Question

我目前正在编程类中教授递归。我注意到我的学生很难掌握递归的概念。有没有一种很好的方法可视化函数对于教学目的的作用？

作为一个例子，这里有一个用于获得第n个斐波纳契数的R函数：

fib_r <- function(n) {
    if(n <= 2) return(1)
    fib_r(n-1) + fib_r(n-2)
}

感谢。

Answer 1

这就是我在R中解释递归函数的方法：

首先，我同意@AEBilgrau的说法，因子是递归的一个很好的例子。（在我的选择中比斐波纳契更好。）

然后我会快速通过理论基础，为什么可以将阶乘定义为递归函数，这很简单，如

4！ = 4 * 3 * 2 * 1 = 4 * 3！

然后你可以向他们展示相应的递归R函数

fact=function(x) if (x==0) return(1) else return(x*fact(x-1))
fact(3)
#6

但还会向他们展示以下输出

#|fact(3) called
#|fact(3) is calculated via 3*fact(2)
#|fact(2) is unknown yet. Therefore calling fact(2) now
#|Waiting for result from fact(2)
#|  fact(2) called
#|  fact(2) is calculated via 2*fact(1)
#|  fact(1) is unknown yet. Therefore calling fact(1) now
#|  Waiting for result from fact(1)
#|  |   fact(1) called
#|  |   fact(1) is calculated via 1*fact(0)
#|  |   fact(0) is unknown yet. Therefore calling fact(0) now
#|  |   Waiting for result from fact(0)
#|  |   |   fact(0) called
#|  |   |   fact(0)=1 per definition. Nothing to calculate.
#|  |   |   fact(0) returning 1 to waiting fact(1)
#|  |   fact(1) received 1 from fact(0)
#|  |   fact(1) can now calculate 1*fact(0)=1*1=1
#|  |   fact(1) returning 1 to waiting fact(2)
#|  fact(2) received 1 from fact(1)
#|  fact(2) can now calculate 2*fact(1)=2*1=2
#|fact(3) received 2 from fact(2)
#|fact(3) can now calculate 3*fact(2)=3*2=6
#[1] 6

源自

#helper function for formatting
tabs=function(n) paste0("|",rep("\t",n),collapse="")

fact=function(x) {
  #determine length of call stack
  sfl=length(sys.frames())-1
  #we need to define tmp and tmp1 here because they are used in on.exit
  tmp=NULL
  tmp1=NULL

  #on.exit will print the returned function value when we exit the function ...
  #... i.e., when one function call is removed from the stack
  on.exit({
    if (sfl>1) {
      cat(tabs(sfl),"fact(",x,") returning ",
          tmp," to waiting fact(",x+1,")\n",sep="")
    }
  })
  cat(tabs(sfl),"fact(",x,") called\n",sep="")
  if (x==0) {
    cat(tabs(sfl),"fact(0)=1 per definition. Nothing to calculate.\n",sep="")
    #set tmp for printing in on.exit
    tmp=1
    return(1)
  } else {
    #print some info for students
    cat(tabs(sfl),"fact(",x,") is calculated via ",x,"*fact(",x-1,")\n",sep="")
    cat(tabs(sfl),"fact(",x-1,
        ") is unknown yet. Therefore calling fact(",x-1,") now\n",sep="")
    cat(tabs(sfl),"Waiting for result from fact(",x-1,")\n",sep="")
    #call fact again
    tmp1=fact(x-1)
    #more info for students
    cat(tabs(sfl),"fact(",x,") received ",tmp1," from fact(",x-1,")\n",sep="")
    tmp=x*tmp1
    cat(tabs(sfl),"fact(",x,") can now calculate ",
        x,"*fact(",x-1,")=",x,"*",tmp1,"=",tmp,"\n",sep="")
    return(tmp)
  }
}

fact(3)

Answer 2

这是我的例子，可能在很多教科书中使用过：

recursive_sum <- function(n){
  if(n == 1) {print("Remember 1, add everything together"); return(n)}
  print(paste0("Remember ", n, ", pass ", n-1, " to recursive function"))
  n + recursive_sum(n-1)
}

输出：

> recursive_sum(4)
[1] "Remember 4, pass 3 to recursive function"
[1] "Remember 3, pass 2 to recursive function"
[1] "Remember 2, pass 1 to recursive function"
[1] "Remember 1, add everything together"
[1] 10

Answer 3

我认为阶乘函数是递归的一个很好的例子。将其与打印输出结合起来（正如其他人建议的那样）似乎是描述正在发生的事情的好方法：

factorial <- function(n) {
  cat("factorial(", n, ") was called.\n", sep = "")
  if (n == 0) {
    return(1)
  } else {
    return(n * factorial(n - 1))
  }
}

factorial(4)
#factorial(4) was called.
#factorial(3) was called.
#factorial(2) was called.
#factorial(1) was called.
#factorial(0) was called.
#[1] 24

然后，您还可以实现非递归因子函数并比较计算效率。或者也许问他们上面的实现有什么问题（例如factorial(-4)会发生什么）。

关于更合适的可视化（而不仅仅是简单的示例），有websites来说明递归树。

修改： Googling recursion也是一个有用的课程。

Answer 4

在 fib_r

中打印变量 n 的值

print("iteraction at: ", n)

如何有效地可视化递归函数？

4 个答案: