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COMP 620 Algorithm Analysis Franklin University Module 3: Advanced Data Structures Trees, Heap, Binomial Heap, Fibonacci Heap 1 COMP 620 Algorithm Analysis Franklin University Trees • Definition: A tree (which is one type of a data structure) is a finite set of one or more nodes such that: There is a specially designated node called root. The remaining nodes are divided into n>=0 disjoint sets T1, T2, ..... Tn where each of these sets is a tree. T1, T2, .... Tn are called the subtrees of the root. • • • Node - represents an item of information stored in the tree. Branches - represent the links between the nodes. Root of the tree or root node - node at the top of the tree, the start of the tree. Degree of a node - the number of subtrees of the node (i.e., the number of children from any one node). Degree of a tree - the maximum degree of any of the nodes in the tree. Leaf or terminal nodes - nodes with degree 0. • • • 2 COMP 620 Algorithm Analysis Franklin University Tree Terminology (contd.) 7. 8. 9. 10. 11. 12. Child/children - the roots of the subtrees of the parent node. Siblings - children of the same parent. Parent - a node that has subtrees (i.e., a node that has children). Level of a node - defined by root = 1, children of root = 2, grandchildren of root = 3, etc. (In some textbooks, the root is defined to be at level 0, children of the root at level 1, etc.) Depth or height of a tree - the maximum level of any node in the tree. Binary trees are trees with no more than two-way branching from each node in the tree. A binary tree is either empty or consists of a root node and two disjoint binary trees called left and right subtrees. 3 COMP 620 Algorithm Analysis Franklin University Tree - Example A C B E K F D G I H J L M N Degree of the tree = O Height (depth) of the tree = Terminal nodes of the tree are: 4 COMP 620 Algorithm Analysis Franklin University Binary Trees: Properties • The maximum number of nodes on level i of a binary tree is 2 (i-1) , where i >= 1. • The maximum number of nodes in a binary tree of depth k is 2k –1,where k >= 1. • For any non-empty binary tree, T, if n0 represents the number of leaf nodes and n2 represents the number of nodes with degree 2, then n0 = n2 + 1. • A full binary tree of depth k is a binary tree of depth k having 2k - 1 nodes, where k >= 1. • A binary tree with n nodes and depth k is complete if and only if its nodes correspond to the nodes numbered from 1 to n in the full binary tree of depth k, filling in from left to right on each level. 5 COMP 620 Algorithm Analysis Franklin University Full and Complete Binary Trees 1 A C B D 2 F E G 4 3 5 Complete Binary Tree Full Binary Tree Array Representing Above Tree A B C D E F G 0 1 2 3 4 5 6 7 •Leftchild(i) is at 2i if 2i >n no left child •Rightchild(i) is at 2i + 1 if 2i +1 > n no right child 6 COMP 620 Algorithm Analysis Franklin University Neither Full nor Complete Trees A B D C F E G Array Representation of the Above Tree A B 0 1 2 C D 3 4 5 E F .. 10 11 G .. 23 The array representation of the tree that is neither full nor complete is very wasteful of memory 7 COMP 620 Algorithm Analysis Franklin University Linked Representation template <class BaseData> Class BtNode { public: BaseData info; BtNode *leftChild, *rightChild; }; 8 COMP 620 Algorithm Analysis Franklin University • Traversing Trees Inorder traversal 1. Traverse the left subtree 2. Visit the root 3. Traverse the right subtree Void BinTree::inord( BtNode *rt) { if (rt != NULL) { inord(rt->leftChild); processNode(rt->info); inord(rt->rightChild); } } 9 COMP 620 Algorithm Analysis Franklin University Traversing Trees • Preorder traversal 1. Visit root 2. Traverse the left subtree 3. Traverse the right subtree Void BinTree::preord( BtNode *rt) { if (rt != NULL) { processNode(rt->info); preord(rt->leftChild); preord(rt->rightChild); } } 10 COMP 620 Algorithm Analysis Franklin University Traversing Trees • Postorder traversal 1. 2. 