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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 K8 M2 a& @5 C: z
    4 ^6 b3 Q0 b. Z) L: l4 k
    解释的不错  s+ d+ y) x4 O+ d4 C
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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。
    4 h( P' ?' o3 i9 R5 Z7 ?! C9 [  N( G1 O
    关键要素
    - j/ D' X8 n- m$ F  e1. **基线条件(Base Case)*** M4 N3 k* _- ]- ^7 @5 S5 F
       - 递归终止的条件,防止无限循环6 H/ f( t4 F9 Y% u' O- ^2 s; l
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: |- R3 }5 i1 F

    . s/ {: ~" Z! n% S2. **递归条件(Recursive Case)**) a- k2 v# n8 h: z8 N! ]/ F1 y
       - 将原问题分解为更小的子问题
    # B5 [$ L6 p6 A! C$ I# G   - 例如:n! = n × (n-1)!
    * l; n( t2 D+ c) u. I" E7 n: W, o- _0 ~2 z' z! v+ N9 a% [
    经典示例:计算阶乘' J, U. j; a4 |8 ~# Y
    python- B5 j, V! R$ T8 G: [; r+ j
    def factorial(n):
    ( F0 Z; ^0 ~7 x+ D) _    if n == 0:        # 基线条件1 t4 X( ^/ [2 Z3 d, L& p! K
            return 1
    5 E0 J+ i! J7 \6 A    else:             # 递归条件7 Q6 h, a8 F! h- x5 O1 D! l. D4 J
            return n * factorial(n-1)
    ( [* N5 [& ~: f) c' E, g: _4 k执行过程(以计算 3! 为例):
    ) O' z; ^: D9 M/ S' j- [factorial(3)# C9 O% O, h; O. z7 M
    3 * factorial(2)  l- v0 K0 p6 l, \; _+ W; _
    3 * (2 * factorial(1))  f+ e$ z; d2 P$ R
    3 * (2 * (1 * factorial(0)))
    - s3 e; K2 v* q6 ]; q3 P( q& [3 * (2 * (1 * 1)) = 6
    + Y( i2 F0 r3 v5 Q
    . ]: b& u6 I' j) B  r& b( ]" |2 Z6 I 递归思维要点
    - ?3 ^% H) M4 z8 Z% a1. **信任递归**:假设子问题已经解决,专注当前层逻辑% O( K1 n; Y  X. w9 i0 [
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    0 n) C% N, T% J7 H" q3. **递推过程**:不断向下分解问题(递)9 T7 g! t& D9 S3 W+ ?
    4. **回溯过程**:组合子问题结果返回(归), ~$ ?9 z8 o" U1 V* B/ @+ K# p- G
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    注意事项
    , b; B5 p7 z# t( k" g必须要有终止条件% N/ Y/ a. I& r2 K. d& Q3 C
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    6 Z3 }) ]- `1 r$ M某些问题用递归更直观(如树遍历),但效率可能不如迭代  S$ H- l8 f/ |
    尾递归优化可以提升效率(但Python不支持)
    7 K" ~) I5 a# u) c$ `# Q. K8 G; w+ W! J# D) o5 ^# V+ l8 B. f
    递归 vs 迭代
    / y" M. ?# V0 H|          | 递归                          | 迭代               |0 j  N6 Z* B3 ?1 y
    |----------|-----------------------------|------------------|4 D" @0 m7 j4 J5 q% W& h' f" m  Q
    | 实现方式    | 函数自调用                        | 循环结构            |+ o' x& z8 J7 {) m  s8 s) r" p
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
      |! X. b$ V2 ^7 e1 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         |
    ! d, h% a; g. g5 h| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |% k; J% Y8 V: n& q
    3 C0 o/ q6 i& Q
    经典递归应用场景- i* U6 H9 J- I; K' H7 F  L# u
    1. 文件系统遍历(目录树结构)
    6 {, W( [7 L5 a: p5 n; U1 A2. 快速排序/归并排序算法
    * R, a# P# a5 r9 C& e( c3. 汉诺塔问题
    ' d0 a- V+ ]2 F) o1 R4. 二叉树遍历(前序/中序/后序)" N8 k# z2 p$ f
    5. 生成所有可能的组合(回溯算法)
    / g* c+ y+ _; X3 V9 J. b8 H6 y. {3 p9 r
    8 s+ k: F- I% W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,
    ! }" O: @4 Z% F: H我推理机的核心算法应该是二叉树遍历的变种。
    ( S' ~: H( A# g& x9 u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:
    7 K2 }# X. [% b/ e! {$ C5 pKey Idea of Recursion, C" R4 ]/ V  H4 n
    4 D' F1 [  ~! j+ m
    A recursive function solves a problem by:
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    + Y7 l; h$ i2 V9 x    Breaking the problem into smaller instances of the same problem.
    ! z1 j% a' y2 U$ w' i! _+ A0 c1 ?) m' E. g
        Solving the smallest instance directly (base case).
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    ; P4 ]0 I- f+ e! G" y7 \    Combining the results of smaller instances to solve the larger problem.8 t( G' p" R1 r/ X

