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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
     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 + S/ i& l- ?6 m5 {; G9 `
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    解释的不错
    * r" w/ F8 w! o* [0 G0 C) Y; b; h, o" b% e+ ^( @. {, _
    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。6 J* o6 I2 i! A2 ~  ?, j
    3 z& X" o) |* E6 W7 G
    关键要素
    0 u8 P& G5 p( ^9 y9 G2 N) I1. **基线条件(Base Case)**
    0 U/ v& I5 d2 ]6 S   - 递归终止的条件,防止无限循环
    7 P+ v' g$ ]/ r( E+ T* O: E   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    : k: \+ c0 U, P' y
    & g7 J' ]2 X2 Z2. **递归条件(Recursive Case)**0 F( ]9 E! g) ~% R0 r- p. C
       - 将原问题分解为更小的子问题8 j: H0 s0 Z7 g. t; a
       - 例如:n! = n × (n-1)!
    / @  Q% ?: f/ v  x" f, d3 |; d" \# ?
    经典示例:计算阶乘7 H6 @6 }6 s* [  n% y+ `
    python5 Q0 a4 r8 X/ Z# A
    def factorial(n):; l, W; F; H5 l* z/ T- h, D" h. Y
        if n == 0:        # 基线条件& M- d- s1 H# I# I
            return 1$ D! E8 U+ N% D% M% q
        else:             # 递归条件
    . L# A4 m& x. a2 W, ?4 Y        return n * factorial(n-1)
    / _; z3 b4 J6 M执行过程(以计算 3! 为例):
    3 L6 ^7 W9 Y4 B& W* Jfactorial(3)  A+ F2 `# i9 w  k& K. n" G
    3 * factorial(2)
    , u8 {7 V$ |* B  F8 o3 * (2 * factorial(1))# o% u! l( \% X  F3 M
    3 * (2 * (1 * factorial(0)))
    ) {1 ]+ g& u9 _' C1 F* N% e3 * (2 * (1 * 1)) = 6
    1 ^8 ?/ v( B; d# ]% @% S) q) p4 \0 d4 i# k7 n; |( t" _% p2 ?, R
    递归思维要点; e7 @4 j- H" k
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑5 @, b3 R- e9 q& ~1 w
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间). v3 u/ d/ V, z( o
    3. **递推过程**:不断向下分解问题(递)3 v% [# y* A0 ^3 K
    4. **回溯过程**:组合子问题结果返回(归)
    , z9 G* m, }- E" y. A: s% t8 I0 ?* K2 n- b8 `& S
    注意事项
    + C. l" r# k7 V# ?' T) l必须要有终止条件
    ; j7 X: a$ R) b8 ]递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ k/ ~4 Y4 U; j; v4 }; m
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ! p, W, w* j; u. f7 e尾递归优化可以提升效率(但Python不支持)& Z% [! P' S& o' s0 X

    % g- `$ v( n4 y* @4 T 递归 vs 迭代* d" B) \. A  o
    |          | 递归                          | 迭代               |( _& v  ]' s+ f! ^9 j8 N( ^6 a( _
    |----------|-----------------------------|------------------|
    ! K7 e5 x  o! N0 ^+ W. T/ C| 实现方式    | 函数自调用                        | 循环结构            |
    1 M  ?: p1 S+ O1 c: ]| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |0 n% O1 |/ F) Y8 w! f& x& k1 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         |7 S9 \" n! N: n, v& `' `; C
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |2 w* I6 @6 v+ n

    ; ]4 w% V& w- ? 经典递归应用场景/ e3 O7 ~& l0 I1 ]7 w* c7 T% v
    1. 文件系统遍历(目录树结构)
    2 R2 H$ X, p& |5 x; t  x2. 快速排序/归并排序算法9 V2 C. }; i0 h
    3. 汉诺塔问题9 B: J7 N1 I* y& f. ]0 |7 i& a" ?$ I  w
    4. 二叉树遍历(前序/中序/后序)
    4 [: e6 ^; E" B8 n5 G* s  u& |5. 生成所有可能的组合(回溯算法)* u* [; W9 H1 T1 z/ C, ^
    ! p8 T; f) d) G  D4 k, f6 b
    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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    昨天 07:10
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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,1 s& d- K4 A8 _7 t% A( Y
    我推理机的核心算法应该是二叉树遍历的变种。' x/ G' G" s! V4 j! v
    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:
    5 j9 u; N, h+ iKey Idea of Recursion
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    A recursive function solves a problem by:7 }& l5 |: q& G2 C5 L" X9 P) a

