设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2600|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
    . N7 n9 x9 {$ ]8 H1 r
    % ^" h% _. w( O- q( V' t$ j解释的不错
    " c! I% y/ D% N, E5 j6 g7 i- v% g0 B  E. a$ i
    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。7 l3 Y7 B, u: j3 p% M! i
    + X9 c+ ~0 J! O1 Z2 K
    关键要素
    ' q# O, M3 w$ g4 d8 f0 ^1. **基线条件(Base Case)**
    * n9 c8 ^/ q. `" \# @- S  f% F' P/ l   - 递归终止的条件,防止无限循环
    ; K. }0 o2 A5 P4 c  e   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& j! U2 \# M8 F. P1 ^8 N+ Y

    4 R% n9 A; T" Y7 H5 m0 }, |& i2. **递归条件(Recursive Case)**( ]/ ]$ A% ]7 b$ d9 H: \- I
       - 将原问题分解为更小的子问题
    ' L! S% Q7 v. p1 f" x   - 例如:n! = n × (n-1)!
    4 ^; z5 }4 y2 N5 P1 P  m# Z( K% y+ u  E0 W( X( p& X
    经典示例:计算阶乘* ]% i5 X! P4 Y3 g" F4 f
    python: O8 U8 `. P- p
    def factorial(n):
    5 j( L, f: B4 M- k. @- v5 g! Z    if n == 0:        # 基线条件
    ; K; Y  P& P3 w        return 1
    ; I  Q3 S2 `& B) m% N# r    else:             # 递归条件" [6 [4 q, h2 ]# T7 v
            return n * factorial(n-1)' f! E; d3 a* }( w: i( J, F
    执行过程(以计算 3! 为例):2 P. ?2 B# q7 T: t' N. Q
    factorial(3)4 ^- f5 U( [2 S+ o: N' h) s
    3 * factorial(2)/ e) ~, R6 r7 w" y
    3 * (2 * factorial(1))
    # N( o- \. y9 D6 P; J3 * (2 * (1 * factorial(0)))
    ! w1 i% g, \8 @6 \3 * (2 * (1 * 1)) = 69 e. B+ ]+ Q8 j  V# J
    - Y* T% O8 U4 ?' e: `9 S
    递归思维要点5 V  }- e; ?1 q: _# e) x6 f
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : z: M% }3 x! d) |; t7 |+ K; `2. **栈结构**:每次调用都会创建新的栈帧(内存空间)! l* J( k3 F2 V& \+ Q8 z, u
    3. **递推过程**:不断向下分解问题(递). x& _: ?( v/ j
    4. **回溯过程**:组合子问题结果返回(归)
    & {/ x+ ~& f) n2 y( u4 F6 F
    * U# A. g0 M" i$ J; }! T 注意事项3 c4 ~" Z' k- m- u- w3 p
    必须要有终止条件: B3 N. ?; x2 |# \+ H
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)# K# C/ n  b" [" f% o
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ( j! s. b) f5 J尾递归优化可以提升效率(但Python不支持)0 K. ~4 ?( z8 _; u+ Z% K) u

    # P; ?6 C$ |5 k( Y 递归 vs 迭代
    % {9 B. A: u0 H/ X6 C+ l|          | 递归                          | 迭代               |
    . E7 |' S/ L3 |5 ]+ r0 i2 R|----------|-----------------------------|------------------|" u* l- b! j2 _' C+ W
    | 实现方式    | 函数自调用                        | 循环结构            |, @/ W, D' ?' T
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 p4 ~$ @) P" ^2 I* \( x
    | 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         |( p' @* F/ M/ W2 `$ v8 [
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    6 W8 }7 g4 u- G  D/ _9 F" y8 v6 P8 L7 h) B6 k2 s
    经典递归应用场景
    / H' d1 e% X. d! B1. 文件系统遍历(目录树结构)
    8 X6 q- y4 [, s/ x4 v8 I' j; O2. 快速排序/归并排序算法7 A# Y& q5 y0 j" P# r- l3 Y
    3. 汉诺塔问题# j" q. _$ l- X- i: d
    4. 二叉树遍历(前序/中序/后序)
    4 ?/ R4 ]+ w+ V( q0 r5. 生成所有可能的组合(回溯算法)+ q% u3 m6 M' l0 Y  s7 M  J
    9 D2 F: r7 f, B6 t2 t( u$ z; X1 r
    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

