list下界_下界理论

list下界

下界理论 (Lower Bound Theory)

Lower bound (L(n)) is a property of the specific problem i.e. the sorting problem, matrix multiplication not of any particular algorithm solving that problem.

下界(L(n))是特定问题(即排序问题)的矩阵，不是解决该问题的任何特定算法的矩阵乘法。

Lower bound theory says that no algorithm can do the job in fewer than that of (L (n)) times the units for arbitrary inputs i.e. that for every comparison based sorting algorithm must take at least L(n) time in the worst case.

下界理论说，没有一种算法能以少于任意输入的单位(L(n))倍的时间完成这项工作，也就是说，在最坏的情况下，每个基于比较的排序算法必须至少花费L(n)时间。

L(n) is the minimum over all possible algorithm which is maximum complete.

L(n)是所有可能算法中的最小值，而最大值是最大完成度。

Trivial lower bounds are used to yield the bound best option is to count the number of item in the problems input that must be processed and a number of output items that need to be produced.

琐碎的下限用于产生绑定最佳选择是计算问题输入中必须处理的项目数和需要产生的输出项数。

The lower bound theory is the technique that has been used to establish the given algorithm in the most efficient way which is possible. This is done by discovering a function g(n) that is a lower bound on the time that any algorithm must take to solve the given problem. Now if we have an algorithm whose computing time is the same order as g(n), then we know that asymptotically we cannot do better.

下限理论是一种已被用来以可能的最有效方式建立给定算法的技术。这是通过发现一个函数g(n)来完成的，该函数在任何算法解决给定问题所需的时间上都有一个下限。现在，如果我们有一种算法，其计算时间与g(n)相同 ，那么我们知道渐近地我们不能做得更好。

If f(n) is the time for some algorithm, then we write f(n) = Ω(g(n)) to mean that g(n) is the lower bound of f(n). This equation can be formally written, if there exists positive constants c and n0 such that |f(n)| >= c|g(n)| for all n > n0. In addition for developing lower bounds within the constant factor, we are more conscious of the fact to determine more exact bounds whenever this is possible.

如果f(n)是某种算法的时间，则我们将f(n)=Ω(g(n))表示为g(n)是f(n)的下限。如果存在正常数c和n0使得| f(n)|成立 ，则该方程式可以正式写成。 > = c | g(n)| 对于所有n> n0 。除了在恒定因子内建立下界外，我们更意识到在可能的情况下确定更精确界限的事实。

Deriving good lower bounds is more difficult than devising efficient algorithms. This happens because a lower bound states a fact about all possible algorithms for solving a problem. Generally, we cannot enumerate and analyze all these algorithms, so lower bound proofs are often hard to obtain.

与设计有效的算法相比，得出良好的下界更加困难。之所以会发生这种情况，是因为下限指出了所有可能解决问题的算法的事实。通常，我们无法枚举和分析所有这些算法，因此通常很难获得下界证明。

The proofing techniques that are useful for obtaining lower bounds are:

对获得下限有用的校对技术是：

Comparison trees:
比较树：

Comparison trees are the computational model useful for determining lower bounds for sorting and searching problems.
比较树是用于确定排序和搜索问题下限的计算模型。
Oracles and adversary arguments:
Oracle和对手的论点：

One of the techniques that are important for obtaining lower bounds consists of making the use of an oracle
获得下界的重要技术之一是使用预言
Lower bounds through reduction:
通过缩减下界：

This is a very important technique of lower bound, This technique calls for reducing the given problem for which a lower bound is already known.
这是一个非常重要的下限技术。该技术要求减少已知下限的给定问题。
Techniques for the algebraic problem:
代数问题的技术：

Substitution and linear independence are two methods used for deriving lower bounds on algebraic and arithmetic problems. The algebraic problems are operation on integers, polynomials, and rational functions.
代换和线性独立性是用于推导代数和算术问题下界的两种方法。代数问题是对整数，多项式和有理函数的运算。