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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
    ' I7 ~6 i7 j8 A  s/ J
    & t( ?1 {1 G0 ]解释的不错$ o" w0 p; S6 ?# \

    - D2 b7 @, h- W+ 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。6 i; N+ c) b5 T

    4 ?7 P' s2 `5 A' M 关键要素! w& w4 x: D3 L1 ^
    1. **基线条件(Base Case)**+ ^$ s( ^' W  y, [. h* A
       - 递归终止的条件,防止无限循环* z. i' J. R# `0 ?1 h' Q
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    , K$ _6 s1 k2 f* K* W
    + H; N* z" \! L7 E0 S+ ]2. **递归条件(Recursive Case)**
    5 e7 S: Y" a7 O* g% n# L- h8 q   - 将原问题分解为更小的子问题
    % A: O0 D: w: _; w8 t. X6 `8 }$ Q   - 例如:n! = n × (n-1)!" R4 B$ u0 Z: l; z/ T

    5 {& }6 {3 w2 E$ ^( N) e& w' U$ ? 经典示例:计算阶乘4 d' ]; A2 x6 S" W. \
    python
    ! [; Y! b5 L4 y+ Q* y) B9 Kdef factorial(n):
    9 J7 m& F' ]4 B$ y    if n == 0:        # 基线条件/ \9 t: L5 {, O$ n; Q$ C
            return 16 n8 N. F/ ~" L) M! m$ s' x
        else:             # 递归条件
    % j, N7 c, }/ E( e3 [7 _        return n * factorial(n-1)
    * e' o% v3 A0 W执行过程(以计算 3! 为例):3 h! Q( \$ I; L: Y  n8 d
    factorial(3)
    8 ~2 B4 J# f2 j1 `, \8 V: E1 ]3 * factorial(2)6 @: h, J! j- A  l  L1 }4 z( h
    3 * (2 * factorial(1))
    9 N: j2 v" h& s5 D3 * (2 * (1 * factorial(0)))
    % I+ [" }: v8 e2 w( g5 Y3 * (2 * (1 * 1)) = 6* K& l1 z5 b6 i
    ; k5 `! E" [7 D6 T; G) Y
    递归思维要点
    1 S' U1 U. o, U# I1. **信任递归**:假设子问题已经解决,专注当前层逻辑# M" o! D. g1 l$ H- W. v
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)1 R1 ~' r3 V, h5 e
    3. **递推过程**:不断向下分解问题(递); r' B! A& p& L0 N
    4. **回溯过程**:组合子问题结果返回(归)
    ; j6 r4 J8 t+ S' z
    + {7 O3 c5 b7 a& C5 [# ], X5 i3 x1 m 注意事项
    $ U; a0 f. D3 H: m( V' j& B必须要有终止条件
      _3 h( f( Q7 G& ]* y7 K递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ! X% R' u: M, l某些问题用递归更直观(如树遍历),但效率可能不如迭代5 s" i2 y+ m8 c7 w# X( i
    尾递归优化可以提升效率(但Python不支持)
    ; {7 K2 F$ [9 W: t  P
    ' G) G2 B5 L9 I, f1 K 递归 vs 迭代
    + F8 B2 Q7 F/ n* ?) L5 D|          | 递归                          | 迭代               |
    3 R/ y8 N& L: F, V* J5 j|----------|-----------------------------|------------------|
    % [5 Z& d" b$ f! k/ L  q8 w+ \; g| 实现方式    | 函数自调用                        | 循环结构            |
    : A- U. r4 ~3 N4 E/ U: a. _, a, B| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |/ \0 [7 z: H/ W( _8 V  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         |
    & N# m5 _. a6 m% o| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    / E2 s" @' d8 K: o3 X
    " O- ]3 ?( H, S 经典递归应用场景
    1 k4 I( h3 m4 ]1. 文件系统遍历(目录树结构)
    2 S! H. w$ e! P. @; {7 t4 e0 J: C0 k2. 快速排序/归并排序算法9 D9 y& o4 E) }. O+ [
    3. 汉诺塔问题  `+ r  O4 p0 L7 W' _. C
    4. 二叉树遍历(前序/中序/后序)
    + J# P+ p* Y- O: ]5. 生成所有可能的组合(回溯算法): s3 [0 W6 u: p$ g7 B

