Data Structures Fibonacci Heaps, Amortized Analysis

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Data Structures Fibonacci Heaps, Amortized Analysis"

Transcription

1 Chapter 4 Data Structures Fibonacci Heaps, Amortized Analysis Algorithm Theory WS 2012/13 Fabian Kuhn

2 Fibonacci Heaps Lacy merge variant of binomial heaps: Do not merge trees as long as possible Structure: A Fibonacci heap consists of a collection of trees satisfying the min heap property. Variables:. : root of the tree containing the (a) minimum key. : circular, doubly linked, unordered list containing the roots of all trees. : number of nodes currently in Algorithm Theory, WS 2012/13 Fabian Kuhn 2

3 Trees in Fibonacci Heaps Structure of a single node : parent left key degree right child mark. : points to circular, doubly linked and unordered list of the children of.,. : pointers to siblings (in doubly linked list). : will be used later Advantages of circular, doubly linked lists: Deleting an element takes constant time Concatenating two lists takes constant time Algorithm Theory, WS 2012/13 Fabian Kuhn 3

4 Example Figure: Cormen et al., Introduction to Algorithms Algorithm Theory, WS 2012/13 Fabian Kuhn 4

5 Simple (Lazy) Operations Initialize Heap :.. Merge heaps and : concatenate root lists update. Insert element into : create new one node tree containing H merge heaps and Get minimum element of : return. Algorithm Theory, WS 2012/13 Fabian Kuhn 5

6 Operation Delete Min Delete the node with minimum key from and return its element: 1.. ; 2. if. 0 then 3. remove. from. ; 4. add.. (list) to. 5.. ; // Repeatedly merge nodes with equal degree in the root list // until degrees of nodes in the root list are distinct. // Determine the element with minimum key 6. return Algorithm Theory, WS 2012/13 Fabian Kuhn 6

7 Rank and Maximum Degree Ranks of nodes, trees, heap: Node : : degree of Tree : : rank (degree) of root node of Heap : : maximum degree of any node in Assumption ( : number of nodes in ): for a known function Algorithm Theory, WS 2012/13 Fabian Kuhn 7

8 Merging Two Trees Given: Heap ordered trees, with Assume: min key of min key of Operation, : Removes tree from root list and adds to child list of 1. Algorithm Theory, WS 2012/13 Fabian Kuhn 8

9 Consolidation of Root List Array pointing to find roots with the same rank: Consolidate: Time: 1. for 0to do null;. 2. while. null do 3. delete and return first element of. 4. while nulldo 5. ; 6. ; 7., Create new. and. Algorithm Theory, WS 2012/13 Fabian Kuhn 9

10 Operation Decrease Key Decrease Key, : (decrease key of node to new value ) 1. if. then return; 2.. ; update. ; 3. if... then return 4. repeat 5.. ; 6.. ; 7. ; 8. until.. ; 9. if. then. ; Algorithm Theory, WS 2012/13 Fabian Kuhn 10

11 Operation Cut( ) Operation. : Cuts s sub tree from its parent and adds to rootlist 1. if. then 2. // cut the link between and its parent ; 4. remove from.. (list) 5.. null; 6. add to Algorithm Theory, WS 2012/13 Fabian Kuhn 11

12 Decrease Key Example Green nodes are marked Decrease Key, Algorithm Theory, WS 2012/13 Fabian Kuhn 12

13 Fibonacci Heap Marks History of a node : is being linked to a node. a child of is cut. a second child of is cut. Hence, the boolean value. indicates whether node has lost a child since the last time was made the child of another node. Algorithm Theory, WS 2012/13 Fabian Kuhn 13

14 Cost of Delete Min & Decrease Key Delete Min: 1. Delete min. root and add. to. time: 1 2. Consolidate. time: length of. Step 2 can potentially be linear in (size of ) Decrease Key (at node ): 1. If new key parent key, cut sub tree of node time: 1 2. Cascading cuts up the tree as long as nodes are marked time: number of consecutive marked nodes Step 2 can potentially be linear in Exercises: Both operations can take time in the worst case! Algorithm Theory, WS 2012/13 Fabian Kuhn 14

15 Cost of Delete Min & Decrease Key Cost of delete min and decrease key can be Θ Seems a large price to pay to get insert and merge in 1 time Maybe, the operations are efficient most of the time? It seems to require a lot of operations to get a long rootlist and thus, an expensive consolidate operation In each decrease key operation, at most one node gets marked: We need a lot of decrease key operations to get an expensive decrease key operation Can we show that the average cost per operation is small? We can requires amortized analysis Algorithm Theory, WS 2012/13 Fabian Kuhn 15

16 Amortization Consider sequence,,, of operations (typically performed on some data structure ) : execution time of operation : total execution time The execution time of a single operation might vary within a large range (e.g., 1, ) The worst case overall execution time might still be small average execution time per operation might be small in the worst case, even if single operations can be expensive Algorithm Theory, WS 2012/13 Fabian Kuhn 16

17 Analysis of Algorithms Best case Worst case Average case Amortized worst case What it the average cost of an operation in a worst case sequence of operations? Algorithm Theory, WS 2012/13 Fabian Kuhn 17

18 Example: Binary Counter Incrementing a binary counter: determine the bit flip cost: Operation Counter Value Cost Algorithm Theory, WS 2012/13 Fabian Kuhn 18

