Big O Notation

9 Key excerpts on "Big O Notation"

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  eBook - PDF

  Programming with C++

  • Kyla McMullen, Elizabeth Matthews, June Jamrich Parsons, , Kyla McMullen, Kyla McMullen, Elizabeth Matthews, June Jamrich Parsons(Authors)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Big-O notation allows programmers to quantify and express that maximum. Big-O notation takes into consideration the time complexity as a data set increases in size. That data might be accessed from a file, dimensioned as an array, stored as a list, or input during runtime. The letter O is used for Big-O notation because the rate of growth of a function is also called its order. The general format for Big-O notation is O(complexity), where complexity indicates the time or space requirements of an algorithm relative to the growth of the data set n. Q What do you think n stands for in Big-O notation? A In Big-O notation, n indicates the number of elements in a data set. The following Big-O metrics are most commonly used to measure the time complexity of algorithms. O(C) Constant time O(n) Linear time O(n 2 ) Quadratic time O(log n) Logarithmic time Let’s take a quick look at each of these metrics and the kinds of algorithms to which they apply. Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Module 22 AlgorithM CoMplexity And Big-o notAtion 399 Constant Time (22.2.4) Suppose you have a list of fast food restaurants sorted in order by the date they were first established. The first element in the list is White Castle because it was established in 1921. You write a program to find the first element in this list. Finding the first element in a list requires one line of code that specifies one operation: get the item at index [0]. first_item = fast_food_list[0] It does not matter how many fast food restaurants are in the data set.

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  Data Structures And Algorithms

  • Shi-kuo Chang(Author)
  • 2003(Publication Date)
  • World Scientific

    (Publisher)

  It is well known that the implementation depends on the choices performed by the compiler but also on the operating system running on the computer. These considerations allow us to conclude that given a program and its data, we are not able to say, for example, that: the following program will perform its job in 8.34 seconds. However, the problem of the measure of the real execution time of a program remains complex to determine even if we know the type of com-puter, the compiler, the program and the test cases used to execute the program. This is the reason why it is necessary to choose a measure that is independent of the implementation details of the compiler (independent of the average number of machine instructions generated) and of the com-puter. This notation is called Big-O. If we recall the code fragment of the BubbleSort program of Fig. 2.1, we say that the execution time of that fragment is 0(m) and we read it Big-O of TO, instead of saying that the program requires an execution time equal to Am — 1 on a vector of length TO. In other words, we are saying that the execution time of that fragment of code is a constant multiplied by TO . This definition not only allows to ignore the implementation details related to the computer and the com-piler but also to formulate a simple hypothesis for the calculation of the execution time. 22 P. Maresca 2.4.2. Big-O Notation The formal definition of the execution time of a program follows. Let T(n) be the execution time of a program and n the dimension of the input data, we assume that a) n is a non-negative integer b) T(n) is non-negative Vn Let f(n) be a function defined over non-negative integer values of n, then T(n) is 0(f(n)) <£> 3n 0 , c > 0 : Vn > n 0 T(n) < cf(n) (1) In other words T(n) is 0(f(n)) if T{n) is at most a constant multiplied by /(n), with the exception for some values of n. We can apply the Big-0 definition to prove that T(n) is 0(f(n)) for particular functions T and /.

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  The Complete Coding Interview Guide in Java

  An effective guide for aspiring Java developers to ace their programming interviews

  • Anghel Leonard(Author)
  • 2020(Publication Date)
  • Packt Publishing

    (Publisher)

  Chapter 7 : Big O Analysis of Algorithms This chapter covers the most popular metric for analyzing the efficiency and scalability of algorithms—Big O Notation—in the context of a technical interview.

  There are plenty of articles dedicated to this topic. Some of them are purely mathematical (academic), while others try to explain it with a more friendly approach. The pure mathematical approach is quite hard to digest and not very useful during an interview, so we will go for a more friendly approach that will be much more familiar to interviewers and developers.

  Even so, this is not an easy mission because besides being the most popular metric for measuring the efficiency and scalability of algorithms, Big O Notation can often also be the thing that you've never been motivated enough to learn about, despite knowing that it's going to show up in every single interview. From juniors to senior warriors, Big O Notation is probably the biggest Achilles heel for everyone. However, let's make an effort to turn this Achilles heel into a strong point for our interviews.

