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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
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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 ( ^$ r* k/ p8 I' m* K" o, A8 `! I

    / u! Z3 i' k* i  s: b9 [; g8 ~解释的不错. \% j; O5 i% F; L; Z

    / O. F* x$ }& m8 Z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。& t  Z5 @5 N) N+ X
    8 g8 R0 |7 w# d; K% P. O
    关键要素
    3 ?5 |  [9 k4 A! L% B5 N. U, ]# B, k1. **基线条件(Base Case)**
    " A9 T: F* w9 p1 }7 k. T   - 递归终止的条件,防止无限循环
    : Q1 f* H$ r7 b. x" F   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    4 r& G  R1 \9 u1 I5 M
    9 I2 u0 f# F# W) j2 l6 o8 t# K2. **递归条件(Recursive Case)**! T/ q6 b/ B* M+ ?* r# _
       - 将原问题分解为更小的子问题0 R0 m' g% C% ^5 i) r
       - 例如:n! = n × (n-1)!" m! ^  x7 L, f7 a; H5 P
    ) Z8 e2 e5 l0 G+ Y' Q2 ~9 x
    经典示例:计算阶乘
    4 S4 }& }; U9 y9 t# rpython/ q& m8 o7 p0 D+ j
    def factorial(n):
    ; T5 N- Y# H9 w0 p6 g  T    if n == 0:        # 基线条件3 B3 N' l# d7 d9 Q1 s. {
            return 1
    ! f2 `, K$ @/ r; W+ W3 W, f# k    else:             # 递归条件
    ; g4 }% s1 ]: o  i        return n * factorial(n-1)7 p$ T( l3 G9 A* P  g( E
    执行过程(以计算 3! 为例):
    * I; b/ m) S- W! Z( Efactorial(3)
    + f2 c8 v& C# H9 O, M. f1 A3 * factorial(2)
    6 o9 g  V+ H7 |6 X3 * (2 * factorial(1))
    . ?; j6 B- [9 M1 z6 @3 h/ B7 ?3 * (2 * (1 * factorial(0)))& }7 N3 e1 g, m( j& z
    3 * (2 * (1 * 1)) = 6/ Y0 k/ N9 m) X' g+ d5 p3 e

      J9 U4 f; y' ]/ s/ \, k 递归思维要点
    ' k7 |. }% J. _0 T$ g9 k; ~+ F1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    1 @/ `5 s1 K. v& i/ i2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ; p+ _$ x. v% [2 y& j* E- X3. **递推过程**:不断向下分解问题(递)
    8 S0 a! J) b) X% |% `+ w( s! s8 U4. **回溯过程**:组合子问题结果返回(归)% k+ U5 O0 {: [
    ! a4 K5 b+ @, ]( }# G
    注意事项; C2 C* H3 n4 Q) r1 e2 D
    必须要有终止条件
    2 Z5 Z( Y/ m/ {3 b7 p, c递归深度过大可能导致栈溢出(Python默认递归深度约1000层)/ \! a) D3 `* `$ Z! S
    某些问题用递归更直观(如树遍历),但效率可能不如迭代1 H9 r( F% a( a0 D& v
    尾递归优化可以提升效率(但Python不支持)4 [/ N* x! W5 D5 }. z$ T

    ( j& z) R4 e% G! R8 s% e* W& Y 递归 vs 迭代
    7 E9 J0 z& v5 B2 ^|          | 递归                          | 迭代               |
    # v* M0 `6 E/ s9 ^9 |  J/ J, q4 Y3 g|----------|-----------------------------|------------------|& q; @, B" Q6 ?. r
    | 实现方式    | 函数自调用                        | 循环结构            |7 K, P1 D  X; m) K' ]8 x
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |2 g3 ^$ [% B" H, T
    | 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         |- H; S3 D# q( ?0 n* [; r( W
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |4 ^2 _# b- G- i# h  @/ o5 s
    0 K2 p3 Y5 v6 E5 S1 B
    经典递归应用场景
    & M7 p6 P8 _6 g  G- ^0 B5 R: c  A1. 文件系统遍历(目录树结构)
    2 \! H; `3 @  f5 a( z2. 快速排序/归并排序算法' O) q9 y8 e/ u+ `1 C! Y. ^
    3. 汉诺塔问题
    2 }( l% S! Z; J. u4 w) M3 }/ [4. 二叉树遍历(前序/中序/后序): x3 e) G) L% Z2 c( Y4 `7 {
    5. 生成所有可能的组合(回溯算法)# U: v  G' _0 S
    $ V3 X; D( G6 u8 G8 ~; Z3 w
    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 07:11
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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,- A& r' `; L6 Y: j7 {; V
    我推理机的核心算法应该是二叉树遍历的变种。8 F+ y& `( Z0 `
    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:
    1 X4 m- C0 D$ A# \" b, \/ lKey Idea of Recursion
    9 a$ v% R7 s3 k. d) j
    - N5 x# }5 }8 f+ B5 \6 VA recursive function solves a problem by:7 @  h( o, ?# F9 b

