Trees

1. Trees Make Money Fast! BankRobbery StockFraud PonziScheme

2. Computers”R”Us Sales Manufacturing R&D US International Laptops Desktops Europe Asia Canada What is a Tree • In computer science, a tree is an abstract model of a hierarchical structure • A tree consists of nodes with a parent-child relation • Applications: • Organization charts • File systems • Programming environments

3. A C D B E G H F K I J Tree Terminology • Subtree: tree consisting of a node and its descendants • Root: node without parent (A) • Internal node: node with at least one child (A, B, C, F) • External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, D) • Ancestors of a node: parent, grandparent, grand-grandparent, etc. • Depth of a node: number of ancestors • Height of a tree: maximum depth of any node (3) • Descendant of a node: child, grandchild, grand-grandchild, etc. subtree

4. Tree ADT (§ 6.1.2) • Query methods: • boolean isInternal(p) • boolean isExternal(p) • boolean isRoot(p) • Update method: • object replace (p, o) • Additional update methods may be defined by data structures implementing the Tree ADT • We use positions to abstract nodes • Generic methods: • integer size() • boolean isEmpty() • Iterator elements() • Iterator positions() • Accessor methods: • position root() • position parent(p) • positionIterator children(p)

5. Preorder Traversal AlgorithmpreOrder(v) visit(v) foreachchild w of v preorder (w) • A traversal visits the nodes of a tree in a systematic manner • In a preorder traversal, a node is visited before its descendants • Application: print a structured document 1 Make Money Fast! 2 5 9 1. Motivations 2. Methods References 6 7 8 3 4 2.3 BankRobbery 2.1 StockFraud 2.2 PonziScheme 1.1 Greed 1.2 Avidity

6. Postorder Traversal AlgorithmpostOrder(v) foreachchild w of v postOrder (w) visit(v) • In a postorder traversal, a node is visited after its descendants • Application: compute space used by files in a directory and its subdirectories 9 cs16/ 8 3 7 todo.txt1K homeworks/ programs/ 4 5 6 1 2 Robot.java20K h1c.doc3K h1nc.doc2K DDR.java10K Stocks.java25K

7. Binary Trees (§ 6.3) • Applications: • arithmetic expressions • decision processes • searching • A binary tree is a tree with the following properties: • Each internal node has at most two children (exactly two for proper binary trees) • The children of a node are an ordered pair • We call the children of an internal node left child and right child • Alternative recursive definition: a binary tree is either • a tree consisting of a single node, or • a tree whose root has an ordered pair of children, each of which is a binary tree A C B D E F G I H

8. +   2 - 3 b a 1 Arithmetic Expression Tree • Binary tree associated with an arithmetic expression • internal nodes: operators • external nodes: operands • Example: arithmetic expression tree for the expression (2  (a - 1) + (3  b))

9. Decision Tree • Binary tree associated with a decision process • internal nodes: questions with yes/no answer • external nodes: decisions • Example: dining decision Want a fast meal? No Yes How about coffee? On expense account? Yes No Yes No Starbucks Spike’s Al Forno Café Paragon

10. Properties of Proper Binary Trees • Properties: • e = i +1 • n =2e -1 • h  i • h  (n -1)/2 • e 2h • h log2e • h log2 (n +1)-1 • Notation n number of nodes e number of external nodes i number of internal nodes h height

11. The BinaryTree ADT extends the Tree ADT, i.e., it inherits all the methods of the Tree ADT Additional methods: position left(p) position right(p) boolean hasLeft(p) boolean hasRight(p) Update methods may be defined by data structures implementing the BinaryTree ADT BinaryTree ADT (§ 6.3.1)

12. Inorder Traversal AlgorithminOrder(v) ifhasLeft (v) inOrder (left (v)) visit(v) ifhasRight (v) inOrder (right (v)) • In an inorder traversal a node is visited after its left subtree and before its right subtree • Application: draw a binary tree • x(v) = inorder rank of v • y(v) = depth of v 6 2 8 1 4 7 9 3 5

13. +   2 - 3 b a 1 Print Arithmetic Expressions AlgorithmprintExpression(v) ifhasLeft (v)print(“(’’) inOrder (left(v)) print(v.element ()) ifhasRight (v) inOrder (right(v)) print (“)’’) • Specialization of an inorder traversal • print operand or operator when visiting node • print “(“ before traversing left subtree • print “)“ after traversing right subtree ((2  (a - 1)) + (3  b))

