DOCSLIB.ORG

A Fast Binary Logarithm Algorithm

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Fast Binary Logarithm Algorithm dsp TIPS&TRICKS [ ] Clay S. Turner A Fast Binary Logarithm Algorithm algorithm’s mathematical development. We see the right-hand side of (6) is a “DSP Tips and Tricks” introduces To start, we simply desire to find product of two factors, where the left fac- practical design and implementation tor is equal to either one or two depend- signal processing algorithms that y 5 log 1x2. (1) 2 ing on the value of y . Thus we have the you may wish to incorporate into 1 your designs. We welcome readers Thus we can invert this and find following two cases: 211 1 211 1 211 1...222 to submit their contributions. 5 y 5 2 5 2 y2 2 y3 2 y4 x 2 . (2) y1 0: x 2 Contact Associate Editors Rick Lyons (7) ([email protected]) or C. Britton Since the iterative part of our log algo- Rorabaugh ([email protected]). rithm assumes x is in 1 # x , 2, we or may need to “normalize” x. This nor- # 211 1 211 1 211 1...222 y 51: x 2 52 22 y2 2 y3 2 y4 . malization is performed by a simple 1 (8) his article presents a com- succession of divides/multiplies by two. putationally fast algorithm These divides/multiplies by two may be Since both (7) and (8) contain the com- 211 1 211 1 211 1 222 for computing logarithms. efficiently effected by binary shifts. The mon factor 22 y2 2 y3 2 y4 ... , and The algorithm is particular- count of the divides/multiplies gives this factor will have a value that is great- ly well suited for implemen- the characteristic of our logarithm er than or equal to one and less than Ttation using fixed-point processors. result. This will be added to the man- two, we see the square of x will the great- tissa (calculated below) to form the er than or equal to two if and only if 5 5 BACKGROUND complete logarithm. The number of y1 1, otherwise y1 0. Long before the technological age, in divides is taken as a positive number Step 2: Compare the result of squar- 1427, the Persian mathematician and and the number of multiplies is taken ing and if x2 is greater than or equal to astronomer al Kashi presented an algo- as a negative number. two, then set mantissa bit to “1” and rithm for efficient integral exponentia- Now with x properly scaled, then we divide x2 by two. Otherwise set mantissa tion via repeated squaring and know from (1) that 0 # y , 1. So let’s bit to “0” and leave x2 unchanged. multiplying [1]. Al Kashi’s method now expand y into a binary series, thus So now at the end of step 2, we have sees much use in modern cryptographic 2 2 2 2 2 11 y 12 11 y 12 11 y 1...222 5 # 21 1 # 22 1 # 23 x 5 2 2 3 4 . (9) systems where its efficiency with respect y y1 2 y2 2 y3 2 2 to very large numbers is paramount. 1 # 24 1 ... Realizing that x is just a number, we see y4 2 . (3) Related but not as well known as al that (9) has the exact same form as (5). Kashi’s method is an algorithm for find- It will be efficacious to put this into nest- Thus we can repeat Steps 1 and 2 to ing binary logarithms via repeated ed parenthetical form extract the remaining bits in y. squaring and dividing. Performing these After the prerequisite number of bits operations in a radix two-number for- 5 21 1 1 21 1 1 21 in the mantissa is obtained, then simply y 2 y1 2 y2 2 mat reduces the divisions to binary add the characteristic to the mantissa. If 3 1 1 21 1 1 ...2222 shifts, thus making the algorithm ame- y3 2 y4 . (4) we require a logarithm having a radix nable to fixed-point implementations on other than two, then we may employ the low-complexity microprocessors and So putting (4) into (2) we have following property of logarithms field-programmable gate arrays. 1 2 2 2 2 2 log2 x 2 11 y 12 11 y 12 11 y 12 11 y 1...2222 1 2 5 x 5 2 1 2 3 4 . (5) loga x 1 2 , (10) THE LOGARITHM ALGORITHM log2 a 1 2 Now that we have revealed the two main Now we are ready to see how to sequen- where the division by log2 a is imple- types of computations for finding the tially extract the bits comprising y. mented as a multiplication by the fixed Step 1: 1 2 binary logarithm, let’s go through our Square x as value 1/log2 a . 