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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 c: a# V5 ?7 x. e" q
      Y, x$ O1 s. W% l  H5 l解释的不错- W( u1 B! C7 b# X$ V7 T: @+ Y
    - }6 w& O0 V* B3 c8 a  P
    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。
    " F# E4 m9 t- r& |
    6 U& O) ?9 t9 G 关键要素
    : f$ J3 X1 W+ X1. **基线条件(Base Case)**- G9 ?( c# v& h; g. P. _3 l
       - 递归终止的条件,防止无限循环' O+ G9 ^4 i7 p; I
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ; @5 l4 T' w$ z. ~" \# E3 I- O1 R4 s) i' s# A3 N9 |6 ]
    2. **递归条件(Recursive Case)**
    5 A3 X2 h# t! o/ c4 i& r) }0 C   - 将原问题分解为更小的子问题
    6 n; t6 P) F7 @# m: R8 m   - 例如:n! = n × (n-1)!$ B. u& \( [% ~8 W+ E) ~0 e) Q

    # ?0 |7 Q- @/ M( j. M 经典示例:计算阶乘" c* }9 \, H& t0 ]7 X: G! `
    python
    ' o8 w7 _% R, l7 ydef factorial(n):' ^* n/ P# Z) O$ ?
        if n == 0:        # 基线条件
    . W: M7 k7 ?" f6 c5 h. B7 T( X        return 1
    5 O! n9 G( a( f& z8 V" l# s    else:             # 递归条件9 Z8 }2 K5 q. b: n3 w
            return n * factorial(n-1)7 K7 h& i' T) O1 x- ~. _4 B1 Q
    执行过程(以计算 3! 为例):
    0 t* m  q9 t. k6 P6 S! {3 m* c1 N& T1 sfactorial(3)
    9 G2 K& U: B) H; g! }2 w3 * factorial(2)' E" N) p! u8 A# P9 _/ D( S! O
    3 * (2 * factorial(1))
    ) a* ~6 U+ w- H3 * (2 * (1 * factorial(0)))
      N  u9 n: C" ~' d  _# m3 * (2 * (1 * 1)) = 66 g0 [" h0 A( q
    . N$ z' |6 C1 r
    递归思维要点7 q' K6 b7 P, F
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! m& w( B: t% q1 J+ K. m2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    - T! M9 p! @  ^  I3. **递推过程**:不断向下分解问题(递)
    8 j; p% e1 U+ u: x4. **回溯过程**:组合子问题结果返回(归)- g  p7 G  W4 t; Y3 K

    # `) x" I0 M  J2 p4 f6 M+ f, X& N6 h 注意事项; W  @9 S' O/ K% t+ ~8 `
    必须要有终止条件
    % Z3 ^* K1 j. ?0 y递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ! f" ?6 e+ u1 V) B' H某些问题用递归更直观(如树遍历),但效率可能不如迭代
    6 g5 _* c9 X* ?4 {+ ]8 {# I1 y尾递归优化可以提升效率(但Python不支持)
    ) ]! T4 y4 I; N) Y" j' s6 Z$ B
    " I: z: k- q  R 递归 vs 迭代
    9 E* M( x  M* d  K( D|          | 递归                          | 迭代               |
    3 M' }: E9 e8 [' ||----------|-----------------------------|------------------|
    " ^" p$ [! w( L9 K, R| 实现方式    | 函数自调用                        | 循环结构            |
    & L( q/ B: F6 D| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |% y6 e& v7 q8 g% V, r
    | 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         |
    ; |& Z5 b1 n; U+ G4 ?# B3 `7 L| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |- v4 U8 n: @% g9 |9 B! J

    1 I' ]5 m2 q! e, @0 e: x 经典递归应用场景3 V" ?3 ?( h$ ?+ U' Q0 }
    1. 文件系统遍历(目录树结构)
    + y4 Y9 z) Z- ~/ K$ U% {6 |. Z" x$ F2. 快速排序/归并排序算法* C& d6 `/ E# b, `" s
    3. 汉诺塔问题
    - N) N! c& q$ e9 @) O$ P# \4. 二叉树遍历(前序/中序/后序)
    % d& a6 U0 C. E8 L" B* N" B5. 生成所有可能的组合(回溯算法)* o* k/ r2 Q% ?

