哈尔滨工业大学深圳 高级算法设计 2017 考试 复习题 重点

Insertion Sort（原理及伪代码学习，第一次出现伪代码）Asymptotic Time Analysis:Big O (upper bound) and Big Ω (lower bound) Merge Sort (Divide-and-Conquer)

Recurrences: three methods.

Substitution method: guess→verify→solve

Recursion-tree method:

Master method: 应用于形如T(n)=aT(n/b)+f(n),a>=1,b>1

证明主方法

Big Integer Multiplication

Strassen matrix multiplication Chessboard Cover

Shell Sort 缩小增量

Comparison Sort ( O(nlgn) the best we can do)

Counting sort: No comparisons between elements.

Counting sort is not a comparison sort.

Counting sort is a stable sort: it preserves the input order among equal elements.

Why don’t we always use counting sort?

Because it depends on range k of elements

Radix Sort

run counting sort on each digit.(从低位往高位排)

Sort on least-significant digit first with auxiliary stable sort

Bucket Sort

In Bucket sort, the range [a,b] of input numbers is divided into m equal sized intervals, called buckets.

Hash Table

给定表M，存在函数f(key)，对任意给定的关键字值key，代入函数后若能得到包含该关键字的记录在表中的地址，则称表M为哈希(Hash）表，函数f(key)为哈希(Hash) 函数。对不同的关键字可能得到同一散列地址，即k1≠k2，而f(k1)=f(k2)，这种现象称为碰撞（英语：Collision）。

Let n be the number of keys in the table, and

let m be the number of slots.

Define the load factor of T to be α= n/m =average number of keys per slot.

The expected time for an unsuccessful search for a record with a given key is = Θ(1 + α) .

Hash function:

Division method h(k) = k mod m.

Don’t pick an m that has a small remainder d. If m= 2r, then the hash

doesn’t even depend on all the bits of k.

Pick m to be a prime not too close to a power of 2 or 10 and not otherwise used prominently

in the computing environment. Sometimes, making the table size a prime is inconvenient

Multiplication Method:

Resolving collisions by open addressing

Binary Search Trees

属性: key[left(x)]≤key[x]≤key[right(x)]

traversa l遍历

Inorder Tree Walk: print left, then root, then right.

Preorder tree walk: print root, then left, then right.

Postorder tree walk: print left, then right, then root.

n个结点子树，时间Θ(n)

Search

第二种迭代的方式更高效。

Insert

The running time of TreeSearch() and TreeInsert() O(h),h is the height of a BST

The worst case of h=O(n) when tree is just a linear string of left or right children.

Successor (顺序遍历当前节点的后继)节点val值大于该节点val值并且值最小的节点

1.x存在右子树，后继是右子树中的最小结点

2.x没有右子树，后继是x的最低的祖先，其左孩子同样是x的祖先。

Delete

1.x has no children→remove x

2.x has one children→splice out(剪接) x

3.x has two children→swap x with successor, then perform case 1 or 2 to delete it.

Sorting with binary search trees

Since time is proportional to the number of comparisons, same time as quicksort O(nlgn)

Red-Black tree

性质1. 节点是红色或黑色。

性质2 每个叶节点（NIL节点，空节点）是黑色的。

性质3 每个红色节点的两个子节点都是黑色。(从每个叶子到根的所有路径上不能有两个连续的红色节点)

性质4. 从任一节点到其每个叶子的所有路径都包含相同数目的黑色节点。

性质5. 根节点是黑色。

Black-height: # of nodes on path to leaf.

定理：拥有n个结点的红黑树高度h≤2lg(n+1)