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.


相关文章:
ACM程序设计竞赛例题
ACM程序设计竞赛例题_计算机软件及应用_IT/计算机_专业资料。备战 ACM 资料 一:...(最短路径) Recursive Search Techniques (回溯搜索技术) Minimum Spanning Tree ...
ACM国际大学生程序设计竞赛指南
ACM国际大学生程序设计竞赛指南_IT/计算机_专业资料。ACM的一些信息ACM...(最短路径) Recursive Search Techniques (回溯搜索技术) Minimum Spanning Tree ...
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)...
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...
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 ...
NSRecursiveLock,递归锁
NSRecursiveLock,递归锁_计算机软件及应用_IT/计算机_专业资料。NSRecursiveLock,递归锁 NSRecursiveLock,多次调用不会阻塞已获取该锁的 线程。 NSRecursiveLock *the...
ACM程序设计-Operand-Code the Tree
百度文库 专业资料 IT/计算机 计算机软件及应用...Obviously, the dividing process is recursive. As ...functions are always valid.Display a blank line ...
17review(Recursion)_计算机软件及应用_IT/计算机_专业资料
17review(Recursion)_计算机软件及应用_IT/计算机_专业资料。Chapter 17 Recursion...(TRUE) Recursive functions are always simpler than non-recursive functions. ...
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 ...
二叉树的部分算法
Uses: The functions recursive_insert, recursive_height */ { if (sub_root == NULL) sub_root = new Binary_node<Entry>(x); else if (recursive_...
更多相关标签: