Algorithms and Data Structures (INF1) Lecture 8/15 Hua Lu

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1 Algorithms and Data Structures (INF1) Lecture 8/15 Hua Lu Department of Computer Science Aalborg University Fall 2007

2 This Lecture Trees Basics Rooted trees Binary trees Binary tree ADT Tree traversal Symbol tables Symbol table ADT Linked lists 2

3 Motivation Problem for Trees Three coins with the identical appearance, two are of the same weight, while the other is different. How can you find out the different one? Number them, and then compare them compare 1, 2 1<2 1=2 1>2 compare 1, 3 answer=3 Symmetric 1<3 1=3 answer=1 answer=2 This is a tree structure Decision tree 3

4 More Examples (1) Containment relationship AD (INF-1) Lecture 1 Lecture 2 Introduction Algorithm Correctness Total Partial 4

5 More Examples (2) Expression (4-3)*(2+1) * Trees are very useful and powerful in organizing and representing hierarchical data 5

6 Formal Definition A tree is a set of nodes (vertices) connected by edges (links) such that there is exactly one way to get from any node to any other node. Examples Which of them are trees? Yes No No We will study cycles later. A general model: Graph Forest: multiple tress 6

7 Theorem Every non-empty tree with n nodes has exactly n-1 edges How to prove? Induction (n 1) Examples We can use the theorem to disprove a given structure is a tree. nodes: 3, edges: 3 nodes: 9, edges: 9 7

8 Rooted Tree A tree is called a rooted tree if one of its node is distinguished as root. Recursive definition root 5 A rooted tree consists of a root node and a finite set of sub-trees, which are themselves rooted trees. root We usually put a root at the top. 5 subtrees 8

9 Terminology r is root y is a parent of x (z); r is a parent of y r y r, y and x are ancestors of x r and y are proper ancestors of x w x z x and z are children of y u x, y, z and u are descendants of r x and z are siblings All ancestors of u form a path from u to the root <u, z, y, r> Leaf is a node without any children w, x and u in our example Others are internal nodes 9

10 Sub-Trees Sub-tree A node n plus all its descendants Example n is the root of the sub-tree y is the root of the sub-tree Order of sub-trees often matters! - - x r y z u Definition An ordered tree consists of a root node and a finite sequence of sub-trees, which are themselves ordered trees. Recursive again! You will see a lot recursions in trees. 10

11 Binary Tree A binary tree is either empty or it consists of a root node and two subtrees (left and right) which are themselves binary trees. root A special type of ordered trees Examples Which are binary trees? left subtree T1 T2 T3 T4 right subtree Are these two binary trees the same? 11

12 Height and Depth in Binary Trees Empty binary trees Represented in blank squares, called external nodes The height of a node n in a binary tree is the number of edges on the longest path from n to an external node The height of a rooted tree is the height of its root The height of an external node is 0 (special leaf node) The depth of a node is the length of the path to the root The root has depth 0; depths of external nodes can be different height depth

13 Binary Tree ADT Operations (1) Mathematical entity: binary trees of entries (nodes) make new_entry(val:value_type):entry_type Returns a new entry with value val new_nil_entry():entry_type Returns a nil entry (an empty tree) nil_entry(x:entry_type):boolean true if and only if x is NIL empty():boolean True if and only if the tree is empty root:entry_type Returns the root of the tree 13

14 Binary Tree ADT Operations (2) put_root(x:entry_type) Makes x the root of the tree. x can be NIL left_child(x:entry_type):entry_type Returns the left child of x, possibly NIL right_child(x:entry_type):entry_type Returns the right child of x, possibly NIL put_left_child(x,y:entry_type):entry_type Replace the left child of x with y put_right_child(x,y:entry_type):entry_type Replace the right child of x with y empty, root, left_child, right_child do not change the tree. 14

15 Examples of Binary Tree ADT t.make t is an empty tree z:=t.empty z=true z:=t.root z=nil x:=t.new_entry(5) x.value=5 y:=t.new_entry(6) y.value=6 t.put_left_child(x,y) z:=t.empty z=true t.put_root(x) t= x y NIL y x NIL z:=t.left_child(t.root).value z=6 y:=t.nil_entry y=nil t.put_left_child(t.root,y) t= NIL x NIL 15

16 Practical Problems t.empty=t.nil_entry(t.root) Count the number of nodes in a tree t count(x:entry_type):int if (t.nil_entry(x)) then sum:=0; else sum:=1 + count(t.left_child(x)) + count(t.right_child(x)); return sum; count(t.root) returns the number of nodes in t Recursion in trees is so natural! 16

17 Implementation of Binary Tree A tree is referenced by its root root Reference to the root or NIL (empty tree) Node x has three parts x.left reference to its left child All nodes can be organized into a tree via their pointers Similar way to the linked lists x.value All operations can be done in O(1) time How to find the parent of a node x in a tree t? O(n), where n is the number of nodes of t Add in node x a reference to its parent : O(1) x.right reference to its right child 17

18 Traversing Binary Trees How to visit each node exactly once given a binary tree? Starting from the root, you can have three choices Visiting the root itself; Visiting the root s left subtree; Visiting the root s right subtree; Three different orders three main paradigms preorder traversal inorder traversal postorder traversal 18

19 Traversals Given a tree t and a node x, traverse all nodes bellow x preorder(x:entry_type) if (NOT t.nil(x)) then visit(x); preorder(t.left_child(x)); preorder(t.right_child(x)); inorder(x:entry_type) if (NOT t.nil(x)) then inorder(t.left_child(x)); visit(x); inorder(t.right_child(x)); postorder(x:entry_type) if (NOT t.nil(x)) then postorder(t.left_child(x)); postorder(t.right_child(x)); visit(x); 19

20 Traversal Example To traverse a tree t Call preorder(t.root), inorder(t.root), postorder(t.root) Arithmetical expression evaluation example 5+(2-1) preorder inorder postorder Reverse Polish notation! Preorder and postorder traversals are depth-first traversal Breadth-first traversal Ambiguous! Parentheses needed

21 Breadth-First Traversal Trees can also be traversed in level-order, where we visit every node on a level before going to a lower level A tree example r Breadth-first traversal? u v r, u, v, w, x, y, z w x y z 21

22 Symbol Tables How to store/access data with a given identification key? Phone company information system key: phone number data: user name, address, plan, balance Kommune information system key: CPR number data: name, address, Common requirement of such problems Fast access to data via key Data structure support Symbol tables (a.k.a. dictionaries) A key can be regarded as a word, while the data is the detailed explanation on that word in a dictionary 22

23 Symbol Table ADT A symbol table is an ADT whose mathematical entity is a set of entries of the form (key, value) such that each key uniquely identifies its entry All keys alone also form a set Given an entry x, its key is x.key and its value is x.value A symbol table is a mapping from keys to values Operations make nil_entry(x:entry_type):boolean new_entry(key:key_type, val:value_type):entry_type insert(x:entry_type) delete(x:entry_type) retrieve(key:key_type):entry_type Returns entry x s.t. x.key=key and NIL otherwise 23

24 Symbol Table ADT Example s:symbol_table s.make s={} x:=s.new_entry(2, Jan ) x.key=2, x.value= Jan y:=s.new_entry(5, Lars ) y.key=5, y.value= Lars s.insert(x) s={(2, Jan )} s.insert(y) s={(2, Jan ), (5, Lars )} z:=s.retrieve(2).value z= Jan z:=s.retrieve(3) z=nil s.delete(s.retrieve(5)) s={(2, Jan )} 24

25 Additional Operations An ordering of all keys (then also all entries) is defined retrieve_first:entry_type Returns the entry with minimum key, NIL otherwise retrieve_last:entry_type retrieve_from(key:key_type):entry_type Returns the entry with minimum key not less than the given key, NIL otherwise retrieve_upto(key:key_type):entry_type Returns the entry with maximum key not greater than the given key, NIL otherwise retrieve_next(x:entry_type):entry_type Returns the next entry after x, NIL otherwise retrieve_prev(x:entry_type):entry_type 25

26 Linked List Implementation s={(k 1, v 1 ), (k 2, v 2 ),, (k n, v n )}} a 1 a 2 a n first entry key value k 1 k 2 k 3 v 1 v 2 v 3 a 1 a 2 a 3 next k n v n a n NIL In an ordered symbol table where k 1 <k 2 < <k n retrieve(key:key_type):entry_type res:=first while res NIL and res.key key do res:=res.next return res Linear search O(n) 26

27 Improvements on Linked List Locality of reference on symbol tables is the tendency for retrieving data of a speific key repeatedly E.g., retrieve(k 1 ), retrieve(k 2 ), retrieve(k 1 ), retrieve(k 1 ) Such tendency can be exploited to improve access speed Heuristic is a method whose advantages seem real but not easily quatifiable or proveble Heuristics on linked list based symbol table Move to front After retrieve(k x ), move its entry to the front of the list Transpose After retrieve(k x ), if it is not the front, exchange it with its predecessor Frequency count Each entry keeps the number of times it has been accessed, and keeps the link sorted in non-increasing order of that number Worst case cost is still O(n) We will study other implementations 27

28 Next Lecture Binary search trees Splay trees B-trees Hashing 28

29 Autumn Break More than half of this course has been finished! What s left? Symbol table implementations (1) Algorithm design techniques (2) Graphs and relevant algorithms (4) Take the chance of break to Finish all undone exercises Mark those you feel difficult for consultation later Take a look at previous exam questions and answers Available on the course website now Go over lecture slides Feedback if any Have fun! 29

Algorithms and Data Structures

Algorithms and Data Structures Lesson 3: trees and visits Luciano Bononi http://www.cs.unibo.it/~bononi/ (slide credits: these slides are a revised version of slides created by Dr. Gabriele D Angelo) International