Algorithm Analysis Sorting II. Fall 2013 Carola Wenk
|
|
|
- Gervais Marshall
- 7 years ago
- Views:
Transcription
1 Algorithm Analysis Sorting II Fall 2013 Carola Wenk
2 Is Selection Sort Practical? What is the running time of selection sort for lists that have thousands of items? The theoretical performance is not particularly promising, neither is the practical performance: Size: n n 2 Selection Sort: seconds ,000 1,000, , ,000, ,000 10,000,000, Can we do better? What is done in practice? How good is the Python library sort function?
3 Revisiting Selection Sort What is the minimum amount of time required to sort a list? Selection sort takes linear time to place just a single element. Is this really necessary? compare
4 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10?,?,?,?,?,?,?,?,?,?,?
5 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1,?,?,?,?,?,?,?,?,?,?
6 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1,?,?,?,?,?,?,?,?,?,?
7 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2,?,?,?,?,?,?,?,?,?
8 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2,?,?,?,?,?,?,?,?,?
9 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3,?,?,?,?,?,?,?,?
10 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3,?,?,?,?,?,?,?,?
11 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4,?,?,?,?,?,?,?
12 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4,?,?,?,?,?,?,?
13 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5,?,?,?,?,?,?
14 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5,?,?,?,?,?,?
15 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6,?,?,?,?,?
16 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6,?,?,?,?,?
17 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7,?,?,?,?
18 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7,?,?,?,?
19 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7, 8,?,?,?
20 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7, 8,?,?,?
21 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7, 8, 9,?,?
22 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7, 8, 9,?,?
23 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,?
24 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,?
25 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
26 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Sorted List A Sorted List B 1, 4, 6, 8, 11 2, 3, 5, 7, 9, 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 The key idea is to scan through both lists, while moving the smallest element to a new list. If we finish scanning either list, the rest of the other list is appended to the result.
27 Merging Lists Suppose that we instead had a list that had two sorted halves. Could we do better? Algorithm: 1. Start at the beginning of both lists. 2. Move the smaller element to the result list, and consider the next element. 3. Repeat until one list is exhausted. 4. Put the other list at the end of the result list.
28 Merging Lists Algorithm: 1. Start at the beginning of both lists. 2. Move the smaller element to the result list, and consider the next element. 3. Repeat until one list is exhausted. 4. Put the other list at the end of the result list. Does this always produce a sorted list? How long does it take?
29 Merging Lists Algorithm: 1. Start at the beginning of both lists. 2. Move the smaller element to the result list, and consider the next element. 3. Repeat until one list is exhausted. 4. Put the other list at the end of the result list. Does this always produce a sorted list? How long does it take? For two lists with a total of items, time.
30 Merging Lists Algorithm: 1. Start at the beginning of both lists. 2. Move the smaller element to the result list, and consider the next element. 3. Repeat until one list is exhausted. 4. Put the other list at the end of the result list. What is the point of doing this? Aren t we trying to sort the list?
31 Merge Sort Suppose that we know how to merge two sorted lists. Then, we can sort recursively: Merge Sort: 1. Split the given list into two equal parts. 2. Recursively sort each half. 3. Merge the sorted halves and return the result.
32 Merge Sort Suppose that we know how to merge two sorted lists. Then, we can sort recursively: def merge_sort (L): n = len(l) #base case: if n<=1: return L #recursive case: Recursively sort each half A = merge_sort(l[:n/2]) # left half, L[0..n/2-1] B = merge_sort(l[n/2:]) # right half, L[n/2..n-1] # merge sorted halves: return merge(a,b)
33 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 Recursive Calls 5, 25, 55, 1 8, 10, 11, 2 5, 25 55, 1 8, 10 11, Actually, not a lot is happening in the recursive calls. So where is the sorting happening?
34 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 5, 25, 55, 1 8, 10, 11, 2 5, 25 55, 1 8, 10 11, The merge step is actually doing all of the work!
35 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 5, 25, 55, 1 8, 10, 11, 2 5, 25 55, 1 8, 10 11, The merge step is actually doing all of the work!