    0 K. q3 _! m4 Q' D! t    Breaking the problem into smaller instances of the same problem.& Z( Q' u- L; w+ B, }: F$ a

    ! c% R1 c- a/ |$ _4 N    Solving the smallest instance directly (base case).* n3 D6 ^, s6 O
    $ {* o% w: C9 J" S- Z
        Combining the results of smaller instances to solve the larger problem.
    % Q# ~) X; i, P4 G0 }7 i$ V9 S- u6 p& t& ~
    Components of a Recursive Function
    0 _0 o9 _& ~0 c+ F1 r, ^
    . q! W: S" f% y5 [, s    Base Case:
    ! D/ F! c1 y) t
    : S$ \* F/ R) c        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* k4 B- p9 N: F! X
    6 i  r* F% [. s7 ]1 N0 \0 e" A
            It acts as the stopping condition to prevent infinite recursion.6 k9 |/ t3 v" M8 k

    : Z' S# i- x2 o8 ~  E/ p        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.9 n: V* w( z. S2 W0 W

    3 R$ R( i# E' J% y7 X5 B    Recursive Case:
    7 E6 b4 ]+ J/ t7 ?8 w$ o  c3 r# j) r' E  A( x5 v" c# |$ D
            This is where the function calls itself with a smaller or simpler version of the problem.0 s& K% s; V& v# j4 H6 O

    # D: `9 K- c( R4 B. i        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ( B& c: I. F  Y! J! ~, a3 x( {$ @
    ( p' m9 ?  h' a: LExample: Factorial Calculation
    ! `, `2 ?1 f  d9 t! \8 j/ ^' u+ A5 q" |
    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:" c  P" J1 z4 j" `
    $ j+ F: j/ K- {" B) t& P: e0 i0 t7 O
        Base case: 0! = 1
    $ E8 G+ [+ ?; m; i5 ~) q
    ' I3 A& Y- t8 \8 D0 h# @% ?, Q+ D# s    Recursive case: n! = n * (n-1)!/ w1 _8 l8 I3 x% V

    # F- D, n. f- ?& O8 R8 p) X0 xHere’s how it looks in code (Python):
    2 a; [8 h  f1 Y8 V' t$ Upython- Y9 p5 ]( |$ _3 k

    0 T; o+ s2 L! C+ v/ S* }7 Z8 k2 J& j' g/ N2 ]. B3 x" p
    def factorial(n):
      u  T( K. @9 B: S' `    # Base case% D- R  o" L9 I$ e' t  h, H
        if n == 0:9 L2 V; N/ V7 [4 W' d: a: V
            return 15 `& ^( w6 I) s5 e4 _3 @
        # Recursive case
    ; G4 i  E  F/ O% g, v+ U/ y    else:9 z6 R9 q( j& w0 M" K+ C
            return n * factorial(n - 1), B( ~7 c: z+ ?' t6 O4 X2 \1 T

      O! ^4 S- T& x* {1 w7 b# Example usage
    4 b, I3 y& x" M& d* c- |print(factorial(5))  # Output: 1207 {) P% @* x$ X2 N6 Z

    ; {3 y$ J1 I- E+ NHow Recursion Works
    2 d. S4 m: r+ R9 S' G( c% O( V* O" ]3 s
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ' S( [* ?$ k4 h$ F0 n6 Z9 k8 Z( a2 {6 `# f0 a0 X4 H
        Once the base case is reached, the function starts returning values back up the call stack.
    , r3 j/ D& A, u/ }& `) p3 z! w/ O* \$ [: P, k3 g5 J0 e& Z
        These returned values are combined to produce the final result.
    9 S, B) e2 z/ j( ?3 U' }
    . Q9 B! W0 e% w0 NFor factorial(5):& j1 c- G) @( k$ a
    / ?8 O$ b% Y5 ~( ~% |
    / Y. q3 m2 W3 c$ z. g6 x
    factorial(5) = 5 * factorial(4)
    % I/ d7 Z( D9 B: s! J) W6 z% rfactorial(4) = 4 * factorial(3)( _" \% Y4 I" y2 o: p
    factorial(3) = 3 * factorial(2)
    6 N1 q& n. r& @factorial(2) = 2 * factorial(1). h. M& _+ e" g( ~5 R! k6 ~  L
    factorial(1) = 1 * factorial(0)9 x. ^6 q: u5 j; I
    factorial(0) = 1  # Base case
    # a$ q7 q5 O& n) G0 u4 K5 r0 c  J
    1 H$ P" |/ P4 e/ _Then, the results are combined:/ n+ Q7 ]0 L3 \: q( ~+ y5 G7 ?* x