19 Accounting Method Observation: Each increment flips exactly one 0 into a Idea: Have a bank account (with initial amount 0) Paying to the bank account costs Take money from account to pay for expensive operations Applied to binary counter: Flip from 0 to 1: pay 1 to bank account (cost: 2) Flip from 1 to 0: take 1 from bank account (cost: 0) Amount on bank account = number of ones We always have enough money to pay! Algorithm Theory, WS 2012/13 Fabian Kuhn 19

20 Accounting Method Op. Counter Cost To Bank From Bank Net Cost Credit Algorithm Theory, WS 2012/13 Fabian Kuhn 20

21 Potential Function Method Most generic and elegant way to do amortized analysis! But, also more abstract than the others State of data structure / system: (state space) Potential function : Operation : : actual cost of operation : state after execution of operation ( : initial state) Φ : potential after exec. of operation : amortized cost of operation : Algorithm Theory, WS 2012/13 Fabian Kuhn 21

22 Potential Function Method Operation : actual cost: amortized cost: Φ Φ Overall cost: Φ Φ Algorithm Theory, WS 2012/13 Fabian Kuhn 22

23 Binary Counter: Potential Method Potential function: : Clearly, Φ 0and Φ 0for all 0 Actual cost : 1 flip from 0 to 1 1flips from 1 to 0 Potential difference: Φ Φ Amortized cost: Φ Φ 2 Algorithm Theory, WS 2012/13 Fabian Kuhn 23

24 Back to Fibonacci Heaps Worst case cost of a single delete min or decrease key operation is Ω Can we prove a small worst case amortized cost for delete min and decrease key operations? Remark: Data structure that allows operations,, We say that operation has amortized cost if for every execution the total time is where is the number of operations of type Algorithm Theory, WS 2012/13 Fabian Kuhn 24,

25 Amortized Cost of Fibonacci Heaps Initialize heap, is empty, get min, insert, and merge have worst case cost Delete min has amortized cost Decrease key has amortized cost Starting with an empty heap, any sequence of operations with at most delete min operations has total cost (time). We will now need the marks Cost for Dijkstra: log Algorithm Theory, WS 2012/13 Fabian Kuhn 25

26 Fibonacci Heaps: Marks Cycle of a node: 1. Node is removed from root list and linked to a node. 2. Child node of is cut and added to root list. 3. Second child of is cut node is cut as well and moved to root list The boolean value. indicates whether node has lost a child since the last time was made the child of another node. 26 Algorithm Theory, WS 2012/13 Fabian Kuhn Winter term 11/12 26

27 Potential Function System state characterized by two parameters: : number of trees (length of. ) : number of marked nodes that are not in the root list Potential function: Example: , 2 Φ 11 Algorithm Theory, WS 2012/13 Fabian Kuhn 27

28 Actual Time of Operations Operations: initialize heap, is empty, insert, get min, merge actual time: 1 Normalize unit time such that,,,, 1 Operation delete min: Actual time: length of. Normalize unit time such that length of. Operation descrease key: Actual time: length of path to next unmarked ancestor Normalize unit time such that lengthof path to next unmarked ancestor Algorithm Theory, WS 2012/13 Fabian Kuhn 28

29 Amortized Times Assume operation is of type: initialize heap: actual time: 1, potential: Φ Φ 0 amortized time: Φ Φ 1 is empty, get min: actual time: 1, potential: Φ Φ (heap doesn t change) amortized time: Φ Φ 1 merge: Actual time: 1 combined potential of both heaps: Φ Φ amortized time: Φ Φ 1 Algorithm Theory, WS 2012/13 Fabian Kuhn 29

30 Amortized Time of Insert Assume that operation is an insert operation: Actual time: 1 Potential function: remains unchanged (no nodes are marked or unmarked, no marked nodes are moved to the root list) grows by 1 (one element is added to the root list), Φ Φ 1 1 Amortized time: Algorithm Theory, WS 2012/13 Fabian Kuhn 30

31 Amortized Time of Delete Min Assume that operation is a delete min operation: Actual time:. Potential function : : changes from. to at most : (# of marked nodes that are not in the root list) no new marks if node is moved away from root list,. is set to false value of does not change!,. Φ Φ. Amortized Time: Algorithm Theory, WS 2012/13 Fabian Kuhn 31

32 Amortized Time of Decrease Key Assume that operation is a decrease key operation at node : Actual time: length of path to next unmarked ancestor Potential function : Assume, node and nodes,, are moved to root list,, are marked and moved to root list,. mark is set to true marked nodes go to root list, 1node gets newly marked grows by 1, grows by 1 and is decreased by 1, 1 Φ Φ Φ 3 Amortized time: Algorithm Theory, WS 2012/13 Fabian Kuhn 32

33 Complexities Fibonacci Heap Initialize Heap: Is Empty: Insert: Get Min: Delete Min: Decrease Key: amortized Merge (heaps of size and, ): How large can get? Algorithm Theory, WS 2012/13 Fabian Kuhn 33

34 Rank of Children Lemma: Consider a node of rank and let,, be the children of in the order in which they were linked to. Then, Proof:. Algorithm Theory, WS 2012/13 Fabian Kuhn 34

35 Size of Trees Fibonacci Numbers: 0, 1, 2: Lemma: In a Fibonacci heap, the size of the sub tree of a node with rank is at least. Proof: : minimum size of the sub tree of a node of rank Algorithm Theory, WS 2012/13 Fabian Kuhn 35

