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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 ' ~% N" n" y( Y" J3 e
    5 Z, Q$ ^5 ?  J$ c  p
    解释的不错5 |6 D( p$ M& a2 V, k9 w

    , ~, T: e- [! }' X0 \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。9 m& d4 e  o" o, \8 T' {" D

    . g3 q5 g8 f: R, L9 |9 A 关键要素
    : v) j! |6 b% ?% v7 o% |" u  l1. **基线条件(Base Case)**) I+ c% H9 Q2 s
       - 递归终止的条件,防止无限循环
    # O4 N' A' ^4 \. m   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1$ j+ Y( G; Q( {, F! |5 b

    + d3 d) g% o) x6 e) Y) v$ f2. **递归条件(Recursive Case)**% w: D+ Y4 [# ^6 P# B/ _
       - 将原问题分解为更小的子问题
    7 g5 c3 @3 y6 M9 X8 o: z   - 例如:n! = n × (n-1)!, {; e8 F' @( G4 o; z0 j

    . l8 f' g* T( \" ^ 经典示例:计算阶乘
    7 u  b7 h1 C* p/ B, l0 o5 Bpython6 _6 d9 e4 V6 \( t: l7 v
    def factorial(n):
    9 D- _5 x, J0 E, c: S    if n == 0:        # 基线条件
    - @# x; b0 t# ?+ D" T        return 1
    8 ]' p3 y0 M' R( C" B    else:             # 递归条件
    1 d2 K) y% O( @2 x! W% J$ v        return n * factorial(n-1)
    ( f! Q5 v3 ]) J" P执行过程(以计算 3! 为例):! B2 a1 h8 o# S" i
    factorial(3)" \+ t( r$ ~$ o  ?1 d% ?5 q
    3 * factorial(2)  C3 G, ]* t8 ~) p6 F
    3 * (2 * factorial(1))8 O8 Y+ v0 z8 g1 j! O
    3 * (2 * (1 * factorial(0)))
    , @+ }2 C) P3 W' ?& I3 * (2 * (1 * 1)) = 6
    7 T2 T4 F2 @' s; X7 }; l1 Z5 n& t- T5 r- ?/ g9 e! c
    递归思维要点
    ( {: [- |- B" B& Y1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : ?/ D: d0 h/ I6 _2 G# m# _2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    8 h7 x! n* m& }5 q/ ^3. **递推过程**:不断向下分解问题(递)
    / s* O. B2 E; q3 ]# s: n7 L( z4. **回溯过程**:组合子问题结果返回(归)1 i" |  z/ d* p5 B7 a
      _. c' t% @" {  Z
    注意事项
    # J1 h- a4 d$ t3 o6 E必须要有终止条件/ q, m5 e, d) E  ?
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    % ?/ I6 O; S  e某些问题用递归更直观(如树遍历),但效率可能不如迭代: u9 |: n- U$ ~& k
    尾递归优化可以提升效率(但Python不支持)) d$ R" q+ j: T5 O2 i9 S
    ' R7 r8 v/ d$ ?5 b- T& P
    递归 vs 迭代. c- _# q) @& j) \" h5 i2 `6 P
    |          | 递归                          | 迭代               |
    / T% s7 I$ d$ n# {; l|----------|-----------------------------|------------------|
    1 b5 s9 O( ?* J$ j- n| 实现方式    | 函数自调用                        | 循环结构            |
    ( v7 W9 V: x, `  W; ]. F4 y| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    6 o4 ^+ a% \( i/ @- ^| 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         |9 R- B7 h' y$ a) a$ x+ o( ~7 v
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |2 \' r! o! p3 J

    3 B; `; b3 K. ` 经典递归应用场景* ~. \, P0 \2 m! [8 Y% ]! V; V8 S
    1. 文件系统遍历(目录树结构)
    ! @) \* L: y' Z3 W! i9 X4 H2. 快速排序/归并排序算法
    8 T! K5 ~  h/ S: U3. 汉诺塔问题
    " w/ o  ~# q4 U& q$ `6 C4. 二叉树遍历(前序/中序/后序)
    8 o; \5 [1 v. o3 _& Y6 {5. 生成所有可能的组合(回溯算法)7 F' N8 f$ G9 u" N, M3 c2 j! t+ @
    * _9 \; W. L! b- t1 c% u2 N" N3 P2 ]0 V* L
    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 M0 g$ J) ^7 G1 ]+ P& I* s我推理机的核心算法应该是二叉树遍历的变种。$ P! y) j6 C$ l) 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:2 f: ~$ P# J( ^) I
    Key Idea of Recursion
    4 e' k8 u3 T  m1 u& O- Z9 p0 g" Q* @5 X/ `, n
    A recursive function solves a problem by:+ b. W4 w) Y4 ]+ X
    8 c+ I6 |- q, ?4 i/ U6 D, g
        Breaking the problem into smaller instances of the same problem.! {8 `* E5 [1 ^; C- H

