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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
  • 签到天数: 3 天

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    楼主
     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
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    解释的不错
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    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。
    3 N1 R6 W0 ~; ?4 {3 F7 c: W7 n+ Y0 _; c5 c4 O0 W! N% m# Y
    关键要素
    & P5 }+ _7 L6 o. k2 W1. **基线条件(Base Case)**
    8 ^! \6 \5 q" c% n3 u/ t   - 递归终止的条件,防止无限循环( m% @( z8 q+ `8 C
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    * [$ n, S) E. K  }) ^3 Z0 n8 ~6 Q3 V+ K, s# Y2 \/ U+ ]" [. O
    2. **递归条件(Recursive Case)**) f+ s# n7 ~- p
       - 将原问题分解为更小的子问题( t' n* o  w5 E( @8 G- y  T7 H5 r6 G
       - 例如:n! = n × (n-1)!
    ; ?. \2 `$ y7 z. q0 C" u
    ; m9 b% @# u8 U: ` 经典示例:计算阶乘
    ) A; B7 Y  A0 A4 c; ?) t5 Hpython3 j4 m' ~/ S/ }( s! S7 \
    def factorial(n):  z$ p& _- g& J
        if n == 0:        # 基线条件
    . I# J/ N$ N6 |* z+ Q% U        return 1
    * y! v/ w( P0 G1 ^' \% H    else:             # 递归条件
    ( {, q9 d: D% }5 X9 m# V/ E        return n * factorial(n-1)$ W0 N' V$ _8 G2 R/ }
    执行过程(以计算 3! 为例):# g5 B& h9 w/ X9 h# w4 ~
    factorial(3)* y; I, a. l, F  K/ Y
    3 * factorial(2)5 A9 Q: }  N' i4 X: C5 j* O7 ~
    3 * (2 * factorial(1))# \. S' K7 |0 E6 z% j+ y0 Q% u
    3 * (2 * (1 * factorial(0)))6 d+ D- h6 o$ _/ `
    3 * (2 * (1 * 1)) = 6
    " g) I, P) L; S9 G7 g0 Y/ x
    ' t5 H; O' W6 p8 L& g- ]8 \1 d 递归思维要点
    , h' k0 ^) f9 S5 j! ^; ?1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 `. W! T% J; X
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)$ x& ~2 a3 s) }/ B0 a
    3. **递推过程**:不断向下分解问题(递)) y8 K6 y. |2 u8 L8 }. f# w9 b* t
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项
    0 }( d/ Q+ O0 ?- {7 k/ _必须要有终止条件
    6 m; [# H+ ^' i( x! v递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    0 W2 \: t! O2 m8 {; M; e- W& P某些问题用递归更直观(如树遍历),但效率可能不如迭代( g$ l8 \# D4 d2 R; `4 q4 j
    尾递归优化可以提升效率(但Python不支持)" J* J. j# m/ F/ \

    * ]4 V& {  Q8 B! \9 P% q 递归 vs 迭代
    - b6 t: U( I/ E% Y8 U" W" T|          | 递归                          | 迭代               |) Y  @: }# V$ x; h
    |----------|-----------------------------|------------------|1 l9 P2 h: i$ L- ?* a+ y& l; t
    | 实现方式    | 函数自调用                        | 循环结构            |
    " @$ D, W0 h+ D1 M| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    6 A& e: z9 h0 p3 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         |
    8 c9 |/ s7 _* S) b) J| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
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    经典递归应用场景
    5 T: P7 `6 Z* w1 Y8 t1. 文件系统遍历(目录树结构). ~" z. U; f! X: a
    2. 快速排序/归并排序算法
    * b. h% G" V. o0 n0 F0 J0 k: d3. 汉诺塔问题
    # m, O, g# p3 _! A( |! u4. 二叉树遍历(前序/中序/后序)( {9 f, D; E& Y+ l+ C
    5. 生成所有可能的组合(回溯算法)( a! u: `  q2 j8 x7 D( N( r! y
    + V1 T9 K- K0 Z6 H4 ?( O2 f4 K
    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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  • TA的每日心情

    昨天 12:17
  • 签到天数: 3175 天

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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,
    7 {: I, l4 v+ q  g3 F5 y我推理机的核心算法应该是二叉树遍历的变种。/ \& K9 }2 x9 b) a# c+ E
    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:4 N1 `# @& e1 b( S3 h
    Key Idea of Recursion' g- H$ N, ]! V