  We will quickly go over Big O Notation and highlight the things that matter the most. Next, we'll jump into examples that have been carefully crafted to cover a wide range of problems, and so by the end of this chapter, you'll be able to determine and express Big O for almost any given snippet of code. Our agenda includes the following:

  • Analogy
  • Big O complexity time
  • The best case, worst case, and expected case
  • Big O examples

  So, let's start our Big O journey!

  Analogy

  Imagine a scenario where you've found one of your favorite movies on the internet. You can order it or download it. Since you want to see it as soon as possible, which is the best way to proceed? If you order it, then it will take a day to arrive. If you download it, then it will take half a day to download. So, it is faster to download it. That's the way to go!

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  C++ High Performance

  Master the art of optimizing the functioning of your C++ code, 2nd Edition

  • Björn Andrist, Viktor Sehr(Authors)
  • 2020(Publication Date)
  • Packt Publishing

    (Publisher)

  3

  Analyzing and Measuring Performance

  Since this is a book about writing C++ code that runs efficiently, we need to cover some basics regarding how to measure software performance and estimate algorithmic efficiency. Most of the topics in this chapter are not specific to C++ and can be used whenever you are facing a problem where performance is an issue.

  You will learn how to estimate algorithmic efficiency using Big O Notation. This is essential knowledge when choosing algorithms and data structures from the C++ standard library. If you are new to Big O Notation, this part might take some time to digest. But don't give up! This is a very important topic to grasp in order to understand the rest of the book, and, more importantly, to become a performance-aware programmer. If you want a more formal or more practical introduction to these concepts, there are plenty of books and online resources dedicated to this topic. On the other hand, if you have already mastered Big O Notation and know what amortized time complexity is, you can skim the next section and go to the later parts of this chapter.

  This chapter includes sections on:

  • Estimating algorithmic efficiency using Big O Notation
  • A suggested workflow when optimizing code so that you don't spend time fine-tuning code without good reason
  • CPU profilers—what they are and why you should use them
  • Microbenchmarking

  Let's begin by taking a look at how to estimate algorithmic efficiency using big O notation.

  Asymptotic complexity and Big O Notation

  There is usually more than one way to solve a problem, and if efficiency is a concern, you should first focus on high-level optimizations by choosing the right algorithms and data structures. A useful way of evaluating and comparing algorithms is by analyzing their asymptotic computational complexity—that is, analyzing how the running time or memory consumption grows when the size of the input increases. In addition, the C++ standard library specifies the asymptotic complexity for all containers and algorithms, which means that a basic understanding of this topic is a must if you are using this library. If you already have a good understanding of algorithm complexity and the Big O Notation, you can safely skip this section.

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  Hands-On Data Structures and Algorithms with Python

  • Dr. Basant Agarwal(Author)
  • 2022(Publication Date)
  • Packt Publishing

    (Publisher)

  T(n) <=c * F(n) .

  Since, 2n+5 ≤ 3n, for all n ≥ 5.

  The condition is true for c =3, n0 =5.

  2n + 5 ≤ O(n) F(n) = n

  1. Find F(n) for the function T(n) = n 2 +n, such that T(n) = O(F(n)) .
     Solution : Using O notation, since, n 2 + n ≤ 2n 2 , for all n ≥ 1 (with c = 2, n 0 =2)
     n2 + n ≤ O(n2 )
     F(n) = n2

  1. Prove that f(n) =2n3 - 6n ≠ O(n2 ).
     Solution : Clearly, 2n3 -6n ≥ n2 , for n ≥ 2. So it cannot be true that 2n3 - 6n ≠ O(n2 ).

  1. Prove that: 20n2 +2n+5 = O(n2 ).
     Solution : It is clear that:
     20n2 + 2n + 5 <= 21n2 for all n > 4 (let c = 21 and n0 = 4)
     n2 > 2n + 5 for all n > 4
     So, the complexity is O(n2 ).

  So, Big-O notation provides an upper bound on a function, which ensures that the function never grows faster than the upper-bounded function. In the next section, we will discuss Omega notation.

  Omega notation

  Omega notation (Ω) describes an asymptotic lower bound on algorithms, similar to the way in which Big O Notation describes an upper bound. Omega notation computes the best-case runtime complexity of the algorithm. The Ω notation (Ω (F (n )) is pronounced as omega of F of n), is a set of functions in such a way that there are positive constants n0 and c such that for all values of n greater than n0 , T (n ) always lies on or above a function to c *F (n ).

