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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
      I% V4 i4 `# G: R" V' V3 k' r9 ?
    解释的不错
    : [* y0 n! n; S4 x8 q5 ^' q
    4 b5 m0 K7 D; S( E6 F2 V- Y9 q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。! [  p8 ?7 @/ y9 l- S
    ) K7 [" i  Q+ V5 c4 U
    关键要素1 i; A& j7 h0 ^3 I  A* T
    1. **基线条件(Base Case)**
    $ @3 _8 D/ G. k   - 递归终止的条件,防止无限循环
    5 l# z9 B) A! Q" m% O2 I   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    $ |: r9 A$ B: }# G9 `% j; Q- C( I6 d8 v2 Z! O
    2. **递归条件(Recursive Case)**
    % M$ K( T- G% C" M7 Q/ ?   - 将原问题分解为更小的子问题
    - m0 p% |8 o1 R  F- h   - 例如:n! = n × (n-1)!
    # `3 D% e' ]/ U5 d
      {% `- H* w. N8 m, i3 v7 K 经典示例:计算阶乘
    3 S5 U3 J& s9 lpython
    , [% k8 ?3 k& m- M; \def factorial(n):2 q7 i1 y4 e  O/ c
        if n == 0:        # 基线条件
    2 j0 u# S* E( D        return 1
    3 c2 I8 B: p8 q4 C% V3 }$ p    else:             # 递归条件
    7 l9 }, E# N  }* ^6 B0 m, N, T        return n * factorial(n-1)
    0 P2 p- |; Q: G$ Z& C4 P执行过程(以计算 3! 为例):3 A5 y3 J1 H+ O* D
    factorial(3)+ o" z3 O7 O; q" ^1 `) [& w- b
    3 * factorial(2)
    2 R" N6 f# u' |, Q3 n3 ?3 * (2 * factorial(1))
    4 a- s4 y2 P* k4 V3 * (2 * (1 * factorial(0)))4 F6 K; h0 @7 s; [; y9 W* j/ I
    3 * (2 * (1 * 1)) = 6! C# E0 y; z( a* K8 T' n/ S% {! K
    + y% E9 c( @, o9 f, d2 H
    递归思维要点
    : k! U, z8 L" U( _1. **信任递归**:假设子问题已经解决,专注当前层逻辑( X  i7 Q, b# w8 G
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' _* [6 w7 H2 p6 c1 V& E: @) i" ~
    3. **递推过程**:不断向下分解问题(递)
    - p! A" l$ G% B& S* h  Q- v  Y2 E4. **回溯过程**:组合子问题结果返回(归)
    ! G/ w8 t3 l5 k0 e: |. E0 m8 S) p; s: ^, y# U4 T6 S8 g
    注意事项. y4 p1 B2 }- Y, i' a( d6 @
    必须要有终止条件
    2 e; f2 G2 d6 j1 f: x% P递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    , w1 d! S, b2 u& }' L某些问题用递归更直观(如树遍历),但效率可能不如迭代
    # I: S% q  k( [! r' W* `尾递归优化可以提升效率(但Python不支持), W& ^2 y* y6 K5 v: r1 U0 K3 J1 q' N
    , y6 }, N. O; F7 Z- g: Y
    递归 vs 迭代
    * B$ z7 i& v( W: x|          | 递归                          | 迭代               |
    , [2 R  [8 n, {- h  a: b; h|----------|-----------------------------|------------------|
    . l" ^5 X4 M0 i| 实现方式    | 函数自调用                        | 循环结构            |
    / G) o' c8 x& v! w. W( w| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 U# G* W8 k$ o8 i! K' n2 R6 H| 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         |
    ; \4 G) H$ {4 H" H4 ~0 u| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ) `/ P5 z6 H/ V: C6 `
    6 \$ n0 p8 n" H4 v% K+ m1 Z% s( ` 经典递归应用场景' h1 Z: F( u2 U
    1. 文件系统遍历(目录树结构)
    4 {( A" X4 u7 G. f; s) ^2. 快速排序/归并排序算法
    * @) j6 O9 L7 f! J, E% R3 Y0 c+ F3. 汉诺塔问题
    4 ]% ]5 z# X* ?1 V$ }0 g* D4. 二叉树遍历(前序/中序/后序)' v7 [& j! K- T2 J
    5. 生成所有可能的组合(回溯算法)2 ~/ ?" u$ T% t8 N" j
    2 I' Y- [4 o8 e
    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,* K+ k4 g  k, d
    我推理机的核心算法应该是二叉树遍历的变种。1 y4 a! O5 i! s8 P
    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:' o( `# T' n4 j  L6 X
    Key Idea of Recursion& d6 `" m# N2 L! m; Z0 b

    $ I8 u3 @6 N0 s5 {, zA recursive function solves a problem by:9 I; h7 v( h& h' p0 H8 o
    7 ]$ D, r4 _- b1 t, T( o
        Breaking the problem into smaller instances of the same problem.9 y0 i) |; y" `4 r; N

    % h' f# `" }9 C, j+ b& b" P: ^* E    Solving the smallest instance directly (base case)., H( v1 ]1 l. ?

      |) q7 j* V% n+ H    Combining the results of smaller instances to solve the larger problem.4 N; s) Z6 q7 L- \6 U6 O

    0 X: h9 }8 b6 Z* J/ m9 \. dComponents of a Recursive Function
    . s% k& w! {2 L7 \, F. b1 j, E0 Z+ L5 V- u6 f% y9 O* R
        Base Case:2 q; |* Q9 u) ?, h$ m* `
      \7 p- N! n- y1 I: y; D$ F
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.9 i9 k6 I, M& [+ H5 J8 `
    # t$ l6 m- x* g2 O- H
            It acts as the stopping condition to prevent infinite recursion.* u7 f/ W4 v) n  c
    0 t9 h- b7 H4 ~- I1 K  f/ k- u
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    1 r# m% U: [, J# o# f. c& `2 ?1 W0 `5 q" k
        Recursive Case:
    6 S( {( R! \- Y' E9 v' F0 J
    8 |3 K( T0 J/ o  g3 G- B1 Q        This is where the function calls itself with a smaller or simpler version of the problem." N& y( Y( P0 W

    6 U) `0 i2 r8 }; b6 v        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).! w; z. X7 M. I! Z4 x
    ) e& r3 n$ `) Y9 v
    Example: Factorial Calculation1 z: {, {2 G' |! |+ }& X

    7 q" n! i1 X, O1 s/ fThe 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:
    ; e; c( T. H& H  ]6 g* C8 c6 h8 H& v1 D# L  f' L' o( k
        Base case: 0! = 1
    3 @# E( m3 y) M) ^" r+ G% Q
    ! \. [+ v* \$ H1 v5 k0 x" h- s9 M1 u    Recursive case: n! = n * (n-1)!
    # W0 K7 Z! @7 n3 P) R5 r
    0 Z4 v+ s0 t# }8 }+ R2 hHere’s how it looks in code (Python):
    & K" O/ E! _, E. F$ r  |9 bpython) i6 Y% t+ q) {/ P+ D. |% A, }! \

    7 P# s0 o/ I$ }8 \! [1 r+ b% q! a$ J2 s- I6 m
    def factorial(n):& X9 {0 N% n& L; Y) B  K
        # Base case
    . n" ^' l% f, ~; w    if n == 0:
    5 `. |+ D) ~3 t, l; t        return 14 u$ S9 |  E1 g0 ?9 z
        # Recursive case
    / J4 F, p* i! y) Q6 S    else:) [/ R( t& J, @  B6 X5 F! f
            return n * factorial(n - 1)
    9 Q/ P: A$ R! v- U* f* a1 _1 t' e2 a) G3 j$ t0 x4 O
    # Example usage" j' `: N* ]3 @2 r  b/ [
    print(factorial(5))  # Output: 120
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    How Recursion Works* C! j# V! I1 c9 `( ?$ {0 g

    + t; R( I! Q8 s- G    The function keeps calling itself with smaller inputs until it reaches the base case.
    . B( ^* |" U1 O" l- ^) |, C  y, O7 c, R/ s' `: I
        Once the base case is reached, the function starts returning values back up the call stack.. _% A# ]% P  ]! v

    # S# I( @4 Y' ~' M* C    These returned values are combined to produce the final result., O. |) K. `% G% s

    ; e0 G) L1 X. p8 J& BFor factorial(5):. j' s- B% C5 m/ B  E  G( \
    - k' _/ s  h/ }5 e6 m4 z4 X& |

