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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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    2025-9-8 05:08
  • 签到天数: 3 天

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     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 1 w- ~& ]) ]" f: |# u& E

    ) W* x; U7 T9 ]5 l- h解释的不错
    / L' _( M' b9 N/ P" ~1 F
    2 V  S8 s! U4 s- k6 p8 l4 f+ R: u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。
    % u# K7 O% L9 ~  X1 C2 @
    ! D1 O( C" z$ Y6 [ 关键要素
    * q/ @  t9 V: f7 r1. **基线条件(Base Case)**
    ( Z4 e' t$ y' N: J0 W3 \" R% a  A   - 递归终止的条件,防止无限循环
    4 V2 V# m- r' R. E8 T   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    - [5 ?" i2 K) P; d" T
    : h& n- j% M: M3 U. ~2 J* `3 R2. **递归条件(Recursive Case)**& Z( O) D# X* }8 h
       - 将原问题分解为更小的子问题* _2 o! i+ K) b8 ~/ w- m& E6 d. A& \
       - 例如:n! = n × (n-1)!
    # M- w+ Z  o+ x* W7 c6 v* \# G0 w  G* |
    经典示例:计算阶乘8 _6 ?; c( |: K3 W9 l  ^
    python- ~" X! M) [( S$ u" v) ?- T7 _
    def factorial(n):! ?# o) j, H* L, W$ K; E6 f1 q
        if n == 0:        # 基线条件
    1 U% }# u% s' ?% W% b        return 1
    3 K4 z* q/ S, A/ ^* t  x    else:             # 递归条件
    2 v5 Y1 l* N! ?" g        return n * factorial(n-1)* `- {7 W, J( V0 `! e8 j
    执行过程(以计算 3! 为例):8 U0 c$ r( ]0 D, G( [3 v3 F7 Y" P
    factorial(3)
    ! X2 k6 V/ r' O( V4 `3 * factorial(2)$ {$ _$ T: X; C$ n; {# E
    3 * (2 * factorial(1))" Y8 ]. O* a6 u  a( t
    3 * (2 * (1 * factorial(0)))$ J( x% _* |6 Y: W- t9 x1 {  x
    3 * (2 * (1 * 1)) = 6( O1 Y/ W, ?' J9 ]& m

    ' L$ }2 X  Z  z# Q5 C 递归思维要点2 S% _/ |7 G' q
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    8 G- s9 W3 a/ j/ h  ^9 t2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    4 t  ?$ v. d4 |, o: J# F3. **递推过程**:不断向下分解问题(递)
    * W/ y2 s) m8 ^& |% D4 T# |% o0 Y4. **回溯过程**:组合子问题结果返回(归)4 R0 A- f4 ?# U6 ?7 S4 a

    , _; o4 q/ w1 t, A) f" \* n9 } 注意事项
    ' R% m# a/ M/ W" s% K- F必须要有终止条件& L& v  U6 W& m' G
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层), _4 M" N7 _' T
    某些问题用递归更直观(如树遍历),但效率可能不如迭代/ r+ Q7 g; E$ f: @# V5 A/ U9 S
    尾递归优化可以提升效率(但Python不支持): [) z$ S2 z: f8 P
    ( z$ K% _0 C4 M/ [) G2 P
    递归 vs 迭代
    ! ~$ B  Y2 }6 C4 {7 z% N- S|          | 递归                          | 迭代               |8 n$ c; J1 f( _+ A. |: O
    |----------|-----------------------------|------------------|; |: j; ]$ T6 Z( L6 D) y
    | 实现方式    | 函数自调用                        | 循环结构            |. g/ w% c! T! Y
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |( |/ ~4 p' o# M( I
    | 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         |# j. Z, S+ M! ~2 x0 w& Q  {5 `+ f
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    . a! I* M) k* h, V' b
    1 E" z) ]$ r) D 经典递归应用场景
    ) O' f# f; \3 ~6 A1. 文件系统遍历(目录树结构)
    . Y& W1 M$ l& s- U/ w' T. z2. 快速排序/归并排序算法3 `& d. c5 @4 z
    3. 汉诺塔问题) C/ B9 a6 O7 W& x5 a  A" {
    4. 二叉树遍历(前序/中序/后序)
    + I+ q* ]& l- W, @% T$ A4 Q% [5. 生成所有可能的组合(回溯算法)
    * q  Z0 u9 w6 N/ ~& H- T4 X# T0 s5 q# d( z
    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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  • TA的每日心情
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    昨天 06:54
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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,6 J1 ^  i& N* A' q& b
    我推理机的核心算法应该是二叉树遍历的变种。( E& u: j0 t" 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。
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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:6 {: M+ B; x2 J$ n" G
    Key Idea of Recursion
    + |9 ^+ ~( h- W
    ! j& S  ]" g2 x. c) vA recursive function solves a problem by:
    " H4 _5 y6 O' j' C; J; e8 a
    5 F& @8 h. D( f1 p6 k    Breaking the problem into smaller instances of the same problem.
    / `8 k  p4 _- f+ C3 |) A& W
    * |' {( i3 @" n/ Q    Solving the smallest instance directly (base case).
    , O; L# @% ~, k& |* @' H" |3 y/ ?7 O5 b4 |
        Combining the results of smaller instances to solve the larger problem.
    + T. S, F( \* E4 A5 ]
    % l* B  ]) v/ ]/ U% W3 H# q. s5 X- tComponents of a Recursive Function
    / {7 F5 M4 `8 c3 ^" v' @3 Z. U
    * D% L  f' I6 `    Base Case:
    3 @1 K, F4 E! d
    ) X6 S8 `: j& [& E; t+ K        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.) q. N( E9 F8 E% P3 ~4 P1 R
    6 E* I0 Y! q9 y+ F$ W- d6 I
            It acts as the stopping condition to prevent infinite recursion.8 U8 ^' C* S, k

