CSC401 – Analysis of Algorithms Lecture Notes 4 Trees and Priority Queues Objectives: General Trees and ADT Properties of Trees Tree Traversals Binary Trees Priority Queues and ADT

2 The Tree Structure 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 Computers”R”Us SalesR&DManufacturing LaptopsDesktops US International EuropeAsiaCanada

3 subtree Tree Terminology 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. A B DC GH E F IJ K Subtree: tree consisting of a node and its descendants

4 Tree ADT We use positions to abstract nodes Generic methods: –integer size() –boolean isEmpty() –objectIterator elements() –positionIterator positions() Accessor methods: –position root() –position parent(p) –positionIterator children(p) Query methods: –boolean isInternal(p) –boolean isExternal(p) –boolean isRoot(p) Update methods: –swapElements(p, q) –object replaceElement(p, o) Additional update methods may be defined by data structures implementing the Tree ADT

5 Depth and Height Depth -- the depth of v is the number of ancestors, excluding v itself –the depth of the root is 0 –the depth of v other than the root is one plus the depth of its parent –time efficiency is O(1+d) Height -- the height of a subtree v is the maximum depth of its external nodes –the height of an external node is 0 –the height of an internal node v is one plus the maximum height of its children –time efficiency is O(n) Algorithm depth(T,v) if T.isRoot(v) then return 0 else return 1+depth(T, T.parent(v)) Algorithm height(T,v) if T.isExternal(v) then return 0 else h=0; for each w  T.children(v) do h=max(h, height(T,w)) return 1+h

6 Preorder Traversal A traversal visits the nodes of a tree in a systematic manner In a preorder traversal, a node is visited before its descendants The running time is O(n) Application: print a structured document Make Money Fast! 1. MotivationsReferences2. Methods 2.1 Stock Fraud 2.2 Ponzi Scheme 1.1 Greed1.2 Avidity 2.3 Bank Robbery Algorithm preOrder(v) visit(v) for each child w of v preorder (w)

7 Postorder Traversal In a postorder traversal, a node is visited after its descendants The running time is O(n) Application: compute space used by files in a directory and its subdirectories Algorithm postOrder(v) for each child w of v postOrder (w) visit(v) cs16/ homeworks/ todo.txt 1K programs/ DDR.java 10K Stocks.java 25K h1c.doc 3K h1nc.doc 2K Robot.java 20K

8 Binary Tree A binary tree is a tree with the following properties: –Each internal node has two children –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 Applications: –arithmetic expressions –decision processes –searching A B C FG D E H I

9 Binary Tree Examples Arithmetic expression binary tree –internal nodes: operators –external nodes: operands –Example: arithmetic expression tree for the expression (2 ( a1)(3 b))    2 a1 3b Decision tree –internal nodes: questions with yes/no answer –external nodes: decisions –Example: dining decision Want a fast meal? How about coffee?On expense account? StarbucksSpike’sAl FornoCafé Paragon Yes No YesNoYesNo

10 Properties of Binary Trees Notation n number of nodes e number of external nodes i number of internal nodes h height Properties: –e  i  1 –n  2e  1 –h  i –h  (n  1)  2 –h+1  e  2 h –h  log 2 e –h  log 2 (n  1)  1

11 BinaryTree ADT The BinaryTree ADT extends the Tree ADT, i.e., it inherits all the methods of the Tree ADT Additional methods: –position leftChild(p) –position rightChild(p) –position sibling(p) Update methods may be defined by data structures implementing the BinaryTree ADT

12 Inorder Traversal In an inorder traversal a node is visited after its left subtree and before its right subtree Time efficiency is O(n) Application: draw a binary tree –x(v) = inorder rank of v –y(v) = depth of v Algorithm inOrder(v) if isInternal (v) inOrder (leftChild (v)) visit(v) if isInternal (v) inOrder (rightChild (v))

13 Print Arithmetic Expressions Specialization of an inorder traversal –print operand or operator when visiting node –print “(“ before traversing left subtree –print “)“ after traversing right subtree Algorithm printExpression(v) if isInternal (v) print( “(’’ ) inOrder (leftChild (v)) print(v.element ()) if isInternal (v) inOrder (rightChild (v)) print ( “)’’ )    2 a1 3b ((2  ( a  1))  (3  b))

14 Evaluate Arithmetic Expressions Specialization of a postorder traversal –recursive method returning the value of a subtree –when visiting an internal node, combine the values of the subtrees Algorithm evalExpr(v) if isExternal (v) return v.element () else x  evalExpr(leftChild (v)) y  evalExpr(rightChild (v))   operator stored at v return x  y   

