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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 0 F. d$ b* G  B0 s- G

      c' t3 `: @4 _" k3 m. W解释的不错3 I8 R3 j1 J" e" r. X

    4 |( Y* V. v7 X7 `, s& e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。
    5 _/ e! ]- y+ \$ A) p" S* v3 S6 ?6 ~+ x
    关键要素
    0 j8 G# X' n3 P" v) [1. **基线条件(Base Case)**
    , w. w+ c! h. L   - 递归终止的条件,防止无限循环
    8 t1 |" T6 B! X, V: _+ A9 G7 a   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ) L' I& K3 [. N4 S
    % M* _+ ^/ B4 S: ~4 {: u* z2. **递归条件(Recursive Case)**
    & `/ X0 l1 X8 N! T   - 将原问题分解为更小的子问题' _: p: v" r3 D6 X8 ~! P& h) t1 n, s* d
       - 例如:n! = n × (n-1)!
    / m$ a. c/ g. U! p, o, P$ W3 k6 D5 L3 d! q# N5 g2 O
    经典示例:计算阶乘1 Y% s8 X$ x' J
    python
    % w: @+ C, ^1 ]. i# M3 Ddef factorial(n):
    $ O% W( N: w+ v    if n == 0:        # 基线条件
    : H* x0 X% _& r3 ]0 {7 {* s        return 1' K; m# H( h: e7 ^9 w3 v' U
        else:             # 递归条件6 [  n+ b+ y  O6 _' q" ?- g1 Y/ J% n
            return n * factorial(n-1)
    4 p% D- U6 |& Y3 |: U8 a1 r执行过程(以计算 3! 为例):
    , P9 a7 b9 |* m( l6 r1 @$ ?% Ufactorial(3)
    * N9 @# _. q  [' \( x7 f4 s8 h3 y3 * factorial(2)
    , E0 Y; L- ^1 f6 G  |1 ^2 n3 j" \3 * (2 * factorial(1))
    9 r2 C  _# @9 u+ F- I# E3 * (2 * (1 * factorial(0)))
    & ^  G# E3 J5 ]5 b6 p" t: G) n; c3 * (2 * (1 * 1)) = 6
    " I( B0 h/ o. e2 [) r3 R; u# r7 N+ B) i- M
    递归思维要点
    2 p8 l5 _1 Q6 ^& \0 {' k; k9 Z1. **信任递归**:假设子问题已经解决,专注当前层逻辑$ ?; e+ ?7 V% [
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' G3 b- ?# D2 x; b, w
    3. **递推过程**:不断向下分解问题(递)
    ; x: R+ z8 x# _  ^) U: P, ]4. **回溯过程**:组合子问题结果返回(归)* c  V2 |- V; Y- @
    8 r& {# B7 ]2 l
    注意事项  Q! M2 C4 h' R9 X5 [
    必须要有终止条件: _. J7 N% b, E* Q) w5 B' [
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& h5 y/ T6 k1 |8 S
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
      n0 k' I& p" O( @尾递归优化可以提升效率(但Python不支持)
    0 U/ N/ ~( D0 E1 A
    8 R  s# t: r5 J, h- H 递归 vs 迭代! t# C% t/ ~5 a1 V
    |          | 递归                          | 迭代               |
    - T% N7 B% p  N$ o|----------|-----------------------------|------------------|9 @6 }9 a' i. G! p2 a
    | 实现方式    | 函数自调用                        | 循环结构            |0 c, _5 ~! t! \( {  I, c
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 _4 a1 f, C% w# P; }5 q
    | 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         |
    . I/ H/ q. G+ x) h# M| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    1 G/ h# {7 o% u: n
    ! T- A1 m0 @2 q" \7 y2 y2 `) @ 经典递归应用场景
    , I9 }0 R; x# Q) r/ G* q1. 文件系统遍历(目录树结构)1 Z; @+ ^6 A2 _% R) b: E; r6 Q; n! A
    2. 快速排序/归并排序算法4 c: d3 n9 w: M2 y1 t
    3. 汉诺塔问题4 C, ]5 R0 T: O
    4. 二叉树遍历(前序/中序/后序)
    * q- }- i1 K9 t: s5 ?. c4 G5. 生成所有可能的组合(回溯算法)2 }6 R! W+ i9 ]( S' {3 U, [

