## On Asymptotic Notation with Multiple Variables (2008)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Howell08onasymptotic,

author = {Rodney R. Howell},

title = {On Asymptotic Notation with Multiple Variables},

year = {2008}

}

### OpenURL

### Abstract

We show that it is impossible to define big-O notation for functions on more than one variable in a way that implies the properties commonly used in algorithm analysis. We also demonstrate that common definitions do not imply these properties even if the functions within the big-O notation are restricted to being strictly nondecreasing. We then propose an alternative definition that does imply these properties whenever the function within the big-O notation is strictly nondecreasing. 1

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Citation Context ...In the following decades, its properties became well-understood, and it has been widely used in the analysis of algorithms. Little-o notation for 1-variable functions was introduced by Landau in 1909 =-=[7]-=-, and big-Ω, big-Θ, and little-ω were defined for single-variable functions by Knuth in 1976 [6]. As with big-O notation for one variable, these asymptotic notations are all well-understood and widely... |

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Citation Context ...ld for each variable, then we can pick N to be the maximum of these individual thresholds. A common definition of O(f(n1,...,nk)) is exactly the set of all functions satisfying Property 1 (see, e.g., =-=[2, 9]-=-). In order to distinguish this definition from others, let us denote it as O∀(f(n1,...,nk)). We will now demonstrate that this definition is too weak. Consider, for example, the algorithm shown in Fi... |

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Citation Context ...ld for each variable, then we can pick N to be the maximum of these individual thresholds. A common definition of O(f(n1,...,nk)) is exactly the set of all functions satisfying Property 1 (see, e.g., =-=[2, 9]-=-). In order to distinguish this definition from others, let us denote it as O∀(f(n1,...,nk)). We will now demonstrate that this definition is too weak. Consider, for example, the algorithm shown in Fi... |

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Citation Context ...g times of most graph algorithms depend on both the number of vertices and the number of edges when the graph is represented by an adjacency list. However, most textbooks on algorithm analysis (e.g., =-=[3, 4, 5, 8]-=-) do not explicitly extend the definition of asymptotic notation to multi-variable functions. Instead, they include analyses using asymptotic notation with multiple variables, applying the properties ... |

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Citation Context ...the analysis of algorithms. Little-o notation for 1-variable functions was introduced by Landau in 1909 [7], and big-Ω, big-Θ, and little-ω were defined for single-variable functions by Knuth in 1976 =-=[6]-=-. As with big-O notation for one variable, these asymptotic notations are all well-understood and widely used in algorithm analysis. Many algorithms have more than one natural parameter influencing th... |

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Citation Context ...g times of most graph algorithms depend on both the number of vertices and the number of edges when the graph is represented by an adjacency list. However, most textbooks on algorithm analysis (e.g., =-=[3, 4, 5, 8]-=-) do not explicitly extend the definition of asymptotic notation to multi-variable functions. Instead, they include analyses using asymptotic notation with multiple variables, applying the properties ... |

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Citation Context ...g times of most graph algorithms depend on both the number of vertices and the number of edges when the graph is represented by an adjacency list. However, most textbooks on algorithm analysis (e.g., =-=[3, 4, 5, 8]-=-) do not explicitly extend the definition of asymptotic notation to multi-variable functions. Instead, they include analyses using asymptotic notation with multiple variables, applying the properties ... |

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Citation Context |

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Citation Context ...at does imply these properties whenever the function within the big-O notation is strictly nondecreasing. 1 Introduction Big-O notation for functions on one variable was introduced by Bachman in 1894 =-=[1]-=-. In the following decades, its properties became well-understood, and it has been widely used in the analysis of algorithms. Little-o notation for 1-variable functions was introduced by Landau in 190... |