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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
    7 D. z! q  y5 C' y: q' r; o* I
    ) |! G* L/ B. f; F  J解释的不错& D  E- Z9 n. a. X" r/ V' A
    $ }& h* N+ x5 f
    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。
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    关键要素, e" q/ Q7 ^! B: P8 F
    1. **基线条件(Base Case)**
    ( h+ \8 F; s5 f$ |* g, n   - 递归终止的条件,防止无限循环
    & [3 D" ?" O9 q, Y; G6 K1 q   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: Z1 Y" O  M1 v3 P! p

    1 t2 C8 d& x! k3 U+ i2. **递归条件(Recursive Case)**
    + ^# t2 g% W: }9 H; x. q$ _   - 将原问题分解为更小的子问题
    % H7 q" T* ~4 |5 c3 ?   - 例如:n! = n × (n-1)!3 M; K; b; }$ k+ D  \

    ( D9 U' T+ i2 x9 m/ @# K, r1 { 经典示例:计算阶乘. ?! Q( o  D! F2 q: W0 J
    python2 \9 U# {2 \$ y6 l7 l9 n
    def factorial(n):
    # A/ v, Q+ T6 ?" }    if n == 0:        # 基线条件# {; y2 m! f. n: J1 o" ~
            return 1* J- a0 n6 x- O, x- g* I$ n
        else:             # 递归条件, Z/ w. x% a9 j, Y3 @; g% L9 F) {
            return n * factorial(n-1)8 k/ O; E. T' U
    执行过程(以计算 3! 为例):
    3 j* t+ ^4 \1 v) Y2 m- F' Y- |factorial(3)
    8 r: Z: U5 r& N7 k; Y8 e3 * factorial(2)
    2 `" D- t8 v1 ?7 R( c* J3 T3 * (2 * factorial(1))! q$ b/ y+ Q$ w
    3 * (2 * (1 * factorial(0)))( q# a# p4 T2 I1 M& A
    3 * (2 * (1 * 1)) = 6
    ' i2 h; k, P" P8 |+ S7 B2 \1 v  ?( V# Y
    递归思维要点
    # ~! t5 ~. D. ]+ l2 Q* e1. **信任递归**:假设子问题已经解决,专注当前层逻辑: y$ T& D. \: u0 ~0 s- N
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    # Q4 X+ I" i( S* ]) Q/ [& m- k0 g; [3. **递推过程**:不断向下分解问题(递)9 Z6 i$ B2 f5 }# O
    4. **回溯过程**:组合子问题结果返回(归)
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    6 ?, h7 N  ]6 j( m! A, J. i' X 注意事项
    . D4 g8 ~" ~1 [% X3 c) @* d2 C必须要有终止条件2 R6 b: \  Z8 X! Y  O
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    9 v6 S0 f% H" M: h7 M" v- O某些问题用递归更直观(如树遍历),但效率可能不如迭代" ^* H: W6 a# V+ l. {% K: E4 y
    尾递归优化可以提升效率(但Python不支持)
    . U+ `: N5 {: O$ a) u% z5 N3 w& Y1 Q* _+ j# O/ B
    递归 vs 迭代$ \" ^9 a6 m/ O
    |          | 递归                          | 迭代               |
    1 \' P4 b% n3 `|----------|-----------------------------|------------------|% L  O( b8 D  ]5 K( O/ k
    | 实现方式    | 函数自调用                        | 循环结构            |9 `6 ^. j: ]+ g& Z0 U1 o  b- [
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    - w1 a" y5 F( U" Z* [| 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         |
    # L8 x8 {1 a8 R5 P6 K| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |$ o% F* g& J. j/ Z/ B) ]' p
    % a& H0 h) l, d7 ], H0 q! h
    经典递归应用场景, D: I4 z- p* E
    1. 文件系统遍历(目录树结构)
    * N; A  D. d: A: L0 ~8 U2. 快速排序/归并排序算法' P6 c5 [' `7 x/ m, i6 ?, n
    3. 汉诺塔问题; p+ b! M  }8 o
    4. 二叉树遍历(前序/中序/后序)
    # E; X" e% K& V( d% v# k6 P' q: i. x5. 生成所有可能的组合(回溯算法)* z- `. |8 f& q+ G* E" Q# @( {% G

