山东建筑大学计算机学院算法分析算法复习题(Yuconan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 and below.

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 = 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 V).

一个矩阵相乘问题的解决可以一个时间复杂度算法的所有路径的最短路径问题，改进后的时间复杂度是。

12.Floyd-Warshall algorithm, runs in Θ(V3) time to solve the all-pairs