    5 W! P( O# z: c7 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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    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,( T' }9 L& i+ q! P
    我推理机的核心算法应该是二叉树遍历的变种。& S' T6 b# g) h. @6 T' `: A- ?
    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) F9 t. {2 R$ R& g( o9 WKey Idea of Recursion$ t: Q) t2 J. F# P8 @3 i5 U
    ! K% @6 ~5 b" N& E
    A recursive function solves a problem by:
    8 `& Z3 ]  y0 ~; c( g( i. ^" R: `* l. V) j" X
        Breaking the problem into smaller instances of the same problem.  U& P1 ]; @$ G  I$ T! T2 A

    5 H( y; l6 N- z    Solving the smallest instance directly (base case).. M/ E4 w9 j8 R% W+ {2 Y; k

    8 D3 L2 G% }; v. P" \- B  t    Combining the results of smaller instances to solve the larger problem.
    : x5 |# }7 ^# }7 s. ]7 B/ u
    # M, T. o( O+ _- C0 h# RComponents of a Recursive Function
    4 j8 L* ?/ C% b+ ?: M
    * c2 u# w" l5 h- @. {    Base Case:/ j  x! e+ ^5 v! H5 X4 Y8 O

    2 W" X" F$ B9 ]7 s        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.2 V0 Z- m2 [+ S! m* z# T3 `

    ' O5 z  S3 }, y" h        It acts as the stopping condition to prevent infinite recursion.
    3 V! Z6 ^( X. `- ~& M/ x# f0 E2 m5 q
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.0 i6 a; M0 O; l3 f

    0 N6 w0 i* y% Y0 F+ N; x0 O( B    Recursive Case:
    8 H6 n7 `7 s0 v) s* W' T
    ( C, X+ F+ N, R        This is where the function calls itself with a smaller or simpler version of the problem.' z3 l: ?# @! K1 k; \

    - d/ J: K1 J0 J! m        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    " E9 _( e9 d4 n  G2 L' _+ ]/ R; t1 L, `8 [
    Example: Factorial Calculation
    8 C- L* F- I4 y& Q
    3 {6 {+ F. C( f8 h0 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:! l5 A2 T7 |: ]: @# \3 D
    5 G. u( \% n: M1 G6 d* [
        Base case: 0! = 1! u) G" B- r+ ^& _' U% r2 W( q9 k
    0 m2 ^1 W- f( l$ @4 l! [2 N# W) B
        Recursive case: n! = n * (n-1)!; i$ r& `1 I5 k* I# `+ _+ ^! X

      h8 T( T/ ?, }- |3 h& sHere’s how it looks in code (Python):1 o5 }! `0 {5 R+ V8 {
    python
    ( J' \6 W! a8 m: ~+ s9 }' i9 v
    & i- {- U# F: v3 j$ s0 r
      R, k, D* q* e' v& qdef factorial(n):
    : N$ D% z! b1 g8 G. l8 P8 }    # Base case) c. e4 H1 c; v( g
        if n == 0:
    7 B# w" f1 \, c& A( |        return 1! C3 j/ Z1 m+ K# q" Q4 o& K9 g$ V
        # Recursive case0 }) l1 T+ S$ S, t8 s* l6 S4 O2 R
        else:' {) T0 h0 }7 u
            return n * factorial(n - 1)& N- U- ^* C4 U6 y( ]- J
    4 }8 E7 v6 G! n) {: a
    # Example usage
    ) j4 r' }0 E% Yprint(factorial(5))  # Output: 120
    ) P6 h5 p/ k/ n- ^) ?( R4 V5 G9 l! A" @/ }; K
    How Recursion Works3 F' V# X/ V0 ]) }& K; O' {0 \
    - x- @4 x# c* `' J
        The function keeps calling itself with smaller inputs until it reaches the base case.
    / J- C; G0 y8 M( Z* [9 g& K( f3 K9 j2 j  W( W
        Once the base case is reached, the function starts returning values back up the call stack.2 O. c, F: i$ m
    / H5 K( D% o. W: R. G7 I. L
        These returned values are combined to produce the final result.. o$ v  o4 h  ?/ \7 X9 z9 ]5 G