  T (n ) = Ω (F (n ))

  Iff constants n0 and c are present, then:

  Figure 2.4 shows the graphical representation of the omega (Ω) notation. It can be observed from the figure that the value of T (n ) always lies above cF(n) for values of n greater than n0 .

  Figure 2.4: The graphical representation of Ω notation

  If the running time of an algorithm is Ω (F (n )), it means that the running time of the algorithm is at least a constant multiplier of F (n ) for sufficiently large values of input size (n). The Ω notation gives a lower bound on the best-case running time complexity of a given algorithm. It means that the running time for a given algorithm will be at least F (n

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  Golang for Jobseekers

  Unleash the power of Go programming for career advancement (English Edition)

  • Hairizuan Bin Noorazman(Author)
  • 2023(Publication Date)
  • BPB Publications

    (Publisher)

  We will be covering various sorting algorithms that are somewhat commonly implemented in the industry and then explore the different time and space complexities each of the algorithms bring. The time and space complexities mentioned here will be further explained in the subsection “Big O Notation.” The sorting algorithms covered in this chapter will generally deal with handling arrays and slices. After having a sorted list, we can then do certain operations in a more efficient way—namely, searching for an element within a list. This will be covered in detail within the Binary Search section of this chapter. The last portion will cover the basics of dynamic programming and how it can be useful in certain scenarios. Big O Notation There are numerous algorithms that have been designed out there, and each has its own peculiarities and properties in order to implement the solution for a computation problem. However, to us as programmers, we will need to be able to understand the benefits and advantages an algorithm brings compared to an alternative algorithm, and we need some sort of common mechanism that will allow us to understand and compare algorithms. With that mechanism, we will be able to argue about why an algorithm is chosen (maybe it can solve the problem in a more timely fashion). Most of the time, we will be concerned with how fast an algorithm solves a problem (a lot of computational problems out there require us to develop algorithms that will ensure that the systems being built are responsive, and this would require the algorithms to complete calculations in as short of a timeframe as possible). This is known as time complexity—and we can try to somewhat estimate the time complexity of the algorithm by going through its implementation to see how many times it has to go through the data and how frequently it accesses a piece of data. Let us say we take the most simplest problem of finding an item in an “unsorted” list.

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  Algorithm Design and Applications

  • Michael T. Goodrich, Roberto Tamassia(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Likewise, we say that f (n) is ω(g(n)) (pronounced “f (n) is little-omega of g(n)”) if g(n) is o(f (n)), that is, if, for any constant c > 0, there is a constant n 0 > 0 such that g(n) ≤ cf (n) for n ≥ n 0 . Intuitively, o(·) is analogous to “less than” in an asymptotic sense, and ω(·) is analogous to “greater than” in an asymptotic sense. Example 1.11: The function f (n) = 12n 2 + 6n is o(n 3 ) and ω(n). Proof: Let us first show that f (n) is o(n 3 ). Let c > 0 be any constant. If we take n 0 = (12 + 6)/c = 18/c, then 18 ≤ cn, for n ≥ n 0 . Thus, if n ≥ n 0 , f (n) = 12n 2 + 6n ≤ 12n 2 + 6n 2 = 18n 2 ≤ cn 3 . Thus, f (n) is o(n 3 ). To show that f (n) is ω(n), let c > 0 again be any constant. If we take n 0 = c/12, then, for n ≥ n 0 , 12n ≥ c. Thus, if n ≥ n 0 , f (n) = 12n 2 + 6n ≥ 12n 2 ≥ cn. Thus, f (n) is ω(n). For the reader familiar with limits, we note that f (n) is o(g(n)) if and only if lim n→∞ f (n) g(n) = 0, provided this limit exists. The main difference between the little-oh and big-Oh notions is that f (n) is O(g(n)) if there exist constants c > 0 and n 0 ≥ 1 such that f (n) ≤ cg(n), for n ≥ n 0 ; whereas f (n) is o(g(n)) if for all constants c > 0 there is a constant n 0 such that f (n) ≤ cg(n), for n ≥ n 0 . Intuitively, f (n) is o(g(n)) if f (n) becomes insignificant compared to g(n) as n grows toward infinity. As previously mentioned, asymptotic notation is useful because it allows us to concentrate on the main factor determining a function’s growth. To summarize, the asymptotic notations of big-Oh, big-Omega, and big-Theta, as well as little-oh and little-omega, provide a convenient language for us to analyze data structures and algorithms. As mentioned earlier, these notations provide con- venience because they let us concentrate on the “big picture” rather than low-level details. 1.1. Analyzing Algorithms 17 1.1.6 The Importance of Asymptotic Notation Asymptotic notation has many important benefits, which might not be immediately obvious.