3. Traverse the left subtree Traverse the right subtree Visit the root • Levelorder traversal Every node is visited in turn from left to right on every level, starting at level 1, then level 2, etc., until level n, where n represents the height (depth) of the tree. 11 COMP 620 Algorithm Analysis Franklin University Binary Search Trees • Binary search tree is of significant importance in Computer Science. • Has better performance than many other data structures especially for operations like insertion, deletion, and searching. • Definition:A binary search tree is a binary tree. It may be empty. If not empty, it satisfies the following properties: Every element has a key, and traditionally no two elements have the same key. Keys in a non-empty left subtree must be smaller than the key in the root of the subtree. Keys in a non-empty right subtree must be larger than the key in the root of the subtree. Left and right subtrees are also binary search trees. 12 COMP 620 Algorithm Analysis Franklin University Binary Search Tree G F A O H E I C B J P M K D N L 13 COMP 620 Algorithm Analysis Franklin University • • Binary Search Trees: Operations Searching a binary search tree: Begin at the root. If the key of the element to be searched = root key, then the search is successful. If the key of the element to be searched < root key, then search the left subtree. If the key of the element to be searched > root key, then search the right subtree. Inserting into a binary search tree: - Search for the key - If the key is not present, locate the parent node - Insert the new node as the left/right child of the parent node All insertions take place at leaf nodes. 14 COMP 620 Algorithm Analysis Franklin University Binary Search Trees: Operations • Deleting from a binary search tree: The deletion process begins with a search to find the node to be deleted from the binary search tree. Three possible scenarios exist in the deletion process: 1. Delete a leaf node. - Make the appropriate pointer in x’s parent a null pointer 2. Delete a node with 1 child. - Set the appropriate pointer in x’s parent to point to this child 3. Delete a node with 2 children. - Replace the value stored in the node x by its inorder successor (predecessor) and then delete the successor (predecessor) 15 COMP 620 Algorithm Analysis Franklin University Binary Search Trees: Variations • AVL (or height-balanced) trees A binary search tree in which the balance factor of each node is 0, 1, or –1, where the balance factor of a node x is defined as the height of the left subtree of x minus the height of x’s right subtree. • 2-3-4 trees: A tree with the following properties: Each node stores at most 3 data values. Each internal node is a 2-node, 3-node, or a 4node. All the leaves are on the same level. 16 COMP 620 Algorithm Analysis Franklin University Motivation for AVL Tree 2 1 2 1 4 3 3 5 4 What is the value of balance of each node? 5 The height of AVL tree is approximately Log2 n 17 COMP 620 Algorithm Analysis Franklin University 2-3-4 Tree 53 27 16 25 38 33 60 36 41 46 48 55 59 65 70 68 73 75 79 18 COMP 620 Algorithm Analysis Franklin University Red Black Trees • Red Black trees A binary search tree with two kinds of nodes, red and black, which satisfy the following properties: • Every node is either red or black. Root is black. • Every leaf (NIL) is black. • If a node is red, both its children are black. • Every simple path from node to descendant leaf contains the same number of black nodes. • A red-black tree with n internal nodes has height at most 2log(n+1). 19 COMP 620 Algorithm Analysis Franklin University 11 14 2 1 15 7 5 8 RED-BLACK TREE 20 COMP 620 Algorithm Analysis Franklin University B-trees • B-trees of minimum degree t A tree with the following properties: - Each node stores at most 2t -1 data values. - Every node other than the root must have at least t1 data values. - n[x] keys in each node x stored in nondecreasing order, so that key1[x] <= key2[x] .. <= key n[x] [x]. - Each internal node contains n[x] + 1 children. X contains n +1 pointers c1[x], c2[x] … c n[x]+1[x]. 21 COMP 620 Algorithm Analysis Franklin University B-trees - The keys keyi [x] separates the ranges of keys stored in each subtree: if ki is any key stored in the subtree with root ci[x] then, then k1 <= key1[x] <= key2[x] <= ….<= key n[x] [x] <= k n[x]+1 - All the leaves are on the same level. - Balanced and designed to work efficiently on disks and other direct access secondary storage devices. Many database systems use B-tree. - the maximum height of a n-key B tree is logt ((n+1)/2) - insert, delete, and search time on a B-Tree is (log n) 22 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 23 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 24 COMP 620 Algorithm Analysis Franklin University B-tree of minimum degree t = 2 Is 2-3-4 tree a B-tree? M Q T D H B C F G J K L N P X R S V W Y Z 25 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 26 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 27 COMP 620 Algorithm Analysis Franklin University B-Tree • Insert - locate the node where the key should be inserted - split any full nodes encountered while descending the tree • Deletion Deletion is similar to Insertion. Make sure that the tree is still a B-Tree after deletion. Detailed discussion of deleting from a B-tree, refer to Section 18.3, pages 450-453, of Cormen, Leiserson, and Rivest. 