    $ ]5 z7 k0 Z9 S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 G; \9 P8 x- W- G  n
    我推理机的核心算法应该是二叉树遍历的变种。
    - o9 Y; S4 j2 @/ z5 A% q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:
    : ]: l" g9 M9 x) cKey Idea of Recursion
    ) q4 L$ P: x  {8 g! ]* x. w# e( C4 O# ]# w& y
    A recursive function solves a problem by:
    7 F. U" y! Q3 j) `3 ^
    ! b6 q& s4 y! s    Breaking the problem into smaller instances of the same problem.& U8 S% e, S, W8 d8 O7 W7 Y  V

    2 n8 S, z4 e" d    Solving the smallest instance directly (base case).6 f) g3 x- I& `0 D& l3 ^
    + |5 w/ Z3 u0 k$ D& W8 K2 j# f8 O4 v  s
        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function
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    1 O1 ~/ n- P3 J. t    Base Case:
    8 k& `2 d9 F4 u6 L6 l4 B) {% |  m$ f* o' U/ f+ h4 ?8 k
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    3 d& x& F' k; J; C1 p6 b3 l; r  ~/ m! i: A6 r9 h: @
            It acts as the stopping condition to prevent infinite recursion.- L$ W5 H' _/ x% m. m5 Z

    + G2 C; C! I3 }6 I( o        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.! i2 X5 L# l+ ^4 c" T% c  h( H. e

    * g* O+ E/ |( \3 a    Recursive Case:
    * E6 S: x: i' J/ G- {. ~) d  G+ x7 W7 n
            This is where the function calls itself with a smaller or simpler version of the problem.
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            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    - O; g4 e1 ^9 E2 H/ n- aExample: Factorial Calculation! H/ t! }, t5 F* q9 B" \) Y
    # _# g4 W' A4 s* i0 Z
    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:; X" S% F) l2 h. c6 e8 H& q, p

    % C! i3 |: D" }8 D: h    Base case: 0! = 1, |7 J. l7 a# D! T! i8 d
    8 h  P9 ^4 \0 `) q( O
        Recursive case: n! = n * (n-1)!
    1 f' A5 L5 L2 R" y0 d9 h$ I3 U' a5 d4 [! c& |( b, f) \4 ~
    Here’s how it looks in code (Python):: g" h( e' _, W  c9 V3 `
    python( F- R+ g4 p% [2 [
    ! @+ Y% \- W2 }- F8 R

    5 P% _) T) o" S" r+ m! Vdef factorial(n):
    , q1 h. F9 `0 I' }7 `' |    # Base case
    * Q8 [$ P7 D. l# V/ U    if n == 0:# b  b* d2 S" h) q* H7 b# }
            return 1, F6 v, b5 z5 G1 k1 s/ X) c! r
        # Recursive case4 G1 _9 ]& h  h2 m2 y6 M+ A
        else:- `2 f2 F% ^% ?" b+ h1 w* U
            return n * factorial(n - 1)6 @' O; m, ]; A9 F2 v+ n
    ) g0 M7 s! B( h- R/ r5 P" h
    # Example usage
    5 }! \* b6 W7 G5 `print(factorial(5))  # Output: 120+ U, c+ u- `) ?5 D4 T6 [

    % ?% f7 N/ z" m; C7 U/ z% UHow Recursion Works' V/ e( `- N* r& J1 E+ a9 Q7 s5 z

    0 H0 @0 i! l# y  b    The function keeps calling itself with smaller inputs until it reaches the base case.
    * T, X! z! N0 M
    ; Y+ s' e9 {- J, J    Once the base case is reached, the function starts returning values back up the call stack.
    7 V8 K! Z/ u2 f" Y: j+ [7 Z( ^2 \
        These returned values are combined to produce the final result.
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    For factorial(5):; {: c' r, j1 ]( r, b
    0 Q# I% T& ?0 e1 Q' ?/ f; R! O