36 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 5, 25, 55, 1 8, 10, 11, 2 5, 25 55, 1 8, 10 11, The merge step is actually doing all of the work!
37 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 5, 25, 55, 1 8, 10, 11, 2 5, 25 55, 1 8, 10 11, The merge step is actually doing all of the work!
38 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 5, 25, 55, 1 8, 10, 11, 2 5, 25 1, 55 8, 10 11, The merge step is actually doing all of the work!
39 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 1, 5, 25, 55 8, 10, 11, 2 5, 25 1, 55 8, 10 11, The merge step is actually doing all of the work!
40 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 1, 5, 25, 55 8, 10, 11, 2 5, 25 1, 55 8, 10 11, The merge step is actually doing all of the work!
41 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 1, 5, 25, 55 8, 10, 11, 2 5, 25 1, 55 8, 10 11, The merge step is actually doing all of the work!
42 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 1, 5, 25, 55 8, 10, 11, 2 5, 25 1, 55 8, 10 11, The merge step is actually doing all of the work!
43 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 1, 5, 25, 55 8, 10, 11, 2 5, 25 1, 55 8, 10 2, The merge step is actually doing all of the work!
44 Merge Sort 5, 25, 55, 1, 8, 10, 11, 2 1, 5, 25, 55 2, 8, 10, 11 5, 25 1, 55 8, 10 2, The merge step is actually doing all of the work!
45 Merge Sort 1, 2, 5, 8, 10, 11, 25, 55 1, 5, 25, 55 2, 8, 10, 11 5, 25 1, 55 8, 10 2, The merge step is actually doing all of the work!
46 Merge Sort Runtime Analysis Runtime T(n) c T(n/2) T(n/2) dn def merge_sort (L): n = len(l) #base case: if n<=1: return L #recursive case: Recursively sort each half A = merge_sort(l[:n/2]) # left half, L[0..n/2-1] B = merge_sort(l[n/2:]) # right half, L[n/2..n-1] # merge sorted halves: return merge(a,b)
47 Runtime Recurrence for Merge Sort T(n) = c if n = 1; 2T(n/2) + dn if n > 1. But what does T(n) solve to? I.e., is it O(n) or O(n 2 ) or O(n 3 ) or?
48 Recursion Tree
49 Recursion Tree
50 Recursion Tree
51 Recursion Tree So, Merge Sort has runtime O(n log n) Is this faster than selection sort? By how much?
52 Divide-And-Conquer Input Solve Base Cases Divide-and-Conquer: 1. If the input is small enough, solve. 2. Otherwise, split input into parts. 3. Recursively solve each part. 4. Merge solutions. Solution Implementing these algorithms is easy because they are recursive.
53 Analysis of Divide-and-Conquer The divide-and-conquer paradigm for algorithms is easy to implement because we can use recursion, but the trick to is have an efficient merge step. How can we analyze these kinds of algorithms? Generalized Divide-and-Conquer Recurrence work to merge number of size of each split recursive calls Because of the divide step, these algorithms will often have a logarithmic term in the running time.
54 Real-World Sorting Size: n n 2 Selection Sort: seconds Merge Sort: seconds Tim Sort: seconds ,000 1,000, , ,000, ,000 10,000,000, Selection Sort does not scale, but Merge Sort can easily handle lists with hundreds of thousands of items. The builtin sort is cleverly optimized to run even faster on many lists, although its theoretical worst-case performance is identical to Merge Sort.
55 Real-World Sorting Size: n n 2 Selection Sort: seconds Merge Sort: seconds Tim Sort: seconds ,000 1,000, , ,000, ,000 10,000,000, So, what is the point of sorting? Is it really so important to do quickly? Yes! Sorting is probably the most commonly used subroutine in software, and the savings in work can add up drastically.
56 Google in a Nutshell jaguar jaguar jaguar jaguar Results Page B Page C Page E Page A... Google processes the entire web and computes PageRank to determine which pages are most authoritative. The PageRank is essentially the chance that a random web-surfer would end up on a particular page.
