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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
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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 X+ S$ T, `0 c- C3 Y  e" \0 e3 k
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    解释的不错
    - m  E5 U" T+ p8 p) G' B! z4 |
    # d' Q- k: Z) w( R9 `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 S+ q  a& F1 p& l0 T" e' K# ^1 J. y% S5 ^2 R1 i
    关键要素& m2 Y. K$ C5 B; x' g# m
    1. **基线条件(Base Case)**% \/ G* i) ^& m% n* P$ K5 \2 Y6 p5 A: Y; Z
       - 递归终止的条件,防止无限循环
    6 a+ }% r5 S9 F& c: q. w! o8 m   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      Q! t9 \; D$ h5 u" [+ ^7 B; a
    7 g4 \6 @" J, {% D, q9 |+ m2. **递归条件(Recursive Case)**1 ?& w/ [8 |2 V' @8 I; q
       - 将原问题分解为更小的子问题
      |; G6 D0 Y' j3 ~   - 例如:n! = n × (n-1)!
    - o) s- j& ?: a* f! R' g( V3 v" E8 n8 D/ j. a  \) o/ P
    经典示例:计算阶乘' H5 P% Y* S; _( j" |, H
    python# U8 d. g4 F" Q6 ]$ D9 `3 @
    def factorial(n):$ l- `9 v+ W' g, v+ N7 a3 g& b$ {
        if n == 0:        # 基线条件
    % M3 |: J1 @+ f. f. n        return 1  p0 z* i$ G& u  M3 F& l: G2 ^$ m
        else:             # 递归条件/ c* L6 e* Q% W  i! {6 o$ H
            return n * factorial(n-1)* z# e- P( ~# P
    执行过程(以计算 3! 为例):
    / Z# _7 U" Q9 Zfactorial(3)  X; N# y8 j$ Y- Z$ G/ Y
    3 * factorial(2)
    ) Q6 t& a5 m# T( n3 * (2 * factorial(1))# ~& R& I0 A7 H- F: v# j* ~
    3 * (2 * (1 * factorial(0)))) w9 U$ |) j* Z! Y* O
    3 * (2 * (1 * 1)) = 6
    & e# \/ z2 |% a* w1 l+ t6 o# V! `, L7 S
    递归思维要点
      I: `, X+ j& w+ }9 T1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ; a- t! Q) s; Y4 |* n$ v+ K& b2. **栈结构**:每次调用都会创建新的栈帧(内存空间)7 t5 s" w3 O9 X" H/ i, e( e0 |
    3. **递推过程**:不断向下分解问题(递). R0 P# ~" Z4 H7 }3 L$ e% K; F8 h
    4. **回溯过程**:组合子问题结果返回(归)6 _3 H, F! S& [4 E0 A& `% p

    + H% U8 w; d5 z  B/ n# T 注意事项- M; m) a0 q4 k( W) z; U
    必须要有终止条件. o6 g1 b1 q- |' p  Y8 W7 j  t
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)1 c0 Y& G, h, z$ }% L4 Q1 N  X
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    % c4 `2 P- x5 }, b尾递归优化可以提升效率(但Python不支持)7 l9 u# _/ r: @
    % }1 e* i1 F- W+ p/ r, @, P
    递归 vs 迭代2 _3 c$ Z) `- d
    |          | 递归                          | 迭代               |
    + S! k; E4 C" p0 ]! t/ [: f|----------|-----------------------------|------------------|
    1 U- m/ x! S( n% n0 r8 T& U| 实现方式    | 函数自调用                        | 循环结构            |
    8 i# b( h5 d6 o6 j( ~: z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    - d9 x. f2 `$ U7 b; P% d| 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         |
    5 T4 Z! X9 \) P5 Y; c7 V| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |) _0 H4 O: U& k. i3 L

    + e  c+ i9 F" h# x4 R, A 经典递归应用场景0 L) l( T; G. }  N' s) \
    1. 文件系统遍历(目录树结构)" {' N) C, G) e; |6 q0 @9 b
    2. 快速排序/归并排序算法
    # H( _3 Q9 ^5 w$ t1 t! Z3. 汉诺塔问题6 w# p# |$ G- X- v* p1 i% C
    4. 二叉树遍历(前序/中序/后序)
    $ ^7 N! y+ N9 o) P3 @5. 生成所有可能的组合(回溯算法)5 L5 T2 y  B7 s; X( C9 l- n9 k