    " O- E4 I! Y. X% ~4 \- Yfactorial(5) = 5 * factorial(4)4 D$ G0 W8 \- O! v, i
    factorial(4) = 4 * factorial(3)9 U3 {. M6 I5 v" I& Q& k3 y
    factorial(3) = 3 * factorial(2)  @: H2 i4 ?" e5 N, `1 ]. M
    factorial(2) = 2 * factorial(1)
    2 e0 F4 \9 f4 |' K  J6 B6 lfactorial(1) = 1 * factorial(0)7 y7 ]' C, R: y0 Y
    factorial(0) = 1  # Base case& K: z( f3 @* N5 D5 v9 M# L& y

    9 m6 ^, D% @  @" |) v; V( U& [Then, the results are combined:
    ) ^9 |5 X* d8 `/ v% h+ b* u8 l2 N8 U, a3 B
    ! x. J2 S  i' d8 V9 V( ]2 W4 |
    factorial(1) = 1 * 1 = 1
    1 i8 j4 |& `( N# `factorial(2) = 2 * 1 = 2% e* h# }+ M/ q  y# g
    factorial(3) = 3 * 2 = 6
    5 ^6 H8 Y2 f5 A; Z3 Sfactorial(4) = 4 * 6 = 24
    2 M9 L3 o5 b7 Z- s) V9 j# `" |8 \factorial(5) = 5 * 24 = 120
    7 _0 y% ~9 q' Y) S/ b" s
    / e. u, \/ f, u6 s' N  y& y% M3 uAdvantages of Recursion
    # D* T) }* I3 c# a
    / h; p. Q# z0 D    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).8 ~* S2 x: B0 J6 z- s

    % @% h/ L+ c. [$ D& N    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    8 [8 f' ^+ l' e5 _5 A' N: J; KDisadvantages of Recursion
    + z7 x# @: o6 ]4 \$ c/ a! w# [0 G+ t3 V* k& `# V0 X5 Q
        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.5 @) Q$ M" @/ y) x! r. V

    # ]/ [) f3 ?, V, O    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    * _8 A3 r: i% ]1 q4 C. I: @' z
    0 |1 H* W/ A) p2 i* r' W6 |When to Use Recursion, ^' a0 u5 }; b& |8 f9 W
    6 D. X6 t- ^9 W3 s
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    2 x0 j7 |9 M2 q. A: L  b1 w, B8 t  W. Z
        Problems with a clear base case and recursive case.
    0 J+ h' U) J7 t* |" {# ]. i7 C5 g4 _0 u  K
    Example: Fibonacci Sequence
    1 M9 P% d& b  u
    . B# y/ Y9 x6 I+ }% nThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:, ^2 @9 r, G4 c8 ^# K

    5 [3 P( z% s: K    Base case: fib(0) = 0, fib(1) = 1
    0 H- L1 H5 f8 ^9 P
    & C$ i7 Y7 c2 I+ f1 K    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! W/ H$ X$ j- |4 Z* E* ~7 \: r5 |) M$ [" j: }& p' e
    python% c3 h# y1 U8 k( v
    0 N3 _5 p3 F* @- r' j0 ]9 d  m& J

    ; {9 o+ ]4 ?# z, rdef fibonacci(n):
    4 t3 q; ?. Q% {: M. |: ^8 @- K    # Base cases; z% @' _! G7 Z7 S2 X3 S
        if n == 0:. l* c0 J2 U* ^& D" j' |& [
            return 0  |, Z/ l# h' I* s% A' F) c
        elif n == 1:2 r4 S  p: }. y& d. B8 ]: s
            return 1
    : y2 S& f" V* H2 T4 V  h+ }" i    # Recursive case
    ! ?8 `+ D) J5 \    else:' S8 q0 \, S) e8 ~$ {! h
            return fibonacci(n - 1) + fibonacci(n - 2)* y$ o* J/ b4 z, D) L
    + s# x' _* ], [# `) q+ h! @' w
    # Example usage
    ) t, X4 B( ?2 S% Bprint(fibonacci(6))  # Output: 8, Q# F) s) p/ k" `1 J' S6 U1 Z

    ' I2 v% F6 o, r4 x/ S, yTail Recursion
    * S0 }" S* k; u) V9 a3 u5 N; [: c; b
    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).
    * i2 _! A) p, d4 O' A* n' ?8 v
    . }2 u9 ^) T3 m0 t9 O/ BIn 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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