tceic.com
学霸学习网 这下你爽了
广告
当前位置:首页 >> 学科竞赛 >>

Recursive Functions计算机竞赛


Recursive Functions A definition that defines an object in terms of itself is said to be recursive. This theoretical mathematical topic serves as an introduction to recursive programming (which is supported by Pascal, but not by most BASICs). It also provides a framework for analyzing algorithms (determining the running time and/or space required) which contain loops or recursive sections. Many expressions may be defined recursively in a very simple and elegant manner. The art of recursive thinking is an important skill. Applications of recursion appear in many areas of mathematics (factorials, compound interest, difference equations, etc.) In this round recursive relationships will be shown not in the context of a programming language but usually in functional notation from mathematics or in an algorithmic description. References Roberts, Eric S. Thinking Recursively, Wiley (1986). Rohl, J.S. Recursion via Pascal, Cambridge University Press (1984). Wirth, Niklaus. Algorithms + Data Structures = Programs, Prentice-Hall (1976), Chapter 3.

Sample Problems Consider the following recursive algorithm for painting a square: 1. Given a square. 2. If the length of a side is less than 2 feet, then stop. 3. Divide the square into 4 equal size squares (i.e., draw a “plus” sign inside the square). 4. Paint one of these 4 small squares. 5. Repeat this procedure (start at step 1) for each of the 3 unpainted squares. If this algorithm is applied to a square with a side of 16 feet (having a total area of 256 sq. feet), how many square feet will be painted? In the first pass, we get four squares of side 8. One is painted; three are unpainted. Next, we have 3*4 squares of side 4: three are painted (area=3*42), nine are not. Next, we have 9*4 squares of side 2: nine are painted (area = 9*22), 27 are not. Finally, we have 27*4 squares of side 1: twenty-seven are painted. Therefore, the total painted is 1*82 + 3*42 + 9*22 + 27*12 = 175.

Evaluate f(12, 6), given: f(x,y) = +2 ? f(x-y,y-1) x+y when x>y otherwise

Evaluate the function as follows: f(12, 6) = f(6, 5)+2 = (f (1, 4)+2)+2 = f(1, 4)+4 = (1+4)+4 =9 Working backwards, we get f(0)= 1 f(1) = 2 f(2) = f(f(0))+1 = f(1)+1 = 2+1 = 3 f(3) = f(f(1))+1 = f(2)+1 = 3+1 = 4 f(4) = f(f(2))+1 = f(3)+1 = 4+1 = 5 f(5) = f(f(3))+1 = f(4)+1 = 5+1 = 6 f(6) = f(f(4))+1 = f(5)+1 = 6+1 = 7 Ackerman’s Function is infamous for its potential growth. In fact, we don’t have room here to give a full explanation of the problem. For details, refer to the 1981-82 ACSL All-Star Contest. By evaluating A(1,0), A(1,1), A(1,2) and A(1,3), we see that in general, A(1, x)=2+x. If we evaluate A(2,0), A(2,1), …, we see that in general, A(2,x)=2x+3. To solve our problem, we substitute x=3 and we get an answer of 9.

Find f(6), given:

f(x) =

?

f (f (x-2))+1 2 1

when x>1 when x=1 when x=0

One of the best known recursive functions, Ackerman’s Function, is defined below. Evaluate A(2, 3). A(M,N) =

?

N+1 if M=0 A(M-1, 1) if M?0, N=0 A(M-1, A(M, N-1)) if M?0, N?0

Challenge for the bored: Evaluate A(n, m) in terms of n and m.


推荐相关:

acm竞赛的题型分析

(最短路径) Recursive Search Techniques (回溯搜索技术) Minimum Spanning Tree ...(LCS) 1080 human gene functions,1159 palindrome,1458 common subsequence,2192...


ACM重要知识点_计算机软件及应用_IT/计算机_专业资料

ACM 竞赛的题型分析 竞赛的程序设计一般只有 16 种类型,它们分别是: Dynamic Programming...(最短路径) Recursive Search Techniques (回溯搜索技术) Minimum Spanning...


计算机全英班09期末

计算机全英班09期末 华南理工大学历年历届C++ 期末考试试卷试题、答案及复习资料大全...B) Recursive functions are the functions that call themselves directly. C)...


大学计算机基础类术语

recursive 递回 recursively 递回地 recycler 回收站 redefinition 重复定义 ...functions 终止函式 termination handler 终止处理常式 terms of use 使用规定 ...


2015数据结构实验手册_电脑基础知识_IT/计算机_专业资料

特别是多文件大型工程的编 程;了解 ACM 竞赛的赛题,掌握参加 ACM 竞赛的基本...Step 2. Write recursive version functions of inOrder, preOrder and postOrder...


数据结构 实验三_计算机软件及应用_IT/计算机_专业资料

数据结构 实验三_计算机软件及应用_IT/计算机_专业资料。设计三 树和二叉树一、...(递归)叶结点 void leafRecursive(Tnode *p, ostream &out) { if (p != ...


递归函数的讨论

The application of recursive functions in the algorithm design will be ...4.1 递归函数时间复杂度分析为了比较算法的效率,计算机科学领域采用计算复杂度指标...


exponential functions and recursive rules.

exponential functions and recursive rules._高一数学_数学_高中教育_教育专区。国外指数函数教学设计exponential functions and recursive rules. In this lesson make ...


C语言程序设计大赛资料

(最短路径) Recursive Search Techniques (回溯搜索技术) Minimum Spanning Tree ...竞赛类的黑宝书) 《青少年国际和全国信息学(计算机)奥林匹克竞赛指导)――组合...


Recursive Patterns OF SNOBOL

In our previous discussion of recursive functions, we said they work because successive calls present the function with progressively simpler problems, until ...

网站首页 | 网站地图
All rights reserved Powered by 学霸学习网 www.tceic.com
copyright ©right 2010-2021。
文档资料库内容来自网络,如有侵犯请联系客服。zhit325@126.com