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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| 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
    : L: A0 g: X5 T' P6 D2 e5 y/ P% [; r+ k& ?7 z
    解释的不错# b( k" \( p% b; ?

    $ x0 V& t  A' u: e' q- h! q: 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。$ x$ Q& K5 X- Q+ b$ W% E% f
    8 Y( @, D' V  I  M: N$ i3 l
    关键要素
    ! K5 s* M+ o# i! [1. **基线条件(Base Case)**
    2 Z. K4 w6 u) }: O  w   - 递归终止的条件,防止无限循环: J9 k  Z& ^  X
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ) O7 s1 Z% d1 z
    7 @6 s3 M; u' u  Z" K4 a* l- V" x2. **递归条件(Recursive Case)**
    ; {  S- S$ {$ i3 ~9 |5 y" M   - 将原问题分解为更小的子问题
    * R; w2 ?& v/ k& w3 J   - 例如:n! = n × (n-1)!, c' ~. q* M, |( g# l

    5 h0 x2 j0 A: W" f. k# S' G5 d 经典示例:计算阶乘+ m) j! Q8 ?# M3 x$ h/ O9 R0 l
    python
    2 Q) M& e: j1 e7 a  K; Xdef factorial(n):
    & K! Z7 B5 }# t0 Y* i    if n == 0:        # 基线条件" z! ~$ ^3 _1 k
            return 1
    ! s2 L2 v6 ?0 h. ?8 M. ?  K% D    else:             # 递归条件
    " R8 a/ H4 P$ p- p# }        return n * factorial(n-1)' w, t: I, F" J4 P. Q6 ^
    执行过程(以计算 3! 为例):
    $ X; i3 N( E2 D" R3 ~9 ^factorial(3)/ n) e3 k+ d$ D- z
    3 * factorial(2)# R8 t! [' u# A+ ~/ t
    3 * (2 * factorial(1))
    / l( x& A2 D) |9 z# P4 [. @" v3 * (2 * (1 * factorial(0)))) v) t% s/ f, a$ E, X' J/ O& E
    3 * (2 * (1 * 1)) = 6" ~4 U7 k* N+ r$ o  S  Y9 N

    : h9 I% I  A: C! Y 递归思维要点+ f3 l0 v3 H1 c+ g  i
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    " G  S. k; d. L/ I4 H7 I2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    , x/ s9 {& l5 C# s3. **递推过程**:不断向下分解问题(递)1 k, ~# L8 y' Q! g! P+ B( o" X
    4. **回溯过程**:组合子问题结果返回(归)
    6 Y3 D" ?/ J. b9 g5 M# O7 h
    2 X! H0 \- {# I7 A 注意事项0 o4 M: o8 ^0 Q# z5 o/ u
    必须要有终止条件. @; `, F- I; N3 L# {/ O1 H
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( R8 p2 O+ ?  T; J; Q- {# [/ |7 l$ v- v
    某些问题用递归更直观(如树遍历),但效率可能不如迭代+ [, j) ]6 D( D
    尾递归优化可以提升效率(但Python不支持)
    8 V) R: D* c. @. g$ L% U" G# [; a* x" R, |9 ]  E6 M
    递归 vs 迭代
    4 x( k7 `; o- L- N|          | 递归                          | 迭代               |
    ) m6 S) L# L: U2 C7 _) Q|----------|-----------------------------|------------------|
    # Y; L5 D, B0 I| 实现方式    | 函数自调用                        | 循环结构            |
    8 i" U6 w% A, |5 ?1 \: F* d| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |! }# |7 t2 n% e( [2 d% H+ N
    | 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         |/ P  p. d% ]% t% p  @6 c
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |/ D+ i* [- {! D

      A* \1 h" B2 D, a0 g 经典递归应用场景
    7 w& a) K5 p# ]$ N; z1. 文件系统遍历(目录树结构)
    ! v7 \1 }+ A+ g) t! _2. 快速排序/归并排序算法7 ~( ?( ?, g$ b6 }0 k" ]! j* F6 o
    3. 汉诺塔问题
    $ n% u. q! \! N/ K- A4. 二叉树遍历(前序/中序/后序)2 T  C: h. n$ B5 u# X! e; {
    5. 生成所有可能的组合(回溯算法)
    # m8 D8 |# k; `9 H- }( S# _+ z) f: }, Y) a
    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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    郁闷
    14 小时前
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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,9 l; N. L, q  `7 G7 m3 |- L7 L
    我推理机的核心算法应该是二叉树遍历的变种。
    - S+ j% w+ p5 [) o: z7 B( m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 Z: |3 T) h: A# z
    Key Idea of Recursion7 q3 d: p. ?* s* i% P1 |; o