    5 e: G4 H% k# I7 K4 R6 _; SComponents of a Recursive Function" g+ U$ G9 N3 s1 C% ]9 u( ^
    8 F$ V4 B. W- X' z, _9 i0 k& m
        Base Case:8 G, j7 H9 t- F: d3 B" R
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.& z7 z  `6 l( n/ k7 x6 L$ c

    ; J' U- s% g; D1 j0 X3 b( d% \        It acts as the stopping condition to prevent infinite recursion." p: k( s$ f5 S7 V
    # p4 ]* k& ?, R
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.6 D# K7 k$ Z6 n

      u7 A  H% z# t# e1 L. q6 a" w    Recursive Case:5 R& m. U+ n3 T
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            This is where the function calls itself with a smaller or simpler version of the problem.
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    - D  E0 {2 ]6 O  z" _5 x        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).- h0 m1 S8 Z0 B+ D/ T7 J
    / G* x; D; Y2 z; o" o
    Example: Factorial Calculation) S/ l3 ]3 k4 G% Z& x: ~" P* @

    * F9 {1 `& h; U+ n/ yThe 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:
    . j/ ]5 l1 v" P- q; Y8 m5 ~3 R
    , g1 G* V# p5 j( E% m' V" l    Base case: 0! = 1* ]6 x7 d/ V* e9 D
    ' D; x/ x1 U% e+ u1 b& |1 c
        Recursive case: n! = n * (n-1)!( b1 m- |( t& T3 b" X" T. H: M& H2 J

    1 W# `/ I% i( OHere’s how it looks in code (Python):. h, X1 @# y) |8 R$ }
    python& W) [* D& j2 {/ Y

    * L7 r( p. x+ C
    + f5 m2 q- _/ |2 N% S; |" xdef factorial(n):
    : @/ G3 G6 A  {+ x& C5 @& K    # Base case
    / d' f; @* \' b  j% c5 `* W( C    if n == 0:& f5 y% l0 O8 B+ L
            return 1
    " s/ X9 u" }9 `4 J    # Recursive case: t& i5 S5 S) {, a5 \
        else:$ @; I4 P4 d/ v/ Q
            return n * factorial(n - 1)( o1 ]1 s5 @: h; E4 l6 k

    4 u% q( C/ D" q# Example usage
    & [6 c# v+ ^! H- _print(factorial(5))  # Output: 120* \, |, G  w4 E4 p; P8 H: x. W

    5 R: ~  c" j8 G8 IHow Recursion Works9 u# n0 a2 W; `
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        The function keeps calling itself with smaller inputs until it reaches the base case.
    " }3 b% J% _- v7 X0 S
    8 c  A3 a) N2 q    Once the base case is reached, the function starts returning values back up the call stack.5 I7 c, `" _/ t# ^0 t7 w9 G! B

    3 Q! s% A. M  d1 y6 R/ T' U' }2 p$ h$ S    These returned values are combined to produce the final result.: D4 w4 d% X- V3 Y& x, x

    * _, W5 }# X$ F3 U7 nFor factorial(5):8 G  m1 g5 C, t* ?! i- @6 d

    7 I& T2 M+ o5 _% ?* x" R5 S9 j& u; o/ {4 B  @
    factorial(5) = 5 * factorial(4)
    3 ]$ o3 U# j7 `4 C/ b% Efactorial(4) = 4 * factorial(3)2 V4 {. U& [/ y" W+ ]7 f( r
    factorial(3) = 3 * factorial(2)
    + [! X# E7 k3 T4 F" ~# zfactorial(2) = 2 * factorial(1)
    # z' Q5 [3 o/ q$ zfactorial(1) = 1 * factorial(0)
    9 ~6 I' `. n. K% {" o- |factorial(0) = 1  # Base case
    $ @% j0 H9 A" A; r' [* t) o( ~5 G8 @2 P' [
    Then, the results are combined:! H  ?0 }& T% r

