Chapter 9 Tables And Information Retrieval

1. Introduction: Breaking the lgn Barrier 2. Rectangular Arrays

3. Tables of Various Shapes

4. Tables: A New Abstract Data Type

5. Application: Radix Sort

Chapter 9 Tables And Information Retrieval

6. Hashing

7. Analysis of Hashing

8. Conclusions: Comparison of Methods 9. Application: The Life Game Revisited 10. Pointers and Pitfalls

9.1 Introduction: Breaking the lgn Barrier

◆By use of key comparisons alone, it is impossible to complete a search of n items in fewer than lg n comparisons, on average(lower bound_search_thm).◆Ordinary table lookup or array access requires only constant time O(1).◆Both table lookup and searching share the same essential purpose, that of information retrieval. The key used for searching and the index used for table lookup have the same essential purpose: one piece of information that is used to locate further information.

◆Both table lookup and searching algorithms provide functions from a set of keys or indices to locations in a list or array.◆In this chapter we study ways to implement and access various kinds of tables in contiguous storage.◆Several steps may be needed to retrieve an entry from some kinds of tables, but the time required remains O(1).It is bounded by a constant that does not depend on the size of the table. Thus table lookup can be more efficient than any searching method.

◆We shall implement abstractly defined tables with arrays. In order to distinguish between the abstract concept and its implementation, we introduce: Convention The index defining an entry of an abstractly defined table is enclosed in parentheses,whereas the index of an entry of an arrayis enclosed in square brackets.

9.2 Rectangular Arrays

1 Row-major and Column-major

Ordering:

Fig 9.1 pg.381

通常行主序存储方式较普遍

2 indexing Rectangular table Pg.382

Suppose position of Entry(0,0) is Loc(0,0), every entry store c cell ， then Entry(i,j) position Loc(i,j)

Loc(i,j)=Loc(0,0)+(i*3+j)*c （ Row-major ordering ） Loc(i,j)=Loc(0,0)+(i+2*j)*c （ Column-major ordering ） In row-major ordering, entry [i, j ] goes to position ni+j .

存储位置(地址 )

序号index

3 Variation: A Access Arrays

Pg.382

Fig 9.2 pg.383

9.3 Tables of Various Shapes

Fig 9.3 pg.384

1 triangular table

Fig 9.4 pg.384

Lower triangular matrix

我们以矩阵的下三角部分为例，来分析矩阵元素 对应的存储空间的地址。在下三角部分，第 k 行有 k 个元素，而 元素之前共有 i-1 行，再加上第 i 行的 j 个元素，所以 排在下三角部分的位序 (index) 是：

2. Jagged Table with

Access Table

Fig 9.5 pg.385

3. Inverted Table

Fig 9.6 pg.387

倒排表

◆ In mathematics a function is defined in terms of two sets and a correspondence from elements of the first set to elements of the second. If f is a function from a set A to a set B, then f assigns to each element of A a unique element of B.◆The set A is called the domain of f , and the set B is called the codomain of f .

9.4 Tables: A New Abstract Data Type

1

Functions

◆The subset of B containing just those elements that occur as values of f is called the range of f .DEFINITION A table with index set I and base type T is afunction from I to T together with the following operations.1. Table access: Evaluate the function at any index in I .2. Table assignment: Modify the function by changing itsvalue at a specified index in I to the new value specifiedin the assignment.3. Creation: Set up a new function.4. Clearing: Remove all elements from the index set I , sothere is no remaining domain.5. Insertion: Adjoin a new element x to the index set I anddefine a corresponding value of the function at x.6. Deletion: Delete an element x from the index set I andrestrict the function to the resulting smaller domain.