    " y- K6 g3 q" _2 t        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.: x0 a, b2 R4 H% `6 W2 d% x* L

    ) E" H. ^  P( u' v' Y# j    Recursive Case:
    # K; p# v: V1 e9 n; @
    1 w1 T/ W/ W; U1 h' z  n# ?        This is where the function calls itself with a smaller or simpler version of the problem.+ ?, `* E3 [; a
      w0 }; i, ]3 d: ^  I' ]7 o
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).3 k9 k) B/ y6 d4 C5 a( W5 _9 |& t) q6 l
    4 H& N& t; D' w
    Example: Factorial Calculation
    $ f) O* {4 k2 d) {$ ^& z0 h1 l$ T; C; Z
    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:& Q3 [! T; B: `" V- |

    1 O' p6 |6 H) h; T* Y- m# L+ k    Base case: 0! = 1, k9 A! C9 d, b4 R# y2 _
    2 B( B4 Z6 S& p: n
        Recursive case: n! = n * (n-1)!
    + R9 x* c/ N8 R/ c6 B, ]/ R/ T* J
    3 I+ u0 p( h9 E& _0 uHere’s how it looks in code (Python):# r1 f# I5 b, j! P- \
    python
    ( H) _8 {8 |" V7 @2 [8 y% J' y& I6 O% {; ]+ R, Y- h7 V
    , G% [! T  G( O3 T+ H( L- c' K# b4 w
    def factorial(n):
    / Z7 t; g/ _' A+ k5 I( K  K; Q    # Base case
    ! |3 r+ \0 j( i    if n == 0:
    & ]  B" D" P' S9 M  S% e1 H        return 1/ ?7 w5 b# z" h9 j% X2 i# e% O# o
        # Recursive case
    + K% `, h4 y6 _* J3 w    else:# v% S1 N+ b7 Q5 N4 L/ R9 V7 z
            return n * factorial(n - 1)$ g; a5 R; b) o5 L- b/ L6 \
    2 b7 q3 G5 k- r$ f& x
    # Example usage
    : E; t5 b5 k% b& w" ]' jprint(factorial(5))  # Output: 120  u8 h- P. @6 I! l, e+ V& f# ~

    5 [" O: p& d5 a' m( d  cHow Recursion Works
    2 Q/ A" _3 N/ `' K1 B1 W' |& O" i; v$ a8 F) L9 w2 i# ]
        The function keeps calling itself with smaller inputs until it reaches the base case.
    # T/ i2 v% N& D+ [1 r. k4 I, n
    . h) }8 V1 q$ K- q- l* v    Once the base case is reached, the function starts returning values back up the call stack.% S7 b% N) T$ q$ I, I5 X2 ^7 h2 @- [
    3 @8 B) c/ K  Z; V
        These returned values are combined to produce the final result.
    ( B. Z" P8 s$ F8 P% f7 k, C/ n- t# A3 ]6 P
    For factorial(5):
    % N6 z. u2 q  I/ R7 b: [# J6 u) e; O& y

