hash tables try inserting some new data items type in three-digit number and watch what happens most items will go into the first cell they trybut some will suffer collisions and need to step along to find an empty cell the number of steps they take is the probe length most probe lengths are only few cells long sometimeshoweveryou may see probe lengths of four or five cellsor even longer as the array becomes excessively full notice which keys hash to the same index if the array size is the keys and so on up to all hash to index try inserting this sequence or similar one such sequences will demonstrate the linear probe the del button the del button deletes an item whose key is typed by the user deletion isn' accomplished by simply removing data item from cellleaving it empty why notremember that during insertion the probe process steps along series of cellslooking for vacant one if cell is made empty in the middle of this sequence of full cellsthe find routine will give up when it sees the empty celleven if the desired cell can eventually be reached for this reason deleted item is replaced by an item with special key value that identifies it as deleted in this applet we assume all legitimate key values are positiveso the deleted value is chosen as - deleted items are marked with the special key *delthe insert button will insert new item at the first available empty cell or in *delitem the find button will treat *delitem as an existing item for the purposes of searching for another item further along if there are many deletionsthe hash table fills up with these ersatz *deldata itemswhich makes it less efficient for this reason many hash table implementations don' allow deletion if it is implementedit should be used sparingly duplicates allowedcan you allow data items with duplicate keys to be used in hash tablesthe fill routine in the hash workshop applet doesn' allow duplicatesbut you can insert them with the insert button if you like then you'll see that only the first one can be accessed the only way to access second item with the same key is to delete the first one this isn' too convenient you could rewrite the find algorithm to look for all items with the same key instead of just the first one howeverit would then need to search through all the cells of every linear sequence it encountered this wastes time for all table accesseseven when no duplicates are involved in the majority of cases you probably want to forbid duplicates