    ( S' y; }" z; o; j7 \# xA recursive function solves a problem by:
    3 @- e- {, w; G3 ?" }. ?
    9 i+ O, C( j% W+ B# s# q+ \    Breaking the problem into smaller instances of the same problem.; r$ j4 Y5 r1 a) q: x$ P4 t* Y- V
    ! k, j6 F$ X. j7 e
        Solving the smallest instance directly (base case).2 e& c# f* \. ~8 [7 l9 x2 q& w, W: j7 s
    % Q8 s) G1 l3 f6 A; o8 ^8 }2 c
        Combining the results of smaller instances to solve the larger problem.0 j( ?) p0 B& n" Q. u. Y4 L$ F3 Q. F
    # \* L8 `1 F" F8 `. h2 y1 d
    Components of a Recursive Function
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        Base Case:1 `2 |; L: u5 ~) T8 q
    : D- q+ D8 d! j. B6 w9 k! \% e
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.5 F3 j) _/ C" G  a7 F
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            It acts as the stopping condition to prevent infinite recursion.! N& q7 ]# Z5 W; ?' _5 \. o) m, t

    ; D, b/ G2 J) v% v( G9 [        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% g# N9 b" ^5 _$ A! o$ G; w

    9 M: A! w! v8 l5 Z3 C7 P6 G    Recursive Case:
    1 P4 q4 k- V6 }1 W* d* ]2 s1 v: p( a- S. z, `  {4 W: {" j
            This is where the function calls itself with a smaller or simpler version of the problem.: r& ?8 k, g6 F

    5 |1 M5 @: }3 v4 g' [        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 H2 Z! e  A2 f. N: z

    5 ^8 ]5 M4 O5 X; |" C" O- s/ Y, MExample: Factorial Calculation' C; ]; @/ e$ |# u0 p. M3 i

    6 B% P9 ~  ~6 b* \' C- i1 X% 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:# x2 k5 t: o1 Z8 D0 `6 O! M+ L
    $ W$ A5 E' b! J' q) @1 k
        Base case: 0! = 1
    ' J1 ~6 x$ Y5 K% C% o
    5 W# T( `8 p1 N% v0 p0 R; A    Recursive case: n! = n * (n-1)!  g/ q4 ^: m# x: R5 W( L& x
    ) J2 l* T) R- Q* y- q
    Here’s how it looks in code (Python):
    , _0 \# ]. \6 q2 gpython  S9 L' x, F: e. _& [

    , y+ y- ~" A. Y" K* _1 U9 R
    , y( z1 z! L, C+ c$ ?def factorial(n):$ W  o* J7 o! N4 [( m
        # Base case. I6 y; \. l1 s4 F3 a
        if n == 0:/ {5 r4 I* \  k
            return 1  @! a! O# C2 J1 C- v+ U4 K
        # Recursive case3 E( m. G0 `+ j) N( u8 S, D* K
        else:- k2 L4 ^* A6 `% k6 L  K% v
            return n * factorial(n - 1)2 B( q. i+ E# Q1 o2 f' K9 E9 t0 d. O
    3 \; \: c9 p, H) Q' B! ~
    # Example usage+ ~: C4 ~# M: m: M
    print(factorial(5))  # Output: 1204 ~& @7 X( B1 w! ~: D  E( F

    7 q- W0 {; A' t0 ?: O- HHow Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.% j) V( x* @+ A# v; B5 @

    8 ]/ u: N$ q0 m. ^6 C    Once the base case is reached, the function starts returning values back up the call stack.) x. E7 i  _+ H0 F4 o

    % L& M$ E" S7 U    These returned values are combined to produce the final result.
    ; k8 d. B: \+ D0 s# `$ ~  T/ _4 g% c# b0 ^
    For factorial(5):) t9 Z0 K, F' M. U
    2 P: g0 e: c+ f' f

    - E- h" }/ L' E7 F) Ifactorial(5) = 5 * factorial(4)
    ( I. R' L0 K3 T  @5 zfactorial(4) = 4 * factorial(3): s3 q' j; `3 n
    factorial(3) = 3 * factorial(2)
    + J* X, @" n8 q' Y1 `  R8 G# Ofactorial(2) = 2 * factorial(1)
    # S0 c) t7 v; }' P+ D0 b1 mfactorial(1) = 1 * factorial(0)6 t( {& Q% r/ C$ V
    factorial(0) = 1  # Base case% L( \8 O$ i+ a4 c
    ) I) C2 f, ]; y  l; p* u% a1 F
    Then, the results are combined:6 W7 v( }$ U( B- K- w! _& ]5 I3 a
    / w1 N  S- H4 F5 |5 L6 Y) G

    9 a* C' l% n6 ~/ t' N6 W/ dfactorial(1) = 1 * 1 = 1
    3 u: [! p5 j5 o& L; ifactorial(2) = 2 * 1 = 2% k. q* g) [. G, s% t0 m: S$ F
    factorial(3) = 3 * 2 = 6
    * P  P8 y7 Y7 Z# X4 L" P) y" qfactorial(4) = 4 * 6 = 246 _  z, u% v8 A& P# h
    factorial(5) = 5 * 24 = 120
    + k5 ?% v( J* @6 c6 |& H- ~2 m1 t3 \3 n2 ?" y" t, R: p
    Advantages of Recursion6 C3 j# X) m- X3 \2 j8 y2 z
    ! |7 M& a* q: k6 o- I" B. Q
        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).
    " g' i3 Z" S7 r& v. W2 V
      m# ^/ L% W# w# e    Readability: Recursive code can be more readable and concise compared to iterative solutions.+ X6 |5 ^5 e) ^4 w% L3 R
    2 A& J( Z. c0 J; D. R0 I
    Disadvantages of Recursion
    * N8 L& P- s" u  {' \9 J2 D3 X% }6 @. F( |; t
        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 i& T+ ^' B% s2 F) D' R* a- Q2 Y- M$ A" w
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ) w% z1 l$ |1 _% D$ {  j5 x: u# J  O4 n/ _/ Y% ~1 r
    When to Use Recursion: O4 [9 @9 D& }  o9 o1 H
    6 I7 `9 E  N- z
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    / y; y3 s" c2 j6 h! D4 f4 q9 @. G' T6 R) b! V
        Problems with a clear base case and recursive case.
    3 {. h, V" k5 Z$ F: k- s7 e5 M1 A9 C1 @( W* M
    Example: Fibonacci Sequence$ X5 b* N! {7 u% P

    . O9 x3 j* X# m: x' TThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    " X3 L3 T8 \+ g0 Z4 B
    ) `% k( T  w. s    Base case: fib(0) = 0, fib(1) = 1  w; g) y" O4 E, [, r

    * s$ W( l5 J9 n/ g" r' ^/ t3 x    Recursive case: fib(n) = fib(n-1) + fib(n-2): P8 }3 ~( j* m& t
    4 b, [- k2 t& S' V
    python
    . F4 \5 z1 F# T0 }6 i0 _( ]6 E
    # i4 h6 u) m+ D7 [* V# D
    . a/ q* h- \/ s  gdef fibonacci(n):
    5 p5 V1 T4 x! z8 c" ?' G  q0 J    # Base cases' ~9 P/ l7 O! L6 ~+ M
        if n == 0:' o6 u; ~0 v( Z4 T' i1 M6 [1 |( ]0 U
            return 0( A: p  z. T- h
        elif n == 1:
    8 N) w$ }7 Q, O3 K& F        return 14 w; z$ V7 R3 s6 X' Z" J
        # Recursive case) B! l; |3 u4 P0 E2 r- `
        else:7 C& j" x  `# |* f7 A
            return fibonacci(n - 1) + fibonacci(n - 2)
    9 Y; K3 [+ I7 I- M9 M; d
    4 F+ |9 B8 h( L# G; }# Example usage
    3 _# ?) b8 i3 _: i6 Dprint(fibonacci(6))  # Output: 8
    , g7 J& Z0 ]+ I5 i7 d' f& n! Q+ f2 L: \! y8 B
    Tail Recursion
    . J1 m2 `# t1 W: X' J" A0 Q# i) d% p# B( w; u2 [' W; x2 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).. ~, k) s$ |3 H  r! u
    6 `. Y' ^7 |6 B- j
    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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