Data Structures and Algorithms Lecture 7 DCI FEEI TUKE. Balanced Trees

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1 Balanced Trees AVL trees reconstruction of perfect balance can be quite expensive operation less rigid criteria of balance (e.g. AVL) [2] inorder, self-balancing binary search tree (BST) Definition: AVL tree is balanced, if the heights of the subtrees of every vertex differs at most by 1. On AVL trees, following operations can be executed in O(log n) time: MEMBER searching for a vertex by key, INSERT inserting the vertex with a given key, DELETE removing the vertex with a given key. Corollary of the lemma proved by the authors (Adelson-Velskii and Landis): the height of an AVL tree with n vertices is at most 2.log n. [3] Inserting vertices [4] three cases can occur (inserting to L-subtree): h L = h R (h L = h L + 1, OK) h L < h R (h L = h L + 1 => h L = h R, OK) h L > h R (h L = h L + 1, AVL criterion can be broken reconstruction needed) only two different cases of AVL unbalance exists (with another two as a mirror image)

2 Case 1: LL (RR) B A A B Case 2: LR (RL) C B A A C B

3 vertical moves allowed only, relative horizontal positions without change balancing operations defined like sequences of pointers interchanges simple rotation (LL, RR) double rotation (LR, RL) Removing vertices leaf simple remove vertex with 1 son exchange, remove vertex with 2 sons replace with rightmost vertex form the left subtree (leftmost from the right subtree) balancing, if needed Notes: inserting can evoke at most one rotation removing can evoke more rotations implementation it is useful to keep explicit balance factor for every vertex e.g. bal = h L h R (figure Algomaster) [1] Aho, A.V., Hopcroft, J.E., Ullman, J.D.: Design and Analysis of Computer Algorithms, Addison-Wesley, [2] Wirth, N.: Algorithms and Data Structures, [3] Töpfer, P.: Algoritmy a programovací techniky, Prometheus, [4] Data Structure Visualizations AVL Tree visualization,

4 Example: inserting elements into AVL tree (4, 5, 7, 2, 1, 3, 6)

5 Example: removing elements from AVL tree (4, 8, 6, 5, 2, 1, 7)

6 2-3 trees instructions MEMBER, INSERT, DELETE hash table O(1) resp. O(n); BVS O(log n) resp. O(n) for 1 instruction balanced trees reducing worst case to O(log n) height of the tree problem preserving the balance of a tree (after INSERT, DELETE) Definition: 2-3 tree is a tree, in which every internal (non-leaf) vertex has 2 or 3 sons, and every path from the root to a leaf has the same length. Tree consisting of 1 vertex also is 2-3 tree [1]. Lemma: Let T is a 2-3 tree of height h. Number of vertices of T is between 2 h+1-1 and (3 h+1-1)/2 and number of leaves is between 2 h and 3 h.

7 Representations of set S (TO) with 2-3 tree elements assigned to leaves of 2-3 tree two basic methods of such assignment dictionary (operations M, I, D) in increasing order from the left to the right every non-leaf vertex v - values L[v], M[v] (searching) L[v] maximal element of subtree, which root is the leftmost son of v M[v] maximal element of subtree, which root is the second (from left) son of v method for UNION operation without restrictions on element ordering DELETE searching for a leaf with given element (auxiliary array...) Types of supported operations: 1. INSERT, DELETE, MEMBER, 2. INSERT, DELETE, MIN, 3. INSERT, DELETE, UNION, MIN, 4. INSERT, DELETE, FIND, CONCATENATE, SPLIT. Types of data structures, which process corresponding instructions: 1. Dictionary, 2. Priority queue, 3. Mergeable heap, 4. Concatenable queue. 2-3 trees suitable DS for implementing structures 1-4, in O(n log n) time for n instructions.

8 Operations on 2-3 trees Inserting an element into 2-3 tree input: nonempty 2-3 tree T with the root r, inserting element a, which is not present in the tree output: updated 2-3 tree with the new leaf labeled a INSERT23(a) 1. If T has only 1 vertex (b), create new root r. Create a new leaf (a). If b<a, b left son, otherwise a left. Label (L[r ],M[r ]). 2. If T has more than 1 vertex, use f=search(a,r), create new leaf l labeled a. a) If f has 2 sons now (b1,b2), create l the third son of f. l is left son, if a<b1, middle, if b1<a<b2, right, if b2<a. b) If f has 3 sons create 4-th son and call ADDSON(f), update L,M values. procedure SEARCH(a,r) if son of r is a leaf then return r else begin let Si is the i-th son of r if a <= L[r] then return SEARCH(a,S1) else if r has 2 sons OR a<=m[r] then return SEARCH(a,S2) else return SEARCH(a,S3) end

9 procedure ADDSON(v) begin create new vertex v ; make 2 rightmost sons of v left and right sons of v ; if v has no father then create new root r; make v left and v right sons of r; else begin let f is the father of v; make v a son immediately right of v if f has now 4 sons then ADDSON(f) end end Removing an element from 2-3 tree DELETE23(a) (where a label of vertex l) 1. if l is the root of the tree remove l (the only element) 2. if l is a son of vertex with 3 sons, remove l 3. if l is a son of f, which has 2 sons (s,l) a) f is root remove l, f, leave s as the root b) f is not the root assume: f has a brother g on the left (right analogically) i. g has 2 sons, make s the rightmost son of g, remove l, call DELETE(f) ii.g has 3 sons, make the rightmost son of g the left son of f and remove l

10 Example:

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12 Example: some of ADT set implementations in Java (HashSet, LinkedHashSet, TreeSet). HashSet class implements the Set interface, backed by a hash table; makes no guarantees as to the iteration order [5] LinkedHashSet Hash table and linked list implementation of the Set interface, with predictable iteration order [6] the linked list defines the iteration ordering, which is the order in which elements were inserted into the set TreeSet elements are ordered using their natural ordering, or by a Comparator provided at set creation time [7] this implementation provides guaranteed log(n) time cost for the basic operations (add, remove and contains) [5] Class HashSet, [6] Class LinkedHashSet, [7] Class TreeSet,