57 Google in a Nutshell jaguar Query Result Google Data Center 1. Search for query keywords in mined pages. 2. Select a set of matching pages and ads. 3. Sort pages by PageRank and return results.
58 Google in a Nutshell jaguar Query Result Google Data Center 1. Search for query keywords in mined pages. 2. Select a set of matching pages and ads. 3. Sort pages by PageRank and return results. This is done 3,000,000,000 times a day.
59 Google Data Center on the Columbia River in Oregon. An average Google query takes.2s. Suppose that 50% of the time was due to sorting, and that we are sorting about 10,000 items. What would happen if we substituted selection sort? Recall that computation is work, and requires electricity. This is a major recurring cost for Google (2 billion kwh in 2010); they attempt to maximize the revenue-per-query.
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
The Tower of Hanoi. Recursion Solution. Recursive Function. Time Complexity. Recursive Thinking. Why Recursion? n! = n* (n-1)!
The Tower of Hanoi Recursion Solution recursion recursion recursion Recursive Thinking: ignore everything but the bottom disk. 1 2 Recursive Function Time Complexity Hanoi (n, src, dest, temp): If (n >
Algorithms. Margaret M. Fleck. 18 October 2010
Algorithms Margaret M. Fleck 18 October 2010 These notes cover how to analyze the running time of algorithms (sections 3.1, 3.3, 4.4, and 7.1 of Rosen). 1 Introduction The main reason for studying big-o
CSC148 Lecture 8. Algorithm Analysis Binary Search Sorting
CSC148 Lecture 8 Algorithm Analysis Binary Search Sorting Algorithm Analysis Recall definition of Big Oh: We say a function f(n) is O(g(n)) if there exists positive constants c,b such that f(n)
CSC 180 H1F Algorithm Runtime Analysis Lecture Notes Fall 2015
1 Introduction These notes introduce basic runtime analysis of algorithms. We would like to be able to tell if a given algorithm is time-efficient, and to be able to compare different algorithms. 2 Linear
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
Many algorithms, particularly divide and conquer algorithms, have time complexities which are naturally
Recurrence Relations Many algorithms, particularly divide and conquer algorithms, have time complexities which are naturally modeled by recurrence relations. A recurrence relation is an equation which
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
Closest Pair of Points. Kleinberg and Tardos Section 5.4
Closest Pair of Points Kleinberg and Tardos Section 5.4 Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Fundamental geometric
Research Tools & Techniques
Research Tools & Techniques for Computer Engineering Ron Sass http://www.rcs.uncc.edu/ rsass University of North Carolina at Charlotte Fall 2009 1/ 106 Overview of Research Tools & Techniques Course What
CSE 421 Algorithms: Divide and Conquer
CSE 421 Algorithms: Divide and Conquer Summer 2011 Larry Ruzzo Thanks to Paul Beame, James Lee, Kevin Wayne for some slides hw2 empirical run times e n Plot Time vs n Fit curve to it (e.g., with Excel)
5.4 Closest Pair of Points
5.4 Closest Pair of Points Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Fundamental geometric primitive. Graphics, computer
CS473 - Algorithms I
CS473 - Algorithms I Lecture 4 The Divide-and-Conquer Design Paradigm View in slide-show mode 1 Reminder: Merge Sort Input array A sort this half sort this half Divide Conquer merge two sorted halves Combine
Closest Pair Problem
Closest Pair Problem Given n points in d-dimensions, find two whose mutual distance is smallest. Fundamental problem in many applications as well as a key step in many algorithms. p q A naive algorithm
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
28 Closest-Point Problems -------------------------------------------------------------------
28 Closest-Point Problems ------------------------------------------------------------------- Geometric problems involving points on the plane usually involve implicit or explicit treatment of distances
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
Learning Example. Machine learning and our focus. Another Example. An example: data (loan application) The data and the goal
Learning Example Chapter 18: Learning from Examples 22c:145 An emergency room in a hospital measures 17 variables (e.g., blood pressure, age, etc) of newly admitted patients. A decision is needed: whether
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
Data Structures. Algorithm Performance and Big O Analysis
Data Structures Algorithm Performance and Big O Analysis What s an Algorithm? a clearly specified set of instructions to be followed to solve a problem. In essence: A computer program. In detail: Defined
Algorithm Design and Recursion