28 COMP 620 Algorithm Analysis Franklin University B-Tree Insertion - Inserts only in leaf node G M P X A C D E J K N O R S TU V Y Z (a) Initial Tree G M P X A B C D E J K N O R S TU V Y Z (b) B Inserted 29 COMP 620 Algorithm Analysis Franklin University B-Tree Insertion - Inserts only in leaf node G M P T X A B C D E J K N O Q R S U V Y Z (C ) Q Inserted P G M A B C D E J K L T X N O Q R S U V Y Z (d ) L Inserted 30 COMP 620 Algorithm Analysis Franklin University B-Tree Insertion - Inserts only in leaf node P C G M A B D E F J K L T X N O Q R S U V Y Z (e ) F Inserted 31 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 32 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 33 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 34 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 35 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 36 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 37 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 38 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. COMP 620 Algorithm Analysis Franklin University 39 COMP 620 Algorithm Analysis Franklin University Heaps • Heaps data structures are mostly used to support the following operations (not efficient to use for search): Insert element x Return min element Delete minimum element Union of two heaps • Application: Dijkstra’s Shortest Path Prim’s MST Huffman Encoding Heap Sort 40 COMP 620 Algorithm Analysis Franklin University Binary Heap A heap is a complete binary tree such that the value of the key in the root is greater than the value of the key in each of its children, and that both subtrees are also heaps (a recursive definition). http://homepages.ius.edu/rwisman/C455/html/notes/Chapte r6/HeapSort.htm Heapsort: (1) Make a heap of the elements to be sorted. (2) Convert the heap into a sorted list. 10 6 9 3 2 5 41 COMP 620 Algorithm Analysis Franklin University Heap Sort HeapSort(A) 1 Build_Max_Heap(A) 2 for i ← length[A] downto 2 do 3 exchange A[1] ↔ A[i] 4 heap-size[A] ← heap-size[A] – 1 5 Max_Heapify(A, 1) Build_Max_Heap(A) 1 heap-size[A] ← length[A] 2 for i ← floor(length[A]/2) downto 1 do 3 Max_Heapify (A, i) 42 COMP 620 Algorithm Analysis Franklin University HEAP SORT Heapify(A, i) 1 l ← LEFT(i) 2 r ← RIGHT(i) 3 if l ≤ heap-size[A] and A[l] > A[i] 4 then largest ← l 5 else largest ← i 6 if r ≤ heap-size[A] and A[r] > A[largest] 7 then largest ← r 8 if largest ≠ i 9 then exchange A[i] ↔ A[largest] 10 Max_Heapify (A, largest) 43 COMP 620 Algorithm Analysis Franklin University HEAPS Operation Linked List Binary Heap Binomial Fibonacci Heap Heap make-heap 1 1 1 1 insert 1 Log N Log N 1 find-min N 1 Log N 1 delete-min N Log N Log N Log N decrease-key 1 Log N Log N 1 delete N Log N Log N Log N union 1 N Log N 1 44 COMP 620 Algorithm Analysis Franklin University BINOMIAL TREE • Recursive Definition : Bk consists of 2 binomial trees B k-1 that are linked together; the root of the one is the leftmost Child of the root of the other. B0 B1 B2 B3 45 COMP 620 Algorithm Analysis Franklin University Binomial Tree Properties • • • • Number of nodes = 2 k Height of the tree = k k There are nodes i nodes at level i The root has degree k and it children are B k-1, B k-2,.. B 0 from left to right. 46 COMP 620 Algorithm Analysis Franklin University Binomial Heap • A binomial heap is a set of binomial trees • Each binomial tree in H is heap-ordered key (x) >= key (parent(x)). • There never exist 2 or more trees with same degree in the heap. • Linked list of roots in order of increasing degree • Binomial tree is stored in a left child-right sibling representation. 