200分高分悬赏急用英译汉(明天早上之前要)
这几个的确句型有些长,翻译习惯也可以参照一下别人的,反正今天好象回答你的人好多.
-------------------------------
overview
The order of growth of the running time of an algorithm, defined in Chapter 2, gives a simple
characterization of the algorithm's efficiency and also allows us to compare the relative
performance of alternative algorithms. Once the input size n becomes large enough, merge
sort, with its Θ(n lg n) worst-case running time, beats insertion sort, whose worst-case running
time is Θ(n2). Although we can sometimes determine the exact running time of an algorithm,
as we did for insertion sort in Chapter 2, the extra precision is not usually worth the effort of
computing it. For large enough inputs, the multiplicative constants and lower-order terms of
an exact running time are dominated by the effects of the input size itself.
When we look at input sizes large enough to make only the order of growth of the running
time relevant, we are studying the asymptotic efficiency of algorithms. That is, we are
concerned with how the running time of an algorithm increases with the size of the input in
the limit, as the size of the input increases without bound. Usually, an algorithm that is
asymptotically more efficient will be the best choice for all but very small inputs.
概要
算法运行时间的成长次序, 被定义在第二章里, 给算法的特性一个简单的描述并允许我们把它与供选择的算法进行比较。一旦输入大小n变得足够大, 合并排序法, 以Θ(nlgn) 最差运行时间,来检验最差运行时间是Θ(n2)的插入排序。虽然我们有时能确定算法确切的运行时间, 但如同在第二章里我们为了插入排序,而努力计算额外精确度通常并没有价值。为了足够大的投入, 常数和一具体运行时间的低秩序期限的积由投入规模本身所控制.
当投入规模大到足够确定相关命令运行时间的情况下, 我们应该学习渐进效率算法。即，当输入大小的增加没有限制，我们就要考虑怎样计算运行时间的增长变化，就像输入大小有限制的时候那样。通常, 除了非常小的输入，渐进算法是更加高效率的算法，将是最佳的选择。
This chapter gives several standard methods for simplifying the asymptotic analysis of
algorithms. The next section begins by defining several types of "asymptotic notation," of
which we have already seen an example in Θ-notation. Several notational conventions used
throughout this book are then presented, and finally we review the behavior of functions that
commonly arise in the analysis of algorithms.
本章讲述了算法渐进分析的几种简化方法。下一部分将首先定义几种“渐进符号”，在这些符号中我们已经见过的如Θ符号。还将给出本书中通篇出现的几种符号约定。最后我们将重温一下出现在算法分析中的常用函数的特性。
3.1 Asymptotic notation The notations we use to describe the asymptotic running time of an algorithm are defined in
terms of functions whose domains are the set of natural numbers N = {0, 1, 2, ...}.
Such notations are convenient for describing the worst-case running-time function T (n), which is
usually defined only on integer input sizes.
It is sometimes convenient, however, to abuse asymptotic notation in a variety of ways.
For example, the notation is easily extended to the domain of real numbers or, alternatively, restricted to a subset of the natural numbers. It is important, however, to understand the precise meaning of the notation so that when it is
abused, it is not misused. This section defines the basic asymptotic notations and also
introduces some common abuses.
3.1 渐进法
我们用来描述连续运行时间算法的一种记数法，其定义是：函数项的域值为自然数N={0，1，2，…}。
这种记数法在描述最简单的连续时间函数T(n)是方便的，因为它的输入范围通常仅取整数。它有时是方便的，然而，我们需要用比较多的方式来描述不规则渐进记法。例如，记法很容易在实数域内取值，或者交替地限定在自然数的一个子集内。尽管记数法重要，但我们需要了解其精确涵义，以便函数不规则时它没有被误用。本节定义了基本的渐进记法并且介绍一些一般的不规则函数。
这几个的确句型有些长,翻译习惯也可以参照一下别人的,反正今天好象回答你的人好多.
-------------------------------
overview
The order of growth of the running time of an algorithm, defined in Chapter 2, gives a simple
characterization of the algorithm's efficiency and also allows us to compare the relative
performance of alternative algorithms. Once the input size n becomes large enough, merge
sort, with its Θ(n lg n) worst-case running time, beats insertion sort, whose worst-case running
time is Θ(n2). Although we can sometimes determine the exact running time of an algorithm,
as we did for insertion sort in Chapter 2, the extra precision is not usually worth the effort of
computing it. For large enough inputs, the multiplicative constants and lower-order terms of
an exact running time are dominated by the effects of the input size itself.