2 y # 2211 y 12211 y 12211 y 1...222 Digital Object Identifier 10.1109/MSP.2010.937503 x 5 2 1 2 2 3 4 . (6) (continued on page 140) IEEE SIGNAL PROCESSING MAGAZINE [124] SEPTEMBER 2010 1053-5888/10/$26.00©2010IEEE [dates AHEAD] Please send calendar submissions to: 2010 IEEE Workshop on Signal General Chair: Masayuki Tanimoto Dates Ahead, c/o Jessica Barragué, Processing Systems (SiPS 2010) URL: http://www.pcs2010.org/ IEEE Signal Processing Magazine 6–8 October, San Francisco, California. 445 Hoes Lane General Cochairs: Shuvra Battacharyya 2010 IEEE Spoken Language Piscataway, NJ 08855 USA, and Jorn Janneck Technology Workshop (SLT’10) e-mail: [email protected] URL: http://www.sips2010.org/ (Colored conference title indicates 12–15 December, Berkeley, California. SP-sponsored conference.) General Chairs: Dilek Hakkani-Tür and 2010 IEEE International Symposium Mari Ostendorf on Phased Array Systems and URL: http://www.slt2010.org/ 2010 Technology (ARRAY’10) 12–15 October, Waltham, Massachusetts. The IEEE International Workshop [SEPTEMBER] Conference Chair: Mark Russell on Information Forensics and URL: http://www.array2010.org/ 2010 International Conference Security (WIFS) on Image Processing (ICIP 2010) 12–15 December, Seattle, Washington. 26–29 September, Hong Kong. 2010 The 10th IEEE International General Cochairs: Darko Kirovski and General Chair: Wan-Chi Siu Conference on Signal Processing Paul Van Oorschot URL: http://www.icip2010.org (ICSP’10) URL: http://www.wifs10.org 24–28 October, Beijing, China. OCTOBER [ ] Conference Chair: Yuan Baozong The 6th IEEE Sensor Array and URL: http://icsp10.bjtu.edu.cn 2011 Multichannel Signal Processing Workshop (SAM ‘10) [JANUARY] [NOVEMBER] 4–7 October, Israel. 2011 IEEE Digital Signal Processing General Cochairs: Hagit Messer 2010 2nd International Conference and Signal Processing Education and Jeffrey L. Krolik on Audio, Language, and Image Workshop (DSP/SPE’10) URL: http://www.sam-2010.org/ Processing 4–7 January, Sedona, Arizona. 23–25 November 2010, Shanghai, China. General Chairs: Lina Karam General Chairs: Fa-Long Luo, Wanggen Wan, The 1st IEEE International Conference and Ronald Schafer and Thomas Sikora on Smart Grid Communications URL: http://www.dspe2011.org 4–6 October, Gaithersburg, Maryland. URL: http://www.icalip2010.cn/ General Cochairs: George Arnold and [MARCH] Stefano Galli [DECEMBER] URL: http://www.ieee-smartgridcomm.org Data Compression Conference 28th Picture Coding 29–31 March, Snowbird, Utah. Symposium (PCS’10) General Chair: James A. Storer Digital Object Identifier 10.1109/MSP.2010.937314 7–10 December, Nagoya, Japan. URL: http://www.cs.brandeis.edu/~dcc/ dsp TIPS&TRICKS continued from page 124 [ ] We list the binary logarithm algo- 8) Scale for next bit: b 5 b/2. AUTHOR rithm’s steps as follows: 9) Go to Step 6 and repeat until Clay S. Turner (Clay.Turner@PaceOMatic. 1) Initialize result to 0: y 5 0. desired number of mantissa bits are com) is chief scientist for Pace-O-Matic, 2) Initialize mantissa-bit decimal found. Inc. He has over 30 years of experience in value to 0.5: b 5 1/2. 10) Final log1x2 value: y. digital signal processing and mathemati- 3) While x , 1, x 5 2x, y 5 y 2 1. A “C” program, and MATLAB code, cal and embedded programming. 4) While x $ 2, x 5 x/2, y 5 y 1 1. demonstrating the algorithm are avail- 5) Go to Step 3 and repeat until able for download at http://www. REFERENCE 1 # x , 2. signalprocessingsociety.org/publications/ [1] D. E. Knuth, The Art of Computer Programming: # Seminumerical Algorithms, 2nd ed. Reading, MA: 6) Square: x 5 x x. periodicals/spm/columns-resources- Addison-Wesley, vol. 2, 1981, pp. 441–466. 7) If x $ 2, x 5 x/2, y 5 y 1 b. archive/2010-columns-resources/#tips. [SP] IEEE SIGNAL PROCESSING MAGAZINE [140] SEPTEMBER 2010.
