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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
    6 F& s  L( k8 V9 }
    / k5 i7 T% p  O; h解释的不错6 p( ^$ Q8 C2 i

    4 j2 t9 z# B4 Y8 [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  \# J! ], q1 J! L6 _
    1. **基线条件(Base Case)**
    5 j  h1 u$ I5 q2 S% p" O* [   - 递归终止的条件,防止无限循环
    2 L& M0 ]( g5 B- Q   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    1 Z, s( z, ]1 I: @; b$ e
    2 M( d  F0 M; D8 M% L2. **递归条件(Recursive Case)**9 n5 Q& G9 y$ ~4 c8 \9 C
       - 将原问题分解为更小的子问题
    . c3 _8 l, B( a( \* \/ C. Q   - 例如:n! = n × (n-1)!, S' ?( w( k/ B( u% M4 i

    , _7 C5 z+ _+ K: `, s! q4 Y 经典示例:计算阶乘
    9 a1 Z# ]5 q5 ^6 C, D6 Dpython0 k* w3 e* y3 i
    def factorial(n):
    9 t( B8 s5 C2 Y+ H$ a4 g/ F    if n == 0:        # 基线条件
    & |! ?, t0 [# R+ B2 S. z        return 1, x* ]8 b4 S! a' W4 o
        else:             # 递归条件
    " s* g) [1 b- y        return n * factorial(n-1)
    6 z; C, k& y( H执行过程(以计算 3! 为例):
    8 u, Y" h$ @+ Qfactorial(3)
    ( l- }+ O* B. h' E8 r3 * factorial(2)# K& z9 l$ L  w! L' S' ]" n! [4 N
    3 * (2 * factorial(1)); d# F1 j) @% ]3 P& X3 x# s
    3 * (2 * (1 * factorial(0)))
    2 H; _1 O. e. f0 m) C3 * (2 * (1 * 1)) = 6. p& @6 r9 i  ^0 m' H- \
    6 [4 W! f/ z# V; Y- y3 L
    递归思维要点
    . v$ o  P6 d/ a* u2 ?1 x1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    / m% z- i% X, \- y& e! f2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    & \0 h0 X2 h: c0 _. o3. **递推过程**:不断向下分解问题(递)5 B+ W4 a( M- ?4 w/ p0 x
    4. **回溯过程**:组合子问题结果返回(归)
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    $ G; [3 F. \! n+ ? 注意事项
    * @; r% F$ u, m7 B+ x% i必须要有终止条件
    5 F" G& Z; T. K: D递归深度过大可能导致栈溢出(Python默认递归深度约1000层)3 f' n7 B! {% S% P
    某些问题用递归更直观(如树遍历),但效率可能不如迭代7 Y$ m" w$ B) g) p$ W- ?6 A
    尾递归优化可以提升效率(但Python不支持)6 `  ]. ?' m$ h3 w
    4 I9 k$ Y. _* \1 u0 ^5 l' p- I
    递归 vs 迭代- T3 }  w; ^# p4 R  d
    |          | 递归                          | 迭代               |
    ( ]2 X! O+ m& W& N: E|----------|-----------------------------|------------------|
    4 a6 o+ @8 c. e8 e9 ?) B| 实现方式    | 函数自调用                        | 循环结构            |
    # X; n1 w! J( v| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ( }8 o& G" N' X2 ~7 O8 F  d1 || 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         |
    ( T+ s- f4 c1 z. o) h1 z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |8 ~. M5 k! y2 N7 y$ a: s$ R
    ' x' c0 ~3 S1 C# W
    经典递归应用场景
    7 T2 a1 A, Q; U! i# ?# c2 x! c1. 文件系统遍历(目录树结构)
    - E' C9 {$ `$ q' W& p% ]. E2. 快速排序/归并排序算法8 ]! v& L" W6 W3 N: Y) A
    3. 汉诺塔问题) v% x. ^( O2 K; s) A9 `5 {
    4. 二叉树遍历(前序/中序/后序)
    6 O. F8 f$ d6 {' O- U  M5. 生成所有可能的组合(回溯算法)4 J# w6 ^( o6 h