Chapter 13 Algorithm Design and Recursion Objectives To understand basic techniques for analyzing the efficiency of algorithms. To know what searching is and understand the algorithms for linear and binary
Introduction to Programming (in C++) Sorting. Jordi Cortadella, Ricard Gavaldà, Fernando Orejas Dept. of Computer Science, UPC
Introduction to Programming (in C++) Sorting Jordi Cortadella, Ricard Gavaldà, Fernando Orejas Dept. of Computer Science, UPC Sorting Let elem be a type with a operation, which is a total order A vector
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
Introduction to Parallel Programming and MapReduce
Introduction to Parallel Programming and MapReduce Audience and Pre-Requisites This tutorial covers the basics of parallel programming and the MapReduce programming model. The pre-requisites are significant
Computational Geometry. Lecture 1: Introduction and Convex Hulls
Lecture 1: Introduction and convex hulls 1 Geometry: points, lines,... Plane (two-dimensional), R 2 Space (three-dimensional), R 3 Space (higher-dimensional), R d A point in the plane, 3-dimensional space,
Data Structures. Chapter 8
Chapter 8 Data Structures Computer has to process lots and lots of data. To systematically process those data efficiently, those data are organized as a whole, appropriate for the application, called a
Quick Sort. Implementation
Implementation Next, recall that our goal is to partition all remaining elements based on whether they are smaller than or greater than the pivot We will find two entries: One larger than the pivot (staring
Greatest Common Factor and Least Common Multiple
Greatest Common Factor and Least Common Multiple Intro In order to understand the concepts of Greatest Common Factor (GCF) and Least Common Multiple (LCM), we need to define two key terms: Multiple: Multiples
Battleships Searching Algorithms
Activity 6 Battleships Searching Algorithms Summary Computers are often required to find information in large collections of data. They need to develop quick and efficient ways of doing this. This activity
A Note on Maximum Independent Sets in Rectangle Intersection Graphs
A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,
APP INVENTOR. Test Review
APP INVENTOR Test Review Main Concepts App Inventor Lists Creating Random Numbers Variables Searching and Sorting Data Linear Search Binary Search Selection Sort Quick Sort Abstraction Modulus Division
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
recursion, O(n), linked lists 6/14
recursion, O(n), linked lists 6/14 recursion reducing the amount of data to process and processing a smaller amount of data example: process one item in a list, recursively process the rest of the list
Programming Using Python
Introduction to Computation and Programming Using Python Revised and Expanded Edition John V. Guttag The MIT Press Cambridge, Massachusetts London, England CONTENTS PREFACE xiii ACKNOWLEDGMENTS xv 1 GETTING
Zabin Visram Room CS115 CS126 Searching. Binary Search
Zabin Visram Room CS115 CS126 Searching Binary Search Binary Search Sequential search is not efficient for large lists as it searches half the list, on average Another search algorithm Binary search Very
Rethinking SIMD Vectorization for In-Memory Databases
SIGMOD 215, Melbourne, Victoria, Australia Rethinking SIMD Vectorization for In-Memory Databases Orestis Polychroniou Columbia University Arun Raghavan Oracle Labs Kenneth A. Ross Columbia University Latest
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
Lecture 18: Applications of Dynamic Programming Steven Skiena. Department of Computer Science State University of New York Stony Brook, NY 11794 4400
Lecture 18: Applications of Dynamic Programming Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Problem of the Day
Quick Tricks for Multiplication
Quick Tricks for Multiplication Why multiply? A computer can multiply thousands of numbers in less than a second. A human is lucky to multiply two numbers in less than a minute. So we tend to have computers
16.1 MAPREDUCE. For personal use only, not for distribution. 333
For personal use only, not for distribution. 333 16.1 MAPREDUCE Initially designed by the Google labs and used internally by Google, the MAPREDUCE distributed programming model is now promoted by several
Big Data Technology Map-Reduce Motivation: Indexing in Search Engines
Big Data Technology Map-Reduce Motivation: Indexing in Search Engines Edward Bortnikov & Ronny Lempel Yahoo Labs, Haifa Indexing in Search Engines Information Retrieval s two main stages: Indexing process
Extreme Computing. Big Data. Stratis Viglas. School of Informatics University of Edinburgh sviglas@inf.ed.ac.uk. Stratis Viglas Extreme Computing 1
Extreme Computing Big Data Stratis Viglas School of Informatics University of Edinburgh sviglas@inf.ed.ac.uk Stratis Viglas Extreme Computing 1 Petabyte Age Big Data Challenges Stratis Viglas Extreme Computing