Load more

Recommended publications
  • Mock In-Class Test COMS10007 Algorithms 2018/2019
    Mock In-class Test COMS10007 Algorithms 2018/2019 Throughout this paper log() denotes the binary logarithm, i.e, log(n) = log2(n), and ln() denotes the logarithm to base e, i.e., ln(n) = loge(n). 1 O-notation 1. Let f : N ! N be a function. Define the set Θ(f(n)). Proof. Θ(f(n)) = fg(n) : There exist positive constants c1; c2 and n0 s.t. 0 ≤ c1f(n) ≤ g(n) ≤ c2f(n) for all n ≥ n0g 2. Give a formal proof of the statement: p 10 n 2 O(n) : p Proof. We need to show that there are positive constants c; n0 such that 10 n ≤ c · n, 10 p for every n ≥ n0. The previous inequality is equivalent to c ≤ n, which in turn 100 100 gives c2 ≤ n. Hence, we can pick c = 1 and n0 = 12 = 100. 3. Use the racetrack principle to prove the following statement: n 2 O(2n) : Hint: The following facts can be useful: • The derivative of 2n is ln(2)2n. 1 • 2 ≤ ln(2) ≤ 1 holds. n Proof. We need to show that there are positive constants c; n0 such that n ≤ c·2 . We n pick c = 1 and n0 = 1. Observe that n ≤ 2 holds for n = n0(= 1). It remains to show n that n ≤ 2 also holds for every n ≥ n0. To show this, we use the racetrack principle. Observe that the derivative of n is 1 and the derivative of 2n is ln(2)2n. Hence, by n the racetrack principle it is enough to show that 1 ≤ ln(2)2 holds for every n ≥ n0, 1 1 1 1 or log( ln 2 ) ≤ n.
    [Show full text]
  • Optimizing MPC for Robust and Scalable Integer and Floating-Point Arithmetic
    Optimizing MPC for robust and scalable integer and floating-point arithmetic Liisi Kerik1, Peeter Laud1, and Jaak Randmets1,2 1 Cybernetica AS, Tartu, Estonia 2 University of Tartu, Tartu, Estonia {liisi.kerik, peeter.laud, jaak.randmets}@cyber.ee Abstract. Secure multiparty computation (SMC) is a rapidly matur- ing field, but its number of practical applications so far has been small. Most existing applications have been run on small data volumes with the exception of a recent study processing tens of millions of education and tax records. For practical usability, SMC frameworks must be able to work with large collections of data and perform reliably under such conditions. In this work we demonstrate that with the help of our re- cently developed tools and some optimizations, the Sharemind secure computation framework is capable of executing tens of millions integer operations or hundreds of thousands floating-point operations per sec- ond. We also demonstrate robustness in handling a billion integer inputs and a million floating-point inputs in parallel. Such capabilities are ab- solutely necessary for real world deployments. Keywords: Secure Multiparty Computation, Floating-point operations, Protocol design 1 Introduction Secure multiparty computation (SMC) [19] allows a group of mutually distrust- ing entities to perform computations on data private to various members of the group, without others learning anything about that data or about the intermedi- ate values in the computation. Theory-wise, the field is quite mature; there exist several techniques to achieve privacy and correctness of any computation [19, 28, 15, 16, 21], and the asymptotic overheads of these techniques are known.
    [Show full text]
  • CS 270 Algorithms
    CS 270 Algorithms Week 10 Oliver Kullmann Binary heaps Sorting Heapification Building a heap 1 Binary heaps HEAP- SORT Priority 2 Heapification queues QUICK- 3 Building a heap SORT Analysing 4 QUICK- HEAP-SORT SORT 5 Priority queues Tutorial 6 QUICK-SORT 7 Analysing QUICK-SORT 8 Tutorial CS 270 General remarks Algorithms Oliver Kullmann Binary heaps Heapification Building a heap We return to sorting, considering HEAP-SORT and HEAP- QUICK-SORT. SORT Priority queues CLRS Reading from for week 7 QUICK- SORT 1 Chapter 6, Sections 6.1 - 6.5. Analysing 2 QUICK- Chapter 7, Sections 7.1, 7.2. SORT Tutorial CS 270 Discover the properties of binary heaps Algorithms Oliver Running example Kullmann Binary heaps Heapification Building a heap HEAP- SORT Priority queues QUICK- SORT Analysing QUICK- SORT Tutorial CS 270 First property: level-completeness Algorithms Oliver Kullmann Binary heaps In week 7 we have seen binary trees: Heapification Building a 1 We said they should be as “balanced” as possible. heap 2 Perfect are the perfect binary trees. HEAP- SORT 3 Now close to perfect come the level-complete binary Priority trees: queues QUICK- 1 We can partition the nodes of a (binary) tree T into levels, SORT according to their distance from the root. Analysing 2 We have levels 0, 1,..., ht(T ). QUICK- k SORT 3 Level k has from 1 to 2 nodes. Tutorial 4 If all levels k except possibly of level ht(T ) are full (have precisely 2k nodes in them), then we call the tree level-complete. CS 270 Examples Algorithms Oliver The binary tree Kullmann 1 ❚ Binary heaps ❥❥❥❥ ❚❚❚❚ ❥❥❥❥ ❚❚❚❚ ❥❥❥❥ ❚❚❚❚ Heapification 2 ❥ 3 ❖ ❄❄ ❖❖ Building a ⑧⑧ ❄ ⑧⑧ ❖❖ heap ⑧⑧ ❄ ⑧⑧ ❖❖❖ 4 5 6 ❄ 7 ❄ HEAP- ⑧ ❄ ⑧ ❄ SORT ⑧⑧ ❄ ⑧⑧ ❄ Priority 10 13 14 15 queues QUICK- is level-complete (level-sizes are 1, 2, 4, 4), while SORT ❥ 1 ❚❚ Analysing ❥❥❥❥ ❚❚❚❚ QUICK- ❥❥❥ ❚❚❚ SORT ❥❥❥ ❚❚❚❚ 2 ❥❥ 3 ♦♦♦ ❄❄ ⑧ Tutorial ♦♦ ❄❄ ⑧⑧ ♦♦♦ ⑧ 4 ❄ 5 ❄ 6 ❄ ⑧⑧ ❄❄ ⑧ ❄ ⑧ ❄ ⑧⑧ ❄ ⑧⑧ ❄ ⑧⑧ ❄ 8 9 10 11 12 13 is not (level-sizes are 1, 2, 3, 6).
    [Show full text]
  • CS211 ❒ Overview • Code (Number Guessing Example)
    CS211 1. Number Guessing Example ASYMPTOTIC COMPLEXITY 1.1 Code ❒ Overview import java.io.*; • code (number guessing example) public class NumberGuess { • approximate analysis to motivate math public static void main(String[] args) throws IOException { int guess; • machine model int count; final int LOW=Integer.parseInt(args[0]); • analysis (space, time) final int HIGH=Integer.parseInt(args[1]); final int STOP=HIGH-LOW+1; • machine architecture and time assumptions int target = (int)(Math.random()*(HIGH-LOW+1))+(int)(LOW); BufferedReader in = new BufferedReader(new • code analysis InputStreamReader(System.in)); • more assumptions System.out.print("\nGuess an integer: "); guess = Integer.parseInt(in.readLine()); • Big Oh notation count = 1; • examples while (guess != target && guess >= LOW && • extra material (theory, background, math) guess <= HIGH && count < STOP ) { if (guess < target) System.out.println("Too low!"); else if (guess>target) System.out.println("Too high!"); else System.exit(0); System.out.print("\nGuess an integer: "); guess = Integer.parseInt(in.readLine()); count++ ; } if (target == guess) System.out.println("\nCongratulations!\n"); } } 12 1.2 Solution Algorithms 1.4 More General • random: • count only comparisons of guess to target - pick any number (same number as guesses) - worst-case time could be infinite • other examples: • linear: range linea binar pattern r y - guess one number at a time 1111→1 - start from bottom and head to top 1–10 10 5 5→7→8→9→10 1–100 100 8 50→75→87→93→96→98→99→100 - worst-case time could
    [Show full text]
  • Binary Search Algorithm Anthony Lin¹* Et Al
    WikiJournal of Science, 2019, 2(1):5 doi: 10.15347/wjs/2019.005 Encyclopedic Review Article Binary search algorithm Anthony Lin¹* et al. Abstract In In computer science, binary search, also known as half-interval search,[1] logarithmic search,[2] or binary chop,[3] is a search algorithm that finds a position of a target value within a sorted array.[4] Binary search compares the target value to an element in the middle of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array. Binary search runs in logarithmic time in the worst case, making 푂(log 푛) comparisons, where 푛 is the number of elements in the array, the 푂 is ‘Big O’ notation, and 푙표푔 is the logarithm.[5] Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are spe- cialized data structures designed for fast searching, such as hash tables, that can be searched more efficiently than binary search. However, binary search can be used to solve a wider range of problems, such as finding the next- smallest or next-largest element in the array relative to the target even if it is absent from the array. There are numerous variations of binary search.
    [Show full text]
  • Introduction to Algorithms
    CS 5002: Discrete Structures Fall 2018 Lecture 9: November 8, 2018 1 Instructors: Adrienne Slaughter, Tamara Bonaci Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. Introduction to Algorithms Readings for this week: Rosen, Chapter 2.1, 2.2, 2.5, 2.6 Sets, Set Operations, Cardinality of Sets, Matrices 9.1 Overview Objective: Introduce algorithms. 1. Review logarithms 2. Asymptotic analysis 3. Define Algorithm 4. How to express or describe an algorithm 5. Run time, space (Resource usage) 6. Determining Correctness 7. Introduce representative problems 1. foo 9.2 Asymptotic Analysis The goal with asymptotic analysis is to try to find a bound, or an asymptote, of a function. This allows us to come up with an \ordering" of functions, such that one function is definitely bigger than another, in order to compare two functions. We do this by considering the value of the functions as n goes to infinity, so for very large values of n, as opposed to small values of n that may be easy to calculate. Once we have this ordering, we can introduce some terminology to describe the relationship of two functions. 9-1 9-2 Lecture 9: November 8, 2018 Growth of Functions nn 4096 n! 2048 1024 512 2n 256 128 n2 64 32 n log(n) 16 n 8 p 4 n log(n) 2 1 1 2 3 4 5 6 7 8 From this chart, we see: p 1 log n n n n log(n) n2 2n n! nn (9.1) Complexity Terminology Θ(1) Constant Θ(log n) Logarithmic Θ(n) Linear Θ(n log n) Linearithmic Θ nb Polynomial Θ(bn) (where b > 1) Exponential Θ(n!) Factorial 9.2.1 Big-O: Upper Bound Definition 9.1 (Big-O: Upper Bound) f(n) = O(g(n)) means there exists some constant c such that f(n) ≤ c · g(n), for large enough n (that is, as n ! 1).
    [Show full text]
  • Comparison of Parallel Sorting Algorithms
    Comparison of parallel sorting algorithms Darko Božidar and Tomaž Dobravec Faculty of Computer and Information Science, University of Ljubljana, Slovenia Technical report November 2015 TABLE OF CONTENTS 1. INTRODUCTION .................................................................................................... 3 2. THE ALGORITHMS ................................................................................................ 3 2.1 BITONIC SORT ........................................................................................................................ 3 2.2 MULTISTEP BITONIC SORT ................................................................................................. 3 2.3 IBR BITONIC SORT ................................................................................................................ 3 2.4 MERGE SORT .......................................................................................................................... 4 2.5 QUICKSORT ............................................................................................................................ 4 2.6 RADIX SORT ........................................................................................................................... 4 2.7 SAMPLE SORT ........................................................................................................................ 4 3. TESTING ENVIRONMENT .................................................................................... 5 4. RESULTS ................................................................................................................