    " w" N2 p: |( Z# }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,$ j% I! j% d* X7 \6 q5 U
    我推理机的核心算法应该是二叉树遍历的变种。
    ; z3 C' P( |6 P2 E0 h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:: q! ~' S8 }5 l% ]0 a% H
    Key Idea of Recursion
    , z- x! F8 j* @4 m; ]# n; O" d3 V/ v. B4 S( W# `: }5 A3 i
    A recursive function solves a problem by:
      ~" v1 ~# J7 ?5 B  D0 w' A! C# }0 [: N: h1 N
        Breaking the problem into smaller instances of the same problem.5 x# F# A" M4 P2 l; P2 K+ g

    ; {# _+ i$ \. Y- f/ [. P    Solving the smallest instance directly (base case).( c- I8 l- W9 h7 X
    1 d7 D4 j8 V3 N2 O
        Combining the results of smaller instances to solve the larger problem.' l6 J  W9 v: I/ Y. `! U
    & L0 b  T9 [' O# i# D% t
    Components of a Recursive Function: r1 |5 D& I9 E7 i! r) X5 }
    ; Z0 m+ K. u1 P6 \7 ?
        Base Case:- I, W5 m' V9 W8 D* k

    4 E$ X$ u% Q5 ]9 O$ V! i        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    $ `1 s, W: L. a% {' t
    ) C1 G* {: w  s* \! T        It acts as the stopping condition to prevent infinite recursion.
    $ O) ^9 G: c) ]1 ~5 ]1 J( z
    ( F1 D, e7 r' I5 x. R; [        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    1 T8 ?) \9 l% [& g
    . Y2 P/ K) @. X0 Q- O$ l    Recursive Case:
    0 e, F4 U, r9 l0 L
    % k7 x' [6 ]2 q' }; n) C        This is where the function calls itself with a smaller or simpler version of the problem.
    1 Q% `$ x. C7 Z1 S1 N8 P7 h3 P) ]: @* c( D+ F
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).( ^+ i7 q6 u! O# g
    2 j; E: E  u7 i  u) f- K4 D# |5 H
    Example: Factorial Calculation
    9 D8 j: {1 r& M- O  \
    : s$ ]- T3 i( W6 kThe 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:1 C- c3 O; `8 g0 |5 N1 e2 o4 E
    0 `. r: }% r* V% B/ p0 C$ }3 T2 a" i0 x
        Base case: 0! = 1/ {! A0 k; K: q% O0 D  b' A) d5 h
    # b2 x. W, Z( t8 \0 c6 O3 O
        Recursive case: n! = n * (n-1)!
    ; C7 P/ o4 d2 Z; `! O9 E2 Q: w! P% i  q% D) h1 @9 ~8 D+ D
    Here’s how it looks in code (Python):! f) D$ ?$ r# m$ e2 |- u
    python
    - H: q: m. Y4 t# f) i, ]7 t
    $ P6 T8 c0 d) M* \: \6 N( S! O) C
    3 o: ]2 X/ z0 F- v* Y4 Bdef factorial(n):+ K) w, }; R; b
        # Base case- @7 g, O( _7 Y9 B
        if n == 0:
    ! p! c* r" _  N        return 1" h" G* |- H% O* {
        # Recursive case
    2 j3 C! a. X/ j/ M6 [+ U    else:
    + [- v- G  F# n        return n * factorial(n - 1)2 O$ m0 W0 M" i. \" A; r& h% C3 e5 U

    3 W9 h, q. n& j2 b# Example usage+ G7 t" M9 {( U! M/ j; C
    print(factorial(5))  # Output: 1209 d4 ~$ A! T+ k4 D) g
    # q8 l- j. a3 n4 H# \' C" }
    How Recursion Works9 E4 m' p3 W- O' U4 `" J. j

    6 o7 U" L1 `2 t, D% l    The function keeps calling itself with smaller inputs until it reaches the base case.
    0 {7 L% ?, }* P1 N2 p1 |) `+ E& X) `4 @3 {8 V
        Once the base case is reached, the function starts returning values back up the call stack.) z/ |3 R  B! c) Z8 m- H
    $ W/ O" b- w+ i$ w# A7 h8 L& m& G
        These returned values are combined to produce the final result.
      z- R5 ?" X. g# `& ?9 S% c& M3 U, T3 `
    For factorial(5):
    5 C" z- `0 x9 c6 N( C- o
    3 B8 w, c! e0 ?  ?
      d7 J$ X" h" |' p6 tfactorial(5) = 5 * factorial(4)8 }/ H, w! p; u/ d0 u8 q5 N4 I7 \6 v
    factorial(4) = 4 * factorial(3)7 r: V; F0 \: @, H
    factorial(3) = 3 * factorial(2)& m+ P  Z# P; o3 ?" |$ I9 n: E0 F. [( F
    factorial(2) = 2 * factorial(1)
    + _4 M$ h  G3 Y) u& ^1 Y* Vfactorial(1) = 1 * factorial(0)
    * A/ w0 \1 [( E7 b! z! t# G+ tfactorial(0) = 1  # Base case" |9 {. T5 _, ^7 G
    7 ^5 ]% V" q* `/ R9 f- O
    Then, the results are combined:8 w, M2 F/ y8 s! V( m

