# CSE 5311 Homework 2 Solution

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CSE 5311 Homework 2 Solution Problem Show that the worst-case running time of MAX-HEAPIFY on a heap of size n is Ω(lg n). (Hint: For a heap with n nodes, give node values that cause MAX- HEAPIFY to be called recursively at every node on a simple path from the root down to a leaf.) If you put a value at the root that is less than every value in the left and right subtrees, then MAX-HEAPIFY will be called recursively until a leaf is reached. To make the recursive calls traverse the longest path to a leaf, choose values that make MAX-HEAPIFY always recurse on the left child. It follows the left branch when the left child is greater than or equal to the right child, so putting 0 at the root and 1 at all the other nodes, for example, will accomplish that. With such values, MAX-HEAPIFY will be called h times (where h is the heap height, which is the number of edges in the longest path from the root to a leaf), so its running time will be Θ(h) (since each call does Θ(1) work), which is Θ(lg n). Since we have a case in which MAX-HEAPIFY s running time is Θ(lg n), its worst-case running time is Ω(lg n). Problem 6-2 Analysis of d-ary heaps A d-ary heap is like a binary heap, but (with one possible exception) non-leaf nodes have d children instead of 2 children. a. How would you represent a d-ary heap in an array? b. What is the height of a d-ary heap of n elements in terms of n and d? c. Give an efficient implementation of EXTRACT-MAX in a d-ary max-heap. Analyze its running time in terms of d and n. d. Give an efficient implementation of INSERT in d-ary max-heap. Analyze its running time in terms of d and n. e. Give an efficient implementation of INCREASE-KEY(A, i, k), which flags an error if k < A[i], but otherwise sets A[i] = k and then updates the d- ary max-heap structure appropriately. Analyze its running time in terms of d and n. 1

2 a. We can represent a d-ary heap in a 1-dimensional array as follows. The root resides in A[1], its d children reside in order in A[2] through A[d + 1, their children reside in order in A[d + 2] through A[d 2 + d + 1], and so on. The following two procedures map a node with index i to its parent and to its j-th child (for 1 j d), respectively. D ARY PARENT( i ) r eturn (i 2)/d + 1 D ARY CHILD( i, j ) r eturn d(i + 1) + j + 1 To convince yourself that these procedures really work, verify that D ARY PARENET(D ARY CHILD( i, j ) ) = i, for any 1 j d. Notice taht the binary heap procedures are a special case of the above procedures when d = 2. b. Since each node has d children, the height of a d-ary heap with n nodes is Θ(log d n) = Θ(lg d/ lg n). c. The procedure HEAP-EXTRACT-MAX given in the text for binary heaps works fine for d-ary heaps too. The change needed to support d-ary heaps is in MAX- HEAPIFY, which must compare the argument node to all d children instead of just 2 children. The running time of HEAP-EXTRACT-MAX is still the running time for MAX-HEAPIFY, but that now takes worst-case time proportional to the product of the height of the heap by the number of children examined at each node (at most d), namely Θ(d log d n) = Θ(d lg d/ lg n). d. The procedure MAX-HEAP-INSERT given in the text for binary heaps works fine for d-ary heaps too, assuming that HEAP-INCREASE-KEY works for d-ary heaps. The worst-case running time is still Θ(h), where h is the height of the heap. (Since only parent pointers are followed, the number of children a node has is irrelevant.) For a d-ary heap, this is Θ(log d n) = Θ(lg d/ lg n). e. The HEAP-INCREASE-KEY procedure with two small changes works for d- ary heaps. First, because the problem specifies that the new key is given by the parameter k, change instances of the variable key to k. Second, change calls of PARENT to calls of D-ARY-PARENT from part (a). In the worst case, the entire height of the tree must be traversed, so the worstcase running time is Θ(h) = Θ(log d n) = Θ(lg d/ lg n). 2

