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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
    3 }8 |$ l  M" @: _& |2 |( U8 b/ K- Q: W+ q8 k7 t) X
    解释的不错
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    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。, z( J( `# u2 R7 o) {) C

    - o# a+ T& O5 U; i: e 关键要素
    7 ~+ {, M% s( z) C( D5 }1. **基线条件(Base Case)**
    4 X* T, M4 E" e" s# `& u2 k: `   - 递归终止的条件,防止无限循环3 H' P9 P: z8 J8 c4 F1 \( T
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1* @8 ^/ R8 ^: A1 H1 |

    ' ]" `5 ]; {+ q/ e$ V- d& v2. **递归条件(Recursive Case)**
    / a# X6 W$ O7 X3 ?2 M5 |   - 将原问题分解为更小的子问题! J* \$ S3 M# M, _0 R' U# a
       - 例如:n! = n × (n-1)!
    + i' y( a, b+ c: H1 S; k5 H" X( o
    经典示例:计算阶乘2 g% P+ @* b* q0 x7 h7 O: x
    python
    6 D& p2 v1 e; O5 q: Z  ^' `def factorial(n):. T  V, G' f% y, }7 Z7 o  s& p
        if n == 0:        # 基线条件
    3 i; G# i! V3 Z* e  T/ D        return 1* ]+ T, a& @# p+ K9 a
        else:             # 递归条件6 `/ F* y: s& O) t" W$ h
            return n * factorial(n-1)
    % [6 t) X0 H$ j2 y: S7 T执行过程(以计算 3! 为例):
    7 p$ ]8 B  x9 P7 }factorial(3)
    ; l1 h& y. m: A$ ]  y3 t7 B7 d% l$ V3 * factorial(2)- R# T$ G& _# p/ \
    3 * (2 * factorial(1))/ d# c" r: y& v# t1 t6 B
    3 * (2 * (1 * factorial(0)))1 w* Y& f. H1 m9 f
    3 * (2 * (1 * 1)) = 6
    . }) N& j, {2 X% s
    9 a  N9 t! J( c/ d 递归思维要点$ I7 H6 w! y: u  y, j/ o5 F  E
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑6 Y' N. i1 C7 [3 _5 q) k
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)5 S. A$ F( ~$ e# _8 X4 ^
    3. **递推过程**:不断向下分解问题(递)3 ^5 `9 }0 y9 |$ q& u. Q! |! Q3 I
    4. **回溯过程**:组合子问题结果返回(归)( I' @6 X$ Q5 d8 ~+ ~: x
    " R- {3 S* P2 L- l: g2 K
    注意事项
    , P- s$ p1 s. _) h" [) v, d/ B必须要有终止条件
    2 V% t9 l  u4 @; R- P递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    " {9 C3 _0 H5 ]& u2 o某些问题用递归更直观(如树遍历),但效率可能不如迭代
    # K4 T) |" d. d( @3 v! ?尾递归优化可以提升效率(但Python不支持)
    4 \! |. c! ]0 h' F8 m" {, M) v8 f, {4 o# ~
    递归 vs 迭代( a6 H' s. P# R/ C+ m$ o' P
    |          | 递归                          | 迭代               |
    , V# J, Z6 ~; B6 n6 p3 c$ b2 M|----------|-----------------------------|------------------|
    ! _' n5 {- G. D, h| 实现方式    | 函数自调用                        | 循环结构            |
    6 ~$ B7 v7 ]) C$ f  f) P| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    7 G7 P( ^* [6 U: k: J. S| 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         |
    2 [/ l* Y/ ?) D5 C1 H/ s| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |7 z8 H" W' M* v; U2 y

    2 i$ J* A, i9 \' l5 w; H 经典递归应用场景
    # g1 l7 a' B5 ?! N4 d1. 文件系统遍历(目录树结构)& b, ~. Y/ h' [9 C
    2. 快速排序/归并排序算法
    + ?' ~' p! G' L' @4 x1 |3. 汉诺塔问题
    & x/ m2 @' d. I$ ]* [& I4. 二叉树遍历(前序/中序/后序)
    8 q6 N$ o* }/ d) [8 o5. 生成所有可能的组合(回溯算法)
    % O: s, u+ T6 `
    , g" x) A" V' j* p/ E2 v6 Z. P/ G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,% F/ z0 k# V2 Y4 x, {
    我推理机的核心算法应该是二叉树遍历的变种。
      x# k7 W2 i4 {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:- @3 i9 P2 ^/ g) H" o6 J
    Key Idea of Recursion
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    2 z( u/ h5 D! p# L- u4 p. BA recursive function solves a problem by:; u4 L: N2 m2 y8 E7 H