    " k: [! }+ N3 K% V# X( c% a
    & \& ~8 f" k0 s5 ^: h6 Ufactorial(1) = 1 * 1 = 1, o9 f% w7 V- I2 g/ B: i
    factorial(2) = 2 * 1 = 2
    / F. O, T  `2 T/ Bfactorial(3) = 3 * 2 = 6
    7 c& s* z) b/ M% s0 xfactorial(4) = 4 * 6 = 24. u( w1 @1 ?) I% }; W! z
    factorial(5) = 5 * 24 = 120) ~5 m) L7 \0 s& ~) t
    ; F0 h! |& I: k! Q
    Advantages of Recursion
    ) q$ @7 n  K, l( ^7 i/ s. V, W, ^" D5 g
        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).% u5 }' |7 T2 K2 e0 n: u
    8 ~( Q2 w$ _- s4 g
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ; M) X2 h# }, X" p, a; L
    $ J! v) ~* m; T( a9 D) Q& ?4 }* T8 t8 uDisadvantages of Recursion
    $ G% Y7 U) g  M2 h8 f/ `9 N$ V! l! Q# \4 Z
        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.2 I8 o0 l2 V1 O' J' i. e1 [
    ; d. F7 G1 X9 ?! }$ S# u0 g
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    . _! _# E) b9 x" g
    % B3 ]0 M0 k5 \+ oWhen to Use Recursion
    # Z3 |9 y6 x7 r+ W  l: t3 Y9 h# E, J& g4 G/ V
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)., s. i8 b% {* w: s, t

    9 t# d8 _' g! `    Problems with a clear base case and recursive case.
    * u( d5 t# o8 A# Z) \- l7 t$ l; i3 c  f- Z" G" A
    Example: Fibonacci Sequence) ], a% o0 Y2 c' m4 S5 {
    - G' y; ]% P& U  @- W" k
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:; S  O3 z; F9 T4 z" I

    / J% }8 d( B0 O8 j' b* M' X9 j% e    Base case: fib(0) = 0, fib(1) = 1
    : O# }9 g5 h7 ?3 l2 j7 }+ _  x2 e  v5 B, G! q' o
        Recursive case: fib(n) = fib(n-1) + fib(n-2)* y' _; l6 N% r. ]! x; m6 J

    $ V9 v' O/ P6 w; dpython  l, z; R+ D3 J8 @9 k
    ' p+ e% P- g% i; ~

    & j, Q3 n- O, K$ Qdef fibonacci(n):
    / e  e1 v8 K4 A8 W$ ?" u' O8 D    # Base cases6 v  n# R! S) f2 f* L& |
        if n == 0:3 H7 ^. l8 j8 \
            return 0
    ' Q% Y1 S- u/ S; ~( {# s3 u    elif n == 1:* w" K" w( k6 c9 ~$ E# x
            return 1
    " m. s7 b& n2 s9 T    # Recursive case' Z0 v! A+ x. E
        else:
    ; c: j# {6 L5 r  G2 G, z        return fibonacci(n - 1) + fibonacci(n - 2)
    : v/ v3 {: b2 P. H) q+ L6 Z- Y
    4 N1 I' B& @6 g: F  i/ b, d# Example usage, a- r( X1 [4 n) S4 d% N9 j
    print(fibonacci(6))  # Output: 80 v! e( B' c/ V6 W$ a+ A
    + g* u. K; A* E: A( W. `* N
    Tail Recursion6 ~" H! Q8 B  b* ~1 x: a, Z
    8 C6 o8 U2 d7 t6 {. y* c/ b. U
    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).3 p$ W: V: o3 O9 D: t* y# n3 }
    - {( R4 y& _# L! G. K; \: V/ o- 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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