3 Problem What is the running time of QUICKSORT when all elements of the array A have the same value? It is Θ(n 2 ), since one of the partitions is always empty (see exercise 7.1.2). Problem 7-1 Hoare partition correctness The version of PARTITION given in this chapter is not the original partitioning algorithm. Here is the original partition algorithm, which is due to C.A.R. Hoare: HOARE PARTITION(A, p, r ) x = A[ p ] i = p 1 j = r + 1 while TRUE r e p e a t j = j 1 u n t i l A[ j ] <= x r e p e a t i = i + 1 u n t i l A[ i ] >= x i f i < j exchange A[ i ] with A[ j ] e l s e return j a. Demonstrate the operation of HOARE-PARTITION on the array A = 13, 19, 9, 5, 12, 8, 7, 4, 11, 2, 6, 21, showing the values of the array and auxiliary values after each iteration of the while loop in lines The next three questions ask you to give a careful argument that the procedure HOARE-PARTITION is correct. Assuming that the subarray A[p..r] contains at least two elements, prove the following: b. The indices i and j are such that we never access an element of A outside the subarray A[p..r]. c. When HOARE-PARTITION terminates, it returns a value j such that p j < r. d. Every element of A[p..j] is less than or equal to every element of A[j +1..r] when HOARE-PARTITION terminates. The PARTITION procedure in section 7.1 separates the pivot value (originally in A[r]) from the two partitions it forms. The HOARE-PARTITION procedure, on the other hand, always places the pivot value (originally in A[p]) into one of the two parititions A[p..j] and A[j + 1..r]. Since p j < r, this split is always nontrivial. Rewrite the QUICKSORT procedure to use HOARE-PARTITION. 3

4 Demonstration At the end of the loop, the variables have the following values: x = 13 j = 9 i = 10 Correctness The indices will not walk of the array. At the first check i < j, i = p and j p (because A[p] = x). If i = j, the algorithm will terminate without accessing invalid elements. If i < j, the next loop will also have indices i and j within the array, (because i r and j p). Note that if one of the indices gets to the end of the array, then i won t be less or equal to j any more. As for the return value, it will be at least one less than j. At the first iteration, either (1) A[p] is the maximum element and then i = p and j = p < r or (2) it is not and A[p] gets swapped with A[j] where j r. The loop will not terminate and on the next iteration, j gets decremented (before eventually getting returned). Combining those two cases we get p j < r. Finally, it s easy to observe the following invariant: Before the condition comparing i to j, all elements A[p..i 1] x and all elements A[j + 1..r] x. Initialization. The two repeat blocks establish just this condition. Maintenance. By exchanging A[i] and A[j] we make the A[p..i] x and A[j..r] x. Incrementing i and decrementing j maintain this invariant. Termination. The loop terminates when i j. The invariant still holds at termination. The third bit follows directly from this invariant. Implementation There s some C code below. #i n c l u d e <s t d b o o l. h> i n t h o a r e p a r t i t i o n ( i n t A[ ], i n t p, i n t r ) { i n t x = A[ p ], i = p 1, j = r, tmp ; while ( true ) { do { j ; } while (! (A[ j ] <= x ) ) ; do { i ++; } while (! (A[ i ] >= x ) ) ; i f ( i < j ) { tmp = A[ i ] ; A[ i ] = A[ j ] ; A[ j ] = tmp ; } e l s e { r eturn j ; 4

5 } } } void q u i c k s o r t ( i n t A[ ], i n t p, i n t r ) { i f ( p < r 1) { i n t q = h o a r e p a r t i t i o n (A, p, r ) ; q u i c k s o r t (A, p, q + 1 ) ; q u i c k s o r t (A, q + 1, r ) ; } } Problem Show that there is no comparison sort whose running time is linear for at least half of the n! inputs of length n. What about a fraction 1/n of the inputs of length n? What about a fraction 1/2 n? If it is linear for at least half of the inputs, then using the same reasoning as in the text, it must hold that: n! 2 2n This holds only for small values for n. Same goes for the other: And: n! n 2n n! 2 n 2n n! 4 n All those have solutions for small n < n 0, but don t hold for larger values. In contrast, insertion sort gets its work done in Θ(n) time in the best case. But this is a 1/n! fraction of the inputs, which is smaller than 1/2 n. Problem 8-3 Sorting variable-length items 1. You are given an array of integers, where different integers may have different number of digits, but the total number of digits over all the integers in the array is n. Show how to sort the array in O(n) time. 2. You are given an array of strings, where different strings may have different numbers of characters, but the total number of characters over all the strings is n. Show how to sort the strings in O(n) time. (Note that the desired order here is the standard alphabetical order; for example; a < ab < b.) 5