      C" z: K1 Y" S. k: @2 DA recursive function solves a problem by:+ E5 p" {7 F* K+ f! h: l! L" T
    - T! G0 W# R) w" F
        Breaking the problem into smaller instances of the same problem.
    $ C$ G. G* I: g8 n: \/ h3 [' P' v
        Solving the smallest instance directly (base case).
    % C- t, M9 |. U8 D7 F$ [# d: Y& w# S
        Combining the results of smaller instances to solve the larger problem.
    3 b  O7 A' \& \/ B) a' ^' x. W+ n; c$ q8 A. N
    Components of a Recursive Function
    4 H3 ~: S& Z. [
    2 i; `% R( F2 G    Base Case:
    7 ~5 Y* B4 C8 N+ p# d7 S/ [& _, l6 i; q7 ~9 [4 E4 X. T& C
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.0 D9 ~  ?* h- P
    ( P- Q' i  o! h8 n$ J6 d- F1 l
            It acts as the stopping condition to prevent infinite recursion.* h9 h2 b0 @( i* L. R
    8 f* R/ \( T$ n! F$ z( e
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ T3 j: @9 @' d/ E' c7 C
    % s# z/ t7 S0 K3 O# p* D% m+ c* h
        Recursive Case:4 Z& {9 {! U" G* }$ d) ~' {
    # i/ m' w6 p- D
            This is where the function calls itself with a smaller or simpler version of the problem.0 s. B+ u  Z) _2 |8 u) P

    - a& e" t+ B5 E# |  P        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., ]' d: J+ K/ r1 B5 f
    & T7 p0 i5 u. J5 x9 V0 i9 E
    Example: Factorial Calculation1 B# H' J6 P; o9 U/ a/ M7 e$ R
    6 x4 s! J( Y7 Z8 a
    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:
      R# r0 b' R$ ^5 {8 h7 \7 j4 @# p! J1 K# ~5 y
        Base case: 0! = 1& R0 V9 j4 x; o  O9 I
    % _& f& [% O$ \, q% A' q* ^
        Recursive case: n! = n * (n-1)!
    3 F% |0 {$ g0 M! G, J9 y" L; r; f9 q: _1 Z. `# H6 t
    Here’s how it looks in code (Python):
      D! D2 E9 m( \8 [! |5 k2 Bpython. d: x- C' P' H; [% g! ~$ w
    . ]# V9 x; [% |' k/ @0 G6 _

    , {! Z5 }$ t  D) \9 L* L5 tdef factorial(n):. o: D7 D+ c2 B. ]
        # Base case
    / q8 b0 s" h8 @- J! V* N' v    if n == 0:1 Q8 j& O$ X2 x& Y( r! G$ {8 O
            return 1; A% i. l0 D; z- [* N! v2 [2 e
        # Recursive case" J+ |* [+ A9 N% j1 j" L
        else:' L8 q9 X" `1 G. Q
            return n * factorial(n - 1)
    7 l" H" m) Q1 x: I+ a2 _0 p- W2 d' G# {. l
    # Example usage
    5 `. _( s  q, D# o! Oprint(factorial(5))  # Output: 120; [, n; a9 T/ p0 p& w

    ! X; k" J5 B( ^2 S3 RHow Recursion Works2 I0 b. ]3 A+ {" l% i: S

    2 ^# ?) v$ q, A. b  e- n    The function keeps calling itself with smaller inputs until it reaches the base case.
    & A" M2 k& ]1 ^; x# o: X$ v+ _: M7 M4 T9 U% D3 \1 H; k
        Once the base case is reached, the function starts returning values back up the call stack.
    0 g6 I( @7 c3 ?/ P  C4 j
    1 [% r# T' I( w    These returned values are combined to produce the final result.
    1 l4 K: a/ D, M
    & R. ?/ y) n1 Y% Q9 {For factorial(5):6 i7 D7 |& s  o