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 B R 

16 Template Method Pattern 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 public abstract class EulerTour { protected BinaryTree tree; protected void visitExternal(Position p, Result r) { } protected void visitLeft(Position p, Result r) { } protected void visitBelow(Position p, Result r) { } protected void visitRight(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.leftChild(p)); visitBelow(p, r); r.rightResult = eulerTour(tree.rightChild(p)); visitRight(p, r); return r.finalResult; } …

17 Specializations of EulerTour 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) public class EvaluateExpression extends EulerTour { protected void visitExternal(Position p, Result r) { r.finalResult = (Integer) p.element(); } protected void visitRight(Position p, Result r) { Operator op = (Operator) p.element(); r.finalResult = op.operation( (Integer) r.leftResult, (Integer) r.rightResult ); } … }

18 Data 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 D A CE F  B  ADF  C  E

19 Data 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 D A CE    B AD CE 

20 Vector-Based Binary Tree Level numbering of nodes of T: p(v) –if v is the root of T, p(v)=1 –if v is the left child of u, p(v)=2p(u) –if v is the right child of u, p(v)=2p(u)+1 Vector S storing the nodes of T by putting the root at the second position and following the above level numbering Properties: Let n be the number of nodes of T, N be the size of the vector S, and PM be the maximum value of p(v) over all the nodes of T –N=PM+1 –N=2^((n+1)/2)

21 Java Implementation Tree interface BinaryTree interface extending Tree Classes implementing Tree and BinaryTree and providing –Constructors –Update methods –Print methods Examples of updates for binary trees –expandExternal( v ) –removeAboveExternal( w ) A  expandExternal( v ) A CB B removeAboveExternal( w ) A v v w

22 Trees in JDSL JDSL is the Library of Data Structures in Java Tree interfaces in JDSL –InspectableBinaryTree –InspectableTree –BinaryTree –Tree Inspectable versions of the interfaces do not have update methods Tree classes in JDSL –NodeBinaryTree –NodeTree JDSL was developed at Brown’s Center for Geometric Computing See the JDSL documentation and tutorials at InspectableTree InspectableBinaryTree Tree BinaryTree

23 Priority Queue ADT A priority queue stores a collection of items An item is a pair (key, element) Main methods of the Priority Queue ADT –insertItem(k, o) -- inserts an item with key k and element o –removeMin() -- removes the item with smallest key and returns its element Additional methods –minKey(k, o) -- returns, but does not remove, the smallest key of an item –minElement() -- returns, but does not remove, the element of an item with smallest key –size(), isEmpty()Applications: –Standby flyers –Auctions –Stock market

24 Total Order Relation Keys in a priority queue can be arbitrary objects on which an order is defined Two distinct items 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

25 Comparator ADT 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 Methods of the Comparator ADT, all with Boolean return type –isLessThan(x, y) –isLessThanOrEqualTo(x, y) –isEqualTo(x,y) –isGreaterThan(x, y) –isGreaterThanOrEqualTo (x,y) –isComparable(x)

26 Sorting with a Priority Queue We can use a priority queue to sort a set of comparable elements –Insert the elements one by one with a series of insertItem(e, e) 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 Algorithm PQ-Sort(S, C) Input sequence S, comparator C for the elements of S Output sequence 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)

27 Sequence-based Priority Queue Implementation with an unsorted sequence –Store the items of the priority queue in a list- based sequence, in arbitrary order Performance: –insertItem takes O(1) time since we can insert the item at the beginning or end of the sequence –removeMin, minKey and minElement take O(n) time since we have to traverse the entire sequence to find the smallest key Implementation with a sorted sequence –Store the items of the priority queue in a sequence, sorted by keyPerformance: –insertItem takes O(n) time since we have to find the place where to insert the item –removeMin, minKey and minElement take O(1) time since the smallest key is at the beginning of the sequence

28 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 n insertItem operations takes O(n) time –Removing the elements in sorted order from the priority queue with n removeMin operations takes time proportional to 1  2  …  n Selection-sort runs in O(n 2 ) time

29 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 n insertItem operations takes time proportional to 1  2  …  n –Removing the elements in sorted order from the priority queue with a series of n removeMin operations takes O(n) time Insertion-sort runs in O(n 2 ) time

30 In-place Insertion-sort 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 swapElements instead of modifying the sequence