14. +   2 - 3 2 5 1 Evaluate Arithmetic Expressions AlgorithmevalExpr(v) ifisExternal (v) returnv.element () else x evalExpr(leftChild (v)) y evalExpr(rightChild (v))  operator stored at v returnx  y • Specialization of a postorder traversal • recursive method returning the value of a subtree • when visiting an internal node, combine the values of the subtrees

15. Euler Tour Traversal • Generic traversal of a binary tree • Includes a special cases the preorder, postorder and inorder traversals • Walk around the tree and visit each node three times: • on the left (preorder) • from below (inorder) • on the right (postorder) +   L R B 2 - 3 2 5 1

16. Template Method Pattern public abstract class EulerTour{ protected BinaryTree tree;protected voidvisitExternal(Position p, Result r) { }protected voidvisitLeft(Position p, Result r) { }protected voidvisitBelow(Position p, Result r) { }protected voidvisitRight(Position p, Result r) { }protected Object eulerTour(Position p) { Result r = new Result();if tree.isExternal(p) { visitExternal(p, r); }else {visitLeft(p, r); r.leftResult = eulerTour(tree.left(p));visitBelow(p, r); r.rightResult = eulerTour(tree.right(p)); visitRight(p, r);return r.finalResult; } … • Generic algorithm that can be specialized by redefining certain steps • Implemented by means of an abstract Java class • Visit methods that can be redefined by subclasses • Template method eulerTour • Recursively called on the left and right children • A Result object with fields leftResult, rightResult and finalResult keeps track of the output of the recursive calls to eulerTour

17. Specializations of EulerTour public class EvaluateExpressionextends EulerTour{ protected voidvisitExternal(Position p, Result r) {r.finalResult = (Integer) p.element(); } protected voidvisitRight(Position p, Result r) {Operator op = (Operator) p.element();r.finalResult = op.operation( (Integer) r.leftResult, (Integer) r.rightResult ); } … } • We show how to specialize class EulerTour to evaluate an arithmetic expression • Assumptions • External nodes store Integer objects • Internal nodes store Operator objects supporting method operation (Integer, Integer)

18.  Linked Structure for Trees • A node is represented by an object storing • Element • Parent node • Sequence of children nodes • Node objects implement the Position ADT B   A D F B A D F   C E C E

19. D C A B E Linked Structure for Binary Trees  • A node is represented by an object storing • Element • Parent node • Left child node • Right child node • Node objects implement the Position ADT   B A D     C E

20. A … B D C E F J G H Array-Based Representation of Binary Trees • nodes are stored in an array 1 2 3 • let rank(node) be defined as follows: • rank(root) = 1 • if node is the left child of parent(node), rank(node) = 2*rank(parent(node)) • if node is the right child of parent(node), rank(node) = 2*rank(parent(node))+1 4 5 6 7 10 11

21. Priority Queues

22. A priority queue stores a collection of entries Each entry is a pair(key, value) Main methods of the Priority Queue ADT insert(k, x)inserts an entry with key k and value x removeMin()removes and returns the entry with smallest key Additional methods min()returns, but does not remove, an entry with smallest key size(), isEmpty() Applications: Standby flyers Auctions Stock market Priority Queue ADT (§ 7.1.3)

23. Keys in a priority queue can be arbitrary objects on which an order is defined Two distinct entries in a priority queue can have the same key Mathematical concept of total order relation  Reflexive property:x  x Antisymmetric property:x  yy  x  x = y Transitive property:x  yy  z  x  z Total Order Relations (§ 7.1.1)

24. An entry in a priority queue is simply a key-value pair Priority queues store entries to allow for efficient insertion and removal based on keys Methods: key(): returns the key for this entry value(): returns the value associated with this entry As a Java interface: /** * Interface for a key-value * pair entry **/ public interface Entry { public Object key(); public Object value(); } Entry ADT (§ 7.1.2)

25. A comparator encapsulates the action of comparing two objects according to a given total order relation A generic priority queue uses an auxiliary comparator The comparator is external to the keys being compared When the priority queue needs to compare two keys, it uses its comparator The primary method of the Comparator ADT: compare(x, y): Returns an integer i such that i < 0 if a < b, i = 0 if a = b, and i > 0 if a > b; an error occurs if a and b cannot be compared. Comparator ADT (§ 7.1.2)

