设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2588|回复: 3
打印 上一主题 下一主题

[科技前沿] 突然想到让deepseek来解释一下递归

[复制链接]
  • TA的每日心情
    开心
    2025-9-8 05:08
  • 签到天数: 3 天

    [LV.2]筑基

    跳转到指定楼层
    楼主
     楼主| 发表于 2025-1-29 14:16:55 | 只看该作者 回帖奖励 |倒序浏览 |阅读模式
    本帖最后由 密银 于 2025-1-29 14:19 编辑 ! h5 O7 m: D! C) C  p
    - \4 M, q% P% x6 ^0 }" r
    解释的不错
    $ t& }0 Q7 K# q4 ^( \
    4 f3 V( Y* o3 @* e2 D; y+ E' A递归是一种通过将问题分解为更小的同类子问题来解决问题的方法。它的核心思想是:**函数直接或间接地调用自身**,直到满足终止条件。1 u+ }0 _' W! A/ {8 {: J8 e, v5 n
    * d$ T4 p3 @& a4 ~! F
    关键要素
    # o+ k/ p. F% V" X& u1. **基线条件(Base Case)*** k0 h  X. P2 W4 |* p( D5 j: X
       - 递归终止的条件,防止无限循环
    * ^) ?- j3 E& Q, b; Q3 v- |   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    " P9 N0 V& ]0 Q+ u0 W$ S  s0 r- V- C5 O5 P. r
    2. **递归条件(Recursive Case)**
    6 o8 n; y5 s4 _. r, q, H6 T   - 将原问题分解为更小的子问题& I* Q5 g! [4 X- P
       - 例如:n! = n × (n-1)!
    3 n# W! Z2 ?5 d# e* k' H- c" x; x
    经典示例:计算阶乘/ }7 Q. Q% @/ v# D7 u3 D! q
    python
    5 U3 M; k; g0 V- T. R$ t) sdef factorial(n):
    ! s9 W" L( j4 V5 s; ]    if n == 0:        # 基线条件
    $ r$ j" n5 m( b0 }3 o4 x. h5 X        return 1
    : T3 n1 _, d0 Y% _    else:             # 递归条件
    5 {' S' @% Z( |        return n * factorial(n-1)  l: s, O" _3 ?6 ^9 W
    执行过程(以计算 3! 为例):- Z, c0 w6 K# `0 o6 M0 D! A2 M8 U
    factorial(3)
    1 U4 f; E/ y; E! s! \9 q% d3 * factorial(2)
    ) Q) F" ^7 r+ x9 F3 * (2 * factorial(1))
    " d, m6 x% ?7 @  S; K3 * (2 * (1 * factorial(0)))
    + @/ V6 l& C' u. ^' A3 * (2 * (1 * 1)) = 6* Q# W3 c0 H9 V. [  v- W- d
    0 M+ I: l: i* @2 ^/ g) |6 K
    递归思维要点3 z8 h* ~* |) S% Q' U6 ^* i
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑, o6 w( y1 c6 \- ^, ]$ R: {4 G
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)) G( r& r. i' L
    3. **递推过程**:不断向下分解问题(递)8 i: _. K( F& _1 T7 ^/ g
    4. **回溯过程**:组合子问题结果返回(归)' f9 L/ V0 e9 h. @5 E! i

    # ?; u+ p" g. [$ _% i 注意事项3 B9 Z  `+ w9 G$ b
    必须要有终止条件
    : b+ Y. ^- ~2 ~  L! D! l递归深度过大可能导致栈溢出(Python默认递归深度约1000层)/ U& w0 v2 N3 k- e  t5 F5 Q
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ) U7 C2 ]7 ~. m, f尾递归优化可以提升效率(但Python不支持)
    3 o& b" Q0 A: [% D* r. c4 V0 A$ m" K* J: H6 f: S0 [. J
    递归 vs 迭代
    ; X  B) ?" d8 b9 [* P  {% O" S|          | 递归                          | 迭代               |
    , C6 s7 o: A. U# X7 T$ A|----------|-----------------------------|------------------|
    8 X8 `  Y, D, r5 A  v/ w| 实现方式    | 函数自调用                        | 循环结构            |: |6 w9 n6 L' u: o0 c# o
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |- @) M6 E/ H- `5 {( p# p. N( \) V
    | 代码可读性  | 对符合递归思维的问题更直观                | 线性流程更直接         |
    3 v3 y, @; ~( M| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |8 T) C2 a; K3 I9 I/ m8 q# y: p( M; u