When we look at input sizes large enough to make only the order of growth of the running
time relevant, we are studying the asymptotic efficiency of algorithms. That is, we are
concerned with how the running time of an algorithm increases with the size of the input in
the limit, as the size of the input increases without bound. Usually, an algorithm that is
asymptotically more efficient will be the best choice for all but very small inputs.
概要
算法运行时间的成长次序, 被定义在第二章里, 给算法的特性一个简单的描述并允许我们把它与供选择的算法进行比较。一旦输入大小n变得足够大, 合并排序法, 以Θ(nlgn) 最差运行时间,来检验最差运行时间是Θ(n2)的插入排序。虽然我们有时能确定算法确切的运行时间, 但如同在第二章里我们为了插入排序,而努力计算额外精确度通常并没有价值。为了足够大的投入, 常数和一具体运行时间的低秩序期限的积由投入规模本身所控制.
当投入规模大到足够确定相关命令运行时间的情况下, 我们应该学习渐进效率算法。即，当输入大小的增加没有限制，我们就要考虑怎样计算运行时间的增长变化，就像输入大小有限制的时候那样。通常, 除了非常小的输入，渐进算法是更加高效率的算法，将是最佳的选择。
This chapter gives several standard methods for simplifying the asymptotic analysis of
algorithms. The next section begins by defining several types of "asymptotic notation," of
which we have already seen an example in Θ-notation. Several notational conventions used
throughout this book are then presented, and finally we review the behavior of functions that
commonly arise in the analysis of algorithms.
本章讲述了算法渐进分析的几种简化方法。下一部分将首先定义几种“渐进符号”，在这些符号中我们已经见过的如Θ符号。还将给出本书中通篇出现的几种符号约定。最后我们将重温一下出现在算法分析中的常用函数的特性。
3.1 Asymptotic notation The notations we use to describe the asymptotic running time of an algorithm are defined in
terms of functions whose domains are the set of natural numbers N = {0, 1, 2, ...}.
Such notations are convenient for describing the worst-case running-time function T (n), which is
usually defined only on integer input sizes.
It is sometimes convenient, however, to abuse asymptotic notation in a variety of ways.
For example, the notation is easily extended to the domain of real numbers or, alternatively, restricted to a subset of the natural numbers. It is important, however, to understand the precise meaning of the notation so that when it is
abused, it is not misused. This section defines the basic asymptotic notations and also
introduces some common abuses.
3.1 渐进法
我们用来描述连续运行时间算法的一种记数法，其定义是：函数项的域值为自然数N={0，1，2，…}。
这种记数法在描述最简单的连续时间函数T(n)是方便的，因为它的输入范围通常仅取整数。它有时是方便的，然而，我们需要用比较多的方式来描述不规则渐进记法。例如，记法很容易在实数域内取值，或者交替地限定在自然数的一个子集内。尽管记数法重要，但我们需要了解其精确涵义，以便函数不规则时它没有被误用。本节定义了基本的渐进记法并且介绍一些一般的不规则函数。
Θ-notation
In Chapter 2, we found that the worst-case running time of insertion sort is T (n) = Θ(n2). Let
us define what this notation means. For a given function g(n), we denote by Θ(g(n)) the set of
functions
Θ(g(n)) = {f(n) : there exist positive constants c1, c2, and n0 such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n)
for all n ≥ n0}.[1]
A function f(n) belongs to the set Θ(g(n)) if there exist positive constants c1 and c2 such that it
can be "sandwiched" between c1g(n) and c2g(n), for sufficiently large n. Because Θ(g(n)) is a
set, we could write "f(n) Θ(g(n))" to indicate that f(n) is a member of Θ(g(n)). Instead, we
will usually write "f(n) = Θ(g(n))" to express the same notion. This abuse of equality to denote
set membership may at first appear confusing, but we shall see later in this section that it has
advantages.
θ-符号
在第2章中,我们看到的最坏的运行排序是T(n)=θ(n2) . 让我们确定这个公式代表了什么. 对于给定函数G(n),我们定义θ(g(n))为g(n)的函数集。
Θ(g(n))=｛f(n): 存在正常数C1、C2和N0，当0≤c1g(n)≤f(n)≤c2g(n),其中n≥ n0} .