26. Lexicographic comparison of 2-D points: /** Comparator for 2D points under the standard lexicographic order. */ public class Lexicographic implements Comparator { int xa, ya, xb, yb; public int compare(Object a, Object b) throws ClassCastException { xa = ((Point2D) a).getX(); ya = ((Point2D) a).getY(); xb = ((Point2D) b).getX(); yb = ((Point2D) b).getY(); if (xa != xb) return (xb - xa); else return (yb - ya); } } Point objects: /** Class representing a point in the plane with integer coordinates */ public class Point2D { protected int xc, yc; // coordinates public Point2D(int x, int y) { xc = x; yc = y; } public int getX() { return xc; } public int getY() { return yc; } } Example Comparator

27. Priority Queue Sorting (§ 7.1.4) AlgorithmPQ-Sort(S, C) • Inputsequence S, comparator C for the elements of S • Outputsequence S sorted in increasing order according to C P priority queue with comparator C whileS.isEmpty () e S.removeFirst () P.insert (e, 0) whileP.isEmpty() e P.removeMin().key() S.insertLast(e) • We can use a priority queue to sort a set of comparable elements • Insert the elements one by one with a series of insert operations • Remove the elements in sorted order with a series of removeMin operations • The running time of this sorting method depends on the priority queue implementation

28. Implementation with an unsorted list Performance: insert takes O(1) time since we can insert the item at the beginning or end of the sequence removeMin and min take O(n) time since we have to traverse the entire sequence to find the smallest key Implementation with a sorted list Performance: insert takes O(n) time since we have to find the place where to insert the item removeMin and min take O(1) time, since the smallest key is at the beginning 4 5 2 3 1 1 2 3 4 5 Sequence-based Priority Queue

29. Selection-Sort • Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence • Running time of Selection-sort: • Inserting the elements into the priority queue with ninsert operations takes O(n) time • Removing the elements in sorted order from the priority queue with nremoveMin operations takes time proportional to1 + 2 + …+ n • Selection-sort runs in O(n2) time

30. Selection-Sort Example Sequence S Priority Queue P Input: (7,4,8,2,5,3,9) () Phase 1 (a) (4,8,2,5,3,9) (7) (b) (8,2,5,3,9) (7,4) .. .. .. . . . (g) () (7,4,8,2,5,3,9) Phase 2 (a) (2) (7,4,8,5,3,9) (b) (2,3) (7,4,8,5,9) (c) (2,3,4) (7,8,5,9) (d) (2,3,4,5) (7,8,9) (e) (2,3,4,5,7) (8,9) (f) (2,3,4,5,7,8) (9) (g) (2,3,4,5,7,8,9) ()

31. Insertion-Sort • Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence • Running time of Insertion-sort: • Inserting the elements into the priority queue with ninsert operations takes time proportional to1 + 2 + …+ n • Removing the elements in sorted order from the priority queue with a series of nremoveMin operations takes O(n) time • Insertion-sort runs in O(n2) time

32. Insertion-Sort Example Sequence S Priority queue P Input: (7,4,8,2,5,3,9) () Phase 1 (a) (4,8,2,5,3,9) (7) (b) (8,2,5,3,9) (4,7) (c) (2,5,3,9) (4,7,8) (d) (5,3,9) (2,4,7,8) (e) (3,9) (2,4,5,7,8) (f) (9) (2,3,4,5,7,8) (g) () (2,3,4,5,7,8,9) Phase 2 (a) (2) (3,4,5,7,8,9) (b) (2,3) (4,5,7,8,9) .. .. .. . . . (g) (2,3,4,5,7,8,9) ()

33. Instead of using an external data structure, we can implement selection-sort and insertion-sort in-place A portion of the input sequence itself serves as the priority queue For in-place insertion-sort We keep sorted the initial portion of the sequence We can use swaps instead of modifying the sequence 5 4 2 3 1 5 4 2 3 1 4 5 2 3 1 2 4 5 3 1 2 3 4 5 1 1 2 3 4 5 1 2 3 4 5 In-place Insertion-sort

34. Heaps 2 5 6 9 7

35. A priority queue stores a collection of entries Each entry is a pair(key, value) Main methods of the Priority Queue ADT insert(k, x)inserts an entry with key k and value x removeMin()removes and returns the entry with smallest key Additional methods min()returns, but does not remove, an entry with smallest key size(), isEmpty() Applications: Standby flyers Auctions Stock market Recall Priority Queue ADT (§ 7.1.3)