      {5 G9 `. g& T( }2 D: A1 Q6 W8 |) h# }- I  Q' A( \8 D
    factorial(1) = 1 * 1 = 18 Q8 h) O% A( O* d- n  U: e+ L. G; a' h
    factorial(2) = 2 * 1 = 2! b  I! L; N0 S: H5 ^; n- E% p& o
    factorial(3) = 3 * 2 = 67 \4 l4 g% e- V# P
    factorial(4) = 4 * 6 = 24
    ' w: ~3 g8 L. T' R. o5 e' Gfactorial(5) = 5 * 24 = 120
    & \( P& C1 Y' T8 R
    2 E6 _  i( x# q, ?Advantages of Recursion
    8 e( L8 ^- v% Z: \" m' ]% l: d6 F* j. l$ O( G! `1 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).
    * `$ S" \: n3 F) D, v
    7 H3 e* a4 c) @- r; q* a* f    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    . \& e; q. ~' \5 D. O$ E& O1 G: ^9 I% f
    Disadvantages of Recursion
    / ^* U! t1 m5 ~* p' b
    , F4 T5 [4 Y' R( _- [- u5 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.1 \9 U; o4 J) I
    3 T. m4 r0 u: B5 Z. P- y5 Z3 w1 J" y9 y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)./ W- n; l7 p3 L5 n4 l" {8 Q
    ! H, I2 ?/ l7 m
    When to Use Recursion- ?9 @* L6 V4 w/ H+ p$ k: p

    ' M0 d. N! f% c+ k    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ( B+ Y; ^# {( @+ p+ {7 G3 z& A1 t9 T+ T+ l+ n
        Problems with a clear base case and recursive case.0 d3 a' ^4 \: \
    ) z/ k" h- h/ Y5 T+ X
    Example: Fibonacci Sequence
    7 c% x. x3 y6 X5 C+ X3 q, }' Z% K. X
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:* @3 H) c6 R" j1 y; Z

    & [8 b0 b/ l8 j! h% o    Base case: fib(0) = 0, fib(1) = 1% x8 ?8 n9 D. A8 i- ?6 v' ^4 l

    2 V+ |2 d2 t! V* K+ U    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    9 W- Y5 i% A, ~& {3 A4 l
    & W2 R9 y% f4 {7 ]# L% f3 tpython
    : X( p( K- E8 O2 X- s; g, m! a8 d. a- ]: a6 S, l( a9 g0 Z
    ; P9 ^/ C1 V" q2 ]# T' z3 S3 [
    def fibonacci(n):0 R( D' f$ _0 L/ K2 E
        # Base cases
    & X) z) b. [3 A* t    if n == 0:; h( ?+ C, i2 e- y
            return 0
    $ Y- _1 l( f5 j9 D9 i    elif n == 1:
    % W; [6 {0 t: D$ n9 f! u* y: o3 l) g        return 12 D+ s5 h' \) D, }9 V7 I. L! ~
        # Recursive case
    : b- a* |9 y& x- v    else:
    % v& `6 Q+ h( F* p        return fibonacci(n - 1) + fibonacci(n - 2)
    ; ~9 C5 m/ `3 c" U$ c
    ! ?) F8 v1 `( A0 r+ z1 z# Example usage, }4 z- ]: e2 p6 V1 r/ X3 ?2 N- Q
    print(fibonacci(6))  # Output: 8$ d" h( r  l: [0 J
    8 {7 \# K6 o! n/ l% G
    Tail Recursion+ \, `/ v. M& ^- P6 V8 W

    ) J& R, g1 B/ `' K, C7 CTail 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)., Q$ C0 L3 I! I0 e3 W# c1 M. V! h
    3 h- J4 k. {3 e# ~8 t
    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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