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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
     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 & x: s$ L9 A. N: B. t; ?

    ( f4 P8 J# P, A7 ?解释的不错6 J0 Z8 E! s' w3 s  q
    , B0 a: e" I: h: [$ c
    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。
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    , ~& q' r5 A7 R  X' I& ~ 关键要素9 P/ W+ H# b! }9 J$ m* q: [
    1. **基线条件(Base Case)**
    & J' \, u! {( s/ A" `" Y   - 递归终止的条件,防止无限循环
    ' S1 r8 D: j' d6 m( K3 |   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    5 W& w+ c% ?6 c% W" o( J  ^
    ' z# ^# o; s, o& J' O( ?2. **递归条件(Recursive Case)**
    " ^# K7 s3 F8 P: z$ Z0 {3 C7 V& w   - 将原问题分解为更小的子问题3 w3 p5 o/ B0 d/ h4 {
       - 例如:n! = n × (n-1)!
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    - z1 J9 K, s0 O- B. ` 经典示例:计算阶乘
    ) z& J$ A; k" Epython. b9 Q* o# N6 K7 A
    def factorial(n):
    ) K2 `& ^1 B2 Q% X    if n == 0:        # 基线条件. e9 Q" o) c7 N8 h1 g) G8 h3 ^
            return 1
    : }- U' Q, t3 K% s+ [1 G6 q  h    else:             # 递归条件
    + h; ?) A% i) I3 `( ~1 A% k  a( ~* L3 z        return n * factorial(n-1)
    - w4 M3 u" }6 B  u" R执行过程(以计算 3! 为例):
    4 R* }! I  U! n5 T& vfactorial(3)
    9 j: y% d1 W; C" h) z3 * factorial(2)
    3 e8 Z5 L2 ?2 V4 N3 v2 S( R# v3 * (2 * factorial(1))! G; t% a+ r- a+ h
    3 * (2 * (1 * factorial(0)))
    4 i, _  R8 J: k6 R, ^3 * (2 * (1 * 1)) = 6
    / T3 h) u7 j. g; s1 E% ]
    1 `3 U7 |8 P& w1 M5 }9 ^ 递归思维要点) D% R$ I5 e  {" ]; R0 L& m
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    " U$ {* Q( w, G4 a* |2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    , k% d. U% M4 S3 \3 W/ X7 h3. **递推过程**:不断向下分解问题(递)
    ( F3 h& s8 M8 h) Q8 |$ C7 p8 h4. **回溯过程**:组合子问题结果返回(归)- ^9 I; e6 t+ D! h& I6 u' L% U: G- ?! P: A

    & s# z) G/ [5 y* J 注意事项% E1 h3 o& q3 i2 k8 d( Y" r8 r
    必须要有终止条件
    0 P8 s+ f" Q$ p! W0 o6 n8 l+ g递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 f, i# f6 M" q" Y
    某些问题用递归更直观(如树遍历),但效率可能不如迭代/ Y/ t6 f( q6 r' N. p
    尾递归优化可以提升效率(但Python不支持)+ W% a  F- }0 M' b
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    递归 vs 迭代0 q( u7 W' v+ b7 E: k0 U9 |
    |          | 递归                          | 迭代               |  U" W5 W0 Y/ ~- k6 d# s7 C
    |----------|-----------------------------|------------------|
    1 C: i  {  {: u3 L| 实现方式    | 函数自调用                        | 循环结构            |* `% ]6 U8 r" u1 c
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    5 F% q% t6 S' z6 F7 R. j| 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         |
    # y! L4 L( J; ~- [3 @6 O| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |' o  Y3 m8 j& F! V& f
    2 ~9 N4 N9 K( s" R6 q5 V$ @
    经典递归应用场景9 n. M$ M( O0 n* s* u( @) Z
    1. 文件系统遍历(目录树结构)
    6 ~/ C5 W' C. Y1 C) l% N5 R1 n9 D' ^  B" F2. 快速排序/归并排序算法
    2 e% W; A3 A; s$ z% c: N3. 汉诺塔问题
    ) v; l7 v7 @. n" X4. 二叉树遍历(前序/中序/后序)" t/ Q* F+ c3 \$ A
    5. 生成所有可能的组合(回溯算法)
    ! ]" P" c: B) J; J! n
    + K* `: \" c: {- n/ ^* Q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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  • TA的每日心情