See pg. 390-391

9.5 Application: Radix Sort

1 The Idea

基数排序和上一章中所述各类排序方法均不同 , 它是一种借助于多排序码排序的思想对单逻辑排序码进行排序的方法。例如：

Fig 9.10 pg.392

前面我们介绍的各种排序方法都是以关键字的比较与数据的移动为主，但基数排序却主要借助“分配”和“收集”两种操作来完成排序任务。

Fig 9.11 pg.393

2 Linked Implementation of Radix Sort◆List definition:

template <class Record>class Sortable_list: public List<Record>{ public: void radix_sort( ); private: // auxiliary functions void rethread(Queue queues[]);};

Inherit from linked Listdiffer from

above RecordContiguous Queue

(chapter 3)

◆Record definition:class Record{ public: char_key letter(int position) const;

Record( ); //default constructor operator Key( ) const; // cast to Key // Add other methods and data members for the class.};

◆Sorting Method, Linked Radix Sortconst int max_chars = 28;

template <class Record>void Sortable_list<Record> :: radix_sort( ){ Record data; Queue queues[max_chars]; for(int position= key_size-1; position >= 0; position--) {while(remove(0,data) == success) { int queue_number=alphabetic_order(data.key_letter(position)); queues[queue_number].append(data); // Queue operation. } rethread(queues); //Reassemble the list. }}

◆Auxiliary Functions, Linked Radix SortSelecting a queue:int alphabetic_order(char c)/* Post: The function returns the alphabetic position of character c , or it returns 0 if the character is blank. */{ if(c==' ') return 0; if('a'<=c && c<='z') return c-'a'+1; if('A'<=c && c<='Z') return c-'A'+1; return 27;}该例子许多部分没有实现，请同学自己练习。也可以不按该例给出的这些类及方法，自己实现之。

3. Analysis of Radix Sort◆The time used by radix sort is (nk), where n is the number of items being sorted and k is the number of characters in a key.◆The relative performance of radix sort to other methods will relate to the relative sizes of nk and nlgn; that is, of k and lgn.◆If the keys are long but there are relatively few of them, then k is large and lg n relatively small, and other methods (such as mergesort) will outperform radix sort.◆If k is small (the keys are short) and there are a large number of keys, then radix sort will be faster than any other method we have studied.

9.6 Hash Tables 在记录的存储位置和它的关键字之间建立一个确定的对应关系 h ，使每个关键字和结构中一个唯一的存储位置相对应。因而在查找时，只要根据这个对应关系 h 找到给定值 k 的映象 h(k) ，若结构中存在关键字和 k 相等的记录，则必定在 h(k) 的存储位置上，因此不需要进行比较便可直接得到所查记录。我们称这个对应关系 h 为散列 (Hash)函数（也译为杂凑函数），用散列函数建立的表称为散列表。1 Sparse Table See

Pg.397

稀疏表

通常，关键字集合比较大，它的元素包括所有可能的关键字，而地址集合的元素仅为散列表中的地址值( 设表长为 n ，则地址为 0 ～ n-1) 。 例如在 C 编译程序中

需对源程序中的标识符建立一张散列表。在设定散列表函数时要考虑的关键字字集合应包含所有可能产生的关键字，按 C 语言的语法规定，标识符可定义为以字母开头的最多 8 个字符的字母、数字、下划线组成的字符串，且大小写字母是可区分的。因此， C 语言中标识的集合大小为：

14763

152

263

152

163

152

152 101.57!CC2!CCCCC

实际上任何源程序都不会出现这么多的标识符，设表长为 1,000 就足够了，地址集合中的元素为 0 ～ 999 。

In practice,however,only a small fraction of the keys

Will actually occur.That is spare.

Hash Table See Pg.398

◆Start with an array that holds the hash table.

◆ Use a hash function to take a key and map it to some

index in the array. This function will generally map

several different keys to the same index.

◆ If the desired record is in the location given by the index, then we are finished; otherwise we must use some method to resolve the collision that may have occurred between two records wanting to go to the same location.◆To use hashing we must (a) find good hash functions and (b) determine how to resolve collisions.

2. Choosing a Hash Function

◆A hash function should be easy and quick to compute.◆ A hash function should achieve an even distribution of thekeys that actually occur across the range of indices.◆The usual way to make a hash function is to take the key, chop it up, mix the pieces together in various ways, and

冲突

thereby obtain an index that will be uniformly distributed over the range of indices.◆ Note that there is nothing random about a hash function. Ifthe function is evaluated more than once on the same key,then it must give the same result every time, so the key canbe retrieved without fail.◆Truncation: Sometimes we ignore part of the key, and use the remaining part as the index.◆Folding: We may partition the key into several parts and combine the parts in a convenient way.◆Modular arithmetic: We may convert the key to an integer,divide by the size of the index range, and take the remainderas the result.◆A better spread of keys is often obtained by taking the size of the table (the index range) to be a prime number.

