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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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    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 q& J3 K" u- W, A: ~

    " Y% _- B  F0 g0 S6 k解释的不错' Y1 g$ @8 A# X7 q4 A0 j/ }
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    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。
      U9 n9 l. {6 C' o: }
    ! t$ ]  W+ E' B5 J8 Y$ S' ~, Q 关键要素0 j/ b4 A9 P/ O  Z/ l$ k- }
    1. **基线条件(Base Case)**
    4 v$ \. j5 l4 J& o" N   - 递归终止的条件,防止无限循环7 r" h$ ~! `. }+ O5 ?5 g' E
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1! d2 r, x* g( q% D8 ^
    3 |+ C; d. p0 O  O9 p8 g
    2. **递归条件(Recursive Case)**) F- u; d0 ?6 }" j
       - 将原问题分解为更小的子问题
    - F: f6 I8 N' `" v; r! L   - 例如:n! = n × (n-1)!
    4 p3 z( V$ U  p$ ]2 m" C% I0 V+ P
    8 U+ W. x, [3 G1 L" @8 \/ n 经典示例:计算阶乘" {4 I2 m! g8 H9 a
    python( V# E; V7 N& g* _
    def factorial(n):/ w) l: Z; [* h/ G
        if n == 0:        # 基线条件$ {: V9 }! b2 N$ I/ S
            return 1$ Y0 {: |7 T* ~
        else:             # 递归条件
    % p0 }1 H4 l6 o! ^; l* _: Z3 ?        return n * factorial(n-1)* J# T0 c, {9 t$ k
    执行过程(以计算 3! 为例):
    , f; p# S3 {: X; ^; L. o( V( [factorial(3)
    8 @" C- O; Y! P  J3 * factorial(2)# ^" J% T. X- ?, `3 v( p
    3 * (2 * factorial(1))
    . `% N5 S5 R& x( U- T, H3 * (2 * (1 * factorial(0)))
    " B) p* R% u* V9 Q* l$ a3 * (2 * (1 * 1)) = 6! O/ q, c% `* \2 c# E3 |# u

    2 t$ a$ `. A; [! |+ @: t& P- x 递归思维要点5 e6 D; y2 ?: S
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 V3 u9 O+ o: N$ a8 N
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    $ @6 i: h2 l* k/ [, k8 F  T4 T3. **递推过程**:不断向下分解问题(递)
    % x& J' h2 _3 _4. **回溯过程**:组合子问题结果返回(归)5 Y* A7 `) l' @! t* L# w$ g
    + M5 u3 A8 ]5 W; g8 Q1 E0 B( u; q
    注意事项/ I; n" I& X% b; I9 d) S1 Y
    必须要有终止条件4 @! |  }7 U# H5 i# X9 v0 L# D
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    4 c$ k) z- _# k" y, @6 B8 U某些问题用递归更直观(如树遍历),但效率可能不如迭代
    3 L) r8 z" b2 a) f0 K* u: Y$ o6 q/ _尾递归优化可以提升效率(但Python不支持)
    - o6 _/ Z" V1 q
    2 o  ]; U  y; l 递归 vs 迭代
    1 V1 J3 D2 n7 W% {. [|          | 递归                          | 迭代               |
    ' U' M, s/ |. w: D3 R' E|----------|-----------------------------|------------------|& k2 _6 F; s2 f
    | 实现方式    | 函数自调用                        | 循环结构            |7 y5 k& Z6 z4 s8 y6 {
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 F: }6 l: b  G5 P
    | 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         |* v$ [, c. X3 b/ v, f
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |; y5 F: g9 M8 G% v: k+ W
    - V( N, X$ B* ]) U" E% e
    经典递归应用场景) j0 c- s) Y& l+ c  V
    1. 文件系统遍历(目录树结构)
    $ X- t' m  ]. b2. 快速排序/归并排序算法8 h6 `( J! j! v2 c
    3. 汉诺塔问题6 H0 w( @" q. M$ {6 O
    4. 二叉树遍历(前序/中序/后序): j! J0 E  D4 \2 a9 k" Y
    5. 生成所有可能的组合(回溯算法), {5 |! l1 g/ w' P0 [