    ' L0 g, x+ `8 b8 Q0 i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,! v: j8 F4 I3 B3 L
    我推理机的核心算法应该是二叉树遍历的变种。' q/ e! B& Q4 A3 c  o( I
    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:- p) l) R9 \, }9 j" ^
    Key Idea of Recursion5 r# g6 [' q4 V' u

    ! {2 F# W: }# s2 W7 P# r- cA recursive function solves a problem by:9 N  _0 `5 a! R
    + m; C: u( j0 l& V2 N# O) U6 h+ S
        Breaking the problem into smaller instances of the same problem.
    ' B  L' l3 O! v) C7 H# q1 k& x9 P
    ( W2 ?8 d2 N8 \1 i  n    Solving the smallest instance directly (base case).& y9 V2 T4 @. F5 l. O! a3 `3 R

    & u8 |8 J! v: _, @0 ]# F% R    Combining the results of smaller instances to solve the larger problem.
    # G3 t: g! S! B% \/ _3 N" e
    : _2 X' ^& P3 H) F# M8 k0 UComponents of a Recursive Function
    3 ~, ^$ Y. O$ _2 D
    , n8 U4 t4 k( D# c6 ]  O    Base Case:+ r0 m9 u: O" ?8 ^. x

    " |- X- E, n# j/ O! ?; H+ h1 D        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    * }$ g3 q1 F, ?# {2 n& H$ c2 K) `! _7 V$ _4 m
            It acts as the stopping condition to prevent infinite recursion.# R9 v9 [8 L9 `: y( I8 b

    $ ?' `( w! w8 V: A4 h+ w        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    - ~/ w7 Z) S2 {: }6 K1 a/ I$ E( L! t7 Q0 ^# e- [
        Recursive Case:
    ' o- Z0 D7 o$ S2 ]" ^4 E  b" |( v( f! l; E8 E/ J- D& G/ }  g3 ?
            This is where the function calls itself with a smaller or simpler version of the problem.
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    + o3 t9 _3 Y) ?0 ^        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    - H: b5 Z7 i, d3 h+ M: A& S$ JExample: Factorial Calculation
    ! j% ^8 Q' }5 r! w# P1 U( \- ^
    ' u1 D: y7 Y: \" d% a* DThe 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:) t0 y3 y" r( j1 K9 O

    ( H! w- @* b  m2 w( G    Base case: 0! = 1
    0 W3 L% ~" H6 l5 f+ ~; b  b/ d# L5 \1 Z/ D  n& q
        Recursive case: n! = n * (n-1)!9 s/ n$ j) P! I( }# D

    ' r' i, p% G6 J' GHere’s how it looks in code (Python):
    $ F  H# t; G! U2 s0 ^1 f9 kpython
    - V$ W4 \- G8 e2 R$ r: O( b) z7 y$ @: K6 g1 M3 E2 f

    ' \  X+ t9 X7 Q$ gdef factorial(n):
    ( R/ E7 y: o% X6 K    # Base case) J, Z9 q, G' V% ?
        if n == 0:5 ~. S9 G3 d3 @( U+ E* U
            return 1
    2 n$ U* i( |) D" x' O/ w* n    # Recursive case
    # M+ R' f: A2 D5 O9 U    else:5 a3 H* M5 B" |* V7 D4 j! L
            return n * factorial(n - 1)' b* D* W8 {$ c; ^8 [. p; E- X

    # i1 s% U. f8 g& T, _! V- B2 Q  X: m# Example usage
    , u2 d0 v$ w0 P4 d7 aprint(factorial(5))  # Output: 120( r  K3 K" Z+ x. ?" O

    2 [6 X% a7 n# K5 Y2 g) o. t! jHow Recursion Works, G; N# N# B  b' |9 ?( D

      h7 ]! H, w  w    The function keeps calling itself with smaller inputs until it reaches the base case.6 {  r+ o1 X( J, q# @
    $ v5 ~8 Y$ W) |# K5 ]; b& c- k
        Once the base case is reached, the function starts returning values back up the call stack.
    % m  a; Q4 f* O% E. @8 P: C. @* @' P5 ^7 C/ w
        These returned values are combined to produce the final result.
    9 e5 ^6 C+ ^7 Y* \* L8 @8 q. y; L1 E
    For factorial(5):
    6 J: F/ O( O) G" R8 u$ W* J3 q/ |9 J+ |2 Y& z

