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[科技前沿] 突然想到让deepseek来解释一下递归

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  • TA的每日心情
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    2015-11-30 11:11
  • 签到天数: 2 天

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    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 / J- r2 w( A# u2 s7 S

    ( g- a% G9 d# j+ N8 ~( _解释的不错3 \3 v3 C3 |: @
    ' d- H& n  {% T" |7 D2 Y
    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。+ N/ h5 ~2 C$ N& I0 c- D# s# a) s

    - `/ R. [, M& q 关键要素1 \5 h) M8 J8 \6 \+ L; T6 R
    1. **基线条件(Base Case)**
    & t- Q4 a, Y5 B9 c( g$ U+ K& Q$ M3 B   - 递归终止的条件,防止无限循环: D! Z6 b- I: D% e
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: X$ e8 M, q: S8 ~! [$ N5 {

    & k  V2 J* f8 m$ `: I4 k2. **递归条件(Recursive Case)**
    - |5 v9 i1 f% s/ Y4 e& v& Q   - 将原问题分解为更小的子问题3 _4 T9 \4 ]6 d7 `
       - 例如:n! = n × (n-1)!
    6 l3 u+ F6 O8 s$ O' W/ b! k5 x2 [- p( m4 T8 C
    经典示例:计算阶乘# T' i, e! `  W' t: x9 D! l% w0 Y  r
    python
    : F0 i+ }3 k- u/ G9 m/ {def factorial(n):6 [8 U" N: _* a& |+ b
        if n == 0:        # 基线条件' r/ ]  [. ]9 K2 P; z$ }
            return 1
    # H, i& g! E* I+ u    else:             # 递归条件
    ) G& f( C4 Q1 w        return n * factorial(n-1)
    + g7 N% O- Z! ]# f" b+ f7 a8 Z执行过程(以计算 3! 为例):
    5 \5 s4 Z9 ^; ffactorial(3)- G8 f3 E& G* t: Z5 k2 Y, B: p
    3 * factorial(2)
    ) d0 K% D1 D; {3 l8 [+ U9 u! H3 * (2 * factorial(1))! ^7 r# D( J/ J7 x# f
    3 * (2 * (1 * factorial(0)))4 N+ W4 N" Z8 X* x9 g. F; p7 o
    3 * (2 * (1 * 1)) = 67 K2 p0 ]5 v& O0 A- K% {9 A) A9 H

    3 M4 W3 U3 L+ J 递归思维要点$ f7 s8 X4 @, N. v3 L: @
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑+ b+ ?0 @6 U$ I2 P1 e& W$ L3 F0 Y8 B- {
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' J: j+ i7 Z/ z* E3 [- [
    3. **递推过程**:不断向下分解问题(递)
    " W7 F5 O( g1 r( E) J, N3 H3 C4. **回溯过程**:组合子问题结果返回(归)
    2 n5 q3 c. h2 ]. O7 k2 I6 N/ }/ z1 ?9 J2 }9 m: _, ?
    注意事项
    6 o' S  o7 U1 P2 \$ ~6 U: i! T必须要有终止条件$ T( M8 O3 I; r0 L* z) @
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)2 j5 g  {; ^. ^, ?1 Q; ?
    某些问题用递归更直观(如树遍历),但效率可能不如迭代/ p) E2 b# O: w4 n2 `
    尾递归优化可以提升效率(但Python不支持)
    . }: _) a# x+ y; \4 p
    & j6 o% b; J1 v& s5 G% O; w1 M 递归 vs 迭代! P: T% Y: r7 r6 r
    |          | 递归                          | 迭代               |: o0 K0 E" b  n. A' g" g
    |----------|-----------------------------|------------------|9 @8 S  u* t, q" w9 A
    | 实现方式    | 函数自调用                        | 循环结构            |0 M" b% u7 B% ^3 I- j5 S
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |6 d2 Y- P' Y8 C+ O
    | 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         |
    * ^- N8 P' p1 b% ^4 y| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |4 Z% L. z9 @5 G; i  {2 e! V
    - t9 y$ C" @# W! l
    经典递归应用场景" b. i  p1 ^, J( n
    1. 文件系统遍历(目录树结构)% ~& l2 V# y- o: {
    2. 快速排序/归并排序算法
    1 T2 c4 E' {# p" H+ c3. 汉诺塔问题
    8 ^% C; w1 p% t8 V4. 二叉树遍历(前序/中序/后序), x' w9 s5 h5 ~5 f8 W0 Y
    5. 生成所有可能的组合(回溯算法), ]' R  H# y; B% v) _
    4 s6 {0 m" j& J1 X* v
    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。

