设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2070|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 4 e8 p9 I- [# U9 H" f" ?: H; Z
    . ?0 W6 I4 T0 e7 d- B8 n1 ~- q
    解释的不错
    . |6 B% n9 g* y: T( u& u
    , y  ?* C: v1 O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。
    4 z+ n( D/ ?% f. v) ?" O9 y( j1 w# X% p" F. q) ?, ?6 t3 Q% k
    关键要素
    / H* c; e% m# P. s; w1. **基线条件(Base Case)**
    ; R) k  R9 s6 K4 ]# R   - 递归终止的条件,防止无限循环: J' h- D, u6 V2 j6 w3 F& X+ [
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 17 i; f9 V2 w" m( j& t
    1 B2 A8 M+ r# d( A3 w9 [) o" B! W
    2. **递归条件(Recursive Case)**! |9 s9 n% m# _1 E5 M* h
       - 将原问题分解为更小的子问题* X; r& V2 v; U, k1 v$ W1 w/ J
       - 例如:n! = n × (n-1)!
    - b; r7 z3 u, q) p/ ^& `3 e9 w# U# s- ], C' b
    经典示例:计算阶乘9 A2 b7 k/ O) N" k6 F
    python2 R% @+ u, f4 K1 d( }/ B' Q
    def factorial(n):
    % M; ~( R9 r( _7 _$ L& @" q    if n == 0:        # 基线条件. I" `) s/ r! V9 C
            return 1
    ; y& ?# Z, ?: o2 I9 v0 ~. |    else:             # 递归条件1 `3 i/ T1 ?  E
            return n * factorial(n-1)) b# T/ Q7 \. @- a
    执行过程(以计算 3! 为例):7 E2 I$ v4 v0 C# K7 O
    factorial(3)
    ( r4 f& v9 z* G1 b; c$ V$ c3 * factorial(2)# z2 T: w9 J7 v7 a8 ^
    3 * (2 * factorial(1))
    7 W3 @/ p6 o) E9 `7 q% X$ B3 * (2 * (1 * factorial(0))); o, ?) l1 F. G$ Y" f- S
    3 * (2 * (1 * 1)) = 66 j9 q% b7 {  x3 m3 i

    0 j" P/ a. x# U, ] 递归思维要点
    " O! k% r7 A9 h( X1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    6 E5 F1 s* x& [% ?/ k0 s1 A2. **栈结构**:每次调用都会创建新的栈帧(内存空间)! q1 S1 v7 r7 r! U/ r
    3. **递推过程**:不断向下分解问题(递)
    0 D; W; H. u2 y+ }; O+ {, \/ ^4. **回溯过程**:组合子问题结果返回(归)$ B0 p; c1 u3 e: u1 e, H7 M

    5 S& Z. q! V+ N- o7 K! h 注意事项
    1 V5 j1 Y, a3 W. X0 c1 C必须要有终止条件
    8 s& I4 W/ \7 L$ |# G7 w0 @递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    # t- Y) d( u( N  m! O8 ~某些问题用递归更直观(如树遍历),但效率可能不如迭代
    7 s- j; f& X1 |! z: b& J, e: m) r尾递归优化可以提升效率(但Python不支持)- v- n  S, k/ P# S& u0 d* j( p

    9 `4 i0 e2 h6 u2 k 递归 vs 迭代
    2 x: i3 N7 L! ]|          | 递归                          | 迭代               |
    * J6 i5 [/ \6 M4 x$ X4 a|----------|-----------------------------|------------------|$ \& A1 |. y, j/ y" i3 T9 P; `
    | 实现方式    | 函数自调用                        | 循环结构            |
    4 P, j' m  Z& y3 I| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    % f. Z' y' y: c: ~1 a% b  ]5 f| 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         |# t6 n- g, P/ M. }; ^) p, z/ N
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
      v4 o  @; Y* s- k0 s! U1 I
    7 J7 n6 l0 y9 D- T/ k% W3 A2 g/ m 经典递归应用场景' F4 @$ A# C; x- `4 h  s3 F$ U% W
    1. 文件系统遍历(目录树结构)! j+ |+ \1 D9 e' [& K9 n
    2. 快速排序/归并排序算法
    # J) c8 ^$ _8 Z# c! b4 Y2 a: ]3. 汉诺塔问题' b5 ~! z, ~" Z* M, f9 `4 [
    4. 二叉树遍历(前序/中序/后序)
    & {9 B9 ~* c0 f; j7 t2 n5. 生成所有可能的组合(回溯算法)
    ' H+ |% h6 i1 O% n: T$ u/ m
    " k% I3 P2 d8 G/ P: 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
    难过
    半小时前
  • 签到天数: 3171 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,5 I6 r9 [6 W/ G6 O* b
    我推理机的核心算法应该是二叉树遍历的变种。
    1 [& N5 t. F0 r2 ~* S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:7 e( {4 N+ F! A+ F/ A: }
    Key Idea of Recursion4 z8 H2 S+ R  L  @
    $ q. a& `$ z; D3 [. O
    A recursive function solves a problem by:* ?1 Z9 b9 Y7 c2 {  E