    2 z" {( V. i2 H  Z 经典递归应用场景' [, i. @+ G+ ]
    1. 文件系统遍历(目录树结构)
    & D9 [1 v) e. L1 v2. 快速排序/归并排序算法; N  X; l6 }6 H- N
    3. 汉诺塔问题
    2 |% p  H; x6 x5 E4. 二叉树遍历(前序/中序/后序)
    1 n, W# I3 W6 p/ C5. 生成所有可能的组合(回溯算法)
    7 I3 }3 P5 L5 w' K! |) l4 H" X' B4 ^$ E4 w% d9 B! M) o
    试着用递归思维想象:你站在一面镜子前,镜子里有无数个逐渐变小的你在照镜子,这就是递归的直观体现。但记住每个"分身"最终都要有结束的时刻,这就是基线条件的重要性。

    评分

    参与人数 3爱元 +26 收起 理由
    pcb + 4
    老票 + 16 涨姿势
    住在乡下 + 6 给力

    查看全部评分

  • TA的每日心情
    慵懒
    8 小时前
  • 签到天数: 3225 天

    [LV.Master]无

    沙发
    发表于 2025-1-30 00:07:50 | 只看该作者
    挺好,递归思维要点与我能够回忆起来我当时写递归程序的思路很一致,,或者被它唤醒,
    ; P# |. k9 g4 O4 I我推理机的核心算法应该是二叉树遍历的变种。) A, X! ]  H% y2 K$ [
    另外知识系统的推理机搜索深度(递归深度)并不长,没有超过10层的,如果输入变量多的话,搜索宽度很大,但对那时的286-386DOS系统,计算压力也不算大。
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    板凳
    发表于 2025-2-2 00:45:59 | 只看该作者
    Recursion in programming is a technique where a function calls itself in order to solve a problem. It is a powerful concept that allows you to break down complex problems into smaller, more manageable subproblems. Here's a detailed explanation:% N8 Q6 {9 i- \6 `5 _
    Key Idea of Recursion7 a5 Z# k& Z. w/ T% X
    3 V- i0 j" Q; F9 j4 J& a5 @# `
    A recursive function solves a problem by:
    / T, ~$ U1 [5 W/ r( L7 Z$ n- _
    5 I( V; W/ C8 ]. F- X# T    Breaking the problem into smaller instances of the same problem.8 w) `# ]* g# ~+ H& K9 }# Q& {
    8 V# M$ b) q6 W
        Solving the smallest instance directly (base case).
    ' e4 g: \9 l! E9 p" S- ~3 K7 V5 j# x0 \  Y# e; Y! J3 i. Y
        Combining the results of smaller instances to solve the larger problem.  u& C& B" c0 \+ I! l
    2 W& C  I6 k9 S. M8 k$ i
    Components of a Recursive Function/ z8 y0 i/ \, Q9 ~! E
    / W- U* M2 {7 k
        Base Case:  e+ V! v: @1 C7 p

    % E$ \, e5 k: [: \, W3 ?* Y        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; c: j/ P4 M' J
    8 v$ P3 L* c( T1 ?$ O9 z        It acts as the stopping condition to prevent infinite recursion.
    3 l7 J, ?5 T2 ^: f, {; H; W: x+ H% _# Q( F
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% q' p! T2 G; Z+ S3 y/ H1 W

      m" W8 m# T5 O8 B    Recursive Case:% V" Z3 Z- m8 a3 k% l, ]4 o7 Z* b
    $ ]; k4 @' Q# {
            This is where the function calls itself with a smaller or simpler version of the problem.
    $ _6 K: k# y; \- A! U5 c; U/ N9 s# R6 L+ ~+ O( d6 c
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).  m' D6 N& I7 x4 K4 u, j- q3 g
    * R; d  M' R' O" b
    Example: Factorial Calculation" v/ M8 `- |) b$ h8 O+ ^
    3 z9 Q! F( l" }8 F! C- ]; S3 _
    The factorial of a number n (denoted as n!) is the product of all positive integers less than or equal to n. It can be defined recursively as:
    5 ^, t" n7 q# ?# _) v( h3 o9 u% O, z8 H2 v  Q4 C' R9 _, y
        Base case: 0! = 15 L: @/ y- F" i  x4 _0 g; f7 p$ ?' Y