    6 e( c" R9 D$ d, R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 08:02
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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,
    ' \: C/ t4 h. o/ r5 N- x0 p. q; F我推理机的核心算法应该是二叉树遍历的变种。
    2 }  L) c" w9 |! R  j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:0 q8 b' e3 T* n8 `
    Key Idea of Recursion* `0 k/ r: ^7 p" K

    + L% E$ j2 t% c, t: j% a* t+ `  X5 \A recursive function solves a problem by:/ w$ M  Z& U3 Z% t, {1 U

      F6 N' J' b+ h; A* y6 [    Breaking the problem into smaller instances of the same problem.1 v3 M% q& W! s' a
    " b! m9 A/ @; h# o2 N
        Solving the smallest instance directly (base case).
    + Q' H/ N2 u% E& }
    6 r% W5 |' K4 a; [    Combining the results of smaller instances to solve the larger problem.
    - J8 c* {8 @8 e- P
    2 m3 ~' P0 Z0 P5 q/ L& n* `Components of a Recursive Function& `; V  U5 q  G2 k' X
    1 k& [' m. J. E8 B1 ?! w, b1 N0 S
        Base Case:; C' q: s1 t& u8 a8 B7 R

    4 t: A0 u# J" c8 N        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.2 {0 C& b) `1 H8 ~" `8 `
    & L! D3 q6 j' N: W; \
            It acts as the stopping condition to prevent infinite recursion.* O$ i" G9 t4 `& J; V' d5 @: C. Q# w

    5 ~) k  N7 K$ E7 F, a+ N( ~) `2 |9 C        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 @7 s; M' q# d7 b

    / H; G/ ?" }$ X4 }& O$ m. {    Recursive Case:
    * K. O: O4 c& j$ A: _( q" r: \: y1 ?- M: Y
            This is where the function calls itself with a smaller or simpler version of the problem.
    ; U5 Z/ p2 V. b& k9 U3 I4 r: i: ^4 j( X' }4 ^
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).. U& ^  I9 @5 `+ Y- T' X

    4 Z# i' F) S- a# x/ TExample: Factorial Calculation- z/ d& s: V* e0 n  x5 P8 a' j; i

    ) ]8 y- |+ Z# H; E: _5 v# v0 V5 AThe 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:8 F+ q2 v: Y7 k* W, O  j* Y
    : `# _8 I8 Z, j8 ]+ V
        Base case: 0! = 1
    0 \' f5 C, r( h! z4 P  u0 g0 o# Y- m, `
        Recursive case: n! = n * (n-1)!, y4 a3 m+ G' `* a. y& k
    # [; y; U, s$ T5 \4 f
    Here’s how it looks in code (Python):
    ) y# t* p. [; Q9 b! ?, Spython
      l; o3 n/ g3 g( ^( u+ }* k' I2 [4 y" w% m: i8 Z. [# p
    : r1 e' N0 T+ U' ?+ d
    def factorial(n):& u! l/ G/ @) ?7 c* _8 U' }  [8 I
        # Base case4 e  ~+ T. K6 G/ ?
        if n == 0:
    ; ^0 w8 C0 _) `3 ]        return 1
    % x6 B$ I1 K  E3 O$ o7 h    # Recursive case5 O3 P# c( M2 r% i# y6 ]
        else:8 i6 H' b6 X5 I5 K
            return n * factorial(n - 1)
    9 h7 K  B5 J3 r8 o* M5 }. v  C
    8 I" G; ?* O" p! a! z# Example usage, e' F; \% Q2 `7 B& i
    print(factorial(5))  # Output: 120
    4 F5 g) ]5 c3 c" N! f
    / t! h. s2 g  I6 O" N+ E0 ~3 U3 O: [How Recursion Works6 [, A' ^. ?3 _2 l$ q$ |& Q9 a

    7 d8 g% T% {, w& B/ h2 _9 }    The function keeps calling itself with smaller inputs until it reaches the base case.2 S/ s8 D: W1 C9 z; K