36 Size of Trees 1, 2, 2: 2 Claim about Fibonacci numbers: 0: 1 Algorithm Theory, WS 2012/13 Fabian Kuhn 36

37 Size of Trees 1, 2, 2: 2, 1 Claim of lemma: Algorithm Theory, WS 2012/13 Fabian Kuhn 37

38 Size of Trees Lemma: In a Fibonacci heap, the size of the sub tree of a node with rank is at least. Theorem: The maximum rank of a node in a Fibonacci heap of size is at most. Proof: The Fibonacci numbers grow exponentially: For, we need nodes Algorithm Theory, WS 2012/13 Fabian Kuhn 38

39 Summary: Binomial and Fibonacci Heaps Binomial Heap Fibonacci Heap initialize insert get min delete min * decrease key * merge is empty amortized time Algorithm Theory, WS 2012/13 Fabian Kuhn 39

CS711008Z Algorithm Design and Analysis

CS711008Z Algorithm Design and Analysis CS711008Z Algorithm Design and Analysis Lecture 7 Binary heap, binomial heap, and Fibonacci heap 1 Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 The slides were

More information

Full and Complete Binary Trees

Full and Complete Binary Trees Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

More information

CSE 326, Data Structures. Sample Final Exam. Problem Max Points Score 1 14 (2x7) 2 18 (3x6) 3 4 4 7 5 9 6 16 7 8 8 4 9 8 10 4 Total 92.

CSE 326, Data Structures. Sample Final Exam. Problem Max Points Score 1 14 (2x7) 2 18 (3x6) 3 4 4 7 5 9 6 16 7 8 8 4 9 8 10 4 Total 92. Name: Email ID: CSE 326, Data Structures Section: Sample Final Exam Instructions: The exam is closed book, closed notes. Unless otherwise stated, N denotes the number of elements in the data structure

More information

Cpt S 223. School of EECS, WSU

Cpt S 223. School of EECS, WSU Priority Queues (Heaps) 1 Motivation Queues are a standard mechanism for ordering tasks on a first-come, first-served basis However, some tasks may be more important or timely than others (higher priority)

More information

Binary Search Trees. Data in each node. Larger than the data in its left child Smaller than the data in its right child

Binary Search Trees. Data in each node. Larger than the data in its left child Smaller than the data in its right child Binary Search Trees Data in each node Larger than the data in its left child Smaller than the data in its right child FIGURE 11-6 Arbitrary binary tree FIGURE 11-7 Binary search tree Data Structures Using

More information

A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called heap-order property

A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called heap-order property CmSc 250 Intro to Algorithms Chapter 6. Transform and Conquer Binary Heaps 1. Definition A binary heap is a complete binary tree, where each node has a higher priority than its children. This is called

More information

The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge,

The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge, The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge, cheapest first, we had to determine whether its two endpoints

More information

1) The postfix expression for the infix expression A+B*(C+D)/F+D*E is ABCD+*F/DE*++

1) The postfix expression for the infix expression A+B*(C+D)/F+D*E is ABCD+*F/DE*++ Answer the following 1) The postfix expression for the infix expression A+B*(C+D)/F+D*E is ABCD+*F/DE*++ 2) Which data structure is needed to convert infix notations to postfix notations? Stack 3) The

More information

Multi-Way Search Trees (B Trees)

Multi-Way Search Trees (B Trees) Multi-Way Search Trees (B Trees) Multiway Search Trees An m-way search tree is a tree in which, for some integer m called the order of the tree, each node has at most m children. If n

More information

CSE Homework 1

CSE Homework 1 CSE 00 - Homework.8 (a) Give a precise expression for the minimum number of nodes in an AVL tree of height h. (b) What is the minimum number of nodes in an AVL tree of height? (a) Let S(h) be the minimum

More information

5. A full binary tree with n leaves contains [A] n nodes. [B] log n 2 nodes. [C] 2n 1 nodes. [D] n 2 nodes.

5. A full binary tree with n leaves contains [A] n nodes. [B] log n 2 nodes. [C] 2n 1 nodes. [D] n 2 nodes. 1. The advantage of.. is that they solve the problem if sequential storage representation. But disadvantage in that is they are sequential lists. [A] Lists [B] Linked Lists [A] Trees [A] Queues 2. The

More information

Questions 1 through 25 are worth 2 points each. Choose one best answer for each.

Questions 1 through 25 are worth 2 points each. Choose one best answer for each. Questions 1 through 25 are worth 2 points each. Choose one best answer for each. 1. For the singly linked list implementation of the queue, where are the enqueues and dequeues performed? c a. Enqueue in

More information

Data Structure [Question Bank]

Data Structure [Question Bank] Unit I (Analysis of Algorithms) 1. What are algorithms and how they are useful? 2. Describe the factor on best algorithms depends on? 3. Differentiate: Correct & Incorrect Algorithms? 4. Write short note:

More information

Binary Search Trees. A Generic Tree. Binary Trees. Nodes in a binary search tree ( B-S-T) are of the form. P parent. Key. Satellite data L R

Binary Search Trees. A Generic Tree. Binary Trees. Nodes in a binary search tree ( B-S-T) are of the form. P parent. Key. Satellite data L R Binary Search Trees A Generic Tree Nodes in a binary search tree ( B-S-T) are of the form P parent Key A Satellite data L R B C D E F G H I J The B-S-T has a root node which is the only node whose parent

