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Inserting a key into a leaf node of the B/B⁺-tree

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• The need to maintain an index file
  • Recall:

    The index file is used to speed up the look up records in a data file using a search key

  • Therefore:

    When we insert a new record in the data file: We must also insert the search key of the new record into the index file (in our case: the B+-tree) When we delete a record from the data file: We must also delete the search key of the deleted record into the index file (in our case: the B+-tree)

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• Recall: record format of an index file
  • Index file:

    Index file = a file that store records of the following format: +------------+----------------------+ | Search key | block/record pointer | +------------+----------------------+ Graphically:

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  • Therefore:

    In an insert operation, we will always insert a (key, record-address) pair in the B+-tree In an delete operation, we will always insert a (key, record-address) pair from the B+-tree

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• Assumption
  • Assumption:

    We assume the search keys in the B-tree are unique I.e.: no duplicate keys stored in the B-tree

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• Overview: Inserting the search key x into a B-tree
  • Overview of the B-tree insert algorithm:

    A (new) search key (and a corresponding record pointer) is (always) inserted into a leaf node The insertion can cause an overflow condition which will split the leaf node into 2 leaf nodes An overflow condition will trigger the execution of the insert algorithm for an internal node The insert algorithm for an internal node is similar (but is not the same) as the insert algorithm for a leaf node !!! (So we will actually discuss 2 insertion algorithms !!!)

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• Definition: the middle key
  • Middle key:

    Let n = the number of keys in a B+ tree node The middle key is the key with the index equal to: m = ⌈ (n+1)/2 ⌉ Examples: the middle key n | m ---------+-------- 3 | 2 1 2 3 4 | 3 1 2 3 4 5 | 3 1 2 3 4 5 6 | 4 1 2 3 4 5 6 ... | ...

    Important note:

    When a node is split (see below), the node is split at the middle node The new right node will contain the middle key plus all the keys that follows the middle key

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  • Note:

    We start counting keys at the number 1:           First key = p1 Second key = p2          Third key = p3 ... (Normally, computer science starts counting at 0 !!! :))

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• Inserting search key x and its record pointer into a B⁺-tree
  • Given:

    A B+-tree Search key value x Block/record pointer of x

    Goal:

    Insert (x, ptr(x)) into the B+-tree

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  • Algorithm to insert ( search key x, ptr(x) ) into a B⁺-tree:

    /* ======================================================== Insert( x, record ptr(x) ) into B+-tree ======================================================== */ Insert( x, px ) { Use the B-tree search algorithm to find the leaf node L where the x belongs Let: L = leaf node used to insert x, px /* ============================================ Insert x, px in leaf node L ============================================ */ if ( L ≠ full ) { Shift keys to make room for x Insert (x, px) into its position in leaf node L return // Done } else // Overflow condition { /* ------------------------------------------- Leaf node L has n keys + n ptrs (= full !!!): L: |p1|k1|p2|k2|....|pn|kn|next| ------------------------------------------- */ Make the following virtual node by inserting px,x in L: |p1|k1|p2|k2|...|px|x|...|pn|kn|next| (There are (n+1) keys in this node !!!) Split this node in two 2 "equal" halves: Let: m = ⌈(n+1)/2⌉ L: |p1|k1|p2|k2|...|pm-1|km-1|R| <--- use the old node for L R: |pm|km|....|px|x|...|pn|kn|next| <--- use a new node for R /* ================================================== We need to fix the information at 1 level up to direct the search to the new node R We know that all keys in R: >= km ================================================== */ if ( L == root node of B+ tree ) { Make a new root node containing (L, km, R) return; } else { /* --------------------------------------------- Use the InsertInternal alg to insert: (km,R) into the parent node of L ---------------------------------------------- */ InsertInternal( (km, R), parent(L) ); // Note: all keys in R are ≥ km // So: R is the right subtree of km } } }