Cloud Computing at Google. Architecture
Cloud Computing at Google Google File System Web Systems and Algorithms Google Chris Brooks Department of Computer Science University of San Francisco Google has developed a layered system to handle webscale
Computer Science 210: Data Structures. Searching
Computer Science 210: Data Structures Searching Searching Given a sequence of elements, and a target element, find whether the target occurs in the sequence Variations: find first occurence; find all occurences
Big Data Processing with Google s MapReduce. Alexandru Costan
1 Big Data Processing with Google s MapReduce Alexandru Costan Outline Motivation MapReduce programming model Examples MapReduce system architecture Limitations Extensions 2 Motivation Big Data @Google:
Big Data and Scripting
Big Data and Scripting 1, 2, Big Data and Scripting - abstract/organization contents introduction to Big Data and involved techniques schedule 2 lectures (Mon 1:30 pm, M628 and Thu 10 am F420) 2 tutorials
Logarithmic and Exponential Equations
11.5 Logarithmic and Exponential Equations 11.5 OBJECTIVES 1. Solve a logarithmic equation 2. Solve an exponential equation 3. Solve an application involving an exponential equation Much of the importance
Distributed Computing over Communication Networks: Maximal Independent Set
Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.
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
So today we shall continue our discussion on the search engines and web crawlers. (Refer Slide Time: 01:02)
Internet Technology Prof. Indranil Sengupta Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture No #39 Search Engines and Web Crawler :: Part 2 So today we
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
technical tips and tricks
technical tips and tricks Assigning resources to tasks Document author: Produced by: Andy Jessop Project Learning International Limited The tips and tricks below are taken from Project Mentor, the smart
FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M.
Mathematics Revision Guides Factors, Prime Numbers, H.C.F. and L.C.M. Page 1 of 16 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier FACTORS, PRIME NUMBERS, H.C.F. AND L.C.M. Version:
Predicting Flight Delays
Predicting Flight Delays Dieterich Lawson jdlawson@stanford.edu William Castillo will.castillo@stanford.edu Introduction Every year approximately 20% of airline flights are delayed or cancelled, costing
csci 210: Data Structures Recursion
csci 210: Data Structures Recursion Summary Topics recursion overview simple examples Sierpinski gasket Hanoi towers Blob check READING: GT textbook chapter 3.5 Recursion In general, a method of defining
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
The LCA Problem Revisited
The LA Problem Revisited Michael A. Bender Martín Farach-olton SUNY Stony Brook Rutgers University May 16, 2000 Abstract We present a very simple algorithm for the Least ommon Ancestor problem. We thus
Computer Algorithms. NP-Complete Problems. CISC 4080 Yanjun Li
Computer Algorithms NP-Complete Problems NP-completeness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order
High Performance Computing MapReduce & Hadoop. 17th Apr 2014
High Performance Computing MapReduce & Hadoop 17th Apr 2014 MapReduce Programming model for parallel processing vast amounts of data (TBs/PBs) distributed on commodity clusters Borrows from map() and reduce()
s = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 50 51. 2
1. Use Euler s trick to find the sum 1 + 2 + 3 + 4 + + 49 + 50. s = 1 + 2 +... + 49 + 50 s = 50 + 49 +... + 2 + 1 2s = 51 + 51 +... + 51 + 51 Thus, 2s = 50 51. Therefore, s = 50 51. 2 2. Consider the sequence
BCSD WebMail Documentation
BCSD WebMail Documentation Outlook Web Access is available to all BCSD account holders! Outlook Web Access provides Webbased access to your e-mail, your calendar, your contacts, and the global address
Why Use Google? Chapter 1
Chapter 1 Why Use Google? Google (www.google.com) is the premier search tool on the Internet today, featuring not only the best Web search engine, but many additional features including a directory, image
Extend Table Lens for High-Dimensional Data Visualization and Classification Mining
Extend Table Lens for High-Dimensional Data Visualization and Classification Mining CPSC 533c, Information Visualization Course Project, Term 2 2003 Fengdong Du fdu@cs.ubc.ca University of British Columbia
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
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
Day 1. This is CS50 for MBAs. Harvard Busines School. Spring 2015. Cheng Gong
This is CS50 for MBAs. Harvard Busines School. Spring 2015. Cheng Gong Table of Contents Gangnam Style... 1 Internet Issues... 2 Course info... 6 Day 0, recap... 7 Peanut butter jelly time... 8 Sorting...