More information

PES Institute of Technology-BSC QUESTION BANK

PES Institute of Technology-BSC QUESTION BANK PES Institute of Technology-BSC Faculty: Mrs. R.Bharathi CS35: Data Structures Using C QUESTION BANK UNIT I -BASIC CONCEPTS 1. What is an ADT? Briefly explain the categories that classify the functions

More information

Converting a Number from Decimal to Binary

Converting a Number from Decimal to Binary Converting a Number from Decimal to Binary Convert nonnegative integer in decimal format (base 10) into equivalent binary number (base 2) Rightmost bit of x Remainder of x after division by two Recursive

More information

10CS35: Data Structures Using C

10CS35: Data Structures Using C CS35: Data Structures Using C QUESTION BANK REVIEW OF STRUCTURES AND POINTERS, INTRODUCTION TO SPECIAL FEATURES OF C OBJECTIVE: Learn : Usage of structures, unions - a conventional tool for handling a

More information

Trees. Tree Definitions: Let T = (V, E, r) be a tree. The size of a tree denotes the number of nodes of the tree.

Trees. Tree Definitions: Let T = (V, E, r) be a tree. The size of a tree denotes the number of nodes of the tree. Trees After lists (including stacks and queues) and hash tables, trees represent one of the most widely used data structures. On one hand trees can be used to represent objects that are recursive in nature

More information

Unordered Linked Lists

Unordered Linked Lists Unordered Linked Lists Derive class unorderedlinkedlist from the abstract class linkedlisttype Implement the operations search, insertfirst, insertlast, deletenode See code on page 292 Defines an unordered

More information

From Last Time: Remove (Delete) Operation

From Last Time: Remove (Delete) Operation CSE 32 Lecture : More on Search Trees Today s Topics: Lazy Operations Run Time Analysis of Binary Search Tree Operations Balanced Search Trees AVL Trees and Rotations Covered in Chapter of the text From

More information

AS-2261 M.Sc.(First Semester) Examination-2013 Paper -fourth Subject-Data structure with algorithm

AS-2261 M.Sc.(First Semester) Examination-2013 Paper -fourth Subject-Data structure with algorithm AS-2261 M.Sc.(First Semester) Examination-2013 Paper -fourth Subject-Data structure with algorithm Time: Three Hours] [Maximum Marks: 60 Note Attempts all the questions. All carry equal marks Section A

More information

6 March 2007 1. Array Implementation of Binary Trees

6 March 2007 1. Array Implementation of Binary Trees Heaps CSE 0 Winter 00 March 00 1 Array Implementation of Binary Trees Each node v is stored at index i defined as follows: If v is the root, i = 1 The left child of v is in position i The right child of

More information

Analysis of Algorithms I: Binary Search Trees

Analysis of Algorithms I: Binary Search Trees Analysis of Algorithms I: Binary Search Trees Xi Chen Columbia University Hash table: A data structure that maintains a subset of keys from a universe set U = {0, 1,..., p 1} and supports all three dictionary

More information

Chapter 18 Indexing Structures for Files. Indexes as Access Paths

Chapter 18 Indexing Structures for Files. Indexes as Access Paths Chapter 18 Indexing Structures for Files Indexes as Access Paths A single-level index is an auxiliary file that makes it more efficient to search for a record in the data file. The index is usually specified

More information

Abstract Data Type. EECS 281: Data Structures and Algorithms. The Foundation: Data Structures and Abstract Data Types

Abstract Data Type. EECS 281: Data Structures and Algorithms. The Foundation: Data Structures and Abstract Data Types EECS 281: Data Structures and Algorithms The Foundation: Data Structures and Abstract Data Types Computer science is the science of abstraction. Abstract Data Type Abstraction of a data structure on that

More information

Tables so far. set() get() delete() BST Average O(lg n) O(lg n) O(lg n) Worst O(n) O(n) O(n) RB Tree Average O(lg n) O(lg n) O(lg n)

Tables so far. set() get() delete() BST Average O(lg n) O(lg n) O(lg n) Worst O(n) O(n) O(n) RB Tree Average O(lg n) O(lg n) O(lg n) Hash Tables Tables so far set() get() delete() BST Average O(lg n) O(lg n) O(lg n) Worst O(n) O(n) O(n) RB Tree Average O(lg n) O(lg n) O(lg n) Worst O(lg n) O(lg n) O(lg n) Table naïve array implementation

More information

Outline BST Operations Worst case Average case Balancing AVL Red-black B-trees. Binary Search Trees. Lecturer: Georgy Gimel farb

Outline BST Operations Worst case Average case Balancing AVL Red-black B-trees. Binary Search Trees. Lecturer: Georgy Gimel farb Binary Search Trees Lecturer: Georgy Gimel farb COMPSCI 220 Algorithms and Data Structures 1 / 27 1 Properties of Binary Search Trees 2 Basic BST operations The worst-case time complexity of BST operations

More information

Algorithms and Data Structures

Algorithms and Data Structures Algorithms and Data Structures Part 2: Data Structures PD Dr. rer. nat. habil. Ralf-Peter Mundani Computation in Engineering (CiE) Summer Term 2016 Overview general linked lists stacks queues trees 2 2

More information

UNIVERSITI MALAYSIA SARAWAK KOTA SAMARAHAN SARAWAK PSD2023 ALGORITHM & DATA STRUCTURE

UNIVERSITI MALAYSIA SARAWAK KOTA SAMARAHAN SARAWAK PSD2023 ALGORITHM & DATA STRUCTURE STUDENT IDENTIFICATION NO UNIVERSITI MALAYSIA SARAWAK 94300 KOTA SAMARAHAN SARAWAK FAKULTI SAINS KOMPUTER & TEKNOLOGI MAKLUMAT (Faculty of Computer Science & Information Technology) Diploma in Multimedia