    ( P3 H- b2 x, m* v3 Q) r; |+ @5 x
    . D! h- G7 T: Q6 p0 `2 kfactorial(1) = 1 * 1 = 1& t7 L* H3 V" ^7 P; c; z* ]2 |5 W
    factorial(2) = 2 * 1 = 2
    " y9 G6 {/ o1 L$ p9 wfactorial(3) = 3 * 2 = 6) v* s" T+ F: s  i2 G% ^0 c) y2 I) d
    factorial(4) = 4 * 6 = 24+ \2 T4 c* S5 K# i1 I
    factorial(5) = 5 * 24 = 1204 p$ f: {+ J- F; q9 ~
    ) t4 _" r: ]5 ]6 {0 w
    Advantages of Recursion
    - f" c/ L, @& ]  S: w* C
    ) w, J2 R+ b" A/ g! ^% i    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).; |/ W# H2 R$ J

    , G8 R  O- q7 u2 k    Readability: Recursive code can be more readable and concise compared to iterative solutions.& U9 T( H, Y$ y4 e9 u- ]& Q

    8 G  V. B. c" V* T: `2 r- Y: cDisadvantages of Recursion0 G% U. c6 s, `& b. b

    6 y$ p5 b  f+ k, M    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$ j) R+ j1 Q% T  e& s7 b/ C

    ; A! T6 s* ?: I. K' v% b% t" F- m# w    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    5 I3 F" `  U5 ^; P1 d
    1 Z+ j/ M! E. v! A5 RWhen to Use Recursion
    5 W, g9 J4 a$ ?0 W; C
    ; Y8 S8 H3 `* L3 }    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    : M* k" {0 m" d1 G& h: m8 F, a0 S" ~+ L; r# C
        Problems with a clear base case and recursive case.
    & G6 L+ P3 K+ L: f2 h4 f% y3 [7 a; Y/ a" i$ ]3 x; C& E' W
    Example: Fibonacci Sequence
    & X  w4 R7 l/ ~: B: N5 h  }' j2 ]% I/ q5 `! A. c8 K
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:. V& w8 r8 G7 I0 n! V0 i5 c
    $ J) Z* N6 c5 F; R0 @6 R
        Base case: fib(0) = 0, fib(1) = 19 r# Z& I+ V/ j, e1 w/ D0 l; M( ~

    $ x) ?9 B1 q. _4 ?% `; ]& O    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python# s: P- F, {! L/ Y( }+ n
    + k9 t2 z+ Y8 h* v' h% f( a$ T! `
    % J. Z3 X2 n1 a6 [" P! q/ y8 k
    def fibonacci(n):
    2 L- v; Y/ ?% A) o/ m+ \3 v- S    # Base cases- a+ D- e) _* u; ~& L( y
        if n == 0:
    % u. l+ {; m" U: `& Z1 l7 H7 r  R        return 0
    3 \/ E; X) \1 ?/ C; B    elif n == 1:
    ) o$ Q2 b5 j# T; _' X        return 1# F; [: G  e+ g# ]6 i8 r
        # Recursive case
    1 l6 ^! f0 M# Z- i, \: P# {9 }$ F5 `    else:  \, D" V$ J/ T+ Y5 ?
            return fibonacci(n - 1) + fibonacci(n - 2)5 [- j0 K" q0 S. e( @

    4 i, J; W# k6 S" p( g9 A& v* h9 Z6 E# Example usage
    * T# p- s, m2 S0 |6 d& F! \+ \print(fibonacci(6))  # Output: 8+ _7 w/ V1 F, g2 ?" g! n" H
    - z( u4 J& K; d  Q  ^
    Tail Recursion
    ( D, u& e4 ]2 L. n9 L) t
    ! b) R5 q' W* j- Z+ Z4 K. jTail 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).# K3 u9 ~/ u. E3 o! @% \* k( \8 o4 g
    * r+ \* ^( [3 X& o9 C% u
    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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