    [Show full text]
  • Calculating Logs Manually
    calculating logs manually File Name: calculating logs manually.pdf Size: 2113 KB Type: PDF, ePub, eBook Category: Book Uploaded: 3 May 2019, 12:21 PM Rating: 4.6/5 from 561 votes. Status: AVAILABLE Last checked: 10 Minutes ago! In order to read or download calculating logs manually ebook, you need to create a FREE account. Download Now! eBook includes PDF, ePub and Kindle version ✔ Register a free 1 month Trial Account. ✔ Download as many books as you like (Personal use) ✔ Cancel the membership at any time if not satisfied. ✔ Join Over 80000 Happy Readers Book Descriptions: We have made it easy for you to find a PDF Ebooks without any digging. And by having access to our ebooks online or by storing it on your computer, you have convenient answers with calculating logs manually . To get started finding calculating logs manually , you are right to find our website which has a comprehensive collection of manuals listed. Our library is the biggest of these that have literally hundreds of thousands of different products represented. Home | Contact | DMCA Book Descriptions: calculating logs manually And let’s start with the logarithm of 2. As a kid I always wanted to know how to calculate log 2 and nobody was able to tell me. Can we guess the logarithm of 19,683.Let’s follow the chain. We are looking for 1225, so to account for the difference let’s round 3.0876 up to 3.088. How to calculate log 11. Here is a way to do this. We can take the geometric mean of 10 and 12.
    [Show full text]
  • Hardware Implementation of the Logarithm Function
    DEPARTMENT OF ELECTRICAL AND INFORMATION TECHNOLOGY,FACULTY OF ENGINEERING, LTH MASTER OF SCIENCE THESIS Hardware Implementation of the Logarithm Function using Improved Parabolic Synthesis Supervisor: Author: Peter Nilsson Jingou Lai Erik Hertz Rakesh Gangarajaiah Lund September 2, 2013 The Department of Electrical and Information Technology Lund University Box 118, S-221 00 LUND SWEDEN This thesis is set in Computer Modern 11pt, with the LATEX Documentation System © Jingou Lai 2013 Abstract This thesis presents a design that approximates the fractional part of the based two logarithm function by using Improved Parabolic Synthesis including its CMOS VLSI implementations. Improved Parabolic Synthesis is a novel methodology in favor of implementing unary functions e.g. trigonometric, logarithm, square root etc. in hardware. It is an evolved approach from Parabolic Synthesis by combining it with Second-Degree Interpolation. In the thesis, the design explores a simple and parallel architecture for fast timing and optimizes wordlengths in computing stages for a small design. The error behavior of the design is described and char- acterized to meet the desired error metrics. This implementation is compared to other approaches e.g. Parabolic Synthesis and CORDIC using 65nm standard cell libraries and it is proved to have better performance in terms of smaller chip area, lower dynamic power, and shorter critical path. iv Acknowledgments This thesis would not have been possible without the support from many people. I would like to express my deepest gratitude to the supervisors, Peter Nilsson, Erik Hertz, and Rakesh Gangarajaiah who patiently, inspiringly guided me through this thesis work. My special thanks would go to my families and dearest friends who always have faith in me and encourage me.
    [Show full text]
  • Comparing Time Complexity of Algorithms from Chapter 3, Standard Notation and Common Functions
    Comparing time complexity of algorithms From Chapter 3, standard notation and common functions: • When the base of a log is not mentioned, it is assumed to be base 2. • Analogy between comparisons of functions f(n) and g(n) and comparisons of real num- bers a and b: f(n) = O(g(n)) is like a ≤ b f(n) = Ω(g(n)) is like a ≥ b f(n) = Θ(g(n)) is like a = b f(n) = o(g(n)) is like a < b f(n) = !(g(n)) is like a > b • A polynomial of degree d is Θ(nd). • For all real constants a and b such that a > 1; b > 0; nb lim = 0 x!1 an so nb = o(an). Any exponential function with a base > 1 grows faster than any poly- nomial function. • Notation used for common logarithms: lgn = log2n (binary logarithm) lnn = logen (natural logarithm) • More logarithmic facts: For all real a > 0; b > 0; c > 0, and n, a = b(logba) // Ex: 2(lgn) = n(lg2) = n n logba = nlogba y logbx = y iff x = b logba = (logca)=(logcb) // the base of the log doesn't matter asymptotically a(logbc) = c(logba) lgbn = o(na) // any polynomial grows faster than any polylogarithm n! = o(nn) // factorial grows slower than nn n! = !(2n) // factorial grows faster than exponential with base ≥ 2 lg(n!) = Θ(nlgn) // Stirling's rule 1 • Iterated logarithm function: lg∗n (log star of n) lg(i)n is the log function applied i times in succession. lg∗n = min(i ≥ 0 such that lg(i)n ≤ 1) x lg∗x (−∞; 1] 0 (1; 2] 1 (2; 4] 2 (4; 16] 3 (16; 65536] 4 (65536; 265536] 5 lg∗2 = 1 lg∗4 = 2 lg∗16 = 3 lg∗65536 = 4 Very slow-growing function.