More information

Previous Lectures. B-Trees. External storage. Two types of memory. B-trees. Main principles

Previous Lectures. B-Trees. External storage. Two types of memory. B-trees. Main principles B-Trees Algorithms and data structures for external memory as opposed to the main memory B-Trees Previous Lectures Height balanced binary search trees: AVL trees, red-black trees. Multiway search trees:

More information

Mathematical Induction

Mathematical Induction Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

More information

, each of which contains a unique key value, say k i , R 2. such that k i equals K (or to determine that no such record exists in the collection).

, each of which contains a unique key value, say k i , R 2. such that k i equals K (or to determine that no such record exists in the collection). The Search Problem 1 Suppose we have a collection of records, say R 1, R 2,, R N, each of which contains a unique key value, say k i. Given a particular key value, K, the search problem is to locate the

More information

DATA STRUCTURES USING C

DATA STRUCTURES USING C DATA STRUCTURES USING C QUESTION BANK UNIT I 1. Define data. 2. Define Entity. 3. Define information. 4. Define Array. 5. Define data structure. 6. Give any two applications of data structures. 7. Give

More information

The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

More information

Balanced search trees

Balanced search trees Lecture 8 Balanced search trees 8.1 Overview In this lecture we discuss search trees as a method for storing data in a way that supports fast insert, lookup, and delete operations. (Data structures handling

More information

K-Cover of Binary sequence

K-Cover of Binary sequence K-Cover of Binary sequence Prof Amit Kumar Prof Smruti Sarangi Sahil Aggarwal Swapnil Jain Given a binary sequence, represent all the 1 s in it using at most k- covers, minimizing the total length of all

More information

COT5405 Analysis of Algorithms Homework 3 Solutions

COT5405 Analysis of Algorithms Homework 3 Solutions COT0 Analysis of Algorithms Homework 3 Solutions. Prove or give a counter example: (a) In the textbook, we have two routines for graph traversal - DFS(G) and BFS(G,s) - where G is a graph and s is any

More information

Two General Methods to Reduce Delay and Change of Enumeration Algorithms

Two General Methods to Reduce Delay and Change of Enumeration Algorithms ISSN 1346-5597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII-2003-004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration

More information

Ordered Lists and Binary Trees

Ordered Lists and Binary Trees Data Structures and Algorithms Ordered Lists and Binary Trees Chris Brooks Department of Computer Science University of San Francisco Department of Computer Science University of San Francisco p.1/62 6-0:

More information

Symbol Tables. IE 496 Lecture 13

Symbol Tables. IE 496 Lecture 13 Symbol Tables IE 496 Lecture 13 Reading for This Lecture Horowitz and Sahni, Chapter 2 Symbol Tables and Dictionaries A symbol table is a data structure for storing a list of items, each with a key and

More information

CMSC 451: Graph Properties, DFS, BFS, etc.

CMSC 451: Graph Properties, DFS, BFS, etc. CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs

More information

CMPS 102 Solutions to Homework 1

CMPS 102 Solutions to Homework 1 CMPS 0 Solutions to Homework Lindsay Brown, lbrown@soe.ucsc.edu September 9, 005 Problem..- p. 3 For inputs of size n insertion sort runs in 8n steps, while merge sort runs in 64n lg n steps. For which

More information

Data Structures and Algorithms Written Examination

Data Structures and Algorithms Written Examination Data Structures and Algorithms Written Examination 22 February 2013 FIRST NAME STUDENT NUMBER LAST NAME SIGNATURE Instructions for students: Write First Name, Last Name, Student Number and Signature where

More information

- Easy to insert & delete in O(1) time - Don t need to estimate total memory needed. - Hard to search in less than O(n) time

- Easy to insert & delete in O(1) time - Don t need to estimate total memory needed. - Hard to search in less than O(n) time Skip Lists CMSC 420 Linked Lists Benefits & Drawbacks Benefits: - Easy to insert & delete in O(1) time - Don t need to estimate total memory needed Drawbacks: - Hard to search in less than O(n) time (binary

More information

Data Structures and Algorithms

Data Structures and Algorithms Data Structures and Algorithms CS245-2016S-06 Binary Search Trees David Galles Department of Computer Science University of San Francisco 06-0: Ordered List ADT Operations: Insert an element in the list

More information

Sample Problems in Discrete Mathematics

Sample Problems in Discrete Mathematics Sample Problems in Discrete Mathematics This handout lists some sample problems that you should be able to solve as a pre-requisite to Computer Algorithms Try to solve all of them You should also read

More information

Analysis of a Search Algorithm

Analysis of a Search Algorithm CSE 326 Lecture 4: Lists and Stacks 1. Agfgd 2. Dgsdsfd 3. Hdffdsf 4. Sdfgsfdg 5. Tefsdgass We will review: Analysis: Searching a sorted array (from last time) List ADT: Insert, Delete, Find, First, Kth,

More information

Persistent Binary Search Trees

Persistent Binary Search Trees Persistent Binary Search Trees Datastructures, UvA. May 30, 2008 0440949, Andreas van Cranenburgh Abstract A persistent binary tree allows access to all previous versions of the tree. This paper presents

