Art of Programming contest chapters:1-4

Art of Programming contest chapters:1-4

Table of Contents

• 1. 学习算法注意点
• 2. 一些规则
• 3. C字符串处理函数(部分)
• 4. 基础数据结构

Mathematics

• Prime Number
• Big Integer
• Permutation
• Number Theory
• Factorial
• Fibonacci
• Sequences
• Modulus

Dynamic Programming

• Longest Common Subsequence
• Longest Increasing Subsequence
• Edit Distance
• 0/1 Knapsack
• Coin Change
• Matrix Chain Multiplication
• Max Interval Sum

Graph

• Traversal
• Flood Fill
• Floyed Warshal
• MST
• Max Bipertite Matching
• Network Flow
• Aritculation Point

Sorting

• Bubble Sort
• Quick Sort
• Merge Sort (DAndC)
• Selection Sort
• Radix Sort
• Bucket Sort

Searching

• Complete Search, Brute Force
• Binary Search (DAndC)
• BST

Simulation

• Josephus

String Processing

• String Matching
• Pattern Matching

Computational Geometry

• Convex Hull

AdHoc

• Trivial Problems

1 学习算法注意点

1. Proof of algorithm correctness (especially for Greedy algorithms)
2. Time/Space complexity analysis for non recursive algorithms.
3. For recursive algorithms, the knowledge of computing recurrence relations and analyze them: iterative method, substitution method, recursion tree method and finally, Master Theorem
4. Analysis of randomized algorithm which involves probabilistic knowledge, e.g: Random variable, Expectation, etc.
5. Amortized analysis.
6. Output-sensitive analysis, to analyze algorithm which depends on output size, example: O(n log k) LIS algorithm, which depends on k, which is output size not input size.

2 一些规则

• Biggest built in data structure "long long" is 2⁶³-1: 9*10¹⁸ (up to 18digits)
• If you have k nested loops running about n iterations each, the program has O(n*k)

complexity

• If your program is recursive with b recursive calls per level and has l levels, the

program O(b*l) complexity

• Bear in mind that there are n! permutations and 2ⁿ subsets or combinations of n

elements when dealing with those kinds of algorithms

• The best times for sorting n elements are O(n log n)
• DP algorithms which involves filling in a matrix usually in O(n³)
• In contest, most of the time O(n log n) algorithms will be sufficient.

3 C字符串处理函数(部分)

1. strdup()The library function strdup() is similar to strcpy(), except that strdup()performs its own memory allocation for the destination string with a call tomalloc().

   Using strdup() to copy a string with automatic memory allocation.

   The prototype for strdup() is

   char *strdup( char *source );

   /* The strdup() function. */
    #include <stdlib.h>
    #include <stdio.h>
    #include <string.h>
     char source[] = "The source string.";
   main() {
   char *dest;
        if ( (dest = strdup(source)) == NULL)
        {
            fprintf(stderr, "Error allocating memory.");
            exit(1);
   }
        printf("The destination = %s\n", dest);
   return(0); }
   The destination = The source string.
   

2. strchr()The strchr() function finds the first occurrence of a specified character ina string.

   The prototype is

   char *strchr(char *str, int ch);strchr函数从左到右寻找字符ch，直到找到ch或者"\0"，如果ch找到就返回指向ch的指针，如果找不到，就返回NULL

   /* Searching for a single character with strchr(). */
    #include <stdio.h>
    #include <string.h>
   main() {
        char *loc, buf[80];
        int ch;
       /* Input the string and the character. */
       printf("Enter the string to be searched: ");
       gets(buf);
       printf("Enter the character to search for: ");
       ch = getchar();
       /* Perform the search. */
       loc = strchr(buf, ch);
       if ( loc == NULL )
           printf("The character %c was not found.", ch);
       else
           printf("The character %c was found at position %d.\n",
                   ch, loc-buf);
       return(0); }
   Enter the string to be searched: How now Brown Cow?
   Enter the character to search for: C
   The character C was found at position 14.
   

3. strcspn()The library function strcspn() searches one string for the first occurrenceof any of the characters in a second string.原型如下：

   size_t strcspn(char *str1, char *str2);

   这个函数从str1寻找str2中任意一个字符第一次出现的位置，如果找到了，就返回str1的偏移量，如果没找到，就返回str1字符创的长度。

   看例子：

   /* Searching with strcspn(). */
    #include <stdio.h>
    #include <string.h>
   main() {
        char  buf1[80], buf2[80];
        size_t loc;
       /* Input the strings. */
       printf("Enter the string to be searched: ");
       gets(buf1);
       printf("Enter the string containing target characters: ");
       gets(buf2);
       /* Perform the search. */
       loc = strcspn(buf1, buf2);
       if ( loc ==  strlen(buf1) )
           printf("No match was found.");
       else
           printf("The first match was found at position %d.\n", loc);
   return(0); }
   Enter the string to be searched: How now Brown Cow?
   Enter the string containing target characters: Cat
   The first match was found at position 14.
   

4. strstr()This function searches for the first occurrence of one string withinanother, and it searches for the entire string, not for individualcharacters within the string.原型如下：char *strstr(char *str1, char *str2);这应该是最有用的了，如果在str1里头找到str2，返回指针，若没找到，返回NULL，如果str2的长度为0，那么返回str1。

   例子如下：

   /* Searching with strstr(). */
   #include <stdio.h>
   #include <string.h>
   main() {
       char *loc, buf1[80], buf2[80];
      /* Input the strings. */
      printf("Enter the string to be searched: ");
    gets(buf1);
        printf("Enter the target string: ");
        gets(buf2);
        /* Perform the search. */
        loc = strstr(buf1, buf2);
   if(loc== NULL)
   printf("No match was found.\n");
        else
            printf("%s was found at position %d.\n", buf2, loc-buf1);
   return(0);}
   Enter the string to be searched: How now brown cow?
   Enter the target string: cow
   Cow was found at position 14.
   