    $ n9 K0 M, N6 m2 h$ O3 u    Solving the smallest instance directly (base case).: {" O5 Q% R0 _# B: M9 R. X# F
    1 c  C1 V' N1 \; N
        Combining the results of smaller instances to solve the larger problem.
    ; E7 I' S' o6 R2 Z  R; S$ P$ i! G8 n0 d+ q
    Components of a Recursive Function
    4 x# A, f& d7 h# d
    & q% Z; @; J' [/ q2 _# q$ u* T    Base Case:
    0 L! C, q; l  X8 k+ g# \# M5 d0 A$ V' T6 i4 h
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; m5 K* w) W, T. m- ^, F: }8 Y$ G( y* Q+ D. O5 U7 ?8 T2 I$ q8 G
            It acts as the stopping condition to prevent infinite recursion., J! l) ~7 I) r7 o' x" o

    ; P. O/ l! @1 V& L' t        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    0 I8 j2 h2 R7 h, F( L8 j" P7 N5 v
    3 F9 G; B5 M8 g2 r0 U+ ]! w    Recursive Case:" [0 M4 |4 }# j& y( J
    6 ^$ ?, n/ w. P7 M( O
            This is where the function calls itself with a smaller or simpler version of the problem.6 A8 C! e& S3 ~2 f3 j7 b

    4 c8 m( ~$ C, @7 W1 O+ ]* g        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., X+ |6 _* f3 t/ S. l/ e

    3 O: t1 w" P- A! x  \9 d9 F- cExample: Factorial Calculation# {' x( J5 @- f2 S5 Z2 Y  e  Q! v6 u
    2 m  n0 s' _5 N, F
    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:
    3 }  T( F2 Q+ P' J2 K1 D
    . A0 t5 B$ @- `7 h3 h1 ?8 _    Base case: 0! = 13 N/ ^" @' ~% b2 N# Q

    - N3 N- e6 H9 o# T6 d    Recursive case: n! = n * (n-1)!  |6 h3 E2 l3 O1 o- E& t2 n$ p

    ! a* d( L0 {! b6 q% ~0 k) \7 oHere’s how it looks in code (Python):6 G4 i9 m2 G1 j( T. Q0 F/ h
    python5 a( P3 W; d+ P  e  {
    3 b, G* [! z3 h3 s* c# h

    & d8 Q/ O/ s. _% j5 M* y  wdef factorial(n):
    . l2 i2 s4 ?) o# T* u7 O5 |    # Base case( _' q9 G( ]1 W7 S8 p
        if n == 0:0 Q* i7 g5 V" \' j, ~
            return 1
    2 X4 j9 d' u3 ^, Y7 _    # Recursive case
      S) J0 k3 g7 m0 u    else:$ K' |% o: k5 V6 ~
            return n * factorial(n - 1)
    * o1 g! x! Q  s$ I, P
    . P$ c) J3 k: Z( |+ [# Example usage! `0 ^3 H+ Z- i, h
    print(factorial(5))  # Output: 120/ n8 x) C, B) z0 c' b) F' Y

    , ?- t6 a4 h* P# z! q9 A; UHow Recursion Works
    $ R. x5 D5 h& c6 c; P
    8 C$ Q, O5 G- ?; s    The function keeps calling itself with smaller inputs until it reaches the base case.7 L+ S$ s& [" q; i- X0 W0 s) ?
    # ~% C9 {& c" `+ v5 D! j
        Once the base case is reached, the function starts returning values back up the call stack.7 S! E9 Z! L0 X& u- {" c
    + c2 y4 v5 J% B9 l7 g
        These returned values are combined to produce the final result.
    5 S4 I  T- z/ |3 S
    ' l8 h& N0 ]1 M: t- h+ ?8 zFor factorial(5):  T, x# H' ?- o* r( c+ X4 W

      W' l; U$ b0 R8 H* ^& z1 I
    ) K2 O: ^. T2 k+ l+ v0 \factorial(5) = 5 * factorial(4)
    9 b8 _* |! f; A" A- t* N! G( Vfactorial(4) = 4 * factorial(3)6 b* H1 L4 ?4 `5 ~" u  i6 ]3 ?
    factorial(3) = 3 * factorial(2)
    2 Z" T: a8 V' s. Y7 |2 l  H1 T, Dfactorial(2) = 2 * factorial(1)5 W' O, O* g  w; U9 |5 T' o9 J$ V) g$ c
    factorial(1) = 1 * factorial(0)2 A0 j8 \, r6 j8 v# L
    factorial(0) = 1  # Base case8 j1 s8 s( C( Q$ d

    ) [8 X8 Q6 C. A9 m5 u1 g% WThen, the results are combined:: s* g4 Y- R% t* l1 ?