6 The numbers For the numbers, we can do this: 1. Group the numbers by number of digits and order the groups Radix sort each group 2. Let s analyze the number of steps. Let G i be the group of numbers with i digits and c i = G i. Thus: T (n) = i=1 nc i i = n The strings For the strings, we can do this: 1. Groups the strings by length and order the groups 2. Starting i on the maximum length and going down to 1, perform counting sort on the ith character. Make sure to include only groups that have an ith character. If the groups are subsequent subarrays in the original array, we re performing counting sort on a subarray ending on the last index of the original array. Problem 9-1 Largest i numbers in sorted order Show that the second smallest of n elements can be found with n + lg n 2 comparisons in the worst case. (Hint: Also find the smallest element.) We can compare the elements in a tournament fashion - we split them into pairs, compare each pair and then proceed to compare the winners in the same fashion. We need to keep track of each match the potential winners have participated in. We select a winner in n 1 matches. At this point, we know that the second smallest element is one of the lg n elements that lost to the smallest each of them is smaller than the ones it has been compared to, prior to losing. In another lg n 1 comparisons we can find the smallest element out of those. This is the answer we are looking for. Problem We can sort a given set of n numbers by first building a binary search tree contain- ing these numbers (using TREE-INSERT repeatedly to insert the numbers one by one) and then printing the numbers by an inorder tree walk. What are the worst- case and best-case running times for this sorting algorithm? 6

7 Here s the algorithm: TREE SORT(A) l e t T be an empty binary search t r e e f o r i = 1 to n TREE INSERT(T, A[ i ] ) ; INORDER TREE WALK(T. root ) Worst case: Θ(n 2 ) occurs when a linear chain of nodes results from the repeated TREE-INSERT operations. Best case: Θ(n lg n) occurs when a binary tree of height Θ(lg n) results from the repeated TREE-INSERT operations. Problem Suppose that we absorb every red node in a red-black tree into its black parent, so that the children of the red node become children of the black parent. (Ignore what happens to the keys.) What are the possible degrees of a black node after all its red children are absorbed? What can you say about the depths of the leaves of the resulting tree? After absorbing each red node into its black parent, the degree of each node black node is 2, if both children were already black, 3, if one child was black and one was red, or 4, if both children were red. All leaves of the resulting tree have the same depth. Problem Show that the longest simple path from a node x in a red-black tree to a descendant leaf has length at most twice that of the shortest simple path from node x to a descendant leaf. In the longest path, at least every other node is black. In the shortest path, at most every node is black. Since the two paths contain equal numbers of black nodes, the length of the longest path is at most twice the length of the shortest path. We can say this more precisely, as follows: Since every path contains bh(x) black nodes, even the shortest path from x to a descendant leaf has length at least bh(x). By definition, the longest path 7

8 from x to a descendant leaf has length height(x). Since the longest path has bh(x) black nodes and at least half the nodes on the longest path are black (by property 4), bh(x) height(x)/2, so length of longest path = height(x) 2 bh(x) twice length of shortest path. 8

### 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

### 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

### 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

### 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

### 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

### Quicksort is a divide-and-conquer sorting algorithm in which division is dynamically carried out (as opposed to static division in Mergesort).

Chapter 7: Quicksort Quicksort is a divide-and-conquer sorting algorithm in which division is dynamically carried out (as opposed to static division in Mergesort). The three steps of Quicksort are as follows:

### 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

### 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

### Algorithms and Data Structures

Algorithms and Data Structures CMPSC 465 LECTURES 20-21 Priority Queues and Binary Heaps Adam Smith S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 1 Trees Rooted Tree: collection

### 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:

### 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

### 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

### 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

### 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

### 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:

### 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

### 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

### Heaps & Priority Queues in the C++ STL 2-3 Trees

Heaps & Priority Queues in the C++ STL 2-3 Trees CS 3 Data Structures and Algorithms Lecture Slides Friday, April 7, 2009 Glenn G. Chappell Department of Computer Science University of Alaska Fairbanks

### 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:

### Binary Heaps. CSE 373 Data Structures

Binary Heaps CSE Data Structures Readings Chapter Section. Binary Heaps BST implementation of a Priority Queue Worst case (degenerate tree) FindMin, DeleteMin and Insert (k) are all O(n) Best case (completely