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    郁闷
    7 小时前
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    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,
    , Y7 u* Y5 o# g7 }# k& [$ H我推理机的核心算法应该是二叉树遍历的变种。
    - `: u& c* Y- u& g3 @( u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。
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    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:4 Z4 ^; B/ V9 z/ W' ~8 J7 s
    Key Idea of Recursion% z- n4 w1 s% }, s; R

    - z/ F( q. S8 E" V' tA recursive function solves a problem by:
    ( S/ s& ^/ p! J2 H$ p" N! V" o: _7 {" g2 u6 a: E8 V
        Breaking the problem into smaller instances of the same problem.
    9 L. [% @6 X/ |3 I. d6 A& d
    8 s, r; n/ g5 E. W6 d8 O$ ~$ a    Solving the smallest instance directly (base case)., [; H# s6 y" @5 g& r
    - t) B6 ~1 B7 j! f
        Combining the results of smaller instances to solve the larger problem.
    6 J; B4 f2 G1 ~5 s' k$ O, i9 A9 K6 H, I$ B$ q
    Components of a Recursive Function
    $ c+ d1 E) O) q/ R. f; T- K, K( e7 g7 R
    ( o( v9 [) P: K; h7 T4 R    Base Case:
    1 Z6 j: R  B: W9 C7 i9 W
    5 L! |& J3 m7 y5 M# Z( X! |1 w        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.8 T! _1 G9 ^) P6 l

    8 t* M. [/ l2 l8 N8 O        It acts as the stopping condition to prevent infinite recursion.
    & E6 |. {! L* I, d4 ?1 w" [( X7 I7 f9 B5 j+ Q4 W' l& o
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.0 K4 f4 A- k5 @) t9 P* R
    : X( q% T5 \  e3 Z
        Recursive Case:
    / P0 I% u1 H' l5 O  }! g2 V3 T. P' t4 J
            This is where the function calls itself with a smaller or simpler version of the problem.
    5 l, J2 ~1 Z" s& {9 b# w+ V5 y! p  `$ g+ S  i
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)./ [* X5 p/ H) t9 k# l- {0 W
    " J5 l  L0 U1 W
    Example: Factorial Calculation
    . V' T7 k3 Z- n3 M1 I" G; ?! V# Y1 y# v: v  T7 ?5 H- E+ w9 i: P
    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:
    " B- b4 U% ?, ^# G; Z' Z
    ) U# F+ D' m, x4 [. w( i    Base case: 0! = 1
    3 X5 q) B7 X! l! U  k- c  C$ F
    7 r1 ?6 o( S, c5 J    Recursive case: n! = n * (n-1)!
    6 h$ G, U, j2 b. \8 h$ v" n& P# q5 i/ z9 `1 u6 X7 h
    Here’s how it looks in code (Python):# Y7 \  v% W& c: a
    python: l  I. e8 ]( R8 X  {( c, H
    ) c+ t6 r  k0 g8 K/ z
    ; N/ X7 }& m# U. M% k1 p
    def factorial(n):4 a, f* H/ C) H& t$ \, G
        # Base case' g' W, h4 Q" @( w- Z- R5 q$ x+ g
        if n == 0:* q4 G. Q2 |' \0 [, V# X; @
            return 10 _* M: o* B4 Y; {- B$ K: q
        # Recursive case& t! n( \$ x1 U8 e4 p9 T
        else:7 O+ _" d) F3 `" Q
            return n * factorial(n - 1)8 Q* E  \" ~5 [& R  t. n2 [; z
    : b( Z" }0 W) K* s5 D
    # Example usage2 {9 t0 \7 j) N7 A
    print(factorial(5))  # Output: 120% f% q+ J7 g1 b+ s; g; |. F

    $ l& s: Y. S5 h" ]1 JHow Recursion Works3 X% b3 r: r) i4 ]& z% v. G
    ; j- b! W' {; {' a# t3 l
        The function keeps calling itself with smaller inputs until it reaches the base case.
    : N" l/ e  _/ h4 a  \9 o, w9 u4 F  C! G
        Once the base case is reached, the function starts returning values back up the call stack.5 H( N) r5 [% V
    ; M. v" M# i/ o' t0 b
        These returned values are combined to produce the final result.  j8 p: A5 u, k( h+ _1 R& ^
    . U! @% r4 S3 f: J) e' H0 W
    For factorial(5):
    % v$ v! C2 w' Q2 p. `/ c) X  ]+ K% J  ^0 g! l& N/ ]

