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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
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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
    6 @. t& l) N) `; G! x6 `: C! |" y( C! @6 w' _9 ]
    解释的不错
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    % g% n1 C8 y* J/ ^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。7 P1 y+ \5 X6 H3 D8 _
    2 b5 i/ A. D* N, _
    关键要素- Z) r+ C! i$ }' a- T) Z
    1. **基线条件(Base Case)**
    " b" \1 j- J5 y5 {- l; t   - 递归终止的条件,防止无限循环
    4 C9 m; l. y$ {, [! ~4 h* X7 n% {; e   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 13 G8 i/ B2 T$ m' T1 B

    $ T2 h0 N* i- N: n# ]2. **递归条件(Recursive Case)**
      z1 k# V4 m. @9 p" M! y   - 将原问题分解为更小的子问题
    " Y. H" N1 G+ G) Q# K   - 例如:n! = n × (n-1)!+ q1 D; D1 E& C) h# V( t
    & j! m  E7 h( L. O
    经典示例:计算阶乘
    6 k0 W9 ~) P' N$ l3 t1 dpython# [; _" o( X4 G- I3 v& Y* A$ e
    def factorial(n):& }! T2 Y* A- o, N" b4 c$ f" V
        if n == 0:        # 基线条件
    - ?3 L& I1 [0 \; T" Y        return 13 K6 k) k3 D) L/ p; k( T
        else:             # 递归条件  _( w% t, ^: V9 ?; P
            return n * factorial(n-1)
    9 O0 m- L! v% U! X  E. A执行过程(以计算 3! 为例):0 H, I! [4 j, L* n* G
    factorial(3)
    * g' c- \7 P/ }7 E* C3 U3 * factorial(2)9 F! Z/ F0 G1 u6 q, y6 o/ l
    3 * (2 * factorial(1))" O$ y2 ?! A% }. U) n
    3 * (2 * (1 * factorial(0)))6 B  V, K" b& t0 u, ]' A. z  O
    3 * (2 * (1 * 1)) = 6: `4 d; j0 i1 t& s% F$ K5 Q' X& e

    - v9 `9 z' {+ @ 递归思维要点1 V9 N) b6 W) ~2 Y! o
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , D& k) a' }7 S. t# w2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( y5 Q4 r5 K8 C$ }4 x( j
    3. **递推过程**:不断向下分解问题(递)- b& J! Q6 g9 I- O1 A
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项/ i+ L0 ?5 O! O4 C+ {! k
    必须要有终止条件9 C# Q+ e7 Z. w# H
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& w4 q) A8 r3 B: K, h
    某些问题用递归更直观(如树遍历),但效率可能不如迭代! f) r% N. G/ \
    尾递归优化可以提升效率(但Python不支持)' v- \( l- e# ^) i0 J8 b
    ; k7 {& O4 a8 B. t- M2 h: }7 X
    递归 vs 迭代
    2 s! I) k, i0 l1 a7 n8 P+ R  K|          | 递归                          | 迭代               |
    / G0 s( N3 |/ j1 M1 l8 s|----------|-----------------------------|------------------|
      K- E; A  J. \% d7 ~! t| 实现方式    | 函数自调用                        | 循环结构            |
    $ `4 s" d- c2 L, I' _% J| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    . ~% [8 z" U+ H2 o1 ~0 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         |
    5 G7 o+ M5 k  a0 M| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |0 W; Q( U0 _+ ~5 L2 _% P) r
    ! W) v- @$ o7 l0 G% Z
    经典递归应用场景2 ]: l- e+ Q1 A, \4 f# ?
    1. 文件系统遍历(目录树结构)5 V3 I) |3 r  c. P
    2. 快速排序/归并排序算法
    * \! F3 V  z2 r; ]3 R3. 汉诺塔问题
    ; R$ w; o( J. H+ H8 q4. 二叉树遍历(前序/中序/后序)
    / m  U) V8 @  {! K5 ]2 J& b5. 生成所有可能的组合(回溯算法)+ c* g$ q& |: J( u6 b% r
      j+ G% X5 C+ b2 s- N- A
    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,4 U7 O! h6 d" j( {5 R, G  I
    我推理机的核心算法应该是二叉树遍历的变种。
    ; L6 K; Y' R: L" `. d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 M4 \& U! u1 q* n! l2 o3 Q
    Key Idea of Recursion) q8 Y: t. e$ n

    1 T1 \: d6 P% B( X# CA recursive function solves a problem by:
    0 F! P; z: d; J3 G' y- s
    % C5 p* w5 t' j9 b) A+ Y% C    Breaking the problem into smaller instances of the same problem.: H9 _/ z$ |2 ]6 ^2 F) J
    7 S& {2 j4 ]5 O- n. V
        Solving the smallest instance directly (base case).
    0 p  h9 @+ N9 _" U4 R8 d8 {# ]6 m
        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function/ g$ l8 k1 k3 ~
    , ?, {! p" Y7 e& _
        Base Case:
    3 x% C4 X6 Y: F" k% F; i+ H  H; _) j. j, u+ ^8 I
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.' n) ^4 m6 `( g* B4 g! Q* v

    6 q( @7 q$ T- }/ ^% F" U        It acts as the stopping condition to prevent infinite recursion.0 Z5 \1 `; ?, N, z/ _5 Q% C
    : ~2 T/ V: X% ~  x
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.* c4 ?$ y1 N% S7 ?+ O4 u1 A* W( a& |

    8 m2 z; W1 _" c( ?    Recursive Case:
    / W1 k; y0 x0 @9 N$ s, j2 B/ ~: f+ A% x
            This is where the function calls itself with a smaller or simpler version of the problem.5 K& L9 k2 S. l0 j
    ; D3 E( z6 b0 Z0 _* b5 M. s# Z8 o
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).9 ]; o$ L" I& m3 _4 s2 ^
    * o  L) w- S# O8 `
    Example: Factorial Calculation
    ; R8 p- T$ r* @1 I) P
    8 U) @$ A+ @' a; YThe 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:; \% ~. P# ?6 i! {4 Q

    ) h+ E2 V" I( R. L) X" Q7 V    Base case: 0! = 1
    0 d/ M! ~+ N1 B1 Q/ K8 e5 `7 W( R* X' q. q: u
        Recursive case: n! = n * (n-1)!
    % i* m+ @2 t7 J( x
    2 W3 A6 m& d; m+ `Here’s how it looks in code (Python):
    9 g5 f' }: f' t5 g/ y: f& h  i1 Bpython
    $ P( ?$ V- m# K' j/ ^/ N; J  p. d, C! }% \5 f

    5 g9 W$ C% ?, [3 {: g; T/ W( t' t' }def factorial(n):' J4 i! b. w: x2 W6 n' R
        # Base case
    ( D% f0 P3 ~) e  `- a- P    if n == 0:. i- |" L- }8 [3 ?$ @( g" \. U& O
            return 1
    ' e% E' B5 R& ^1 n/ j! g( s    # Recursive case
    : A1 \- e7 J& L+ Z* |& c    else:7 [. g& U$ P, J+ J6 a6 y  r
            return n * factorial(n - 1)
      ?- h7 Q  h6 x1 c, C
    0 h8 \6 g+ @/ R0 L! Y7 V' U# Example usage
    9 {( F+ J  L( r& @4 Eprint(factorial(5))  # Output: 120
    # }9 w5 U& y7 C6 y3 N5 A2 U, ?; x4 F+ ^" @1 X9 h) R/ {
    How Recursion Works" W* g1 C* t/ k/ c
    4 P6 ^0 K9 n. o* g  @( N3 J/ P! y1 U
        The function keeps calling itself with smaller inputs until it reaches the base case.8 p$ J5 ~) |: g% \0 x+ I* K

    % p% N# w/ a' W7 J3 j: X  a    Once the base case is reached, the function starts returning values back up the call stack.2 V/ b/ x! i9 [( J8 B2 D
    7 z/ g( D  A* B1 M6 T- b+ [
        These returned values are combined to produce the final result.
    5 V: ~% \0 j# l" T. m5 m
      n8 K  W( \& D) ~" Y  i' w7 |. ?For factorial(5):. p1 O; M6 s% y