36. Recall Priority Queue Sorting (§ 7.1.4) • We can use a priority queue to sort a set of comparable elements • Insert the elements with a series of insert operations • Remove the elements in sorted order with a series of removeMin operations • The running time depends on the priority queue implementation: • Unsorted sequence gives selection-sort: O(n2) time • Sorted sequence gives insertion-sort: O(n2) time • Can we do better? AlgorithmPQ-Sort(S, C) • Inputsequence S, comparator C for the elements of S • Outputsequence S sorted in increasing order according to C P priority queue with comparator C whileS.isEmpty () e S.remove (S.first ()) P.insertItem(e, e) whileP.isEmpty() e P.removeMin() S.insertLast(e)

37. A heap is a binary tree storing keys at its nodes and satisfying the following properties: Heap-Order: for every internal node v other than the root,key(v)key(parent(v)) Complete Binary Tree: let h be the height of the heap for i = 0, … , h - 1, there are 2i nodes of depth i at depth h- 1, the internal nodes are to the left of the external nodes The last node of a heap is the rightmost node of depth h Heaps (§7.3) 2 5 6 9 7 last node

38. Height of a Heap (§ 7.3.1) • Theorem: A heap storing nkeys has height O(log n) Proof: (we apply the complete binary tree property) • Let h be the height of a heap storing n keys • Since there are 2i keys at depth i=0, … , h - 1 and at least one key at depth h, we have n1 + 2 + 4 + … + 2h-1 + 1 • Thus, n2h, i.e., hlog n depth keys 0 1 1 2 h-1 2h-1 h 1

39. Heaps and Priority Queues • We can use a heap to implement a priority queue • We store a (key, element) item at each internal node • We keep track of the position of the last node • For simplicity, we show only the keys in the pictures (2, Sue) (5, Pat) (6, Mark) (9, Jeff) (7, Anna)

40. Method insertItem of the priority queue ADT corresponds to the insertion of a key k to the heap The insertion algorithm consists of three steps Find the insertion node z (the new last node) Store k at z Restore the heap-order property (discussed next) Insertion into a Heap (§ 7.3.3) 2 5 6 z 9 7 insertion node 2 5 6 z 9 7 1

41. Upheap • After the insertion of a new key k, the heap-order property may be violated • Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node • Upheap terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k • Since a heap has height O(log n), upheap runs in O(log n) time 2 1 5 1 5 2 z z 9 7 6 9 7 6

42. Removal from a Heap (§ 7.3.3) 2 • Method removeMin of the priority queue ADT corresponds to the removal of the root key from the heap • The removal algorithm consists of three steps • Replace the root key with the key of the last node w • Remove w • Restore the heap-order property (discussed next) 5 6 w 9 7 last node 7 5 6 w 9 new last node

43. Downheap • After replacing the root key with the key k of the last node, the heap-order property may be violated • Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root • Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k • Since a heap has height O(log n), downheap runs in O(log n) time 7 5 5 6 7 6 w w 9 9

44. Updating the Last Node • The insertion node can be found by traversing a path of O(log n) nodes • Go up until a left child or the root is reached • If a left child is reached, go to the right child • Go down left until a leaf is reached • Similar algorithm for updating the last node after a removal

45. Consider a priority queue with n items implemented by means of a heap the space used is O(n) methods insert and removeMin take O(log n) time methods size, isEmpty, and min take time O(1) time Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time The resulting algorithm is called heap-sort Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort and selection-sort Heap-Sort (§2.4.4)

46. 2 5 6 9 7 0 1 2 3 4 5 Vector-based Heap Implementation (§2.4.3) 2 • We can represent a heap with n keys by means of a vector of length n +1 • For the node at rank i • the left child is at rank 2i • the right child is at rank 2i +1 • Links between nodes are not explicitly stored • The cell of at rank 0 is not used • Operation insert corresponds to inserting at rank n +1 • Operation removeMin corresponds to removing at rank n • Yields in-place heap-sort 5 6 9 7

47. Merging Two Heaps 3 2 • We are given two two heaps and a key k • We create a new heap with the root node storing k and with the two heaps as subtrees • We perform downheap to restore the heap-order property 8 5 4 6 7 3 2 8 5 4 6 2 3 4 8 5 7 6

48. 2i -1 2i -1 Bottom-up Heap Construction (§2.4.3) • We can construct a heap storing n given keys in using a bottom-up construction with log n phases • In phase i, pairs of heaps with 2i -1 keys are merged into heaps with 2i+1-1 keys 2i+1-1

49. Example 16 15 4 12 6 7 23 20 25 5 11 27 16 15 4 12 6 7 23 20

50. Example (contd.) 25 5 11 27 16 15 4 12 6 9 23 20 15 4 6 23 16 25 5 12 11 9 27 20