    0 T, y/ l' x) i2 {% o# |factorial(5) = 5 * factorial(4)2 C5 Z* {' A' v# b) Q/ r- h9 e
    factorial(4) = 4 * factorial(3)
      f7 a" ~% j8 ]8 ^# ?5 O: J' d. ^factorial(3) = 3 * factorial(2): l& [% V, h' p* p4 j& M) R2 w
    factorial(2) = 2 * factorial(1); _% Q' x) ^! c# N
    factorial(1) = 1 * factorial(0)9 R1 S0 ?: {, o
    factorial(0) = 1  # Base case- e. i* u" l, U" Z
    5 o9 k9 ~; K( O: W7 G
    Then, the results are combined:
    3 [3 X8 L+ U2 }- @4 B/ X( X5 h+ N0 H+ q
    * O! _4 F# B* H% a
    factorial(1) = 1 * 1 = 1" O& c* U' Z7 a4 i$ }
    factorial(2) = 2 * 1 = 2
    % }' X3 x/ T$ n2 Kfactorial(3) = 3 * 2 = 6
    9 m( e# M& P8 F1 v; V% e8 y+ @factorial(4) = 4 * 6 = 24- Y/ }# h6 G: N* e8 L$ ~8 P& J4 m2 d
    factorial(5) = 5 * 24 = 120
    8 |% C. e9 D1 Z. Y# i3 y3 `# ?$ |8 @; e" O! T# Q/ W) ^( [
    Advantages of Recursion  u5 \( @% |# h- E

    5 v) [' F  Q2 T9 H: r; g    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)./ X+ v' D( w' o5 s- y# S

    4 v! y5 F$ B  [; ^    Readability: Recursive code can be more readable and concise compared to iterative solutions.: j6 L9 g+ H% e/ T8 d
    8 n8 S& E3 E; T( y0 G  ~; R
    Disadvantages of Recursion, Z3 F( b* V3 w1 v

    4 a/ U4 o, e. K7 y; x1 e' 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.
    * B0 }' O$ B0 z8 }9 f/ v7 t' F  d$ f9 D
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 y" h- Y* U- e* l9 P! B7 L2 d( o# L! @. b6 w. g
    When to Use Recursion
    / {0 D; G7 c: Q: s% D( e& Y$ W6 F4 Q0 H5 o6 z- [
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).( A1 c; ^3 H- ]
    & N4 F7 R# ~9 v7 {
        Problems with a clear base case and recursive case.
    9 G0 ]1 b2 T& [0 h4 a- G4 Q1 A
    Example: Fibonacci Sequence7 u2 m; Q  I9 w

    . C2 N+ P5 V) Y; ^: B7 G& u/ {The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . r; i4 P$ J. m; E! j9 Z- X% U8 }' C! D
        Base case: fib(0) = 0, fib(1) = 1
    * l! W1 i4 Z" C' c* C, y0 d
    % J/ _: j( u6 u" Y& m    Recursive case: fib(n) = fib(n-1) + fib(n-2)/ y5 R' l3 t9 H- u% d' ?$ [9 Y
    8 N) B3 W, ?  w" l
    python
    , Q, G4 B3 h) u7 F4 |2 K/ `+ w1 l1 g' ]
    4 H/ K- j) l0 A! p* D  A1 F5 j$ b
    def fibonacci(n):5 _' w- q; q- k2 o) _6 ^" ^
        # Base cases- U6 I0 p, F4 T3 h
        if n == 0:$ N8 y, y* }$ d5 O* b3 X
            return 0. |. j% {# y; g) J
        elif n == 1:( ?: j" U3 w) j7 @* y+ Q6 `& q; a6 l5 H
            return 11 f5 t! V4 A% c& a* w
        # Recursive case
    $ \7 b# v/ @& R5 R: [  M    else:
    $ p/ T* c7 D4 p        return fibonacci(n - 1) + fibonacci(n - 2), w  c* Z8 G  H8 X

    * M9 A6 |8 q. {& y, l# Example usage
    ( @5 g% ^3 ^/ b- cprint(fibonacci(6))  # Output: 8+ `) J1 }$ }1 |) m: _! i
    + g! V1 I  H* @% t1 j1 e- X
    Tail Recursion
    + W6 J/ @% Z# \. s( }9 u  c7 ~* n  Q* L$ n2 P! {( \# J- z
    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)." ?  E! z! u# b( q- ?% x& S

    + R5 |, e+ a* |$ n3 bIn 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.
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    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。
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