均匀分布

C++ Example of a Hash Function pg.400-pg.401

3. Collision Resolution with Open AddressingLinear probing:Linear probing starts with the hash address and searches sequentially for the target key or an empty position. The array should be considered circular, so that when the last location is reached, the search proceeds to the first location of the array.

线性探测聚集

Clustering: See pg.401

当表中 i,i+1,i+2 位置上已填有记录时，下一个散列地址为 i ，i+1 ， i+2 和 i+3 的记录都将填入 i+3 的位置，这种散列地址不同的记录争夺同一个后继散列地址的现象称作“聚集”，即在处理同义词的冲突过程中又增加了非同义词的冲突，显然，这种现象对查找不利。

Quadratic Probing:If there is a collision at hash address h, quadratic probing goes to locations h+1, h+4, h+9, , that is, at locations h+i2 (mod hashsize) for i ＝ 1, 2, . 二次探二次探测Other methods:◆ Key-dependent increments;◆ Random probing.

Hash Table Specificationsconst int hash_size=997; class Hash_table { public: Hash_table( ); void clear( ); Error_code insert(const Record &new entry); Error_code retrieve(const Key &target, Record &found) const;

a prime number of appropriate size

private: Record table[hash_size];};

Hash table :: Hash_table( );Post: The hash table has been created and initialized to be empty.

void Hash table :: clear( );Post: The hash table has been cleared and is empty.

Error_code Hash_table :: retrieve(const Key &target, Record &found) const;Post: If an entry in the hash table has key equal to target,then found takes on the value of such an entry, and success is returned. Otherwise, not_present is returned.

Hash Table Insertion

Error_code Hash_table::insert(const Record &new_entry)/* Post:If the Hash_table is full, a code of overflow is returned. If the table already contains an item with the key of new_entry a code of duplicate error is returned. Otherwise: The Record new_entry is inserted into the Hash_table and success is returned. Uses:Methods for classes Key,and Record. The function hash . */{ Error_code result=success; int probe_count, increment, probe; Key null; null.make_blank( ); probe=hash(new_entry); probe_count = 0; increment=1;

Counter to be surethat table is not fullIncrement used forquadratic probing

Position currently probed in the

hash tableNull key for

comparison purposes

while( table[probe]!=null && table[probe]!=new_entry && probe_count<(hash_size+1)/2) { probe_count++; probe=(probe+increment)%hash_size; increment += 2; } if(table[probe]==null) table[probe]=new_entry; else if(table[probe]==new_entry) result=duplicate_error; else result=overflow; return result;}

Is the location empty?Duplicate key?Has overflow occurred?

Prepare incrementfor next iteration

Insert new_entryhe table is full

4. Collision Resolution by chaining

◆ The linked lists from the hash table are called chains.◆ If the records are large, a chained hash table can save space.◆Collision resolution with chaining is simple; clustering is no problem.

◆ The hash table itself can be smaller than the number of records; overflow is no problem.◆ Deletion is quick and easy in a chained hash table.◆If the records are very small and the table nearly full, chaining may take more space.

Code for Chained Hash Tables

class Hash_table { public: // Specify methods here. private: List<Record> table[hash_size];};

Constructor:The implementation of the constructor simply calls the constructor for each list in the array.

Clear:To clear the table, we must clear the linked list in each of the table positions, using the List method clear( ).

Retrieval:Sequential_search(table[hash(target)], target, position);Insertion:table[hash(new entry)].insert(0, new entry);

Deletion:Error_code Hash table :: remove(const Key type &target, Record &x);

Example:设一个散列表包含 hash_Size = 13 个表项，其下标从 0 到 12 ，散列函数采用除留余数法，用 %hash_Size 将各关键码映像到表中，采用线性探查法解决冲突。给定的关键码集合是： {400,126,45,32,58, 100,3,29,200,10,36 } 。假设每一个关键码被查找的概率相同，请计算出 ASL=？

1.454511

163)3226(1

11

1ASL

9.7 Analysis of Hashing

1. The Birthday Surprise

How many randomly chosen people need to be in a roombefore it becomes likely that two people will have the samebirthday (month and day)?