[1]若存在正常数C1、C2等以使之满足c1g(n)≤f(n)≤c2g(n),如果n足够大，那么函数f(n)属于函数集θ(g(n))。因为θ(g(n))是一个集合,我们可以用"f(n) θ(g(n))"来表明f(n)包含于θ(g(n)). 但是,我们通常会写"f(n)=θ(g(n))"来表达这一概念. 这种等号定义的滥用使得工作组在开始时会出现混乱, 但是,我们应当看到,在这一节梢后,它有许多优点.
Figure 3.1(a) gives an intuitive picture of functions f(n) and g(n), where we have that f(n) =
Θ(g(n)). For all values of n to the right of n0, the value of f(n) lies at or above c1g(n) and at or
below c2g(n). In other words, for all n ≥ n0, the function f(n) is equal to g(n) to within a
constant factor. We say that g(n) is an asymptotically tight bound for f(n).
Figure 3.1: Graphic examples of the Θ, O, and Ω notations. In each part, the value of n0
shown is the minimum possible value; any greater value would also work. (a) Θ-notation
bounds a function to within constant factors. We write f(n) = Θ(g(n)) if there exist positive
constants n0, c1, and c2 such that to the right of n0, the value of f(n) always lies between c1g(n)
and c2g(n) inclusive. (b) O-notation gives an upper bound for a function to within a constant
factor.
图3.1(a):给出函数f(n)和g(n)的曲线，其中，f(n)=Θ(g(n))。当n≥n0时，c1g(n)≤f(n)≤c2g(n)。也就是说，当n≥n0时，f(n)等于g(n)乘以一个常数。我们说，g(n)是f(n)的同类型曲线。
图3.1：Θ, O,和Ω的图例。在每个部分，n0取最小可能值；任何比它大的值都可以。
(a)Θ限制了一个和常数有关的函数。如果存在正常数n0、c1和c2，当 n≥n0时，c1g(n)≤f(n)≤c2g(n)，那么，我们记作f(n) = Θ(g(n))。
(b)符号O表示了一个与常数有关的函数的上限。
We write f(n) = O(g(n)) if there are positive constants n0 and c such that to the right of
n0, the value of f(n) always lies on or below cg(n). (c) Ω-notation gives a lower bound for a
function to within a constant factor. We write f(n) = Ω(g(n)) if there are positive constants n0
and c such that to the right of n0, the value of f(n) always lies on or above cg(n).
The definition of Θ(g(n)) requires that every member f(n) Θ(g(n)) be asymptotically
nonnegative, that is, that f(n) be nonnegative whenever n is sufficiently large. (An
asymptotically positive function is one that is positive for all sufficiently large n.)
Consequently, the function g(n) itself must be asymptotically nonnegative, or else the set
Θ(g(n)) is empty. We shall therefore assume that every function used within Θ-notation is
asymptotically nonnegative. This assumption holds for the other asymptotic notations defined
in this chapter as well.
In Chapter 2, we introduced an informal notion of Θ-notation that amounted to throwing away
lower-order terms and ignoring the leading coefficient of the highest-order term. Let us
briefly justify this intuition by using the formal definition to show that 1/2n2 - 3n = Θ(n2). To
do so, we must determine positive constants c1, c2, and n0 such that
c1n2 ≤ 1/2n2 - 3n ≤ c2n2
for all n ≥ n0. Dividing by n2 yields
c1 ≤ 1/2 - 3/n ≤ c2.
当有正常数n0和大于n0的常数c，我们就记作 f(n) = O(g(n)) ，其中f(n)≤cg(n)。
(c) Ω符号表示在给定常数范围内的函数下限。当有正常数n0和大于n0的常数c，我们记作f(n) = Ω(g(n)) ，其中f(n)≥cg(n)。
Θ(g(n)) 的定义要求每一个 f(n) Θ(g(n)) 都是渐近正函数，就是说，无论n多大，f(n) 都为正的(渐近的正函数就是一个对所有足够大的n来说都为正的函数)；因此，函数 g(n) 本身必须为渐近非负函数，否则Θ(g(n)) 就是空集。因此我们应该假设Θ的所有子函数都为渐近非负函数。这种假设在这章中其他的渐近符号定义中也成立。
在第二章，我们介绍了一个非正式的符号 Θ-符号，这种符号总计丢弃低项，并且忽略高次项的首项系数。让我们简要地通过正式地定义来表示1/2n2 - 3n = Θ(n2)的方法来证明这种直觉是对的，为了这样做，我们必须规定正常数c1，c2和n0满足：
c1n2≤1/2n2-3n≤c2n2
对于所有的n ≥ n0，用n2除后，
c1≤1/2-3/n≤c2.
