Multi-Way Search Tree ( ) (2,4) Trees. Multi-Way Inorder Traversal. Multi-Way Search Tree ( ) Multi-Way Searching. Multi-Way Searching

Transcription

1 Mlti-Way Search Tree ( 0..) (,) Trees (,) Trees (,) Trees Mlti-Way Search Tree ( 0..) A mlti-ay search tree is an ordered tree sch that Each internal node has at least to children and stores d key-element items (k i, o i ), here d is the nmber of children For a node ith children d storing keys k k k d : keys in the sbtree of are less than k keys in the sbtree of i are beteen k i and k i (i =,, d-) keys in the sbtree of d are greater than k d The leaes store no items and sere as placeholders (,) Trees Mlti-Way Inorder Traersal We can extend the notion of inorder traersal from binary trees to mlti-ay search trees Namely, e isit item (k i, o i ) of node beteen the recrsie traersals of the sbtrees of rooted at children i and i + An inorder traersal of a mlti-ay search tree isits the keys in increasing order (,) Trees Mlti-Way Searching : Mlti-Way Searching : (,) Trees 5 (,) Trees 6

2 Mlti-Way Searching Mlti-Way Searching : : (,) Trees 7 (,) Trees 8 Mlti-Way Searching Similar to search in a binary search tree A each internal node ith children d and keys k k k d k = k i (i =,, d ): the search terminates sccessflly k < k : e contine the search in child k i < k < k i (i =,, d ): e contine the search in child i k > k d : e contine the search in child d Reaching an external node terminates the search nsccessflly (,) Trees ( 0..) A (,) tree (also called - tree or -- tree) is a mlti-ay search ith the folloing properties Node-Size Property: eery internal node has at most for children Depth Property: all the external nodes hae the same depth Depending on the nmber of children, an internal node of a (,) tree is called a -node, -node or - node (,) Trees 9 (,) Trees 0 (,) Trees ( 0..) Height of a (,) Tree Theorem: A (,) tree storing n items has height O() Proof: Let h be the height of a (,) tree ith n items As in proper binary trees, there are at least i items at depth i: n h Ths, h log (n + ) Searching in a (,) tree ith n items takes O() time depth items 0 h h h 0 (,) Trees (,) Trees

3 Insertion We insert a ne item (k, o) at the parent of the leaf reached by searching for k We presere the depth property bt We may case an oerflo (i.e., node may become a 5-node) Example: inserting key cases an oerflo Oerflo and Split We handle an oerflo at a 5-node ith a split operation: let 5 be the children of and k k be the keys of node is replaced by nodes ' and " ' is a -node ith keys k k and children " is a -node ith key k and children 5 key k is inserted into the parent of (a ne root may be created) The oerflo may propagate to the parent node (,) Trees (,) Trees Oerflo and Split Analysis of Insertion ' 8 7 " 5 5 Let T be a (,) tree ith n items Tree T has O() height Finding insertion point takes O() time becase e isit O() nodes Inserting the ne entry takes O() time Dealing ith oerflo takes O() time becase each split takes O() time and e perform O() splits in the orst case. Ths, an insertion in a (,) tree takes O() time (,) Trees 5 (,) Trees 6 Starting ith an empty (,) tree, insert items ith keys,,,,5,6,7 (,) Trees 7 (,) Trees 8

4 inserting cases a split: inserting cases a split: (,) Trees 9 7 cases a split (,) Trees 0 Deletion We redce deletion of an entry to the case here the item is at the node ith leaf children Otherise, e replace the entry ith its inorder sccessor (or, eqialently, ith its inorder predecessor) and delete the latter entry Example: to delete key, e replace it ith 7 (inorder sccessor) Underflo Deleting an entry from a node may case an nderflo, here node becomes a -node ith one child and no keys (,) Trees (,) Trees Underflo and Fsion To handle an nderflo at node ith parent, e consider to cases Case : the adjacent siblings of are -nodes Fsion operation: e merge ith an adjacent sibling and moe an entry from to the merged node ' After a fsion, the nderflo may propagate to the parent ' 0 Underflo and Transfer Case : an adjacent sibling of is a -node or a - node Transfer operation:. e moe a child of to. e moe an item from to. e moe an item from to After a transfer, no nderflo occrs (,) Trees (,) Trees

5 Analysis of Deletion Let T be a (,) tree ith n items Tree T has O() height In a deletion operation We isit O() nodes to locate the node from hich to delete the entry We handle an nderflo ith a series of O() fsions, folloed by at most one transfer Each fsion and transfer takes O() time Ths, deleting an item from a (,) tree takes O() time delete, (,) Trees 5 (,) Trees 6 delete,7 delete, delete : replace by delete : replace by delete 7: nderflo 5 (,) Trees 7 (,) Trees 8 delete,7 Complexity comparison delete : replace by delete 7: nderflo Comparison of hash tables and (,) trees Hash Table Search Insert Delete Notes simple to implement Reqires allocating a lot of memory in adance 5 fsion 5 6 (,) Tree complex to implement dynamic memory se (,) Trees 9 (,) Trees 5