    7 Q9 N2 Z, t- g% w6 v/ n5 cfactorial(5) = 5 * factorial(4)1 `1 @5 z2 n$ T& e
    factorial(4) = 4 * factorial(3)
    ! {, z% [0 J$ Sfactorial(3) = 3 * factorial(2)
    : @- U6 d: V) ~& Dfactorial(2) = 2 * factorial(1)3 l/ Q1 r# t! @+ f% n' B1 v& a/ R
    factorial(1) = 1 * factorial(0)
    % k, _; ]9 Y7 @9 `/ ?  b) G, @* bfactorial(0) = 1  # Base case
    7 c6 p/ F' q* @# w; {$ c- J9 p6 D5 X5 j+ ?+ h* p- {2 t) f
    Then, the results are combined:
    ! C( i% C- H2 c" R/ g) \6 s6 g& z4 K" y; K6 F* |
    ) [; H) c4 W* u% F
    factorial(1) = 1 * 1 = 1
      h7 {0 L& O, b/ k6 x3 efactorial(2) = 2 * 1 = 2
    6 M- {6 i6 X7 G3 _  y6 t: ofactorial(3) = 3 * 2 = 62 i+ A7 p% z: D5 D  P3 T
    factorial(4) = 4 * 6 = 249 G( k* l; K3 w* ?: P
    factorial(5) = 5 * 24 = 120! w5 j( m( g" B) ^) |

    0 v6 f8 i2 o, m1 d  A6 ^5 CAdvantages of Recursion" O/ B/ J  O) ?% \% Z
    0 o+ |( Y7 J& u8 X5 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).
    , E9 X) _" ]% H: N. r8 G
    & e4 ^" V" L7 N; @; N8 f9 J6 Y8 R( z; }    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    0 K2 ]) M" b- F0 R4 z& U" }: N. v7 N1 W2 W! H3 k; Z& |
    Disadvantages of Recursion
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        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.( w6 C' q" i8 o" Y5 F; b" b
      `: a) D4 k. y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    , M( V/ i. w5 ?% A* O  t0 f( u, _7 z. Z# }
    When to Use Recursion' H& ]5 q6 g' h
    ; X2 t6 W" B6 ]
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& K$ [$ W) l3 V9 P% o

    / {9 Q8 F) ~5 w' t4 k- T+ t/ T    Problems with a clear base case and recursive case.
    * O2 w" t  D# E! M: c. j; I4 U% F
    Example: Fibonacci Sequence# g* I3 v3 `3 {0 i

    2 w: \4 P( c6 H6 t6 `4 ?' b$ r. F! QThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ( A* p* _# z7 |5 w$ \5 K/ F) |. C
        Base case: fib(0) = 0, fib(1) = 1
    1 v% d6 b* Q' h  \7 Y+ J0 f  g' j- R! |( v
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    5 i8 Y) n$ D* i; g$ n2 Q; z- Ppython
    ; }; u3 |# ?8 L9 z3 }1 _& L
    . S9 i2 G, N9 V6 |8 A0 \) c
    9 L( }% A/ B4 U7 e; Edef fibonacci(n):+ u" J  G' S  F  N) s6 h. g
        # Base cases
    2 }4 A, P- q- R! H- v! x    if n == 0:
    1 w4 [# o, s) a; e2 n% h: Y6 {        return 03 ~7 Q5 i  j6 C! e0 [# t
        elif n == 1:4 _8 C$ H6 \( g0 }: p" p
            return 1  @  T  P. Z6 e" B* b2 ^+ h
        # Recursive case; J# X2 e# w5 J) o! S" W/ D1 r
        else:) g$ q8 n& b; Q# J4 }
            return fibonacci(n - 1) + fibonacci(n - 2)
    0 f* t+ Z( p, K& N6 r9 J2 y) f6 j( w' _( Z% W. ?/ t. M. |
    # Example usage) e' g0 \. |2 i6 C0 o/ [
    print(fibonacci(6))  # Output: 8
    5 I/ P1 e" P3 V% Z0 H  u3 d5 o  z* c
    Tail Recursion
      J! Z7 M: s: R: N" u, E6 I. |) Y# w& [& b, n
    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).
    : E8 E0 s+ p% b1 }
    2 Z2 J0 V% {# ^) X5 L* G4 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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