    : \$ q4 C& {# l  E, e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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    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 f3 y  Q9 J. z( ?1 N4 L$ u
    我推理机的核心算法应该是二叉树遍历的变种。, f* `0 e# {6 D/ A8 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:
    # ]  |8 g% T+ M2 O( j+ [Key Idea of Recursion- T0 Y* S. |3 T
    * j/ P' R+ f0 v: g+ w& x
    A recursive function solves a problem by:5 ?+ j  \' l; z$ ^, l# j

    ' ^9 ^) `6 ?* V7 G  k9 P; I( r    Breaking the problem into smaller instances of the same problem.8 @- _5 V2 |* w  d

    " t! F% t5 Z) }! d4 P7 |" a; L: t    Solving the smallest instance directly (base case).# P- d0 |1 C/ b  \

    8 A' o; x) u: r4 a    Combining the results of smaller instances to solve the larger problem.' l+ U0 o8 N' x; F
    + D1 n8 B( Z+ V9 h* m( ]
    Components of a Recursive Function
    : J; n$ n' j  c: `( n/ l
    * E! Y+ _% X, w" L/ j    Base Case:. n; P. l7 Q! J' R; |" p4 f

    7 Q$ z/ c% Y( E  F' ?+ Q* O3 T' C        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    6 p# g% Y, t4 t3 x: e1 p3 |9 J$ j! W/ H- r; R+ A$ s7 X& I
            It acts as the stopping condition to prevent infinite recursion.
    # ~' @' ?8 i' d9 M" i  }" ^7 y! \0 ?" P  N8 Z( {- e
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    5 H& r# j' Y$ ~: X/ N3 R4 R) s    Recursive Case:
    " k# w- ^$ g  F! E( W( f! p; ?4 Z  ~4 X! ?* I
            This is where the function calls itself with a smaller or simpler version of the problem.8 _7 o; G* T5 ^( r0 {1 [

      Q! q0 M5 `, a7 }2 P5 W& k3 t9 {        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    0 z" _% _( F5 L/ ^& M, E  P; `
    , K* a$ r- C9 ^Example: Factorial Calculation% Y* M$ T; J5 I8 o
    8 T4 c5 M; `5 z. @6 X4 W) U
    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:
    ' d/ |( T0 J/ \
    / Q2 @7 h" Y; S* e' n    Base case: 0! = 1
    4 a0 X( Z0 l/ D  {. ?1 O3 o8 @& U3 S6 K" l" j
        Recursive case: n! = n * (n-1)!  }) d+ H: q4 c3 _3 S

    ' G% p$ D2 `  O7 E8 \; kHere’s how it looks in code (Python):1 T; M4 n2 {: N3 s
    python4 a! u3 i4 E8 u( d) D7 w2 b
    ! r$ \* r, A" _: c: \  z. ~
    9 h: i% _/ W. `  Q
    def factorial(n):  @, r  v" C3 j' E6 F9 m& ?
        # Base case
    3 v( J. t* J) p9 z! q) [    if n == 0:
    , h1 X% s1 e3 s! D        return 1
    / A" Z9 L5 y# z    # Recursive case
    - Q1 Z5 R) q' Q    else:
    4 o9 f) F3 x( e1 v0 k        return n * factorial(n - 1)
    ) b) K. O% g. h" j( ~" R( {8 z( y9 C8 S+ O; }. h3 x, T
    # Example usage) H9 [8 C6 B! @* G; }; Y
    print(factorial(5))  # Output: 120
    - w  w" u  t( m* _. n- U7 j# ?; O+ p1 w6 q: {3 N, U4 _4 W" E' J
    How Recursion Works
    " H; A. A- `8 W; s
      b! p+ J' k4 b1 W0 h7 u- ~4 f    The function keeps calling itself with smaller inputs until it reaches the base case.
    # z8 T% {9 I$ q6 h. @, S$ r" \
    % B+ q4 A, A7 i    Once the base case is reached, the function starts returning values back up the call stack.& [+ ?5 `) J+ t  @* G
      n9 p" l8 j0 K" E8 o
        These returned values are combined to produce the final result.
    8 o" G- j8 s% `  Q+ B2 P* ?$ \6 \' O% ^' z
    For factorial(5):
    9 s# ^' d/ j+ t4 b" t1 j  t4 S# G; [
    3 c7 e  r/ n; [% o% s1 ]9 t( C
    factorial(5) = 5 * factorial(4)
    % X' M' Z' a1 o& e3 v2 V" W( F" sfactorial(4) = 4 * factorial(3)/ {" N6 {* z) O8 K) w, w
    factorial(3) = 3 * factorial(2)1 X, G# h' A6 |- ~# K: ?8 {) P# D
    factorial(2) = 2 * factorial(1)% o) U1 ?3 o5 D0 g% ~' A8 i: ~" D( s
    factorial(1) = 1 * factorial(0)  s1 ]5 Z! G8 y- n, n3 I4 _& Y  [6 U
    factorial(0) = 1  # Base case- F7 g. f* I2 {7 X' ~( c2 Q7 a