    ' |0 o0 k+ I) |- w! ]factorial(5) = 5 * factorial(4)
    " z! E3 k5 f  N+ ]factorial(4) = 4 * factorial(3)0 b/ y, N- R5 ~3 p) A$ Z
    factorial(3) = 3 * factorial(2)
    . t! Q7 T! a; F0 {2 X7 Y* b* O% ffactorial(2) = 2 * factorial(1)
    * {1 |6 S5 U/ B! `factorial(1) = 1 * factorial(0); k8 ^& s8 H( t. R/ P* \8 u) _- V
    factorial(0) = 1  # Base case1 X0 E2 ]) e  k  v1 I( |. x$ z

    2 Z  d( g1 g2 a5 D8 \Then, the results are combined:
    ! P+ k0 v# X2 E4 Q; m  a5 s+ R1 k2 G' R& B+ S  u, s) i

    $ I' S6 S0 F7 L7 x) _& v' O6 Pfactorial(1) = 1 * 1 = 1
    ( A' b" V; X, \. N$ s0 Sfactorial(2) = 2 * 1 = 2
    3 K' i5 F( U; Q3 d% ffactorial(3) = 3 * 2 = 6
    ! s& Y7 s! w0 T5 \7 e) a9 h! mfactorial(4) = 4 * 6 = 24
    / E8 n$ X! M* W5 m' U. ^9 Ufactorial(5) = 5 * 24 = 120# H5 C' `5 V& ]% }2 V

    1 a- i& h6 n3 i  R2 m" h! aAdvantages of Recursion  R' g; [- w& z" i; H) k4 _" p

    # l- n8 ?' t: D$ N! y6 z% 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).
    + q; e; x3 o, ^. Y" x# m9 d& P' s+ S* L2 ]1 f& u' o. ~5 g
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    & k$ b$ O6 e  k' x4 f" Q6 U# T- J; t" ~7 E
    ' Q" K. {4 q3 X% p' G" \# X' JDisadvantages of Recursion& E' N6 W6 K5 O# C1 L6 a. c

    8 G9 a! Y/ q% A, W    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.
    : p5 i' F+ C0 g9 d: B5 K7 |" P+ |# L7 S; u9 G
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ! C( _' c9 Y7 ?0 ~
    ; @1 o+ W2 ^. QWhen to Use Recursion
    * E! h& m8 L8 i0 I
    , Y+ z- r: i8 N4 U8 j2 L) z! Q    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    1 l5 u+ s8 w( q1 P& u/ y! X, q
    $ E9 n  X  H4 T4 \5 E* m    Problems with a clear base case and recursive case.! l+ _$ f3 a' d' V/ U
    8 F) ]: N% |: ^( Z) u( m
    Example: Fibonacci Sequence" N$ i0 |$ p5 i3 j* ^! p) O

    % y( V& q, w( e9 R1 ZThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ( b- w  f9 V" T. A9 N* ~! |
    + }: u+ R' l0 r% r& A- D9 ~    Base case: fib(0) = 0, fib(1) = 1+ E6 r; L& R5 |8 F4 f# R4 r$ \6 F
    # }# ~  f9 X. M, s- m0 v' \: a
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    + e5 p; H! Z) Y
    7 e. \' ?, s  rpython
    ' u0 Y' G9 x# {8 c
    3 H- B' T  f9 ]9 h! C. _% |9 ?
    " B, V& l/ l' f* r( wdef fibonacci(n):0 L0 k3 @6 F& J* b- D
        # Base cases
    : H, `0 B: t$ P- t+ E; O7 r& }6 p- _    if n == 0:( o. f( u! ^$ B# ]4 [
            return 0
    7 W/ Y/ f2 V/ J) r7 a' @    elif n == 1:. A9 u, F2 c/ p& [5 M' p' W
            return 1, H7 f- \% W% L% @' _( B
        # Recursive case* g5 G3 u+ J9 \. Z2 \" g
        else:# b- G2 s" t$ _' y8 ]% ]
            return fibonacci(n - 1) + fibonacci(n - 2)( J% P0 {* z5 U

    " j) W  ?8 L- m' j# Example usage3 w* x, O% W; [# X) s
    print(fibonacci(6))  # Output: 8
    # m6 B& u+ z7 z, E* _* J& e0 t
    , s% A; X  N: I0 P7 S6 }- \Tail Recursion7 L4 o) X8 q4 Z+ x& D) H
    , z, Z+ Y' b# X/ l8 k1 ]3 O% S: X: [
    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).
      l: H4 {: V; I3 a; L9 R8 `, a9 ]  |6 {3 P4 C3 }/ @
    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.
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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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