More information

Symbol Tables. Introduction

Symbol Tables. Introduction Symbol Tables Introduction A compiler needs to collect and use information about the names appearing in the source program. This information is entered into a data structure called a symbol table. The

More information

Learning Outcomes. COMP202 Complexity of Algorithms. Binary Search Trees and Other Search Trees

Learning Outcomes. COMP202 Complexity of Algorithms. Binary Search Trees and Other Search Trees Learning Outcomes COMP202 Complexity of Algorithms Binary Search Trees and Other Search Trees [See relevant sections in chapters 2 and 3 in Goodrich and Tamassia.] At the conclusion of this set of lecture

More information

1 2-3 Trees: The Basics

1 2-3 Trees: The Basics CS10: Data Structures and Object-Oriented Design (Fall 2013) November 1, 2013: 2-3 Trees: Inserting and Deleting Scribes: CS 10 Teaching Team Lecture Summary In this class, we investigated 2-3 Trees in

More information

Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

More information

B-Trees. Algorithms and data structures for external memory as opposed to the main memory B-Trees. B -trees

B-Trees. Algorithms and data structures for external memory as opposed to the main memory B-Trees. B -trees B-Trees Algorithms and data structures for external memory as opposed to the main memory B-Trees Previous Lectures Height balanced binary search trees: AVL trees, red-black trees. Multiway search trees:

More information

Guessing Game: NP-Complete?

Guessing Game: NP-Complete? Guessing Game: NP-Complete? 1. LONGEST-PATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple

More information

Multiway Search Tree (MST)

Multiway Search Tree (MST) Multiway Search Tree (MST) Generalization of BSTs Suitable for disk MST of order n: Each node has n or fewer sub-trees S1 S2. Sm, m n Each node has n-1 or fewer keys K1 Κ2 Κm-1 : m-1 keys in ascending

More information

Priority Queues. Client, Implementation, Interface. Priority Queues. Abstract Data Types

Priority Queues. Client, Implementation, Interface. Priority Queues. Abstract Data Types Client, Implementation, Interface Priority Queues Priority Queue ADT Binary heaps Heapsort Reference: Chapter 6, Algorithms in Java, 3 rd Edition, Robert Sedgewick. Separate interface and implementation

More information

1. The memory address of the first element of an array is called A. floor address B. foundation addressc. first address D.

1. The memory address of the first element of an array is called A. floor address B. foundation addressc. first address D. 1. The memory address of the first element of an array is called A. floor address B. foundation addressc. first address D. base address 2. The memory address of fifth element of an array can be calculated

More information

Section IV.1: Recursive Algorithms and Recursion Trees

Section IV.1: Recursive Algorithms and Recursion Trees Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller

More information

Balanced Binary Search Tree

Balanced Binary Search Tree AVL Trees / Slide 1 Balanced Binary Search Tree Worst case height of binary search tree: N-1 Insertion, deletion can be O(N) in the worst case We want a tree with small height Height of a binary tree with

More information

Krishna Institute of Engineering & Technology, Ghaziabad Department of Computer Application MCA-213 : DATA STRUCTURES USING C

Krishna Institute of Engineering & Technology, Ghaziabad Department of Computer Application MCA-213 : DATA STRUCTURES USING C Tutorial#1 Q 1:- Explain the terms data, elementary item, entity, primary key, domain, attribute and information? Also give examples in support of your answer? Q 2:- What is a Data Type? Differentiate

More information

Trees LOGO STYLE GUIDE Schools within the University 1 19

Trees LOGO STYLE GUIDE Schools within the University 1 19 Trees 1 Example Tree 2 A Generic Tree 3 Tree ADT n Tree definition q A tree is a set of nodes which may be empty q If not empty, then there is a distinguished node r, called root and zero or more non-empty

More information

Binary Heaps * * * * * * * / / \ / \ / \ / \ / \ * * * * * * * * * * * / / \ / \ / / \ / \ * * * * * * * * * *

Binary Heaps * * * * * * * / / \ / \ / \ / \ / \ * * * * * * * * * * * / / \ / \ / / \ / \ * * * * * * * * * * Binary Heaps A binary heap is another data structure. It implements a priority queue. Priority Queue has the following operations: isempty add (with priority) remove (highest priority) peek (at highest

More information

Complexity of Union-Split-Find Problems. Katherine Jane Lai

Complexity of Union-Split-Find Problems. Katherine Jane Lai Complexity of Union-Split-Find Problems by Katherine Jane Lai S.B., Electrical Engineering and Computer Science, MIT, 2007 S.B., Mathematics, MIT, 2007 Submitted to the Department of Electrical Engineering

More information

1. What s wrong with the following proofs by induction?

1. What s wrong with the following proofs by induction? ArsDigita University Month : Discrete Mathematics - Professor Shai Simonson Problem Set 4 Induction and Recurrence Equations Thanks to Jeffrey Radcliffe and Joe Rizzo for many of the solutions. Pasted

More information

Class Notes CS 3137. 1 Creating and Using a Huffman Code. Ref: Weiss, page 433

Class Notes CS 3137. 1 Creating and Using a Huffman Code. Ref: Weiss, page 433 Class Notes CS 3137 1 Creating and Using a Huffman Code. Ref: Weiss, page 433 1. FIXED LENGTH CODES: Codes are used to transmit characters over data links. You are probably aware of the ASCII code, a fixed-length