47 COMP 620 Algorithm Analysis Franklin University Binomial Heap Roots of the trees connected with singly linked list Head points to the first node in the linked list Head 7 5 3 6 B0 Parent 10 4 Key degree B1 17 B2 Sibling Child V i e View the operations on binomial heap at the class web site 48 COMP 620 Algorithm Analysis Franklin University Binomial Heap MAKE_BINOMIAL_HEAP() allocate(H) head[H] = NIL Insert(H, x) H’ = MAKE-BINOMIAL-HEAP() set x’s field appropriately head[H’] = x n[H’] = 1 H = BINOMIAL-HEAP-UNION(H, H’] 49 COMP 620 Algorithm Analysis Franklin University Binomial Heap Binomial-Heap-Minimum(H) y = NIL x = head[H] min = inf while x <> NIL do if key[x] < min then min = key[x] y=x x = sibling[x] return y; 50 COMP 620 Algorithm Analysis Franklin University Binomial Heap BINOMIAL-HEAP-UNION (H1, H2) Merge the root lists of binomial heaps H1 and H2 into a single linked linked list H that is sorted by degree into monotonically increasing order. Links roots of equal degree until at most one root remains of each degree. Binomial-Link (y,z) p[y] = z sibling[y] = child[z] child[z] = y degree [z] = degree[z] + 1 51 COMP 620 Algorithm Analysis Franklin University Binomial Heap BINOMIAL-HEAP-EXTRACT-MIN(H) find the root x with the minimum key in the root list of H, and remove x from the root list of H H’ = MAKE-BINOMIAL-HEAP() reverse the order of the linked list of x’s children and set head[H’] to point to the head of the resulting list H = BINOMIAL-HEAP-UNION (H, H’) return (x) 52 COMP 620 Algorithm Analysis Franklin University Fibonacci Heap • A set of min-heap-ordered trees • Roots of trees are connected with circular doubly linked list. Children of a node are connected with circular doubly linked list. • Pointer to root of tree with minimum element. Parent Key Mark: Newly created nodes are unmarked. This field becomes true if the node has lost a child Since the node became a child Of another node. Degree Left Right Child 53 COMP 620 Algorithm Analysis Franklin University Fibonacci Heap View the operations on Fibonacci heap at the class web site 23 Min[H] 3 7 18 39 52 24 17 38 30 41 26 46 35 54 COMP 620 Algorithm Analysis Franklin University Fibonacci Heap Make_heap() allocate(H) min[H] = NIL n[H]= 0 Insert(H, x) set x’s field appropriately add x to root list of H reset min[H] if needed n[H] = n[H] + 1 55 COMP 620 Algorithm Analysis Franklin University Fibonacci Heap Union(H1, H2) H = new heap whose root list contains roots from H1 and H2 Min[H] = min[H1] If ( min[H1] == NIL) or (min [H2] <> NIL and min[H2] < min[H1]) then min [H] = min[H2] N[H] = n[H1] + n[H2] free the objects H1 and H2 return H 56 COMP 620 Algorithm Analysis Franklin University Fibonacci Heap Extract-min (H) z = min[H] Add z’s children to root list Remove z from root list If root list <> {} then Consolidate H else min[H] = NIL n[H] = n[H] - 1 57 COMP 620 Algorithm Analysis Franklin University Fibonacci Heap Consolidate (H) While 2 trees in H (T1, T2) have the same degree: if root(T1) < root (T2) then make T1 child of T2 else make T2 child of T1 for i = 0 to D(n[H] ) // D is max possible trees if tree T of degree i has root-key < min[H] then min[H] = T 58 COMP 620 Algorithm Analysis Franklin University • • • • • • • • • • • • • • • • • • • • • • Fibonacci Heap Decrease Key FIB-HEAP-DECREASE-KEY(H, x, k) 1 if k > key[x] 2 then error "new key is greater than current key" 3 key[x] ← k 4 y ← p[x] 5 if y ≠ NIL and key[x] < key[y] 6 then CUT(H, x, y) 7 CASCADING-CUT(H, y) 8 if key[x] < key[min[H]] 9 then min[H] ← x CUT(H, x, y) 1 remove x from the child list of y, decrementing degree[y] 2 add x to the root list of H 3 p[x] ← NIL 4 mark[x] ← FALSE CASCADING-CUT(H, y) 1 z ← p[y] 2 if z ≠ NIL 3 then if mark[y] = FALSE 4 then mark[y] ← TRUE 5 else CUT(H, y, z) 6 CASCADING-CUT(H, z) 59 COMP 620 Algorithm Analysis Franklin University Fibonacci Heap Decrease Key 60 COMP 620 Algorithm Analysis Franklin University Fibonacci Heap Decrease Key a. b. The initial Fibonacci heap The node with key 46 has its key decreased to 15. The node becomes a root, and its parent ( with key 24) which had previously been unmarked, becomes marked. c-e The node with key 35 has its key decreased to 5. In part (c) , the node now with key 5, becomes a root. Its parent, with key 26, is cut from its parent and made an unmarked root in (d). Another cascading cut occurs, since the node with key 24 is marked as well. This node is cut from its parent and made an unmarked root in part (e). The cascading cuts stop at this point, since the node with key 7 is a root. 61 COMP 620 Algorithm Analysis Franklin University Fibonacci Heap Delete Node • FIB-HEAP-DELETE(H, x) • 1 FIB-HEAP-DECREASE-KEY(H, x, -∞) • 2 FIB-HEAP-EXTRACT-MIN(H) 62