    4 小时前
  • 签到天数: 3226 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,
    : B6 f9 k# F9 l我推理机的核心算法应该是二叉树遍历的变种。
    - f( K+ D1 t& H# r: ]- f9 i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:
    ( h9 k# u6 Y% X% t' ]; iKey Idea of Recursion
    ' V; Z, h- Q6 p4 I  H" A9 p' k1 H* P# w( p
    A recursive function solves a problem by:
    ( e0 r: [5 ~: B/ `+ U
    ! f3 T7 D2 U2 s, `# t    Breaking the problem into smaller instances of the same problem.
    0 l+ H. ^/ F7 Y( F/ `  n; D" a% N# o# y4 |- Z
        Solving the smallest instance directly (base case).2 e- r* F2 v+ b+ x* x1 S

    2 o' m# W$ }9 O! a7 X/ K4 j    Combining the results of smaller instances to solve the larger problem.
    6 ?. a# D* C( ]$ k( y
    ' H' L. O3 h9 W1 U4 ], k1 b: iComponents of a Recursive Function$ P0 ]: H/ a) t  {5 A* v% j* t

    : G3 v' Z6 _0 J. J  }    Base Case:
    * y6 ~/ H; _5 w; q( {/ Q7 T
    % T' r5 q5 J5 U" Q( z" [' q, J        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
      y& Y( K  h; X8 q% Z+ p! I/ _' v  d6 z0 T) b$ I
            It acts as the stopping condition to prevent infinite recursion.5 L( f( e1 h1 X; M$ M

    / d& @" D( T% S, g        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    & p* b8 ^- E# g1 G6 _2 D  Y, a9 k- f
        Recursive Case:+ {9 ~. [6 T) i& |
    : D& j# l6 A9 ]; ^& F
            This is where the function calls itself with a smaller or simpler version of the problem.+ q# m, m; s) c3 V4 f

    * [6 l$ \# A5 f0 O        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    - k6 ?& m5 s2 N  A3 N  i4 z
    8 f) x+ ^2 ^, Z8 \$ UExample: Factorial Calculation
    4 J. [! m; i; @, J8 h8 N* \0 O8 w0 {$ U1 |( J
    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:
    ) W% N7 E. F" P: O
    ' n- a; Y/ D$ ], B4 w# Q# h    Base case: 0! = 1
      r% g( W9 }9 a* `% _! p
    & e) A: f- t4 V* [$ {$ B) l    Recursive case: n! = n * (n-1)!
    * b0 \6 J# o$ T! {/ c) y2 R' O4 \" n3 C- [/ N! B
    Here’s how it looks in code (Python):5 a( e2 `% W' t+ r+ x  N4 J
    python
      @' W8 T* B' @. U: S
    9 ^' L0 H' V/ o! M# s1 e$ ~
    . `0 m/ r' U9 k; T2 ^def factorial(n):
    5 Y% b& r7 I5 U. y! C    # Base case
    % U) i0 l9 Q% F# v8 i0 a- N    if n == 0:) X/ b0 q: N" I  ~
            return 1
    . V5 U( C3 @2 U1 h, ^, M& Z: v+ e    # Recursive case
    9 [6 d% v1 U, X1 [/ }$ _    else:
    - k6 \* Y0 C: g& R* q$ |        return n * factorial(n - 1)9 M- ^5 M8 s# e+ [/ z. r
    ; T. w, y5 N7 h; F
    # Example usage
    % x) q& a% n; ~; O6 dprint(factorial(5))  # Output: 120
    - P% ?; ?0 M2 v
    & R* E  x3 H- b: N- u0 I6 f3 AHow Recursion Works
    7 Z$ a% e- ^, r$ B) b* }( N  p0 Z/ i& r
        The function keeps calling itself with smaller inputs until it reaches the base case.
    6 d- x) e8 f; A( B# ^8 y( l+ e- g; E( W' ]1 c4 p, {
        Once the base case is reached, the function starts returning values back up the call stack.: D( {# D  Q! j9 _6 z' Q- o% o
    3 s& e/ x3 o8 ^' C
        These returned values are combined to produce the final result.* N. X( m6 W$ C! ~