    3 W5 H. ?# A. {( G' ]/ ?factorial(5) = 5 * factorial(4); I4 X  O1 X) L% }% A/ x2 t! x
    factorial(4) = 4 * factorial(3)
    & c+ e. @6 b/ F% y+ P9 q3 j& b+ efactorial(3) = 3 * factorial(2)9 R  e. r  t' P, x: L* M) k/ y
    factorial(2) = 2 * factorial(1)9 T6 B4 Y; {; P" Q
    factorial(1) = 1 * factorial(0)
    + r3 C. z) ^; g* A9 x# Rfactorial(0) = 1  # Base case3 ]) M" \1 K- b( u4 P3 z7 s6 a

    . q' b# [* H# n* [" y4 }9 \, DThen, the results are combined:
    3 a& K7 |* s$ b, `5 Z' O( Y4 W! V. Y/ j8 a

    ! m; O) o& D! l3 Afactorial(1) = 1 * 1 = 19 h( g3 f1 E/ k/ F
    factorial(2) = 2 * 1 = 2
    - n2 X* l4 U  |7 vfactorial(3) = 3 * 2 = 6
    . v/ D7 }7 w* D7 \, y' s' h4 m0 |factorial(4) = 4 * 6 = 248 {5 o; e5 U1 ]/ w. x$ R% ~: b; X
    factorial(5) = 5 * 24 = 120
    % m, e- h( O9 O# F
    ) q2 Y4 s' M; yAdvantages of Recursion
    % [9 N8 f- q+ @2 r4 M
    ; P! ?* f; u& O7 i1 D2 v    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).: f( v; ~8 H+ ^: M5 H, {
    , o* @. @# n, G: `  `+ `
        Readability: Recursive code can be more readable and concise compared to iterative solutions.+ M9 ^" i7 u/ g6 z$ k0 }7 x
    3 }9 x( o6 Y) p8 M: X" C
    Disadvantages of Recursion: ~/ M% C+ J- {

    1 `7 y: Y6 a# T1 o9 r    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.+ A7 ]8 G0 e  ~# z2 ~
    ; a$ e5 ^! c2 E
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).8 Z- P* V8 u& D2 m: s1 ?- Y2 l( e

    , Y& t. d  l! P/ G" J: RWhen to Use Recursion5 Y$ w( x% e' _

    & B8 P$ [& M: n    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    1 B+ D3 E) q- ?& L. W) h3 i2 L1 _. ]4 Y% M7 n0 Z
        Problems with a clear base case and recursive case.1 b3 L3 y+ K5 E! t6 Q7 z
    % O2 @: B, {7 D9 r) j# Z3 L
    Example: Fibonacci Sequence
    2 b- P4 c& a8 C& X: {2 u* V
    2 G* h: o0 z" z2 i; GThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:* \" L% G8 _" l

    2 Q6 H6 |! W0 p- P/ Q, S    Base case: fib(0) = 0, fib(1) = 14 y) a1 y( `, G- e+ n' A
    0 H  P8 j$ c1 ~: _( ^
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ; D$ {7 u$ @$ l2 i% P
    8 h1 m( Z* U; E* Z- qpython
    0 [8 J: r4 r9 v! p8 z3 \
    5 G$ x2 z9 D' E2 p$ o" P" Q
    5 r( }5 k  x8 {7 n: s4 o2 Qdef fibonacci(n):. Z. }- K; h, T3 C! r2 ^4 K
        # Base cases
    / e9 J, S6 ]+ D8 d- {    if n == 0:4 t$ l; d! O( H6 i0 n, A- C
            return 0
      B; I! A, g- _/ {8 X    elif n == 1:
    1 y1 b3 s" t  n- s# W4 v9 H        return 1% E0 e, s2 x, b4 X  ?
        # Recursive case
    3 |4 O7 L  \! H0 G    else:
    * ~! L# |, v# R8 r$ B3 v0 w        return fibonacci(n - 1) + fibonacci(n - 2)  h8 a/ I2 C* J

    9 D1 P) h5 ?  T1 y4 t# Example usage) o, o) j* G& D; Y2 P
    print(fibonacci(6))  # Output: 8: d0 n9 \: D; C: L9 R/ ?. n

    1 V; O. J' v7 G+ o  |# z4 CTail Recursion
    / V" K! {1 e* c) G$ |' f! C2 b" z; @# O
    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).6 S0 g+ C$ A* b) H/ _8 I) b6 T
    ) x3 [  _4 N3 l" Y
    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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