    & \) I6 f, |. t    Breaking the problem into smaller instances of the same problem.- u1 R2 x# V$ k& W
    - ]% C  I9 }0 K) }/ C
        Solving the smallest instance directly (base case).
    2 [7 ?! b* d2 s5 q7 n+ I
    $ U' O. Y& m% `9 Q    Combining the results of smaller instances to solve the larger problem.- H2 u6 ~1 c' V+ B

    , d# F' k0 }1 Z* ~& L- _" P% y6 f; fComponents of a Recursive Function
    & D: V& f7 C7 [) a4 V( n) Y  A: p1 S* d0 |* Q; M, w% L
        Base Case:; n! V2 d: ^  t) f5 v" v- }
    ( X9 d% f5 y+ a+ G1 \0 ~: w
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; j5 X. {- U( y4 ]
    3 J0 r4 u2 P7 [/ B3 ]        It acts as the stopping condition to prevent infinite recursion.- s! U+ Y/ l8 E, g* n

    : h, A* P# y- b6 a; g7 p        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    # Y- g! x4 {5 }. n# e
    ' }7 [9 B4 x5 F1 ?) W    Recursive Case:, k0 @2 l$ ~$ `" [/ e
    . o+ M) c0 D% r# |' ~; w. [
            This is where the function calls itself with a smaller or simpler version of the problem.
    ; S5 f, Z9 u7 d  d! {. K- d2 M9 p7 h$ N* b5 R% a
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    " v+ K) ^1 c3 ~) L8 ~
    & k- Y% w$ L# n- kExample: Factorial Calculation
    ' _6 Z+ }4 ?6 b/ c9 J0 B
    2 O2 ?0 m( ]: x( g: cThe 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:
    1 a; U# a' d2 S. J( }- _. ~5 v* S* t' I  L% }- q2 `4 A3 w# x
        Base case: 0! = 1
    $ r  y: ]' @  r! x+ I8 u8 f, d+ {0 B/ [/ Q! K, U
        Recursive case: n! = n * (n-1)!) N2 @8 w! i: I1 r+ _
    9 F$ n+ F  Y: G9 e2 D
    Here’s how it looks in code (Python):$ T9 j) `$ V4 G2 I
    python$ i2 a- {! _1 b  Z* I# l  A

    2 o6 o. ], ]$ f( e# Y2 C' Y, T" E6 g9 n; S; }
    def factorial(n):  L) W+ x; V/ a: Y" `
        # Base case- a! s; i( R9 ^
        if n == 0:
    * _; I/ o0 s5 J" }: L        return 1
    5 n4 V* T! ^) E2 A/ ]3 L    # Recursive case
    0 a/ s3 }$ c, c    else:
    2 [3 B( q1 f/ S. }        return n * factorial(n - 1)
    + K' Q, Q; N5 n& ^1 Z$ I2 I1 k
    8 b* k, W& i) f5 U+ ?. B/ E! _# Example usage3 A' Y: q" {+ W: }$ {/ I
    print(factorial(5))  # Output: 120) ]) n" p! g" x( {9 X7 u1 B2 [
    * t" l# i. L5 R/ e
    How Recursion Works
    ) a& D' Q* ?3 ^- j& Z/ K0 e8 H$ n  F. q& e4 k
        The function keeps calling itself with smaller inputs until it reaches the base case.% x' r8 ]. T8 G( J1 g1 n8 B! o
    9 ?0 p# N, C' d/ h& n1 |& p
        Once the base case is reached, the function starts returning values back up the call stack.
    ' x' A3 K; \2 A& u7 o8 c) R) z8 z8 Q( b8 N
        These returned values are combined to produce the final result.4 f+ r6 t: L) L5 `- D7 W2 Z6 Q
    6 Y1 H2 G9 v7 v  |
    For factorial(5):: |, w4 _( n, d