    + x7 z! _# x5 \    Breaking the problem into smaller instances of the same problem.; W0 z8 S6 j0 B% N, s' j7 ~
    - n' w) C& o4 ]; ]
        Solving the smallest instance directly (base case)." ?: P9 I( V, p3 x* ^7 S1 M
    , f  d1 T$ @5 F1 q7 ~$ @
        Combining the results of smaller instances to solve the larger problem.% E: m% a0 e* }- K

    0 ?0 h/ w$ a" ?2 {4 t: AComponents of a Recursive Function
    ; s# W2 l6 n/ E( N; D6 b. a, C8 m; B2 ~9 h, x3 g
        Base Case:7 u! [2 R! ^; w6 m4 V
    ( X+ }0 a1 I$ {0 g
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    $ x/ K2 W$ A+ q
    1 z5 ?* L, T* h+ q) o        It acts as the stopping condition to prevent infinite recursion.
    7 c) x0 _1 s- G; l; Y6 M
    3 X3 }- `# j) V9 n* s; t  F7 X$ v  n        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% d; `- r$ y0 m4 J# E
    7 \4 B: M! ^0 \7 J. @& C. s
        Recursive Case:
      `% C- Z7 u: I5 @' q
    5 X$ l1 J* |, T; ?3 G9 c        This is where the function calls itself with a smaller or simpler version of the problem.) ^' H! i' k. ]& i. E, k* t, v/ A

    ! F, n* U9 ^- V" D        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    " d7 \& j: S' `
    7 }% s0 z! c5 B' ~- ~Example: Factorial Calculation4 f+ Z6 [. ~1 V/ ^. M$ R9 D* ]
    0 g- f7 h: O8 v5 S2 d
    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:
    , A* v3 d$ D. p( E  R; o* c$ t! ~! g# ~
        Base case: 0! = 14 |1 g3 U7 K% b1 A* g; K& \

    # A/ N, C# [5 {9 I    Recursive case: n! = n * (n-1)!
    - k& O/ n8 _" Q) m
    " L, a; s9 ~- Q6 z% S: k. uHere’s how it looks in code (Python):1 R! E# u3 X: A$ c; c9 x; P  ~9 ^4 J
    python; ~+ H7 B4 k3 |: C/ P+ y. ~6 W$ e: e
    * m& m3 o6 f! h- N' h: a
    , Q: l9 F: r' R& J
    def factorial(n):
    * L" G9 `! A/ o# ^9 F5 c    # Base case
    , V( L5 Y0 y& Z6 H( J: F  j    if n == 0:6 f: s$ k' p, |! E6 Q3 W( W
            return 1; r) ]7 E; a, {* y1 D3 V8 Q
        # Recursive case; c7 \) ?& s# a. \; N  m  S
        else:
    ! _: J1 u; H- x6 i' I        return n * factorial(n - 1)) s$ |/ t5 |# o8 a

    7 v7 w4 ?% h: G! A# Example usage
    ( d- {$ g+ L. I# r( l3 Kprint(factorial(5))  # Output: 120: w  H, x3 M6 J2 p5 v

    0 a1 U( m  p5 F9 uHow Recursion Works4 J2 B1 e$ q2 D8 j% a' D) Z
    " D. f( H4 G1 d4 v! Q7 P, e4 |
        The function keeps calling itself with smaller inputs until it reaches the base case.
    " w, L4 u+ Y1 D; d. H; I2 E
    ) t3 @$ h' r: [% m; F7 i    Once the base case is reached, the function starts returning values back up the call stack.
    7 b3 U2 b4 D' w* B
    0 N* N5 l0 E* A: N    These returned values are combined to produce the final result.
    / q- U; X% A+ v# l3 T; i5 u& L
    8 @) F3 v$ _' O1 x7 {For factorial(5):3 O7 \6 Y' f. x" M/ E
    ' P9 W3 J% N0 }8 ~- l! E