    ( M/ y( y& H+ D6 `" |
    : K+ D$ s  p* E$ Ifactorial(5) = 5 * factorial(4)8 j" C0 g& w% w! R; F
    factorial(4) = 4 * factorial(3)( ^/ L& G; S  {& v8 N% H) Z/ J
    factorial(3) = 3 * factorial(2)
    / Y$ r! E/ w. Tfactorial(2) = 2 * factorial(1)
    ; v. T" K) A. V4 wfactorial(1) = 1 * factorial(0)
    ! T5 O- u7 }2 _# @' D4 N* nfactorial(0) = 1  # Base case
    4 ]0 U; e0 i* ?6 f7 n+ ]  W
    * ~4 \! R% v8 {4 H7 DThen, the results are combined:* A" h- m. m0 A) q

      _/ z" E* y& ?0 S9 `, e  a6 e  l0 X% x4 T. d) m" y3 P
    factorial(1) = 1 * 1 = 1
    5 r; q! I/ a$ ofactorial(2) = 2 * 1 = 2# a8 m$ Q. ^$ {3 s/ s
    factorial(3) = 3 * 2 = 6& E; T  D# V, _& {9 c# i
    factorial(4) = 4 * 6 = 24- |& O6 q0 P! c) q, l
    factorial(5) = 5 * 24 = 1203 a$ k) j* s4 |# O

    9 D- l, w) s1 s' h# `  kAdvantages of Recursion$ b6 J2 z3 P9 [, L
    4 l" \/ V( p$ |8 h7 K" f" l
        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).! y- |( n( L$ X* e6 d+ B

    " e+ b/ S4 X3 h  _0 V4 K1 y' ~4 r    Readability: Recursive code can be more readable and concise compared to iterative solutions.: o( {8 `$ X  T' O+ u
    : s  C, ~1 g( [$ c
    Disadvantages of Recursion% m- c8 k% k: G6 c8 {: T  ?1 N
      E2 \" _" j# M1 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.
    & N6 G0 X4 D* I0 i- b# V- t, k2 a8 h, L$ @  A! y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 m% H" @' P, ]- [6 \- N

    " f2 \# f2 `: s: {; h# lWhen to Use Recursion
    / c; V2 o, U4 r3 w2 Z
    $ S' [- r' y' v6 O    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)., ]3 T  U: Y& H+ l  D. v
    & y1 g! a+ ]3 Q
        Problems with a clear base case and recursive case.
    + I5 ]9 S! Z- \) Z" |, {- w
    0 j$ S% ?3 ?2 f" G" Z, @Example: Fibonacci Sequence$ v$ m% h8 N. U

    5 [6 _5 S! A/ r# lThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:. x4 z( d/ B. `7 e; C4 @; w
    2 v0 o- E, l' j4 ?3 G
        Base case: fib(0) = 0, fib(1) = 15 H- K  O: h: @; p
    1 u; s. C$ g; [$ m% s; |( w/ r
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    / W' B7 a! p2 U% e+ A
    3 M& F5 D5 |/ r' lpython- s6 r4 N# v( t( h' x5 Q& M% x

    0 k+ U2 ^3 ]' F8 Y2 a/ U: F3 ^) I4 M0 ~7 r0 K6 r0 Z
    def fibonacci(n):- _& [! s& u" y) j# I+ B
        # Base cases8 b6 l' Q8 T0 X. r1 k
        if n == 0:6 h7 V0 y$ _  \2 t
            return 0
      M1 w; p3 U7 R& K9 b2 g    elif n == 1:* V" S( w* _! R1 m0 ~
            return 1- J( L& h' Q5 P: i$ Y% W; M# q
        # Recursive case
    7 ]0 l+ e- t5 B1 F& @6 f    else:
    5 T7 C9 ^9 @) T. W        return fibonacci(n - 1) + fibonacci(n - 2)( Z# z# B- P! r$ r
    / @% c( o, L2 H
    # Example usage1 ]' m5 y) Q5 J( s# E
    print(fibonacci(6))  # Output: 8
    $ d% N5 H" w4 w  n( a. H8 G! V& p- O" N7 e: D7 k8 U) d" p, }
    Tail Recursion6 c9 ]' d# z- l
    , C( J) n+ m: L
    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).
    . z9 s  L! k6 q& E4 e" h  V* x) a, G8 b' |
    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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