If there is space, put the new key in, rearrange keys and exit. If there is no space for the newly arrived key, then allocate a new leaf node. Split the keys between the two nodes. ceil(n/2) keys/values go into the 1st block and the remaining into the 2nd. Propagate upwards the first key/value of the 2nd block. If insertion in the internal node is possible then do the insertion (of the just propagated key), re-arrange pointers properly, and exit. Otherwise, The internal node has to be split up: - Acquire a new block. - All the keys less than the MEDIAN go to the 1st block and all the keys greater than the median go to the 2nd (WITHOUT the median). - Rearrange and keep track of pointers properly. - The median needs to be up-propagated Goto to 4. 2. 3. 4. Find the entry to be deleted. Delete entry from leaf. If this leaf, call it Z, is left with less than the required # of entries go to 3. Otherwise, Exit. If neither the right nor the left sibling of Z (off the same parent) has more than ceil[(n-1)/2] then GoTo 4. Otherwise, redistribute the entries from ONE sibling with those in Z, so that half of the total are in sibling and half in Z. * If one of the siblings has more records than the other use it. * If both have the same # of keys, use the left one (arbitrary choice) Change the new distinguishing key of Z (from its sibling) in the parent and Exit. Combine the entries of one sibling with those in Z. /* this happens only when BOTH ADJACENT siblings have exactly ceil[n/2] entries; 1. 2. 3. 4. ** B+ Nodes maintain n-1 key values and n pointers B+Tree DELETION ALGORITHM: Rewritten by ad-at+di Find the leaf page in which the insertion should take place. 1. ** B+ Nodes maintain n-1 key values and n pointers B+Tree INSERTION ALGORITHM: Rewrite by ad-at+di.uoa.gr internal node with the larger of its siblings (or left if both Internal Node Redistribution: If an internal node is left sparse, temporarily concatanate 6. The (n-1) keys are now merged to become one internal node. The key and addresses in the parent are deleted. If the parent is too sparse GoTo 5. Otherwise, Exit. If there are exactly (n-1) keys in this concatanation GoTo 6. Otherwise (there are more than [n-1] keys), redistribute values by placing the new middle value in the parent in the place of the previous distinguishing value and Exit. of equal size) ALONG WITH the key distinguishing it from its sibling. are this 5. If the parent is the root, it cannot become sparse unless no value is left. In this case, the present node with the records of Z and its sibling, become the root which is now the entire tree. If the parent is left with less than the necessary entries (sparse) GoTo 5. Otherwise, Exit. u may choose the left sibling if there is one */ Delete from parent the distinguishing key of Z (from its siblings ) and delete the address of Z.