### 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

### 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

### 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

### Solutions for Introduction to algorithms second edition

Solutions for Introduction to algorithms second edition Philip Bille The author of this document takes absolutely no responsibility for the contents. This is merely a vague suggestion to a solution to

### Merge Sort. 2004 Goodrich, Tamassia. Merge Sort 1

Merge Sort 7 2 9 4 2 4 7 9 7 2 2 7 9 4 4 9 7 7 2 2 9 9 4 4 Merge Sort 1 Divide-and-Conquer Divide-and conquer is a general algorithm design paradigm: Divide: divide the input data S in two disjoint subsets

### 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

### 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

### 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

### 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

### Divide-and-Conquer Algorithms Part Four

Divide-and-Conquer Algorithms Part Four Announcements Problem Set 2 due right now. Can submit by Monday at 2:15PM using one late period. Problem Set 3 out, due July 22. Play around with divide-and-conquer

### Sorting Algorithms. Nelson Padua-Perez Bill Pugh. Department of Computer Science University of Maryland, College Park

Sorting Algorithms Nelson Padua-Perez Bill Pugh Department of Computer Science University of Maryland, College Park Overview Comparison sort Bubble sort Selection sort Tree sort Heap sort Quick sort Merge

### Mergesort and Quicksort

Mergesort Mergesort and Quicksort Basic plan: Divide array into two halves. Recursively sort each half. Merge two halves to make sorted whole. mergesort mergesort analysis quicksort quicksort analysis

### 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

### Analysis of Binary Search algorithm and Selection Sort algorithm

Analysis of Binary Search algorithm and Selection Sort algorithm In this section we shall take up two representative problems in computer science, work out the algorithms based on the best strategy to

### 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

### CS473 - Algorithms I

CS473 - Algorithms I Lecture 9 Sorting in Linear Time View in slide-show mode 1 How Fast Can We Sort? The algorithms we have seen so far: Based on comparison of elements We only care about the relative

### Divide and Conquer. Textbook Reading Chapters 4, 7 & 33.4

Divide d Conquer Textook Reading Chapters 4, 7 & 33.4 Overview Design principle Divide d conquer Proof technique Induction, induction, induction Analysis technique Recurrence relations Prolems Sorting

### Data Structure with C

Subject: Data Structure with C Topic : Tree Tree A tree is a set of nodes that either:is empty or has a designated node, called the root, from which hierarchically descend zero or more subtrees, which

### 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

### 12 Abstract Data Types

12 Abstract Data Types 12.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: Define the concept of an abstract data type (ADT).

### The following themes form the major topics of this chapter: The terms and concepts related to trees (Section 5.2).

CHAPTER 5 The Tree Data Model There are many situations in which information has a hierarchical or nested structure like that found in family trees or organization charts. The abstraction that models hierarchical

### Data Structures Fibonacci Heaps, Amortized Analysis

Chapter 4 Data Structures Fibonacci Heaps, Amortized Analysis Algorithm Theory WS 2012/13 Fabian Kuhn Fibonacci Heaps Lacy merge variant of binomial heaps: Do not merge trees as long as possible Structure:

### CS 253: Algorithms. Chapter 12. Binary Search Trees. * Deletion and Problems. Credit: Dr. George Bebis

CS 2: Algorithms Chapter Binary Search Trees * Deletion and Problems Credit: Dr. George Bebis Binary Search Trees Tree representation: A linked data structure in which each node is an object Node representation:

### 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

### Exam study sheet for CS2711. List of topics

Exam study sheet for CS2711 Here is the list of topics you need to know for the final exam. For each data structure listed below, make sure you can do the following: 1. Give an example of this data structure

### CSE 326: Data Structures B-Trees and B+ Trees

Announcements (4//08) CSE 26: Data Structures B-Trees and B+ Trees Brian Curless Spring 2008 Midterm on Friday Special office hour: 4:-5: Thursday in Jaech Gallery (6 th floor of CSE building) This is

### Binary Trees and Huffman Encoding Binary Search Trees

Binary Trees and Huffman Encoding Binary Search Trees Computer Science E119 Harvard Extension School Fall 2012 David G. Sullivan, Ph.D. Motivation: Maintaining a Sorted Collection of Data A data dictionary

### Data Structures. Level 6 C30151. www.fetac.ie. Module Descriptor

The Further Education and Training Awards Council (FETAC) was set up as a statutory body on 11 June 2001 by the Minister for Education and Science. Under the Qualifications (Education & Training) Act,