    [Show full text]
  • Algorithms and Data Structures CS 372
    Algorithms and Data Structures CS 372 Asymptotic notations (Based on slides by M. Nicolescu) Algorithm Analysis • The amount of resources used by the algorithm –Space – Computational time • Running time: – The number of primitive operations (steps) executed before termination • Order of growth – The leading term of a formula – Expresses the behavior of a function toward infinity 2 Asymptotic Notations • A way to describe behavior of functions in the limit – How we indicate running times of algorithms – Describe the running time of an algorithm as n grows to ∞ • O notation: asymptotic “less than”: f(n) “≤”g(n) • Ω notation: asymptotic “greater than”: f(n) “≥”g(n) • Θ notation: asymptotic “equality”: f(n) “=” g(n) 3 1 Logarithms • In algorithm analysis we often use the notation “log n” without specifying the base y y log x Binary logarithm lgn = log2 n log x = Natural logarithm ln n = loge n log xy = log x + log y x lgk n = (lg n)k log = log x − log y y lglgn = lg(lgn) loga x = loga blogb x alogb x = xlogb a 4 Asymptotic notations •O-notation • Intuitively: O(g(n)) = the set of functions with a smaller or same order of growth as g(n) 5 Examples 2 3 2 3 –2n = O(n ): 2n ≤ cn ⇒ 2 ≤ cn ⇒ c = 1 and n0= 2 2 2 2 2 –n= O(n ): n ≤ cn ⇒ c ≥ 1 ⇒ c = 1 and n0= 1 – 1000n2+1000n = O(n2): 2 2 2 2 1000n +1000n ≤ 1000n +1000n =2000n ⇒ c=2000 and n0 = 1 2 2 –n = O(n): n ≤ cn ⇒ cn ≥ 1 ⇒ c = 1 and n0= 1 6 2 Asymptotic notations (cont.) • Ω -notation • Intuitively: Ω(g(n)) = the set of functions with a larger or same order of growth as g(n) 7 Examples – 5n2
    [Show full text]
  • THE NUMBER of BIT COMPARISONS USED by QUICKSORT: an AVERAGE-CASE ANALYSIS 1. Introduction and Summary Algorithms for Sorting
    THE NUMBER OF BIT COMPARISONS USED BY QUICKSORT: AN AVERAGE-CASE ANALYSIS JAMES ALLEN FILL AND SVANTE JANSON Abstract. The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the rep- resentation of the keys involved as bit strings and so count the number of bit comparisons. On the other hand, the standard analyses of many other algorithms (such as Quicksort) are performed in terms of the num- ber of key comparisons. We introduce the prospect of a fair comparison between algorithms of the two types by providing an average-case anal- ysis of the number of bit comparisons required by Quicksort. Counting bit comparisons rather than key comparisons introduces an extra loga- rithmic factor to the asymptotic average total. We also provide a new algorithm, “BitsQuick”, that reduces this factor to constant order by eliminating needless bit comparisons. 1. Introduction and summary Algorithms for sorting and searching (together with their accompanying analy- ses) generally fall into one of two categories: either the algorithm is regarded as comparing items pairwise irrespective of their internal structure (and so the anal- ysis focuses on the number of comparisons), or else it is recognized that the items (typically numbers) are represented as bit strings and that the algorithm operates on the individual bits. Typical examples of the two types are Quicksort and digital search trees, respectively; see [7]. In this extended abstract we take a first step towards bridging the gap between the two points of view, in order to facilitate run-time comparisons across the gap, by answering the following question posed many years ago by Bob Sedgewick [personal communication]: What is the bit complexity of Quicksort? More precisely, we consider Quicksort (see Section 2 for a review) applied to n distinct keys (numbers) from the interval (0, 1).
    [Show full text]