Hadoop and Map-Reduce. Swati Gore
Hadoop and Map-Reduce Swati Gore Contents Why Hadoop? Hadoop Overview Hadoop Architecture Working Description Fault Tolerance Limitations Why Map-Reduce not MPI Distributed sort Why Hadoop? Existing Data
100 SEO Tips. 1. Recognize the importance of web traffic.
1. Recognize the importance of web traffic. 100 SEO Tips SEO or Search Engine Optimization is one of the techniques to use in order to achieve more volume of traffic to your site. Without a good number
How to fill every seat in the house. An event manager s guide to SMS Marketing
How to fill every seat in the house An event manager s guide to SMS Marketing - Why should you use SMS messaging? When was the last time you didn t have your mobile? Chances are you can t remember (because
Randomized algorithms
Randomized algorithms March 10, 2005 1 What are randomized algorithms? Algorithms which use random numbers to make decisions during the executions of the algorithm. Why would we want to do this?? Deterministic
Apache HBase. Crazy dances on the elephant back
Apache HBase Crazy dances on the elephant back Roman Nikitchenko, 16.10.2014 YARN 2 FIRST EVER DATA OS 10.000 nodes computer Recent technology changes are focused on higher scale. Better resource usage
ADVERTISING RESOURCES
ADVERTISING RESOURCES - Great Ways To Increase Your Web Traffic - What Is Web Traffic? Web traffic is the amount of data sent and received by visitors to a web site. Since the mid-1990s, web traffic has
Expanding the CASEsim Framework to Facilitate Load Balancing of Social Network Simulations
Expanding the CASEsim Framework to Facilitate Load Balancing of Social Network Simulations Amara Keller, Martin Kelly, Aaron Todd 4 June 2010 Abstract This research has two components, both involving the
Binary search tree with SIMD bandwidth optimization using SSE
Binary search tree with SIMD bandwidth optimization using SSE Bowen Zhang, Xinwei Li 1.ABSTRACT In-memory tree structured index search is a fundamental database operation. Modern processors provide tremendous
Outline. Introduction Linear Search. Transpose sequential search Interpolation search Binary search Fibonacci search Other search techniques
Searching (Unit 6) Outline Introduction Linear Search Ordered linear search Unordered linear search Transpose sequential search Interpolation search Binary search Fibonacci search Other search techniques
Taxonomy Enterprise System Search Makes Finding Files Easy
Taxonomy Enterprise System Search Makes Finding Files Easy 1 Your Regular Enterprise Search System Can be Improved by Integrating it With the Taxonomy Enterprise Search System Regular Enterprise Search
In-Memory Database: Query Optimisation. S S Kausik (110050003) Aamod Kore (110050004) Mehul Goyal (110050017) Nisheeth Lahoti (110050027)
In-Memory Database: Query Optimisation S S Kausik (110050003) Aamod Kore (110050004) Mehul Goyal (110050017) Nisheeth Lahoti (110050027) Introduction Basic Idea Database Design Data Types Indexing Query
Optimization of SQL Queries in Main-Memory Databases
Optimization of SQL Queries in Main-Memory Databases Ladislav Vastag and Ján Genči Department of Computers and Informatics Technical University of Košice, Letná 9, 042 00 Košice, Slovakia lvastag@netkosice.sk
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
Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
Clustering UE 141 Spring 2013
Clustering UE 141 Spring 013 Jing Gao SUNY Buffalo 1 Definition of Clustering Finding groups of obects such that the obects in a group will be similar (or related) to one another and different from (or
Big Data Storage, Management and challenges. Ahmed Ali-Eldin
Big Data Storage, Management and challenges Ahmed Ali-Eldin (Ambitious) Plan What is Big Data? And Why talk about Big Data? How to store Big Data? BigTables (Google) Dynamo (Amazon) How to process Big
RSA Question 2. Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true?
RSA Question 2 Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p-1)(q-1) = φ(n). Is this true? Bob chooses a random e (1 < e < Φ Bob ) such that gcd(e,φ Bob )=1. Then, d = e -1
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
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
IMPROVING PERFORMANCE OF RANDOMIZED SIGNATURE SORT USING HASHING AND BITWISE OPERATORS
Volume 2, No. 3, March 2011 Journal of Global Research in Computer Science RESEARCH PAPER Available Online at www.jgrcs.info IMPROVING PERFORMANCE OF RANDOMIZED SIGNATURE SORT USING HASHING AND BITWISE
4.2 Sorting and Searching
Sequential Search: Java Implementation 4.2 Sorting and Searching Scan through array, looking for key. search hit: return array index search miss: return -1 public static int search(string key, String[]
Biostatistics 615/815
Merge Sort Biostatistics 615/815 Lecture 8 Notes on Problem Set 2 Union Find algorithms Dynamic Programming Results were very ypositive! You should be gradually becoming g y g comfortable compiling, debugging
Hadoop Fair Scheduler Design Document
Hadoop Fair Scheduler Design Document October 18, 2010 Contents 1 Introduction 2 2 Fair Scheduler Goals 2 3 Scheduler Features 2 3.1 Pools........................................ 2 3.2 Minimum Shares.................................
An Overview of Predictive Analytics for Practitioners. Dean Abbott, Abbott Analytics
An Overview of Predictive Analytics for Practitioners Dean Abbott, Abbott Analytics Thank You Sponsors Empower users with new insights through familiar tools while balancing the need for IT to monitor
Lecture 10: Regression Trees
Lecture 10: Regression Trees 36-350: Data Mining October 11, 2006 Reading: Textbook, sections 5.2 and 10.5. The next three lectures are going to be about a particular kind of nonlinear predictive model,
Classroom Demonstrations of Big Data
Classroom Demonstrations of Big Data Eric A. Suess Abstract We present examples of accessing and analyzing large data sets for use in a classroom at the first year graduate level or senior undergraduate
Logs Transformation in a Regression Equation
Fall, 2001 1 Logs as the Predictor Logs Transformation in a Regression Equation The interpretation of the slope and intercept in a regression change when the predictor (X) is put on a log scale. In this
Vector storage and access; algorithms in GIS. This is lecture 6
Vector storage and access; algorithms in GIS This is lecture 6 Vector data storage and access Vectors are built from points, line and areas. (x,y) Surface: (x,y,z) Vector data access Access to vector
Near Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
Email Etiquette for Lotus Notes Users
Email Etiquette for Lotus Notes Users Email was originally used as an informal means of communication. However in recent times it has become an essential business communication tool used within the HSE.
HW4: Merge Sort. 1 Assignment Goal. 2 Description of the Merging Problem. Course: ENEE759K/CMSC751 Title:
HW4: Merge Sort Course: ENEE759K/CMSC751 Title: Merge Sort Date Assigned: March 24th, 2009 Date Due: April 10th, 2009 11:59pm Contact: Fuat Keceli keceli (at) umd (dot) edu 1 Assignment Goal The final
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
Search Taxonomy. Web Search. Search Engine Optimization. Information Retrieval
Information Retrieval INFO 4300 / CS 4300! Retrieval models Older models» Boolean retrieval» Vector Space model Probabilistic Models» BM25» Language models Web search» Learning to Rank Search Taxonomy!