### Divide And Conquer Algorithms

CSE341T/CSE549T 09/10/2014 Lecture 5 Divide And Conquer Algorithms Recall in last lecture, we looked at one way of parallelizing matrix multiplication. At the end of the lecture, we saw the reduce SUM

### Lecture Notes on Binary Search Trees

Lecture Notes on Binary Search Trees 15-122: Principles of Imperative Computation Frank Pfenning Lecture 17 March 17, 2010 1 Introduction In the previous two lectures we have seen how to exploit the structure

### A binary search tree or BST is a binary tree that is either empty or in which the data element of each node has a key, and:

Binary Search Trees 1 The general binary tree shown in the previous chapter is not terribly useful in practice. The chief use of binary trees is for providing rapid access to data (indexing, if you will)

### 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

### 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

### 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

### 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:

### Binary Search Trees (BST)

Binary Search Trees (BST) 1. Hierarchical data structure with a single reference to node 2. Each node has at most two child nodes (a left and a right child) 3. Nodes are organized by the Binary Search

### DATABASE DESIGN - 1DL400

DATABASE DESIGN - 1DL400 Spring 2015 A course on modern database systems!! http://www.it.uu.se/research/group/udbl/kurser/dbii_vt15/ Kjell Orsborn! Uppsala Database Laboratory! Department of 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

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

### , 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

### Lecture Notes on Binary Search Trees

Lecture Notes on Binary Search Trees 15-122: Principles of Imperative Computation Frank Pfenning André Platzer Lecture 17 October 23, 2014 1 Introduction In this lecture, we will continue considering associative

### Binary Search. Search for x in a sorted array A.

Divide and Conquer A general paradigm for algorithm design; inspired by emperors and colonizers. Three-step process: 1. Divide the problem into smaller problems. 2. Conquer by solving these problems. 3.

### 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

### Laboratory Module 6 Red-Black Trees

Laboratory Module 6 Red-Black Trees Purpose: understand the notion of red-black trees to build, in C, a red-black tree 1 Red-Black Trees 1.1 General Presentation A red-black tree is a binary search tree

### Introduction to Algorithms March 10, 2004 Massachusetts Institute of Technology Professors Erik Demaine and Shafi Goldwasser Quiz 1.

Introduction to Algorithms March 10, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Quiz 1 Quiz 1 Do not open this quiz booklet until you are directed

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### Sample Questions Csci 1112 A. Bellaachia

Sample Questions Csci 1112 A. Bellaachia Important Series : o S( N) 1 2 N N i N(1 N) / 2 i 1 o Sum of squares: N 2 N( N 1)(2N 1) N i for large N i 1 6 o Sum of exponents: N k 1 k N i for large N and k

### 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

### 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)

### Tree Properties (size vs height) Balanced Binary Trees

Tree Properties (size vs height) Balanced Binary Trees Due now: WA 7 (This is the end of the grace period). Note that a grace period takes the place of a late day. Now is the last time you can turn this

### Motivation Suppose we have a database of people We want to gure out who is related to whom Initially, we only have a list of people, and information a

CSE 220: Handout 29 Disjoint Sets 1 Motivation Suppose we have a database of people We want to gure out who is related to whom Initially, we only have a list of people, and information about relations

### 1/1 7/4 2/2 12/7 10/30 12/25

Binary Heaps A binary heap is dened to be a binary tree with a key in each node such that: 1. All leaves are on, at most, two adjacent levels. 2. All leaves on the lowest level occur to the left, and all

### 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

### 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

### 6. Standard Algorithms

6. Standard Algorithms The algorithms we will examine perform Searching and Sorting. 6.1 Searching Algorithms Two algorithms will be studied. These are: 6.1.1. inear Search The inear Search The Binary

### Binary Search Trees > = Binary Search Trees 1. 2004 Goodrich, Tamassia

Binary Search Trees < 6 2 > = 1 4 8 9 Binary Search Trees 1 Binary Search Trees A binary search tree is a binary tree storing keyvalue entries at its internal nodes and satisfying the following property:

### 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

### Quiz 1 Solutions. (a) T F The height of any binary search tree with n nodes is O(log n). Explain:

Introduction to Algorithms March 9, 2011 Massachusetts Institute of Technology 6.006 Spring 2011 Professors Erik Demaine, Piotr Indyk, and Manolis Kellis Quiz 1 Solutions Problem 1. Quiz 1 Solutions True

### B+ Tree Properties B+ Tree Searching B+ Tree Insertion B+ Tree Deletion Static Hashing Extendable Hashing Questions in pass papers

B+ Tree and Hashing B+ Tree Properties B+ Tree Searching B+ Tree Insertion B+ Tree Deletion Static Hashing Extendable Hashing Questions in pass papers B+ Tree Properties Balanced Tree Same height for paths

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### Lecture 6: Binary Search Trees CSCI 700 - Algorithms I. Andrew Rosenberg

Lecture 6: Binary Search Trees CSCI 700 - Algorithms I Andrew Rosenberg Last Time Linear Time Sorting Counting Sort Radix Sort Bucket Sort Today Binary Search Trees Data Structures Data structure is a

### The ADT Binary Search Tree

The ADT Binary Search Tree The Binary Search Tree is a particular type of binary tree that enables easy searching for specific items. Definition The ADT Binary Search Tree is a binary tree which has an

### Output: 12 18 30 72 90 87. struct treenode{ int data; struct treenode *left, *right; } struct treenode *tree_ptr;

50 20 70 10 30 69 90 14 35 68 85 98 16 22 60 34 (c) Execute the algorithm shown below using the tree shown above. Show the exact output produced by the algorithm. Assume that the initial call is: prob3(root)

### Dynamic Programming. Lecture 11. 11.1 Overview. 11.2 Introduction

Lecture 11 Dynamic Programming 11.1 Overview Dynamic Programming is a powerful technique that allows one to solve many different types of problems in time O(n 2 ) or O(n 3 ) for which a naive approach

### International Journal of Advanced Research in Computer Science and Software Engineering

Volume 3, Issue 7, July 23 ISSN: 2277 28X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com Greedy Algorithm:

### 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

### Data Structures, Practice Homework 3, with Solutions (not to be handed in)

Data Structures, Practice Homework 3, with Solutions (not to be handed in) 1. Carrano, 4th edition, Chapter 9, Exercise 1: What is the order of each of the following tasks in the worst case? (a) Computing

### S. Muthusundari. Research Scholar, Dept of CSE, Sathyabama University Chennai, India e-mail: nellailath@yahoo.co.in. Dr. R. M.

A Sorting based Algorithm for the Construction of Balanced Search Tree Automatically for smaller elements and with minimum of one Rotation for Greater Elements from BST S. Muthusundari Research Scholar,

### 8. Query Processing. Query Processing & Optimization

ECS-165A WQ 11 136 8. Query Processing Goals: Understand the basic concepts underlying the steps in query processing and optimization and estimating query processing cost; apply query optimization techniques;

### Topological Properties

Advanced Computer Architecture Topological Properties Routing Distance: Number of links on route Node degree: Number of channels per node Network diameter: Longest minimum routing distance between any

### 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

### 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

### A binary search tree or BST is a binary tree that is either empty or in which the data element of each node has a key, and:

1 The general binary tree shown in the previous chapter is not terribly useful in practice. The chief use of binary trees is for providing rapid access to data (indexing, if you will) and the general binary

### Why Use Binary Trees?

Binary Search Trees Why Use Binary Trees? Searches are an important application. What other searches have we considered? brute force search (with array or linked list) O(N) binarysearch with a pre-sorted

### CompSci-61B, Data Structures Final Exam

Your Name: CompSci-61B, Data Structures Final Exam Your 8-digit Student ID: Your CS61B Class Account Login: This is a final test for mastery of the material covered in our labs, lectures, and readings.

### 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

### Persistent Data Structures

6.854 Advanced Algorithms Lecture 2: September 9, 2005 Scribes: Sommer Gentry, Eddie Kohler Lecturer: David Karger Persistent Data Structures 2.1 Introduction and motivation So far, we ve seen only ephemeral

### 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

### Chapter 13: Query Processing. Basic Steps in Query Processing

Chapter 13: Query Processing! Overview! Measures of Query Cost! Selection Operation! Sorting! Join Operation! Other Operations! Evaluation of Expressions 13.1 Basic Steps in Query Processing 1. Parsing