Multi-Way Search Tree

Transcription

1 Presentation for se with the textbook Data Strctres and Algorithms in Jaa, 6 th edition, by M. T. Goodrich, R. Tamassia, and M. H. Goldwasser, Wiley, 204 (2,4) Trees Goodrich, Tamassia, Goldwasser (2,4) Trees Mlti-Way Search Tree! A mlti-way search tree is an ordered tree sch that n Each internal node has at least two children and stores d - key-element items (k i, o i ), where d is the nmber of children n For a node with children 2 d storing keys k k 2 k d- n w keys in the sbtree of are less than k w keys in the sbtree of i are between k i- and k i (i = 2,, d - ) w keys in the sbtree of d are greater than k d- The leaes store no items and sere as placeholders Goodrich, Tamassia, Goldwasser (2,4) Trees 2

2 Mlti-Way Inorder Traersal! We can extend the notion of inorder traersal from binary trees to mlti-way search trees! Namely, we isit item (k i, o i ) of node between the recrsie traersals of the sbtrees of rooted at children i and i +! An inorder traersal of a mlti-way search tree isits the keys in increasing order Goodrich, Tamassia, Goldwasser (2,4) Trees 3 Mlti-Way Searching! Similar to search in a binary search tree! A each internal node with children 2 d and keys k k 2 k d- n k = k i (i =,, d - ): the search terminates sccessflly n k < k : we contine the search in child n k i- < k < k i (i = 2,, d - ): we contine the search in child i n k > k d- : we contine the search in child d! Reaching an external node terminates the search nsccessflly! Example: search for Goodrich, Tamassia, Goldwasser (2,4) Trees 4 2

3 (2,4) Trees! A (2,4) tree (also called 2-4 tree or tree) is a mlti-way search with the following properties n Node-Size Property: eery internal node has at most for children n Depth Property: all the external nodes hae the same depth! Depending on the nmber of children, an internal node of a (2,4) tree is called a 2-node, 3-node or 4-node Goodrich, Tamassia, Goldwasser (2,4) Trees 5 Height of a (2,4) Tree! Theorem: A (2,4) tree storing n items has height O() Proof: n Let h be the height of a (2,4) tree with n items n Since there are at least 2 i items at depth i = 0,, h - and no items at depth h, we hae n h- = 2 h - n Ths, h log (n + )! Searching in a (2,4) tree with n items takes O() time depth 0 items h- h 2 2 h Goodrich, Tamassia, Goldwasser (2,4) Trees 6 3

4 Insertion! We insert a new item (k, o) at the parent of the leaf reached by searching for k n We presere the depth property bt n We may case an oerflow (i.e., node may become a 5-node)! Example: inserting key 30 cases an oerflow Goodrich, Tamassia, Goldwasser (2,4) Trees 7 Oerflow and Split! We handle an oerflow at a 5-node with a split operation: n let 5 be the children of and k k 4 be the keys of n node is replaced nodes ' and " w ' is a 3-node with keys k k 2 and children 2 3 w " is a 2-node with key k 4 and children 4 5 n key k 3 is inserted into the parent of (a new root may be created)! The oerflow may propagate to the parent node ' " Goodrich, Tamassia, Goldwasser (2,4) Trees 8 4

5 Analysis of Insertion Algorithm pt(k, o). We search for key k to locate the insertion node 2. We add the new entry (k, o) at node 3. while oerflow() if isroot() create a new empty root aboe split()! Let T be a (2,4) tree with n items n Tree T has O() height n Step takes O() time becase we isit O() nodes n Step 2 takes O() time n Step 3 takes O() time becase each split takes O() time and we perform O() splits! Ths, an insertion in a (2,4) tree takes O() time 204 Goodrich, Tamassia, Goldwasser (2,4) Trees 9 Deletion! We redce deletion of an entry to the case where the item is at the node with leaf children! Otherwise, we replace the entry with its inorder sccessor (or, eqialently, with its inorder predecessor) and delete the latter entry! Example: to delete key 24, we replace it with 27 (inorder sccessor) Goodrich, Tamassia, Goldwasser (2,4) Trees 0 5

6 Underflow and Fsion! Deleting an entry from a node may case an nderflow, where node becomes a -node with one child and no keys! To handle an nderflow at node with parent, we consider two cases! Case : the adjacent siblings of are 2-nodes n Fsion operation: we merge with an adjacent sibling w and moe an entry from to the merged node ' n After a fsion, the nderflow may propagate to the parent w ' Goodrich, Tamassia, Goldwasser (2,4) Trees Underflow and Transfer! To handle an nderflow at node with parent, we consider two cases! Case 2: an adjacent sibling w of is a 3-node or a 4-node n Transfer operation:. we moe a child of w to 2. we moe an item from to 3. we moe an item from w to n After a transfer, no nderflow occrs w w Goodrich, Tamassia, Goldwasser (2,4) Trees 2 6

7 Analysis of Deletion! Let T be a (2,4) tree with n items n Tree T has O() height! In a deletion operation n We isit O() nodes to locate the node from which to delete the entry n We handle an nderflow with a series of O() fsions, followed by at most one transfer n Each fsion and transfer takes O() time! Ths, deleting an item from a (2,4) tree takes O() time 204 Goodrich, Tamassia, Goldwasser (2,4) Trees 3 Comparison of Map Implementations Hash Table Search Insert Delete Notes expected expected expected o no ordered map methods o simple to implement Skip List high prob. high prob. high prob. o randomized insertion o simple to implement AVL and (2,4) Tree worst-case worst-case worst-case o complex to implement 204 Goodrich, Tamassia, Goldwasser (2,4) Trees 4 7