    ; J! b% F; O  e3 x- j; y% g' O1 a6 YFor factorial(5):
    4 G. ?- F+ ~5 f3 L
    , {; B) }* d7 n6 n: q" l$ g8 W1 ~+ o( J0 q! M2 D
    factorial(5) = 5 * factorial(4)
    ' ~$ s# `4 i; T% C2 w. F5 p; @factorial(4) = 4 * factorial(3)
    7 Z/ y' G4 P( V& _6 [3 {: gfactorial(3) = 3 * factorial(2)
    , y2 Z4 O, a# X. K2 j+ s0 H. B0 efactorial(2) = 2 * factorial(1). B+ g8 r9 v$ E0 S& y
    factorial(1) = 1 * factorial(0)3 l- z6 R  _% H7 H, p; E
    factorial(0) = 1  # Base case5 x3 B( a8 t! j

    . P8 @- ^, o- xThen, the results are combined:4 o1 ^# H6 u7 B  w* t1 Z- _. l

    - y, |) a3 `, T( C
    8 n: B! i' [2 e! G& \% ~- t/ Kfactorial(1) = 1 * 1 = 1
      @6 ]% l2 V7 J  E: {factorial(2) = 2 * 1 = 2$ N6 D3 ^) z  d- L8 v+ r2 Y
    factorial(3) = 3 * 2 = 6
    $ o2 A: _. `( d& H$ Qfactorial(4) = 4 * 6 = 24
    - t% K5 m- b: A: z% I$ Cfactorial(5) = 5 * 24 = 120' N' |' L1 Y( ]2 ~; f$ }* K! C

    6 `) x1 n# u' n  P% XAdvantages of Recursion
    ; W! U  B3 O1 U" t+ e# R  H4 u2 k/ r* v9 N7 }+ k3 \$ |
        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).' s9 O! l  V$ c% k% U

    : g3 E  h* s) z7 i8 s- `    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    / F3 t5 g0 m8 v  E6 A& z0 \' L2 B$ ~) d1 `) ?) k
    Disadvantages of Recursion6 Q2 ^7 I) F3 X7 D9 l: L' A% j  g

    4 K7 _* n, I" t    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.+ {4 F7 l& d( s+ ?8 R$ j, U

    % `+ c% h3 z& D9 j3 u    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).2 @4 K; b* D) q% X1 l2 B. u0 T
    , q6 g3 Z. h- Y, H$ t; c
    When to Use Recursion& Q- w' N4 F* |% P% c) I6 |3 T+ D% q

    # Y$ M/ c1 x5 f4 }' I3 ~# p6 f    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).6 B6 U# e' t* Y, t5 b

    + t$ H" y7 o; {2 E2 s; G# j3 `" n5 \3 a    Problems with a clear base case and recursive case.
    6 r# W6 g' ^: l8 ?/ I8 R0 y# }6 H8 U9 U' K; x  D
    Example: Fibonacci Sequence
    ; I9 E+ K- f* _0 z+ J! ?6 Y  i
    % M  b6 w# ^1 ^8 K6 gThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    / J& v7 d/ p: Y( r3 V+ X+ e1 g- I) X/ p
        Base case: fib(0) = 0, fib(1) = 18 @0 b6 Q  N- f  J9 K

    1 @& w, H' `" q! c' v    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    + ~( F* n5 H9 a; f% a2 ?5 V: x2 M
    / M8 f9 I  |8 ?4 h7 R3 ?+ {python
    3 {8 E. J+ y9 T2 v8 ]6 O) U( ]6 R
    # F/ A. `. H) C+ [. Z4 U) j
    8 N: A: i7 A; g/ m7 Ndef fibonacci(n):2 Y. [. L( F  a6 z
        # Base cases
    ; V! J1 @$ H" e) }( b    if n == 0:8 z- c( W/ a4 c+ ~* O
            return 0+ S% v8 k% ?7 @8 b/ r+ H
        elif n == 1:6 t6 I5 A" ~+ ]( w- d+ E3 F
            return 1
    1 L" U' [: |& X4 l* x    # Recursive case
    " Q4 F( b0 g& c! c2 g4 Q    else:
    2 L* l6 N1 [6 h3 E7 u9 f7 M        return fibonacci(n - 1) + fibonacci(n - 2). e4 }  u/ k% }
    7 Q+ @/ o8 j' ]: Z2 f3 w6 @  H
    # Example usage
    * x2 s8 n5 R5 z& Kprint(fibonacci(6))  # Output: 81 r- ~# L- J3 D

    # N8 n3 s) c1 b: O: e" bTail Recursion: m) y) g6 f( |

    ( D0 _$ j6 d: j# \( oTail 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).
    0 ^0 @" i. [+ F+ @2 Z6 J- L6 x7 e8 n, d9 E2 u7 {' 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, 2026-4-29 19:36 , Processed in 0.059647 second(s), 19 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表