted farther for exampleif three items are deletedthen items beyond the last deletion will need to be shifted three spaces to see how this worksset the applet to dups ok and insert three or four items with the same key then try deleting them table shows the average number of comparisons and moves for the three operationsfirst where no duplicates are allowed and then where they are allowed is the number of items in the array inserting new item counts as one move you can explore these possibilities with the array workshop applet table duplicates ok versus no duplicates no duplicates duplicates ok search / comparisons comparisons insertion no comparisonsone move no comparisonsone move deletion / comparisonsn/ moves comparisonsmore than / moves the difference between and / is not usually considered very significantexcept when fine-tuning program of more importanceas we'll discuss toward the end of this is whether an operation takes one stepn stepslog(nstepsor steps not too swift one of the significant things to notice when using the array applet is the slow and methodical nature of the algorithms with the exception of insertionthe algorithms involve stepping through some or all of the cells in the array different data structures offer much