    ; A' \' Y3 p' n) `; \  x/ Q    Breaking the problem into smaller instances of the same problem.$ A- }0 O; T2 F/ H! Y: {# a
    . P3 O) D- H6 C+ ~$ C+ x9 S
        Solving the smallest instance directly (base case).
    # O: A/ b( b- k" L& M# i  p/ T
    * q) h, r1 Z0 A( s( X* C    Combining the results of smaller instances to solve the larger problem.+ K0 S1 c9 ?( j2 {( Q

    : e/ w! z" n. d  i/ B' {* Z8 aComponents of a Recursive Function
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      |, m$ x' T( H5 j1 Y8 M- v    Base Case:5 T) h# t8 M6 H. g7 n

      [# i: E/ e( p. E, [2 k7 ^$ z        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. ^$ {! W0 `, i4 }7 K
    0 K( S% M6 a0 _
            It acts as the stopping condition to prevent infinite recursion.3 h* N: o4 I  K6 g, k

    7 Y2 W+ |* e! q: w0 M2 X& o        Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ ?+ @) g9 y/ m: w' C0 k

    3 f  g/ w7 Y1 d    Recursive Case:- i3 u0 ^+ ~8 d5 F& N

    " U0 w0 @3 G$ a2 H' W        This is where the function calls itself with a smaller or simpler version of the problem.
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    / m+ ]$ b4 F+ n& I        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).3 ?* l% T. R1 ^

    ! @0 Z9 H+ k" ~0 pExample: Factorial Calculation3 u/ D& y/ q! n

    . c  m; D8 M, b$ q3 S8 M+ VThe 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:
    & Z. N: B8 o" [! D
    ; d$ J) H# B. f% ]4 D4 C    Base case: 0! = 1; D$ W3 D) |2 a

    0 e; w) p$ Q/ k! c    Recursive case: n! = n * (n-1)!
    & f: g) I" `& p  O2 N/ k& l; J  S; I$ p
    Here’s how it looks in code (Python):( |- U. A0 Z2 \  E" K: ?+ Q; E
    python4 I& e! c4 H; _/ f% d: m5 V

    1 a" o) C8 b+ w( s* S* w
    + x1 q" {# q. w2 ?: jdef factorial(n):
    - U$ \% n/ J6 R/ U7 ?0 o    # Base case
    9 h5 }. F  }8 u/ c& [8 ~    if n == 0:& t, @1 ?% I" M8 }* P6 A
            return 1
    " S! m0 Q0 c" j' [9 a, f5 `    # Recursive case' d' D: ?5 ~3 A& y
        else:7 j# i0 N# x% V
            return n * factorial(n - 1)
    - P+ v+ A6 @1 m2 o" Q; e; a( r9 d8 O" c
    # Example usage7 g& m) v* U* t- R% ^* z
    print(factorial(5))  # Output: 120
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    How Recursion Works2 a1 F$ B6 H- _

    9 f5 e7 a9 x1 ^! p4 m+ C    The function keeps calling itself with smaller inputs until it reaches the base case.
    ! [6 Z/ N! R- ?! M4 ~* B" H  A# I' W  ~' J5 C% L3 x/ K
        Once the base case is reached, the function starts returning values back up the call stack.
    : M) Z: K1 b+ Z7 t7 ?9 ?
    7 c; I. `1 [4 B    These returned values are combined to produce the final result.$ N0 a3 v8 d! x1 r
    3 F3 X& y: T+ j: D8 i
    For factorial(5):
    # G, h$ D3 N1 M- N0 W' n: }  x% d+ Y8 K8 G
    . R* @. g2 s: b2 b1 t' x0 y
    factorial(5) = 5 * factorial(4)
    1 A7 m; I4 \1 `  Hfactorial(4) = 4 * factorial(3)
    1 ~& p- I8 Q5 Q8 Zfactorial(3) = 3 * factorial(2): C# x0 n; i! K5 i; y
    factorial(2) = 2 * factorial(1)
    , e6 M, c, f7 C$ x, U0 M% [! Efactorial(1) = 1 * factorial(0)
    7 E$ `  g2 @% z2 y+ ^0 E) S% P$ Ofactorial(0) = 1  # Base case8 d8 I/ K# c6 c5 y1 o
    4 y7 m$ \2 x" k, n  t3 m! \
    Then, the results are combined:' p4 N: [9 o: i- l1 n0 d# t
    7 \+ E( t: v  T0 P" Y