    - n4 c2 C# a1 i* b$ z' k% z; P/ u& M& x' Y$ e2 n. f
    factorial(1) = 1 * 1 = 1
    % s- z7 `/ h% b' t/ nfactorial(2) = 2 * 1 = 2
    . {5 U' h% F4 Hfactorial(3) = 3 * 2 = 69 a' d! K+ b. Q% |4 `& N: v
    factorial(4) = 4 * 6 = 24' V0 e& b0 g9 h6 P$ l8 C
    factorial(5) = 5 * 24 = 120& d5 Q0 L" ]7 N& {6 Z$ G( ^/ o

    9 N. b$ W) J2 L3 q9 a: yAdvantages of Recursion2 D  O3 o4 _/ @3 w4 ~" e$ q$ l

    2 I9 ~  H5 Q; ~' l( D! S7 E    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).
    5 s9 X# ~4 E3 Z5 Z- g. C7 U
    ! F; j, B  t1 T    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    - b# g4 s  @( [, \2 Q( z
    ! _0 J; V; H. [# K5 _Disadvantages of Recursion# h; T8 l* d) K) J) i+ y8 W
    ( ?0 _' ~, X6 |( W
        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./ r% U1 b! y* {/ P0 q

    % W) D7 o$ {- ?" Y    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    # ^% {( K7 U3 u2 B: V7 g' F& b. B1 c9 V
    When to Use Recursion
    % J: ?% F* ~9 b& v; _% l6 O( f0 I0 Z. K# b% V: K9 u9 m$ b
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).  o! N1 g; Z( K" L% ^, G

    ( R5 O2 ?7 ?7 I    Problems with a clear base case and recursive case.
    8 S4 q- ]+ K  ]' K9 C3 h! v; J6 \! K8 p& g6 D
    Example: Fibonacci Sequence
    3 G4 V% Y1 W& s: ?9 }
    3 m! i7 b( m- H' O. o3 m0 [& ?2 f& pThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:5 j# J( t' s& s- o

      U: M# s0 C! X( N! h6 n0 ?5 {    Base case: fib(0) = 0, fib(1) = 1
    6 A8 C5 I1 N- H7 v7 o( n' [, I9 }) h
        Recursive case: fib(n) = fib(n-1) + fib(n-2)* d: g; H+ @; m5 L! Z8 G2 g' s
    % m$ E: |2 x6 T% B4 ]; F
    python
    / Q0 I/ L! }0 f; n" B1 i  z8 `# ]8 l6 {0 t& O8 \; `- ^$ Y
    0 \- ~5 @: f: o) j
    def fibonacci(n):
    5 ?$ x/ X' @2 d! F& \) P    # Base cases
    6 t0 q- j. b  P6 n    if n == 0:
    # Y& g5 {5 A0 s  H        return 0
    + j/ _# t) ?5 ]7 y2 ~' n! r0 n    elif n == 1:
    ; Y7 s2 P+ E7 U* r$ i& _: p0 m        return 1: P. I5 w. W3 _5 S8 ]) Q
        # Recursive case
    1 ^2 ~) h5 L/ m% u. N0 G/ n! i/ c    else:$ Y& r0 ~$ J' ~% N7 ^1 ]
            return fibonacci(n - 1) + fibonacci(n - 2)
    1 F: R3 G5 V- b+ H, |# N- Q' t& a' D! b% K+ s) p# a
    # Example usage9 B. B/ ]' t% N5 z+ u- c, p
    print(fibonacci(6))  # Output: 87 b/ R9 R' F8 R% s8 d2 Z5 ~2 v
    ! t. V& Z& o* ^. V2 U
    Tail Recursion
    ) ^2 a. R* {; x+ x6 ^% ]! f
    ; T# m" P" l# l. z. d8 f* GTail 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)." \8 p1 D) h# ~6 b$ ]- c" y& x2 M

    1 N' Z0 ~% `; |/ B4 x+ xIn 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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