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  Algorithms: Design Techniques And Analysis

  Design Techniques and Analysis

  • M H Alsuwaiyel(Author)
  • 1999(Publication Date)
  • World Scientific

    (Publisher)

  1.8.2 The O-notation We have seen before (Obse~ation 1.4) that the number of elementary o p erations performed by Algorithm INSERTIONSORT is at most m2, where c is some appropriately chosen positive constant. In this case, we say that the running time of Algorithm INSERTIONSORT is O(n2) (read “Oh of n2” or “big-Oh of n2,’). This can be interpreted as follows. Whenever the num- ber of elements to be sorted is equal to or exceeds some threshold no, the running time is at most cn2 for some constant c > 0. It should be empha- sized, however, that this does not mean that the running time is always as large as ma, even for large input sizes. Thus, the O-notation provides an upper bound on the running time; it may not be indicative of the actual running time of an algorithm. For example, for any value of n, the running time of ~ g o r i t h m INSERTIO~SORT is O(n) if the input is already sorted in nondecreasing order. In general, we say that the running time of an algorithm is O(g(n)), if whenever the input size is equal to or exceeds some threshold no, its running time can be bounded a b ~ ~ e by some positive constant c times g(n). The formal definition of this notation is as follows.* Definition 1.2 Let f (n) and g(n) be two functions from the set of natural *The more formal definition of this and subsequent notations is in terms of sets. We prefer not to use their exact formal definitions, as it only complicates things unnecessarily. 26 Basic Concepts in Algorithmic Analysis numbers to the set of nonnegative real numbers. f(n) is said to be O(g(n)) if there exists a natural number no and a constant c > 0 such that Consequently, if limndm f ( n ) / g ( n ) exists, then f (n) lim - # 00 implies f ( n ) = O ( g ( n ) ) . .n-+03 s(n) Informally, this definition says that f grows no faster than some constant times g. The 0-notation can also be used in equations as a simplification tool.

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  Essential Discrete Mathematics for Computer Science

  • Harry Lewis, Rachel Zax(Authors)
  • 2019(Publication Date)
  • Princeton University Press

    (Publisher)

  To denote the set of functions that, asymptotically, grow no faster than g ( n ) , we write O ( g ( n )) . This is pro-nounced “big O of g ( n ) .” (That’s the letter “ O ,” not a zero!) An algorithm with runtime function f ( n ) in O ( g ( n )) is said to have runtime complexity that is O ( g ( n )) . In mathematical terms, f ( n ) = O ( g ( n )) if and only if lim n →∞ f ( n ) g ( n ) < ∞ . (21.4) The “ f ( n ) = O ( g ( n )) ” notation is far from ideal, but it has been used in mathematics since the end of the nineteenth century, so we are stuck with it. To be clear about its meaning: f ( n ) is a function and O ( g ( n )) is a set of functions, so it would make more sense to write “ f ( n ) ∈ O ( g ( n )) ,” or perhaps “ f ∈ O ( g ) .” Unfortunately, the standard notation uses “ = ,” though not to mean equality! It is incorrect, for example, to reverse the expres-sion and write something like “ O ( g ( n )) = f ( n ) ” (since this would mean O ( g ( n )) ∈ f ( n ) , which makes no sense). An equivalent way to define big-O notation is to say that f ( n ) = O ( g ( n )) if and only if there exists a constant c and a minimum value n 0 such that for all n ≥ n 0 , f ( n ) ≤ c · g ( n ) . (21.5) See Figure 21.3. Intuitively, f ( n ) = O ( g ( n )) suggests a “less than or equal to” relation between the growth rates of f and g —like “ f ≤ g ” but ignoring constant factors, for sufficiently large arguments. g ( n ) f ( n ) (a) c.g(n) f ( n ) n 0 (b) Figure 21.3. (a) Functions f and g . The value of g ( n ) is less than the value of f ( n ) , at least for all n in the range that is shown. (b) Functions f ( n ) and c · g ( n ) , where c is some constant greater than 1. Now c · g ( n ) is still less than f ( n ) for small values of n ; in fact, since g ( 0 ) = 0, c · g ( 0 ) will be less than f ( 0 ) no matter how big c is. But for all n ≥ n 0 , f ( n ) ≤ c · g ( n ) , so f ( n ) = O ( g ( n )) .

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