    . T2 G) \7 Y7 xThen, the results are combined:- Z. l! p$ N: X3 A0 R+ i* N
    7 ]( g& v+ D: S* [% U2 X+ O
    7 y( p: w9 f$ q7 P. y
    factorial(1) = 1 * 1 = 1
    2 j) m8 I$ l! u1 o  Wfactorial(2) = 2 * 1 = 2/ j# x" [2 ?2 f3 k2 |7 d( ]7 N
    factorial(3) = 3 * 2 = 6
    7 i) N( n2 I- f* C' W# _factorial(4) = 4 * 6 = 24. J% v+ i# y* u2 K
    factorial(5) = 5 * 24 = 120: i* Y% L. F, _

    5 r. V$ V0 n. C6 \7 R9 i& {Advantages of Recursion  N5 X; a2 `+ M7 R9 ^

    ; s" }3 p8 D7 x4 t# `6 W    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)., P" O- ?. X9 x

    . X5 c1 K. R1 L' `3 ^    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ! L5 T1 A  _# X: ^  s7 A; r  }% `4 L  h
    Disadvantages of Recursion  o, B/ K! x( A, P- m

    5 \6 u# X: T1 B. |) j8 u    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 p2 ]- k, B% n7 @+ C" c& W
    # z/ z: L1 k* W7 Z  C
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).! P& I6 _' P4 T. l* c
    / W8 g# @6 w+ x4 I  z
    When to Use Recursion
    ; e5 [# k7 |( o7 m& O# Y7 ?3 t/ O* R1 N5 U( j' @% }! S! B
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).$ w' f5 J, J- S- B
    / l4 M1 _7 c6 Q/ J! i: v6 x
        Problems with a clear base case and recursive case.2 p! b& X7 Y* M1 \7 [
    + B. \; q" T2 V) S$ @. t8 v
    Example: Fibonacci Sequence
    , O) Q: L5 I9 O" M% a0 S* l4 a# N' ?) h2 l
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    7 f8 Y& B% d4 S# E! b5 s1 }
    2 G  y, o" n# o. M; ?7 \/ H# R6 }$ t    Base case: fib(0) = 0, fib(1) = 1; e: T; G0 r) \  k

    ; Q, {7 x1 r' B+ _    Recursive case: fib(n) = fib(n-1) + fib(n-2)! V( c6 a( l% q) r" C

    % B8 z4 R0 c# C; ?python
    ) _% U$ I2 }/ ]9 e& g( p! S( I7 w) w

    8 h- Z+ v! ]7 Y& B( p) U; Idef fibonacci(n):) W/ [. X: P( n2 ^1 [' N" ^
        # Base cases
    - N) y' Z4 z7 A1 @    if n == 0:9 S( j# ^1 m7 Y2 v4 c9 s1 v# f
            return 0) z* {0 D+ H0 H# {. D$ A8 `8 Y
        elif n == 1:
    ( N9 N) W0 X* }) _) n  i        return 1
    0 \/ E0 W" i( K' G* J1 M    # Recursive case
    4 o' I) k& ]( G+ o( B    else:6 |! D) Q+ m# S5 [$ s
            return fibonacci(n - 1) + fibonacci(n - 2)2 R: m& X# |: A4 l2 p$ b

    6 }$ o9 y6 ]3 s, F# Example usage
    ( M3 U8 X$ C- S# c1 tprint(fibonacci(6))  # Output: 8! X+ W! X* Z: |! r
    : i, J2 F3 d0 l: N. W% a
    Tail Recursion. g* n7 l; j/ q& i

    ! ^0 S" ?* g) U) g0 ATail 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).2 `* u% Z8 }; V" n+ \+ w

    - O5 d  h( n3 ]: d$ g7 S1 ~4 e5 c3 dIn 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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