The probability that m people all have different birthdays is

This expression becomes less than 0.5 whenever m 23.For hashing, the birthday surprise says that for any problem of reasonable size, collisions will almost certainly occur.

A probe is one comparison of a key with the target.

The load factor of the table is = n /t , where n positions are occupied out of a total of t positions in the table.

Retrieval from a chained hash table with load factor requires approximately 1+ /2 probes in the successful case and probes in the unsuccessful case.

Retrieval from a hash table with open addressing, random probing, and load factor requires approximately

probes in the successful case and probes in theunsuccessful case , respectively.

Retrieval from a hash table with open addressing, linearprobing, and load factor requires approximately

probes in the successful case and in the unsuccessful case, respectively.

Theoretical comparisons:

Empirical comparisons:

经验主义的

9.8 Conclusions: Comparison of Methods

We have studied four principal methods of information retrieval,the first two for lists and the second two for tables. Often we can choose either lists or tables for our data structures.◆ Sequential search is (n).

Sequential search is the most flexible method. The data may be stored in any order, with either contiguous or linked representation.

◆ Binary search is ( ㏒ n).

Binary search demands more, but is faster: The keys must be in order, and the data must be in random-access representation (contiguous storage).

可随机访问的

◆ Table lookup is (1).

Ordinary lookup in contiguous tables is best, both in speed and convenience, unless a list is preferred, or the set of keys is sparse, or insertions or deletions are frequent.

◆ Hash-table retrieval is (1).

Hashing requires the most structure, a peculiar ordering of the keys well suited to retrieval from the hash table, but generally useless for any other purpose. If the data are to be available for human inspection, then some kind of order is needed, and a hash table is inappropriate.

9.9 Application: The Life Game Revisited

◆The Life cells are supposed to be on an unbounded grid, not a finite array as used in Chapter 1.

◆In the class Life, we would like to have the declaration

class Life { public: // methods private: bool map[int][int]; // other data and auxiliary functions};

which is, of course, illegal.See Pg. 418

◆Use a hash table to represent the data member map (sparse array).

◆The main function remains unchanged from Chapter 1.◆Rewrite the method update so that it uses a hash table to look up the status of cells.

◆For any given cell in a configuration, we can determine the number of living neighbors by looking up the status of each neighboring cell.

◆Candidates that might live in the coming generation are those that are alive and their (dead) neighbors. Method update will traverse these cells, determine their neighbor counts by using the hash table, and select those cells that will live in the next generation.

See Pg. 420 - 426

有 100 个待排序的整数，其值在区间 [0,999] 而且互不相同 ,试利用散列表结构 ( 设表长为 100 ，处理冲突采用链地址法 ) 实现对上述整数集合的排序。 要求自行设计散列函数，并分析最好情况及最坏情况下的时间复杂度。 答：由于是排序，将散列函数设计为 i / 10( 必须按大小排，因此要根据高位数排列 ) ，发生冲突时按整数值从小到大插入链表。 最好情况出现在这 100 个待排序的整数均匀分布在区间 [0,999]；散列表每项仅一个整数； 最坏情况是：每项有 10 个数在链表中，共 10项不空。

Pointers and Pitfalls

◆Use top-down design for your data structures, just as you do for your algorithms.◆Before considering detailed structures, decide what operations on the data will be required.◆For the design and programming of lists, see Chapter 6.◆Use the logical structure of the data to decide what kind of table to use.◆Let the structure of the data help you decide whether an index function or an access table is better for accessing a table of data.

◆In using a hash table, let the nature of the data and the required operations help you decide between chaining and open addressing.◆Hash functions must usually be custom-designed for the kind of keys used for accessing the hash table.◆Recall from the analysis of hashing that some collisions will almost inevitably occur.◆For open addressing, clustering is unlikely to be a problem until the hash table is more than half full.

◆Sequential search is slow but robust.

Use it for short lists or if there is any

doubt that the keys in the list are

properly ordered.

◆Be extremely careful if you must

reprogram binary search. Verify that your

algorithm is correct and test it on all the

extreme cases.

◆Drawing trees is an excellent way both

to trace the action of an algorithm and to

analyze its behavior.

◆Rely on the big-O analysis of algorithms

for large applications but not for small

applications.