算法分析算法复习题中英文

（有翻译）

1.The O-notation provides an asymptotic upper bound. The Ω-notation provides an

asymptotic lower bound. The Θ-notation asymptotically a function form above

2.To represent a heap as an array，the root of tree is A[1], and given the index i of a

node, the indices of its parent Parent(i) { return ⎣i/2⎦; }，left child, Left(i)

{ return 2*i; }，right child, right(i) { return 2*i + 1; }.

代表一个堆中的一个数组，树的根节点是A[1]，并且给出一个节点i，那么该节点的父节点是左孩子右孩子

3.Because the heap of n elements is a binary tree, the height of any node is at most

Θ(lg n).

因为n个元素的堆是一个二叉树，任意节点的树高最多是

4.In optimization problems, there can be many possible solutions. Each solution

has a value, and we wish to find a solution with the optimal (minimum or maximum) value. We call such a solution an optimal solution to the problem.

在最优化问题中，有很多可能的解，每个解都有一个值，我们希望找到一个最优解（最大或最小），我们称这个解为最优解问题。

5.optimal substructure if an optimal solution to the problem contains within it

optimal solutions to subproblems.

最优子结构中问题的最优解，至少包含它的最优解的子问题。

6. A subsequence of X if there exists a strictly increasing sequence of

indices of X such that for all j = 1, 2, ..., k, we have x i j= z j .

Let X = and Y = be sequences, and let Z =

z2, ..., z k> be any LCS of X and Y.

(1). If x m = y n, then z k = x m = y n and Z k-1 is an LCS of X m-1 and Y n-1.

(2). If x m ≠ y n, then z k ≠ x m implies that Z is an LCS of X m-1 and Y.

(3). If x m ≠ y n, then z k ≠ y n implies that Z is an LCS of X and Y n-1.

7. A greedy algorithm always makes the choice that looks best at the moment. That

is, it makes a locally optimal choice in the hope that this choice will lead to a

globally optimal solution.

贪心算法经常需要在某个时刻寻找最好的选择。正因如此，它在当下找到希望中的最优选择，以便引导出一个全局的最优解。

8.The greedy-choice property and optimal sub-structure are the two key ingredients

of greedy algorithm.

贪心选择和最优子结构是贪心算法的两个重要组成部分。

9.When a recursive algorithm revisits the same problem over and over again, we

say that the optimization problem has overlapping subproblems.

当一个递归算法一遍一遍的遍历同一个问题时，我们说这个最优化问题是重叠子问题。

10.greedy-choice property is a globally optimal solution can be arrived at by making

a locally optimal (greedy) choice.

贪心选择性质是一个全局的最优解，这个最优解可以做一个全局的最优选择。

11.An approach of Matrix multiplication can develope a Θ(V4)-time algorithm for

the all-pairs shortest-paths problem and then improve its running time to Θ(V3lg