    . a/ v2 ]& N! x6 k& R, ^& R# O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,
    1 ]+ \0 @& a% B3 _我推理机的核心算法应该是二叉树遍历的变种。- N- D$ c# M6 y- {4 t
    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:3 x, T1 K2 Q: d5 R2 t7 ^
    Key Idea of Recursion
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    A recursive function solves a problem by:) _, _5 _0 P8 g2 x& j

    , O* U) A- V! k' A0 S, ~    Breaking the problem into smaller instances of the same problem./ h+ C  f' }" E! Q( O8 O* r  ~
    + [( g% i; s' h* c" v
        Solving the smallest instance directly (base case).
    5 L( u, \% _. [+ {, i6 x3 M& ]0 H" k
        Combining the results of smaller instances to solve the larger problem.
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    0 R" ~& Y: x1 i( cComponents of a Recursive Function
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        Base Case:- X# j3 N. f3 w2 {) P0 C& X* V

    . I  V) N$ _9 c. C9 L4 p        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.  m; E/ O* d  j7 r1 W3 G: ?
    8 M2 I( `! a) ~- y
            It acts as the stopping condition to prevent infinite recursion.
    4 `% ^1 W; x1 s3 j- ^2 ]$ ?. f" _& \. E
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    1 j2 j9 i$ S+ \) J    Recursive Case:2 [* S- o  P5 V* w

    9 M( `( ?! t, u        This is where the function calls itself with a smaller or simpler version of the problem./ f! Q' b7 r: b
    , S* k' n, k' |. E8 b+ c1 c
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).! i0 k+ d+ i' g0 q6 L1 M" L

    + z4 {; H! G# n: VExample: Factorial Calculation
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    ; {# g* x" j9 \* U( T( MThe 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:# q/ r4 n( {' R2 F$ x

    ! V( M* @( y# u& R, \. P    Base case: 0! = 15 ]) |$ s$ L1 v8 ]& Z
    / m1 B0 ~9 }: _
        Recursive case: n! = n * (n-1)!
    ; h' O) p% t$ n: {- P5 Q0 n& b# Y, H2 q0 u& R$ C/ T
    Here’s how it looks in code (Python):
    9 q# ]% @8 Z" Z5 b( \python! [& M/ O2 Z- g7 T: s% [% R
    0 ]* V: ?7 D2 s% B7 }  _
    2 a' j' R4 a2 @+ ^7 J) ^3 j
    def factorial(n):; d- _3 {3 d* ^  R2 Q# J2 E
        # Base case
    5 J7 m& W- u6 b* z  Z+ q    if n == 0:
    4 C: u' s: \, a8 }% E: [/ P/ O  m. J        return 1+ z' q1 [# L( k
        # Recursive case
    * d- c! G4 s9 w# W' X" d* V    else:) A% s% ?0 A, Z4 j- `) M( W
            return n * factorial(n - 1)
    ( O+ l! @/ v5 R& F5 S/ K. D8 r1 f2 K4 Q) v0 k+ r5 C
    # Example usage
    9 T- j8 `1 d+ f, p; {7 Oprint(factorial(5))  # Output: 120
    5 Y6 Y2 N0 u) @. V' z% O: N
    % Q! N" X: I: |& ^8 GHow Recursion Works
    ' X. G' C( B; D  _6 i" {$ L8 `6 g2 Z' l7 Y/ v: _
        The function keeps calling itself with smaller inputs until it reaches the base case.) k! {. r2 v$ {$ j8 s. q7 O

    ! N4 L/ N, U, `7 o5 f    Once the base case is reached, the function starts returning values back up the call stack.
    * \; t3 b7 e( w8 ?- J/ _, p, c  g7 u6 E& o) N
        These returned values are combined to produce the final result.
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    For factorial(5):
    5 G  }  I: M& J3 F$ A5 w
    7 u" N+ d4 h1 w0 f/ p) Z4 ~+ ~/ i. T! s
    factorial(5) = 5 * factorial(4)* O' y& M5 B2 R* }; F6 N& z* _9 a
    factorial(4) = 4 * factorial(3)
    7 h) f1 Y( t- Z; H/ k; {  Gfactorial(3) = 3 * factorial(2)
    % f$ y2 q6 U- ?1 Z) {7 ?8 efactorial(2) = 2 * factorial(1)1 q" f0 j8 T/ x# ~. ?9 K+ Q. \
    factorial(1) = 1 * factorial(0)
    ; L) K2 G/ n( b6 e: y1 Sfactorial(0) = 1  # Base case8 ]* w0 k  K( O; M& B