    0 i4 G/ f- E$ s( I5 p8 t, N( f5 {factorial(1) = 1 * 1 = 1
    + r5 N0 I! J% {4 Nfactorial(2) = 2 * 1 = 2
    % V# h0 x4 G* u0 `  bfactorial(3) = 3 * 2 = 69 E5 l- d9 |  q/ F) n  ]
    factorial(4) = 4 * 6 = 24: d$ m1 q, L& [3 Q+ Y/ i) j
    factorial(5) = 5 * 24 = 120
    ( K+ [- H; r9 W% y) w
    2 ]" `7 p( M$ j/ P9 z+ @Advantages of Recursion
    , @, C6 ]2 o  L
      s, F8 i9 i" j9 Z* K3 F0 R8 u    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).
    1 z1 t0 F- U5 ~9 P; @* k$ t( p/ r0 ~
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    2 i  m4 E! \1 E+ Z
    9 L' y4 w0 M6 b" f9 Y* C# Q" S1 IDisadvantages of Recursion; ?# [, v: C3 d. V

      R5 b. ]% h) c7 ~; p! k, b5 g+ d    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.9 s0 U9 s) K% [" B& H

    ! a9 L* o+ @/ [1 L- x% R0 K, c; F    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ; M! f2 U0 t( P6 p
    3 M! C6 t7 R( Z) u4 e; ?+ V" r8 HWhen to Use Recursion0 d$ v/ I- I8 A1 n8 S; y
    3 B1 o6 H5 C% P  k1 @) ~
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).+ ^) ?6 J! P4 M6 _! w0 V! Z$ `
    6 Y8 v* |8 Z5 @1 i! e  ?2 F
        Problems with a clear base case and recursive case.
    * e3 i: _4 @- t! Y: ^( c# u
    - e- r$ T; G) r. B- c" N- |Example: Fibonacci Sequence: A( x2 m8 b, L& _) }- x+ a/ I  M
    6 J& n  X9 b) Z1 S4 F0 p" |
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ' @* ~: o# \9 w! O0 h
    ! ?) E5 _* ~' [6 Y; |  C% q( g    Base case: fib(0) = 0, fib(1) = 10 z, u  P! q3 ]/ i. I2 q% s7 z
    * P# j8 F1 F2 h* r
        Recursive case: fib(n) = fib(n-1) + fib(n-2), [$ H: u( p, ]5 b" H* o
    % ?; P, O) S% p
    python
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    3 I' k3 j) P- o  G) `5 Y3 g
    def fibonacci(n):  t( o( y- J/ l* o8 w3 G, h' h
        # Base cases6 [+ Z* E# c, i% ]' i
        if n == 0:
    3 H/ _6 C+ }  x4 S( u% p8 B        return 0
    8 n" b! \# n& H! b9 e$ }    elif n == 1:
    ; M4 O( ^! S2 O0 C  m9 y6 C        return 1: C0 j4 v) o0 _5 H/ F4 D# H
        # Recursive case
    - Q# `3 Q5 o8 Q1 S9 D0 c    else:) o; Y. v) B! X8 v# q' w
            return fibonacci(n - 1) + fibonacci(n - 2)& m) c! S1 N6 @# u" s9 o3 w" s

    5 Y5 v1 X' ?4 R  \# Example usage( H$ M& d( [, z: g  `' G2 T$ w
    print(fibonacci(6))  # Output: 8
    ' y  N7 X& U: ?
    # v$ E! @: H" k% ]3 z, DTail Recursion
    8 H! |; f) ^9 S  |, }* r! [1 b& F. \) P
    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).5 _  W+ ~) d6 g, R

    ; P2 j* e# e: O: G1 m4 Q1 e  sIn 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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