    ( \* n) Y  s; r5 x9 q5 ~) {0 r3 M5 F1 q8 S' n5 n3 Z
    factorial(5) = 5 * factorial(4)& T) S, A( W: h% a9 ]& [1 C  d. p
    factorial(4) = 4 * factorial(3)
    0 m1 s- l* g! j' F" u8 {4 sfactorial(3) = 3 * factorial(2): k% y7 X' |$ `8 ~2 }
    factorial(2) = 2 * factorial(1). T+ z: M( h8 P- z' s
    factorial(1) = 1 * factorial(0). J) B/ t) s& o4 z. s/ p
    factorial(0) = 1  # Base case
    - Y4 ]0 {: b: j$ H! I7 A3 u, Q: a' u' V% R6 e3 k+ {- \% d/ K  q
    Then, the results are combined:
    5 `3 J8 u/ v$ G& O! L
    + `$ e% T3 O3 l1 j2 i5 ?
    ( G8 k3 V* q( r7 M! sfactorial(1) = 1 * 1 = 16 _5 m( d2 w) h4 h9 m5 x, a4 w7 C1 ^
    factorial(2) = 2 * 1 = 2
    2 Q8 l+ q  W) k" g+ qfactorial(3) = 3 * 2 = 6" T" [- n- W  N2 K: |) U& n4 o6 i+ V6 {
    factorial(4) = 4 * 6 = 24
    0 g. z, P6 O9 c8 wfactorial(5) = 5 * 24 = 120$ H" R% i' t% D4 ]9 z

    - T/ U7 u: O; f7 X9 tAdvantages of Recursion2 Y( [! l1 ~) F9 S# C4 E( G7 m# z

    ' b7 W8 O4 ?' S) S: {, N    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).7 i2 \; F6 A" S3 C2 Z
    7 o  q! k% H# B% g/ O9 w, `& e
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ( t% }& Z) y' w( C4 O* d9 g" y1 a! P- s; [7 ^
    Disadvantages of Recursion
    7 s  H4 r- a0 |7 L" f- ?! ^( y6 ~: ?# k1 @2 J; C. c
        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.( d/ k# c9 a% w

    / ]6 Z! T/ S2 R    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).% x0 f+ P$ L3 o8 ~+ W; ?0 O

    * @/ J  w: ~, |( OWhen to Use Recursion* L' g0 I3 y. _1 N* O. D
    ' z9 [  _- Y. E% P: t, }( W* I
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).3 G+ J$ k3 ~  c% R
    # x( v8 L/ U+ h# k& {& x
        Problems with a clear base case and recursive case.0 c# c* g6 ?0 d9 l, p& v
    6 T4 F, u- Z7 D6 m! \0 D
    Example: Fibonacci Sequence9 `' X2 Z  e2 B/ l2 C

    : w) H8 o" v# g( m: K& n6 h' BThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
      N2 V1 b7 ~8 \
    . S# M9 {4 ]* y" T0 i- W    Base case: fib(0) = 0, fib(1) = 1  D) z$ I# p1 K$ n$ V
    ( [0 J% }' J' t/ W0 O; l
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    & V3 T6 g" z& P3 [1 m# s6 M7 |- C/ u+ E1 Z6 @! y) }
    python
      G# I1 ]+ @. Y+ t/ x2 S1 F! A! p7 R" Y! d% Q1 U8 N
    ' M  y7 e0 o4 e8 c! a" S8 b
    def fibonacci(n):6 G+ N; m' L# A* @7 {
        # Base cases
    , M6 Y5 h$ t+ c4 |+ r+ p4 @5 w6 h    if n == 0:
    3 Z2 E+ r3 D1 g% x+ v; J4 H        return 0
    7 [- X( S5 a9 n    elif n == 1:7 K( |9 E: U/ K) x/ }" ^
            return 1
    - {( T0 }% r1 J' k; Q: G    # Recursive case; O( z! W, U/ \( y
        else:' R( l5 A4 x' E2 I
            return fibonacci(n - 1) + fibonacci(n - 2)7 @  i1 s2 E! E( `' m4 p0 _- a5 [0 H

    : ]' ]( I% G. f- c( \0 c1 V# Example usage7 A  f4 U$ t7 l
    print(fibonacci(6))  # Output: 8+ Z7 S' I# B! E) Y1 l5 m$ X  Q

    % k# k/ Z& A- QTail Recursion& q" ]8 Y+ E4 M

      ^: F( _; G+ k- s2 s! m3 \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).2 }4 @/ ?; \1 M1 ~  v9 f! R" H- ]4 \/ b
    ; ], N% V* a2 |) E* ]% ]6 g
    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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