    3 ]4 A, w6 r( h+ z! M6 y7 R8 n    Recursive case: n! = n * (n-1)!
    0 O0 _1 h0 k" {3 i( T$ d9 `
    8 H; U1 }! `* k- x4 QHere’s how it looks in code (Python):$ V7 j2 W; m) b+ U, @2 `$ K# d- d
    python. _/ @, }% d* R+ f6 R
    5 w1 o/ u3 _  d7 s& Q: ]

    / I3 r) b$ z4 t9 bdef factorial(n):
    " A! k8 y* V) g  ?  I    # Base case
    & B9 p& z# q  u- n, E% a2 e  n% K    if n == 0:
    ( S) L/ m6 N6 u2 U( b  x8 y" i, O        return 1
    7 q  {8 I/ a6 a% S% ?: g5 r' x    # Recursive case
    & ?2 U% l; g  M    else:
    9 w1 ]! K/ W' }        return n * factorial(n - 1)
    6 S2 t9 L+ R# K8 e: w) l0 B/ Y: D) `" z. a
    # Example usage( b8 u9 V" z) s+ H. \1 l5 `
    print(factorial(5))  # Output: 120  H  w  }& x6 ~- J5 v! A

    # H- o( }$ }; O% c- WHow Recursion Works- i- G7 ~. y6 n" \$ j

    ; t& K$ X- f3 y/ y- `    The function keeps calling itself with smaller inputs until it reaches the base case.
    : G3 R' C. B7 q- M7 C6 y$ A$ L3 X. v2 v0 y! N! i. ]$ K
        Once the base case is reached, the function starts returning values back up the call stack.6 h9 v6 d+ u' b( L$ \/ @7 @. @
    , s  d) s9 m, Z- O! Q5 ~- _- {
        These returned values are combined to produce the final result.
      _& u  L$ i+ ~/ s( z. ]8 m2 _) d% T% E! b0 T
    For factorial(5):4 H/ E- r  [2 ?% {3 r* [" U

    " Y3 r$ m) v6 c5 d8 T$ e: w% Z5 Z$ A3 q- L" \2 Y5 H$ V
    factorial(5) = 5 * factorial(4)
    6 S8 P: i# l/ B$ ifactorial(4) = 4 * factorial(3)4 P- o9 M+ [& U5 ]' w1 L. B8 ~1 n
    factorial(3) = 3 * factorial(2)7 M( [( _1 d. Z3 _+ ]- @4 n) U! _# g
    factorial(2) = 2 * factorial(1)$ ]; f8 U( ?7 E$ I9 V: Q6 x
    factorial(1) = 1 * factorial(0)
    6 G) C$ [+ q  E2 s7 {# kfactorial(0) = 1  # Base case
    4 s1 q4 u# {8 y; o/ W' K5 ~- v
    6 ^2 H, g6 w% ~+ Q# ~% k) tThen, the results are combined:
    ! S" }, t1 s& ^% ?% C& [/ S/ \8 H1 @& ?
    + c1 A/ F% x$ H% L& t
    factorial(1) = 1 * 1 = 1
    ' z4 v5 X; Z) S- C/ z0 d& xfactorial(2) = 2 * 1 = 2) n  C" E/ v5 t$ `* m+ X
    factorial(3) = 3 * 2 = 6
    0 B; N$ [3 ?( F1 m1 R! C( zfactorial(4) = 4 * 6 = 24; B- G3 ]6 j) C; G( {$ a$ l9 G
    factorial(5) = 5 * 24 = 120
    $ I8 `2 t8 d/ n
    " R  O9 b! E: y! ]5 {& V: ^Advantages of Recursion
    $ x" l: ?9 Q, |4 g/ k* m" a+ i( D* M
        Simplicity: Recursive solutions are often more intuitive and easier to write for problems that have a natural recursive structure (e.g., tree traversals, divide-and-conquer algorithms).
    1 c3 C. M4 z1 l
    8 L5 |9 b4 C; \# h' T3 K    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    8 Q. C, P2 F6 h# ~# j6 E  ~# j2 W' J* R& v% x5 i
    Disadvantages of Recursion, W; I: V3 L% H4 f) k* g
    # R+ s7 Q, f1 s2 O
        Performance Overhead: Each recursive call adds a new layer to the call stack, which can lead to high memory usage and potential stack overflow for deep recursion.
    * ?+ r" A6 Z( e3 H$ j6 R* }: l2 ]0 _
    5 b, p/ O' B. x- [4 P    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)., B' F  M9 @* T, v( E! m2 M