More information

Algorithms. Theresa Migler-VonDollen CMPS 5P

Algorithms. Theresa Migler-VonDollen CMPS 5P Algorithms Theresa Migler-VonDollen CMPS 5P 1 / 32 Algorithms Write a Python function that accepts a list of numbers and a number, x. If x is in the list, the function returns the position in the list

More information

Data storage Tree indexes

Data storage Tree indexes Data storage Tree indexes Rasmus Pagh February 7 lecture 1 Access paths For many database queries and updates, only a small fraction of the data needs to be accessed. Extreme examples are looking or updating

More information

Sequential Data Structures

Sequential Data Structures Sequential Data Structures In this lecture we introduce the basic data structures for storing sequences of objects. These data structures are based on arrays and linked lists, which you met in first year

More information

root node level: internal node edge leaf node CS@VT Data Structures & Algorithms 2000-2009 McQuain

root node level: internal node edge leaf node CS@VT Data Structures & Algorithms 2000-2009 McQuain inary Trees 1 A binary tree is either empty, or it consists of a node called the root together with two binary trees called the left subtree and the right subtree of the root, which are disjoint from each

More information

Intro. to the Divide-and-Conquer Strategy via Merge Sort CMPSC 465 CLRS Sections 2.3, Intro. to and various parts of Chapter 4

Intro. to the Divide-and-Conquer Strategy via Merge Sort CMPSC 465 CLRS Sections 2.3, Intro. to and various parts of Chapter 4 Intro. to the Divide-and-Conquer Strategy via Merge Sort CMPSC 465 CLRS Sections 2.3, Intro. to and various parts of Chapter 4 I. Algorithm Design and Divide-and-Conquer There are various strategies we

More information

4.4 The recursion-tree method

4.4 The recursion-tree method 4.4 The recursion-tree method Let us see how a recursion tree would provide a good guess for the recurrence = 3 4 ) Start by nding an upper bound Floors and ceilings usually do not matter when solving

More information

Algorithms Chapter 12 Binary Search Trees

Algorithms Chapter 12 Binary Search Trees Algorithms Chapter 1 Binary Search Trees Outline Assistant Professor: Ching Chi Lin 林 清 池 助 理 教 授 chingchi.lin@gmail.com Department of Computer Science and Engineering National Taiwan Ocean University

More information

Data Structures Using C++ 2E. Chapter 5 Linked Lists

Data Structures Using C++ 2E. Chapter 5 Linked Lists Data Structures Using C++ 2E Chapter 5 Linked Lists Doubly Linked Lists Traversed in either direction Typical operations Initialize the list Destroy the list Determine if list empty Search list for a given

More information

Data Structures and Data Manipulation

Data Structures and Data Manipulation Data Structures and Data Manipulation What the Specification Says: Explain how static data structures may be used to implement dynamic data structures; Describe algorithms for the insertion, retrieval

More information

Persistent Data Structures and Planar Point Location. Inge Li Gørtz

Persistent Data Structures and Planar Point Location. Inge Li Gørtz Persistent Data Structures and Planar Point Location Inge Li Gørtz Persistent Data Structures Ephemeral Partial persistence Full persistence Confluent persistence V1 V1 V1 V1 V2 q ue V2 V2 V5 V2 V4 V4

More information

! Overview! Measures of Query Cost! Selection Operation! Sorting! Join Operation! Other Operations! Evaluation of Expressions

! Overview! Measures of Query Cost! Selection Operation! Sorting! Join Operation! Other Operations! Evaluation of Expressions Chapter 13: Query Processing Chapter 13: Query Processing! Overview! Measures of Query Cost! Selection Operation! Sorting! Join Operation! Other Operations! Evaluation of Expressions Basic Steps in Query

More information

AVL Trees. CSE 373 Data Structures

AVL Trees. CSE 373 Data Structures AVL Trees CSE 373 Data Structures Readings Reading Chapter 1 Section 1.2 AVL Trees 2 Binary Search Tree - Best Time All BST operations are O(d), where d is tree depth minimum d is = log N for a binary

More information

Lecture 2: The Complexity of Some Problems

Lecture 2: The Complexity of Some Problems IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 2: The Complexity of Some Problems David Mix Barrington and Alexis Maciel July 18, 2000

More information

Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours)

Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours) Discrete Mathematics (2009 Spring) Induction and Recursion (Chapter 4, 3 hours) Chih-Wei Yi Dept. of Computer Science National Chiao Tung University April 17, 2009 4.1 Mathematical Induction 4.1 Mathematical

More information

Binary Search Trees CMPSC 122

Binary Search Trees CMPSC 122 Binary Search Trees CMPSC 122 Note: This notes packet has significant overlap with the first set of trees notes I do in CMPSC 360, but goes into much greater depth on turning BSTs into pseudocode than

More information

Homework Exam 1, Geometric Algorithms, 2016

Homework Exam 1, Geometric Algorithms, 2016 Homework Exam 1, Geometric Algorithms, 2016 1. (3 points) Let P be a convex polyhedron in 3-dimensional space. The boundary of P is represented as a DCEL, storing the incidence relationships between the

More information

Binary Heap Algorithms

Binary Heap Algorithms CS Data Structures and Algorithms Lecture Slides Wednesday, April 5, 2009 Glenn G. Chappell Department of Computer Science University of Alaska Fairbanks CHAPPELLG@member.ams.org 2005 2009 Glenn G. Chappell