当有正常数n0和大于n0的数c，我们就记作 f(n) = O(g(n)) ，f(n) 的值总是取决于cg(n)或者小于cg(n)，
(c) Ω-符号表示在常数因子范围内给函数一个下界。当有正常数n0 和大于n0的正常数c，我们就记作f(n) = Ω(g(n)) ，f(n)的值总是取决于cg(n)或是比cg(n)大。
Θ(g(n)) 的定义要求每一个 f(n) Θ(g(n)) 都是渐近的非负数，就是说，只要n足够大，
f(n) 就为非负的（渐近的正函数就是一个对所有足够大的n来说都为正的函数）；因此，函数 g(n) 本身必须为渐近的非负函数，否则Θ(g(n)) 就无意义。因此我们应该假设每个用Θ-符号的函数都为渐近非负函数。这种假设在这章中其他的渐近符号定义中也成立。
在第二章，我们介绍了一个非正式的符号 Θ-符号，这种符号总计丢弃低项，并且忽略高次项的首项系数。让我们简要地通过正式地定义来表示1/2n2 - 3n = Θ(n2)的方法来证明这种直觉是对的，为了这样做，我们必须决定正常数c1，c2和n0满足：
c1n2 ≤ 1/2n2 - 3n ≤ c2n2
对于所有的n ≥ n0，用n2除，c1 ≤ 1/2 - 3/n ≤ c2.
本来准备把你的另外几个问题也翻译一下的，可是回答的人太多了，看不到问题了，我就没翻译了
我们写f ( n )的=澳度( g ( n ) )中,如果有积极常数为N0和C等,以权n0时, 价值f ( n )的总是谎言或低于对照组(氮) . (三)Ω-五线谱给出一个下界函数在一个常数因子. 我们写利用F ( n ) =Ω(七( n ) )中,如果有积极常数为N0和C等,以权n0时, 价值f ( n )的总是谎言之上或对照组(氮) . 定义θ度( g ( n ) )的规定,每个会员f ( n )的θ度( g ( n ) )的渐近负,即 函数f ( n )是负每当N足够大. (一渐近正面功能之一,是积极为所有足够大, 2000 ) , 该函数G ( n )本身必须负渐近,否则定θ度( g ( n ) )是空的. 因此,我们会假定每个功能使用θ-五线谱是渐近负. 这种假设认为,为其他渐近252本章定义等. 第2章 我们介绍了一份非正式的概念θ-五线谱,相当于扔掉低阶条款,而忽略了领导系数 最高秩序. 让我们简略地说明这种直觉用正式定义显示1/2n2硝态氮θ= (氮气) . 这样做,我们就必须确定阳性常数C1 , C2组, 并且N0等,酵母≤1/2n2硝态氮≤c2n2对一切n≥n0时. 除以氮气产量的C1≤1 / 2 - 3 /≤的C2
我们写f (n) = O(g (n))如果有积极的常数n0和c这那在右面
n0,在(n)上或在重心下面f (n)的价值总是躺。
(c)Ω-标志给较低的界限为一
职责到在一个经常的因素以内。
我们给f (n) =写Ω(g (n))如果有积极的常数n0
并且c这那在n0右面，在左右重心(n)上f (n)的价值总是躺。
Θ的定义(g (n))要求那每成员f (n)Θ(g (n))渐近地是
非负,是,那f (n)无论何时n是足够地大的非负。
(An
渐近地积极的函数是为所有的足够地大的n是积极的一个。)
从而,函数g (n)自己一定是渐近地非负的,否则集合
Θ(g (n))是空的。
我们将因此假设在Θ以内每函数使用了-标志是
渐近地非负。
这个假设适用于定义的另外的渐近的标志
在这个章也。
在第2章，我们介绍了Θ的一个不正式的观点-标志到丢弃的amounted
降低顺序术语并且忽略高顺序术语的最主要的系数。
让我们
由使用正式的定义显示出那1/2n2简短认为这直觉正当-3n =Θ(n2)。
这样做,我们必须决定积极的常数c1，c2，并且n0这
c1n2≤1/2n2-3n≤c2n2
为所有的n≥n0。
由n2划分让步
c1≤1/2-3/n≤c2。
真是讨厌那些用软件翻译的人，好像只有他们会用一样。
我也在百度知道发了一个翻译，被一个烂小子用软件翻译了。真是垃圾
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