    + a) d& D' B) a' ^  GThen, the results are combined:
    " `$ A$ V: E$ E
    . T; P' O- P- v
    , T8 i# y6 f* w9 bfactorial(1) = 1 * 1 = 1) o! w) x# N8 Q# V  T2 k
    factorial(2) = 2 * 1 = 26 ?5 B  D/ c* q% ?& W- h; D. v
    factorial(3) = 3 * 2 = 6
    ' K2 {; `! w6 i- [! f2 L4 l, ofactorial(4) = 4 * 6 = 24
    0 ~+ L8 G9 o+ \: w: R. ?factorial(5) = 5 * 24 = 120
    7 i# u& V* R) z& D, U
    ) X9 e/ p! ?4 S1 iAdvantages of Recursion
    $ l7 e4 r/ g0 W: R( M* I! C8 e  ]. s4 Z# L0 o' D; B! F/ C
        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).% w7 i$ l. j9 \. }/ y, @) q4 k7 {
    5 q8 f  O- x4 C1 d; b
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion% ~( \8 p$ G: R2 f) s" i

    4 L: D& q  z" c" h8 ?' [    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.6 U* [' y. h% i. w  W

    0 W1 h9 m2 b3 R% F( e& N0 ]    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    , z3 ~0 A8 E5 V) ^" L9 m+ W  }5 t: \1 i( ]: \3 h' i
    When to Use Recursion
    ( f* F2 C( T+ V. i0 |
    ' W' s/ _  y1 a1 J9 t& }+ k- k    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    * b8 i3 Q6 N; m8 }3 ^
    ' O2 f4 m+ o* J9 b; w3 }+ ^; p1 U    Problems with a clear base case and recursive case.) y4 U. I( C% D; R; z
    ( c: B6 N9 y. ^$ P9 ]
    Example: Fibonacci Sequence
    % u2 V/ g5 t1 V% e$ t# F
    & v; E6 L  y; S, OThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    3 ?0 {# r  g% n: Z! _7 B2 p
    . C* F7 |, \+ S/ ~6 b2 r. [; A    Base case: fib(0) = 0, fib(1) = 1* q7 y% ]4 r$ V

    2 b5 ]& [1 _% p& }; u    Recursive case: fib(n) = fib(n-1) + fib(n-2)9 o' z' B. v, s! R
    $ \( h1 S7 `; `& E
    python
    1 `- \4 Z, O8 ?' c5 F# b% U2 D" @+ N6 `6 P0 {

    , s$ b; d" ~5 X7 N9 J8 G* v& pdef fibonacci(n):
    0 Y2 `& F& @9 a9 ~    # Base cases" X9 G/ T) z' s; U
        if n == 0:) k3 j; Q$ X9 {/ b& t  _( Y0 s& E
            return 06 H) t6 s0 D3 V1 T8 q6 J1 X$ c1 q
        elif n == 1:, n1 I) L6 g, q! M; F# S9 z! ]: K0 X
            return 14 m/ J& x% ~, b! r% i% w* O) F* `# l
        # Recursive case
    & k. A+ k" V1 r0 U2 c    else:2 ~5 u1 ^1 R/ T0 G% M4 U
            return fibonacci(n - 1) + fibonacci(n - 2)- {7 z& ^% C! a& h
    4 e4 o/ F5 _  o  w0 x
    # Example usage7 K6 \" X/ j% a7 _9 B
    print(fibonacci(6))  # Output: 8
    9 V; N7 w9 x- p2 d1 r6 ?- G. P1 A8 ~) _! v# y
    Tail Recursion1 I2 L# i% [/ M- i, q- M
    . W+ [- E. @0 M+ M7 ?# L/ 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).% Q$ d/ |& G/ E; |% Y

    , V1 {. z: R6 E3 A$ e" cIn 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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