      U8 \$ _9 b  |, v9 M+ x: Q+ A  o9 D; F; U3 `
    factorial(5) = 5 * factorial(4)$ |( F+ w: I8 y  a$ f
    factorial(4) = 4 * factorial(3)
    5 K8 s4 w- u: X+ m  n7 L' b1 F: Bfactorial(3) = 3 * factorial(2)" N1 ^. ^9 E! b5 B* Y. {7 U% m/ B
    factorial(2) = 2 * factorial(1)
    & p; O5 j) P% k; U+ V  g8 Kfactorial(1) = 1 * factorial(0)+ t% R' z8 z: Z/ w+ o' c' ~
    factorial(0) = 1  # Base case2 h) v: b& h% D% v/ h
    ; e9 L5 @9 g/ g: ]/ \  ~
    Then, the results are combined:
    4 M- H5 t# Q7 \4 N: S* R$ }; M4 |1 `% ^: S# P9 t

    - X/ i/ r7 [+ kfactorial(1) = 1 * 1 = 16 X0 n# a1 D( u! G( g  X
    factorial(2) = 2 * 1 = 2$ F* [0 K7 N" C2 c( W
    factorial(3) = 3 * 2 = 67 X% P  ]9 B2 l! z$ u
    factorial(4) = 4 * 6 = 24
    % N0 ?5 d- g6 `" \# I3 nfactorial(5) = 5 * 24 = 120
    , M7 |/ c$ M! c: F! Q
    8 ]' e# n+ N5 T. {3 YAdvantages of Recursion2 r; ]! C" P' A& \# ^' `
    ! B" Q/ b2 C1 L# l- ^# n# k: _
        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 |: g- l7 E. t4 M; \4 H3 q

    # [6 O" ^& ~' p- w/ g$ i    Readability: Recursive code can be more readable and concise compared to iterative solutions.0 W% H3 A2 c+ ?, d1 K; v  P- N7 ~
    ' f5 n' o$ |, z8 q6 R; R  i
    Disadvantages of Recursion
    # T6 A( Q$ ^; \% R! z& k
    8 }+ h9 O! F7 R$ ?+ Q2 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.0 z/ _5 e4 Y+ H' w/ l# z7 Z: S, n
    $ K, k, Q! o5 Q! O% Y4 ^8 ~
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
      P; U+ m+ ~$ S6 g( a: q6 W) |2 l: n# s7 m: x5 a' a
    When to Use Recursion
    8 w- ~0 @; L+ T/ K, `7 I& _3 B
    * C% j7 L  X* L9 q    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ! H8 t6 Y: k+ Y0 m0 R4 B
    - G  n- d% n1 }0 Y5 Y* t    Problems with a clear base case and recursive case.
    " l- e1 H1 k$ ?/ f1 h' Z3 l8 e/ J' l. |, z4 I, R5 b4 p- G! Y$ V$ q; V
    Example: Fibonacci Sequence
    ) c1 r4 N! l6 o1 R- \# N7 C) \4 `, P  G& y& `9 ]" o7 Q
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:- i" |( u6 r, s3 v& ]
    " G3 t- x0 q2 x' q3 J" _* j
        Base case: fib(0) = 0, fib(1) = 1
    8 U# P9 U" F& h+ N
    $ t' k. ^  J+ E' x/ j( e  H    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ; k1 _, a; u4 `* Y, r" z: W, n; @2 p7 {# e1 k! O
    python) q7 R/ T& z9 Z* ~
    6 {6 g7 k* ]1 a; j; E
    + z4 x, w3 r. N$ M, C
    def fibonacci(n):$ P* x2 h1 z7 o( N
        # Base cases
    5 R8 D4 _! @4 x( [$ r    if n == 0:$ z  z5 K( C* m6 n8 ?# _9 n! P6 M
            return 0! N, M/ d2 V2 n; b+ l
        elif n == 1:. A6 K% H4 ?4 J; L
            return 1- R  @# E" r9 H! x5 T2 P8 A
        # Recursive case
    2 m) w% ]6 J. [/ h, t    else:
    $ C* B0 W2 b6 m. d        return fibonacci(n - 1) + fibonacci(n - 2)6 K9 S  j" z* o  {
    ( R9 L  W1 n; l0 [# r$ _: a
    # Example usage5 T* l% [! l7 Z% J/ L8 I! u
    print(fibonacci(6))  # Output: 8
    6 d  q1 n& C$ Y8 R( t
    # R# E5 K. V5 G9 bTail Recursion4 H4 J6 \5 d7 A# p

    7 ?4 w. C$ O& u- XTail 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)./ z- ]6 f- C$ y# d, p& a8 V

    % u3 R9 S/ z9 F( lIn 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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