5. atoi() atol() atof()原型如下：int atoi(char *ptr);long atol(char *ptr);double atof(char *str);

   String	value Returned by atoi()
   "157"	157
   "-1.6"	-1
   "+50x"	50
   "twelve"	0
   "x506"	0

   atol和atoi类似，只是返回longatof如下

   String	Value Returned by atof()
   "12"	12.000000
   "-0.123"	-0.123000
   "123E+3"	123000.000000
   "123.1e-5"	0.001231

4 基础数据结构

一共有五种基础数据结构：数组，链表，栈，队列和双向队列。

1. 数组

   数组是最有用的数据结构，事实上，这种数据结构也是在竞赛中最常使用的。先看看优点：如果下表已知，搜索元素非常快,O(1)的复杂度，非常适合循环/迭代.除此之外，数组常常被用来实现其他的数据结构，比如栈，队列，哈希表。然而，最简单的数据结构并不意味着高效，在有些时候，数组也并不高效。并且，在标准数组中，它的大小事给定的。如果你实现不知道输入的大小，更好的作法应该是使用vector(一个可变大小的数组)。数组的插入操作很慢，在未排序的数组中搜索数据也很慢，删除操作要移动几乎所有的元素。

   • 可变大小的数组 vector

     #include <stdio.h>
     #include <vector>
     #include <algorithm>
     using namespace std;
     void main() {
     // just do this, write vector<the type you want,
     // in this case, integer> and the vector name
     vector<int> v;
     // try inserting 7 different integers, not ordered
     v.push_back(3); v.push_back(1); v.push_back(2);
     v.push_back(7); v.push_back(6); v.push_back(5);
     v.push_back(4);
     // to access the element, you need an iterator...
     vector<int>::iterator i;
     printf("Unsorted version\n");
     // start with 'begin', end with 'end', advance with i++
     for (i = v.begin(); i!= v.end(); i++)
       printf("%d ",*i); // iterator's pointer hold the value
     printf("\n");
     sort(v.begin(),v.end()); // default sort, ascending
     printf("Sorted version\n");
     for (i = v.begin(); i!= v.end(); i++)
       printf("%d ",*i); // iterator's pointer hold the value
     printf("\n");
     }
     

   • 链表为了解决数组插入删除不高效的问题，引出了链表，除此之外数组不能变大小(vector变大小也比较慢)。链表的插入删除非常快，但是链表的副作用是，当一个节点被删除的时候，会浪费内存，之所以会这样是因为被删除的节点只是被标记为"已删除",这些被浪费的内存只有被垃圾回收期清理了才能使用(取决于编译器/操作系统)。链表通常由数据和指针组成，查找数据得从头往后挨个找。STL中的链表:

     #include<stdio.h>
     #include<list>
     using namespace std;
     list<int>l;
     list<int>::iterator i;
     void print(){
       for(i=l.begin();i!=l.end();i++)
         printf("%d ",*i);
       printf("\n");
     }
     void main()
     {
       l.push_back(3);l.push_back(1); l.push_back(2);
       l.push_back(7); l.push_back(6); l.push_back(5);
       l.push_back(4); l.push_back(7);
       print();
       l.sort();
       print();
       l.remove(3);
       print();
       l.unique(); //删除重复的元素
       print();
       i=l.begin();
       i++;
       l.insert(i,1,10);//插入1个'10'
       print();
     }
     

   • 栈一个只允许在顶部插入和删除的数据结构，这种行为叫后进先出。重要的栈操作：
     1. push(C++ STL:push())

        给栈顶部添加新元素

     2. pop(C++ STL:pop())

        得到并移除栈顶部元素

     3. peek(C++ STL:top())

        得到栈顶元素，但不删除

     4. IsEmpty(C++ STL:empty())

        测试栈是否为空

     栈的一些应用场合：

     1. 计算机世界中真实的栈：递归，调用过程等等
     2. 测试回文(虽然使用队列和栈检查回文是很笨的方法)
     3. 翻转输入数据(也是一个笨的方法)
     4. 测试括号匹配
     5. 后缀计算
     6. 转换数学表达式：前缀，中缀或是后缀

     STL实现的STACK

     #include<iostream>
     #include<stack>
     using namespace std;
     void main()
     {
     stack<int>s;
     s.push(3); s.push(1); s.push(2);
     s.push(7); s.push(6); s.push(5);
     s.push(4);
     while (!s.empty()) {
     printf("%d ",s.top());
     s.pop();
     }
     printf("\n");}
     }
     

   • 队列一个只允许从后插入数据，从前删除数据的数据结构，这种行为叫做先进先出，和真实世界的队列一样。重要的队列操作：
     1. enqueue(C++ STL:push())

        给队列末尾添加新元素

     2. dequeue(C++ STL:pop())

        得到并移除队列顶部元素

     3. peek(C++ STL:top())

        得到队列顶部元素，但不删除

     4. IsEmpty(C++ STL:empty())

        测试队列是否为空

     #include<iostream>
     #include<queue>
     using namespace std;
     void main(){
       queue<int>q;
       q.push(3); q.push(1); q.push(2);
       q.push(7); q.push(6); q.push(5);
       q.push(4);
       while(!q.empty()){
         printf("%d ",q.front());
         q.pop();
       }
       printf("\n");
     }
     

Author: kirchhoff

Created: 2014-11-16 Sun 21:18

Emacs 24.4.1 (Org mode 8.2.10)

Validate