    , r3 A' y; V" y( C% V    Once the base case is reached, the function starts returning values back up the call stack.
    & G, f( S) f: D- K( e
    % U3 y; f- _4 h8 G1 ~    These returned values are combined to produce the final result.
    8 P6 b' f+ a$ d# y( m' P
    % [9 ~- w( J1 O! CFor factorial(5):
    ) O. N. h1 v( }4 k( M/ {" ?; ~. X1 G. r0 _. F
    ( X6 C1 J8 o% G; B, N4 }8 L+ @
    factorial(5) = 5 * factorial(4)- g0 P" N) ?6 Y  ~! \
    factorial(4) = 4 * factorial(3)
    # f( g1 d% u, B5 W3 nfactorial(3) = 3 * factorial(2)
    - X3 ~9 E4 H3 ?3 ^" ufactorial(2) = 2 * factorial(1)2 i- z9 d+ o! h# ~; \' q6 T+ O! w
    factorial(1) = 1 * factorial(0)' R) t$ x  n) w: C2 Q% S
    factorial(0) = 1  # Base case
    4 j6 w; y4 {$ v9 ~$ h) Z/ T- Y: e" m3 e7 [6 v
    Then, the results are combined:
    # j% w- W6 ?6 ?
    + G( ~3 p* L3 Z7 Z$ v
    9 v  k8 b2 ~+ N1 {, ^- A% i" ?factorial(1) = 1 * 1 = 1# s' U8 f  H# n5 w3 l2 u' `7 M
    factorial(2) = 2 * 1 = 2
    + a  R6 X5 w1 D+ n  O$ M* V3 E5 hfactorial(3) = 3 * 2 = 6- [4 g# F8 o( _
    factorial(4) = 4 * 6 = 24
    7 I* ~% x: O# M5 y  \6 E6 xfactorial(5) = 5 * 24 = 120
    # {$ n% b7 t; p* ?
    . c5 f" n) ~9 VAdvantages of Recursion' u' A( l; {( O2 y5 b9 [) k+ n/ L5 _
    , Q; e$ x9 \) j. Y+ @3 b2 U" _  O
        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).& @0 W# N- q* Z  o* V% U

    5 v6 W1 w9 l, L, A7 w+ h    Readability: Recursive code can be more readable and concise compared to iterative solutions.1 j  N2 y+ {/ Q' w

    " d& a1 D$ f. i& n# P. oDisadvantages of Recursion8 c8 F/ y8 B2 q- `) {
    " f! k/ ]: u/ \0 w0 z& v: D' y
        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.
    ! C8 v% {& U! Q  j# U8 h/ d+ d/ \
    , g* k4 V' G, ?/ E    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).+ H: \2 X$ f9 q$ k% ^& ]

    # M  u, A( U. Z2 I3 M- u$ kWhen to Use Recursion
    ! Y0 e/ G% a; U, e
    7 c# s7 c6 D* N" r' W, ^) C, _    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    8 F+ h4 z' _: W2 k- Z3 J) [8 p6 h5 e" @
        Problems with a clear base case and recursive case.
    $ y0 v9 g% B; S
    ' v% a& }6 e9 f! x* X5 ?Example: Fibonacci Sequence
    : `6 q3 g& M/ a% O
    6 ]! ^6 k" n) j- E+ PThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ) F. P: h% {/ _* u/ Q5 A0 S- q% S) H+ T2 {
        Base case: fib(0) = 0, fib(1) = 1
    + q- H( ^9 C& J% Q
    : k7 n) J. a9 m    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # D# A4 \( V  }6 I
    0 d( W8 Q( v- V1 f" O) T1 _8 ]python
    2 O% X2 r; s; x8 V- [/ Z
    4 m. Q. ?7 R# F5 ^, L
    : {# k: ?. {+ h: J7 j+ Y5 Sdef fibonacci(n):
    * S+ B2 B0 M; R/ n, E& E2 \* {    # Base cases0 w: z9 P0 ^2 Q# }! S( Y& k
        if n == 0:
    + f. c7 x) t# m- U# ~( p" q0 \        return 0
    % z; F: o) T. r    elif n == 1:
    , m' f7 I% m. \4 L        return 1- [# B' T& e' _) k4 b' X
        # Recursive case" F7 n$ E, C6 ^) m1 H9 D3 N
        else:
    2 ~' ^8 z: `- g2 l' k/ ?        return fibonacci(n - 1) + fibonacci(n - 2)# c9 S8 P" v, H, R

    - a" Q; z" q! ^( @# Example usage
      R' m" Z- P, E& y4 }* w- X  u% @7 Mprint(fibonacci(6))  # Output: 8' {7 g- A6 W2 @2 r. N2 Q

    6 m: ?* {  Z0 W3 a9 U: PTail Recursion( q* c" h* Y+ B" \7 U9 d
    ( \$ g/ P7 u2 p) i/ t
    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).( [% f- M5 z1 e" b+ q
    ; Y) |' ~& f, @, l
    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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