    6 ~$ o5 T2 m# cWhen to Use Recursion( I  B2 }8 O9 U3 u
    2 R+ H- k# e2 Q- ~( l( L/ g
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ( X- M  z! c1 {# p% \/ {. B* A+ M; D( G
        Problems with a clear base case and recursive case.
    ! a1 ^; p1 t# R0 ^; C7 k' ^2 i3 Z7 s0 g9 J( W
    Example: Fibonacci Sequence7 U: C# T/ L8 k" R: X

    * X* ]; d9 U0 N4 ^4 gThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * \5 x& x1 @/ F0 `8 l9 r. L% x" Z
    : P" v5 H7 }) F2 M- }, j    Base case: fib(0) = 0, fib(1) = 1/ [& u1 o4 z9 @
    7 p7 u1 R9 k& h" w% i& d( m
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ( v8 c% ?5 ?5 x- D4 ~" n, U( q
    - F4 e" O; [+ jpython$ r. R' x# \5 V! B+ t

    5 G9 ^, O4 V. V5 Y2 N" d
    5 X9 ^/ P  ]1 `def fibonacci(n):1 G" ?' O* b! C. l! c6 K) U$ ~! G) F
        # Base cases5 k1 @+ J- j- e
        if n == 0:( q# g3 q3 K6 v, D: ?" [
            return 0
    2 z' d. M  D3 ^4 a    elif n == 1:
    # `5 I7 G2 F# B& n- [7 S        return 1
    + t& w) t5 f; {$ |3 `9 s, q    # Recursive case
    2 ~" W0 X5 b6 _- B1 }    else:
    9 M/ ?1 ^9 M( R        return fibonacci(n - 1) + fibonacci(n - 2)
    9 {! R" h3 ~6 X. R! l
    3 @! Z  ]+ D- B- A# Example usage/ H# d8 o  h6 q& u* C
    print(fibonacci(6))  # Output: 8
    / J2 ^0 c* @! A( L) e0 y
    5 P' y7 u$ \; a; j# E2 u9 i7 fTail Recursion0 f- e' z* d5 r0 t7 r: W

    " z" e: _- L! ]1 H" mTail recursion is a special case of recursion where the recursive call is the last operation in the function. Some programming languages optimize tail-recursive functions to avoid stack overflow, but not all languages (e.g., Python does not optimize tail recursion).
    " b  O0 O8 l& o- o
      ~8 ]% I% v2 U/ w( }9 x8 R( l) e. X: [In summary, recursion is a fundamental concept in programming that allows you to solve problems by breaking them into smaller, self-similar subproblems. It’s important to define a base case to avoid infinite recursion and to understand the trade-offs between recursion and iteration.
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    地板
    发表于 2025-2-2 00:47:27 | 只看该作者
    我还让Deepseek 给我讲讲Linux Kernel Driver 现在的开发流程,让一个老同志复习复习,快忘光了。
    回复 支持 反对

    使用道具 举报

    手机版|小黑屋|Archiver|网站错误报告|爱吱声   

    GMT+8, 2026-4-28 17:47 , Processed in 0.059769 second(s), 18 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表