      L1 M% O, _3 u2 Rfactorial(5) = 5 * factorial(4), |: u8 o0 [+ s4 h
    factorial(4) = 4 * factorial(3)5 f8 r/ [; y$ e) m; s* A
    factorial(3) = 3 * factorial(2)/ X+ @. k6 T' ^
    factorial(2) = 2 * factorial(1)% H3 e$ l9 ~+ u; u- c
    factorial(1) = 1 * factorial(0)3 Q& Y2 k0 X; p6 U1 q
    factorial(0) = 1  # Base case
    ! \1 W0 S& H6 d% N1 C! i
    * T- j6 a+ }6 b# \Then, the results are combined:8 A) C. j( l" s- t% k- ^. l  j# t! w  G
    / t9 Q' c* N$ x: {+ r5 m9 ~

    8 C: X% B, ~+ o, r) ^! o+ Sfactorial(1) = 1 * 1 = 1' w3 t! w4 |9 i) `( }
    factorial(2) = 2 * 1 = 2
    4 {6 ]# N. {2 q( h3 {factorial(3) = 3 * 2 = 6. J$ m6 q7 F0 B
    factorial(4) = 4 * 6 = 24
    5 S; M8 ~1 g% n6 F" x- L6 hfactorial(5) = 5 * 24 = 120
    , j: ?: J! `4 X7 M5 F* C8 s- L- s
    + M9 s% p( Q; l  zAdvantages of Recursion- m. F* ^& _4 A& X1 R7 S

    5 J5 N. j$ ^% N    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).
    3 U8 X& g1 v- q3 g* K# S% z+ J& m1 g& o5 v
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    - x$ F0 R$ o! f2 y
    % [" M0 `8 O* i" ^; NDisadvantages of Recursion2 t" y5 ?, }3 M4 y+ O

    8 j: h  V( h/ h" G    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.
    & @8 L  h( _0 p# m* g6 J: S9 l& @0 W6 N+ k$ h! X0 w$ g0 D1 |- P8 y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 \; I( Q  F( T6 h3 W1 q2 d3 X4 Z

    ; H0 r4 g. O5 g5 a$ dWhen to Use Recursion
      Y# j6 q: R% `* U* s+ i! A4 i
    ' z! E) J3 c* [7 |0 D    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    % m1 H; H9 q: h# Q) K4 y: b5 A. X# B" T6 X" y; }: {/ y6 T2 I
        Problems with a clear base case and recursive case.
    9 U( b5 Z; m! u% I1 T
    & M& K4 J9 T: s' p8 n5 }7 ]Example: Fibonacci Sequence: V* ?( z. t" O+ S  R
    , ]( g, f1 K4 @. _  d/ k
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    - @- i: N& k6 ~
    $ v9 i# T- j. t* L5 t0 q    Base case: fib(0) = 0, fib(1) = 1
    * ?" X2 B& O) b& R% L$ G' s1 v
        Recursive case: fib(n) = fib(n-1) + fib(n-2)0 j& f# D8 n2 `( l

    + y2 ~4 P/ ?9 W6 [: w& v- }- Ppython' ?- [* W$ K8 F/ H9 t

    8 I" T% x, a9 S
    ( ]- [8 d9 D& v/ S3 s2 cdef fibonacci(n):! n( `6 P: ^9 }0 v; X" ^
        # Base cases
    & _8 Z) j2 V9 d8 q    if n == 0:: N; c& C) H0 E( b$ g* y* |
            return 0
    8 [" ^! B) v2 t! g. t' y9 L- o    elif n == 1:7 M/ M2 r4 W$ T. Y1 e. z: e% ]
            return 1( t, P7 D. L2 n0 j; i# U7 i# i/ U8 D( P
        # Recursive case# H/ u. f9 L% |; R, N
        else:
    % o) ]* R+ }3 c* r$ f* O        return fibonacci(n - 1) + fibonacci(n - 2)
    $ ]# p. B3 H! ?- A) D# g- S
    5 T7 G; ~  }; y) `5 u# Example usage; r9 T7 y" L7 J) P8 B5 N
    print(fibonacci(6))  # Output: 8
    # Y, v0 ^2 o/ Z; `/ H' Z
    0 F) u7 u& k) m. STail Recursion3 n/ M$ Y1 S8 W0 ~5 O" r2 n
    / Q% q+ }/ y7 q2 H1 E6 O; k2 m8 B
    Tail 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).- S+ e; \/ U1 y  \  i

    ( u$ a9 i4 N5 {0 z$ j1 R9 eIn 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, 2026-2-12 08:51 , Processed in 0.058453 second(s), 20 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表