    7 X$ O% I& {6 }3 u2 W( n/ RFor factorial(5):
    0 _4 r/ D7 Y, u
    : y4 Z1 l$ T5 F5 q& ?) s
    7 c/ P( P' ~$ J- N) ?: b3 Jfactorial(5) = 5 * factorial(4)
    : {2 R- B0 F8 O( Lfactorial(4) = 4 * factorial(3)1 B2 K+ c# S6 k: c
    factorial(3) = 3 * factorial(2)) Y7 f6 `# Q* O6 A- g
    factorial(2) = 2 * factorial(1)
    # o0 m& q3 a1 ?factorial(1) = 1 * factorial(0): i0 K4 m; |. p
    factorial(0) = 1  # Base case
    7 ~: O" j0 I: y6 ~
    # ^5 ^8 \8 L' o' d8 X% L- i+ P5 h* gThen, the results are combined:0 D4 b; ~0 w* I( D
      D- J; l$ O" N; x

    % N' v% Q: S/ I; S3 I; nfactorial(1) = 1 * 1 = 1
    # |  _# P- a) X0 c2 s7 \8 L( H! ]factorial(2) = 2 * 1 = 28 m( @2 g" k6 Q3 S9 t
    factorial(3) = 3 * 2 = 6% q8 F7 o/ X8 A3 e9 F  U
    factorial(4) = 4 * 6 = 24
    8 v- j. |7 k. C& J* }factorial(5) = 5 * 24 = 120
    / C8 w& g- O& u* O0 N4 o4 q8 k! G
    Advantages of Recursion' }0 R/ }& c1 F  p$ d
    1 D& b2 R' k. U6 y* j1 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).
    & ?) n; k# U' F0 X. Q
    1 }8 B/ e2 c6 e* _; X* ]    Readability: Recursive code can be more readable and concise compared to iterative solutions.6 l; y$ M3 o) [2 S7 v4 f

    9 r5 r. C/ n0 a& n( B, fDisadvantages of Recursion8 ~& m/ L0 c9 N* f% X2 w; m

    ( N- _1 n# _, w* a1 ?4 I, |0 g    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 n- J3 j; x! A8 ?4 u

    % M4 v! N; `- I7 \5 u+ p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ( b6 t% j6 E' _8 W* Z6 R) D- r: a6 I; z8 L! D6 J" c
    When to Use Recursion  S: n4 G5 V; G2 `: j1 u
    - J- @9 i8 u3 h# K. r
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    + u0 \4 _+ I# O8 g' |
    $ O: F* S1 J- V% W. G9 E. [    Problems with a clear base case and recursive case.
    * i- [) O' k% ]% E. L- ]5 x6 V3 d. I% k' r0 @* w4 t
    Example: Fibonacci Sequence) E6 G* _7 u( b. s  U4 U9 h

    & d! K8 k7 Q3 C4 lThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:) E5 ^' n( R) v9 N( e

    # D" x& t# p& C) E" z9 @    Base case: fib(0) = 0, fib(1) = 1' m$ e( y) G% n0 J/ K$ q
    ( x' t, [. `$ ~7 d
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ' \$ X' P7 s* K9 ~1 D% p7 b* {
    8 I! F8 m; j: [, x( Vpython4 k' e( M; ~' U
    / ]( X. n6 s( `% ?

    8 v( l  ]9 V& L! W, sdef fibonacci(n):4 w. k. h  T/ R
        # Base cases
    ! ^6 I  E; w& ~5 A" f& m" Q# D" y# A5 z    if n == 0:, v9 f) G( e. {4 A9 k( r+ t1 X$ G3 V
            return 0* \7 q# z9 @3 Q: s2 A# g9 g( K
        elif n == 1:
    5 t" W9 H# Y9 t0 t5 P( ~        return 1! J, O$ ?3 V& @) k3 K4 s# ?
        # Recursive case, N  |1 m  N0 X/ n* X7 a
        else:
    & ?; _. _( H  D. O6 E! w' o        return fibonacci(n - 1) + fibonacci(n - 2)
    * M5 ?" z' ?+ `# b
    2 o- s' M4 ]( ~5 P# R! t# Example usage- E3 S4 K; j' A
    print(fibonacci(6))  # Output: 8
    3 S/ M' p  A* @4 ?
    6 t" k# Q. B# X6 m: s6 `- ATail Recursion
    & d" |/ a* Z4 V- J: g/ x
    ( f2 s$ P- [5 j, lTail 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)." r( S6 r, s9 C+ R" v
    - X8 q& b( V, E
    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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