More information

Sorting revisited. Build the binary search tree: O(n^2) Traverse the binary tree: O(n) Total: O(n^2) + O(n) = O(n^2)

Sorting revisited. Build the binary search tree: O(n^2) Traverse the binary tree: O(n) Total: O(n^2) + O(n) = O(n^2) Sorting revisited How did we use a binary search tree to sort an array of elements? Tree Sort Algorithm Given: An array of elements to sort 1. Build a binary search tree out of the elements 2. Traverse

More information

Satisfiability Checking

Satisfiability Checking Satisfiability Checking SAT-Solving Prof. Dr. Erika Ábrahám Theory of Hybrid Systems Informatik 2 WS 10/11 Prof. Dr. Erika Ábrahám - Satisfiability Checking 1 / 40 A basic SAT algorithm Assume the CNF

More information

School of Informatics, University of Edinburgh Computer Science 1 Ah [03 04]

School of Informatics, University of Edinburgh Computer Science 1 Ah [03 04] School of Informatics, University of Edinburgh Computer Science 1 Ah [03 04] CS1Ah Lecture Note 25 Binary Search Trees This lecture studies binary trees in more detail, in particular their application

More information

CS112 Lecture: Searching and Sorting

CS112 Lecture: Searching and Sorting CS112 Lecture: Searching and Sorting Objectives: 1. To introduce the linear search algorithm. 2. To introduce the binary search algorithm. 3. To show superiority of O(log n) vs O(n) 4. To take note of

More information

Find-The-Number. 1 Find-The-Number With Comps

Find-The-Number. 1 Find-The-Number With Comps Find-The-Number 1 Find-The-Number With Comps Consider the following two-person game, which we call Find-The-Number with Comps. Player A (for answerer) has a number x between 1 and 1000. Player Q (for questioner)

More information

Balanced Binary Trees

Balanced Binary Trees Reminders about Trees Balanced Binary Trees With pictures by John Morris (ciips.ee.uwa.edu.au/~morris) A binary tree is a tree with exactly two sub-trees for each node, called the left and right sub-trees.

More information

TREE BASIC TERMINOLOGIES

TREE BASIC TERMINOLOGIES TREE Trees are very flexible, versatile and powerful non-liner data structure that can be used to represent data items possessing hierarchical relationship between the grand father and his children and

More information

CSCI Trees. Mark Redekopp David Kempe

CSCI Trees. Mark Redekopp David Kempe 1 CSCI 104 2-3 Trees Mark Redekopp David Kempe 2 Properties, Insertion and Removal BINARY SEARCH TREES 3 Binary Search Tree Binary search tree = binary tree where all nodes meet the property that: All

More information

Heap. Binary Search Tree. Heaps VS BSTs. < el el. Difference between a heap and a BST:

Heap. Binary Search Tree. Heaps VS BSTs. < el el. Difference between a heap and a BST: Heaps VS BSTs Difference between a heap and a BST: Heap el Binary Search Tree el el el < el el Perfectly balanced at all times Immediate access to maximal element Easy to code Does not provide efficient

More information

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 14 Non-Binary Huffman Code and Other Codes In the last class we

More information

Binary Search Trees. Yes! So, a binary tree T is a binary search tree if T is a single node.

Binary Search Trees. Yes! So, a binary tree T is a binary search tree if T is a single node. Which are Binary Search Trees (BST)? Binary Search Trees So, a binary tree T is a binary search tree if T is a single node. Yes! No! All the values on the left subtree must be less than the value at the

More information

GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

More information

Strategic Deployment in Graphs. 1 Introduction. v 1 = v s. v 2. v 4. e 1. e 5 25. e 3. e 2. e 4

Strategic Deployment in Graphs. 1 Introduction. v 1 = v s. v 2. v 4. e 1. e 5 25. e 3. e 2. e 4 Informatica 39 (25) 237 247 237 Strategic Deployment in Graphs Elmar Langetepe and Andreas Lenerz University of Bonn, Department of Computer Science I, Germany Bernd Brüggemann FKIE, Fraunhofer-Institute,

More information

Planar Tree Transformation: Results and Counterexample

Planar Tree Transformation: Results and Counterexample Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem

More information

CS104: Data Structures and Object-Oriented Design (Fall 2013) October 24, 2013: Priority Queues Scribes: CS 104 Teaching Team

CS104: Data Structures and Object-Oriented Design (Fall 2013) October 24, 2013: Priority Queues Scribes: CS 104 Teaching Team CS104: Data Structures and Object-Oriented Design (Fall 2013) October 24, 2013: Priority Queues Scribes: CS 104 Teaching Team Lecture Summary In this lecture, we learned about the ADT Priority Queue. A

More information

Chapter 4. Trees. 4.1 Basics

Chapter 4. Trees. 4.1 Basics Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

More information

Problem. Indexing with B-trees. Indexing. Primary Key Indexing. B-trees. B-trees: Example. primary key indexing

Problem. Indexing with B-trees. Indexing. Primary Key Indexing. B-trees. B-trees: Example. primary key indexing Problem Given a large collection of records, Indexing with B-trees find similar/interesting things, i.e., allow fast, approximate queries Anastassia Ailamaki http://www.cs.cmu.edu/~natassa 2 Indexing Primary

More information

Overview of Data Structures

Overview of Data Structures UNIT 3 Concrete Data Types Classification of Data Structures Concrete vs. Abstract Data Structures Most Important Concrete Data Structures Arrays Records Linked Lists Binary Trees Overview of Data Structures

More information