    昨天 07:45
  • 签到天数: 3217 天

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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,
      X9 ]& m4 g3 g7 i我推理机的核心算法应该是二叉树遍历的变种。( H# L2 a  X  b3 i* z% r3 t
    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 m- Z8 t4 E0 |, {( Y2 H4 XKey Idea of Recursion
      ~7 y( E* D( [% L9 e( m* `* H  B* L, a. @
    A recursive function solves a problem by:
    ; Y: N/ H; w7 T) G# V2 q. ^- h4 W5 y( P, U# L
        Breaking the problem into smaller instances of the same problem.
    ' O) }  T5 F1 F* q% U
    , f, q9 D8 J9 k, d$ `3 L    Solving the smallest instance directly (base case).
    4 S- c9 F) ~# o! `3 n3 a, Q0 p, j7 _, c8 f7 @. D
        Combining the results of smaller instances to solve the larger problem.
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      L2 q/ Q, g. L: e8 ^! R, e) fComponents of a Recursive Function/ S# Q1 r4 ], }  P: t1 |* s9 Y7 f
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        Base Case:, [+ g& K, k% D- B9 C
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* l" r+ N- |4 c) [2 ?/ M1 T
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            It acts as the stopping condition to prevent infinite recursion.
    ( e+ N7 N- x9 J' G8 G; w- [7 r$ R( a1 Y, k: U: N2 f: r
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:( C- h, N+ y6 L3 N- T& I0 a
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            This is where the function calls itself with a smaller or simpler version of the problem." }1 c; P$ {+ c( O3 d5 L" e
      v6 G/ ^6 n( _7 n. c8 L! m
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation
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    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:
    7 P9 [: |8 F  L- W/ R7 {4 _- _- [5 E: z/ c' s& e8 `
        Base case: 0! = 1
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        Recursive case: n! = n * (n-1)!# U( I/ H2 e, C) a3 P  N( G% u

    2 Z' i$ }2 W/ I: I: j$ L% X4 yHere’s how it looks in code (Python):; m2 q. y, ]4 v% {9 _( @' Q
    python- w& _6 `  q2 t' x

    5 j! M1 T( t6 {$ u$ k: B! s$ b4 `5 i3 ~5 S+ P5 @
    def factorial(n):
    9 G- n: S5 Y: |' D    # Base case
    2 {5 p  u8 g# w5 k, r    if n == 0:
    ) z  L$ @" T" K+ x" w. c        return 1; Y! ^' i0 E- B+ A* y& ?& ^( u, \
        # Recursive case+ ]: I6 r, g, X* T, p% w4 n( T) ~* w
        else:
    4 O& Q: o9 G8 b$ I( R        return n * factorial(n - 1)
    0 T& b7 {) |) v2 ]. i8 y" ^- c7 B2 m! o" l) F( X
    # Example usage6 o! T8 L% ~6 x# ]8 H, C! ^+ C" W
    print(factorial(5))  # Output: 120
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    ) o) l; u6 j8 U5 z& RHow Recursion Works
    4 c! x5 t' ]; v5 U5 [
    % }* A& \3 _0 C2 x& Z    The function keeps calling itself with smaller inputs until it reaches the base case.
    ; d3 ^" P. f0 @7 ~. G6 w' D
    0 c2 K" K# o/ F6 {    Once the base case is reached, the function starts returning values back up the call stack.
      c8 L) Z/ U; b9 U0 J
    , D  a  K. z' P2 I* V- X- U" K    These returned values are combined to produce the final result.2 T) w3 O; v( E% U6 m
    & A. S+ V. |$ }
    For factorial(5):
    + J1 F- a) D$ k* i9 W( I
    ; x. f* i+ h: k$ U9 w1 ?1 |
    ; i7 y9 L4 l( t3 O$ {+ Ffactorial(5) = 5 * factorial(4)
    . q% c* g* T8 Z1 J+ U% {factorial(4) = 4 * factorial(3)
    0 M, U7 r/ z: Gfactorial(3) = 3 * factorial(2)1 D9 r, x/ E% [' l$ W4 L4 O  S4 t
    factorial(2) = 2 * factorial(1)- W. f  S2 \  D: `2 l
    factorial(1) = 1 * factorial(0)" P! u+ G8 _( B+ e0 p; Z# m/ g9 Q
    factorial(0) = 1  # Base case
    4 Z7 p( Q" c& ?1 L4 N0 e
    4 [+ K% q( C) z* ?6 sThen, the results are combined:
    ( V8 b- F) b' C4 N' ^6 j
    5 m- P0 b- p/ z1 E0 [5 _
    ; x8 `6 q  {) t7 m, I5 Cfactorial(1) = 1 * 1 = 1
    ( W6 k4 v$ ^# Q* tfactorial(2) = 2 * 1 = 2
    2 j; B8 F. L9 ffactorial(3) = 3 * 2 = 6
    $ F2 U5 M9 s, x9 x$ Zfactorial(4) = 4 * 6 = 24
    $ ]5 Z& Q8 N- ufactorial(5) = 5 * 24 = 120
    1 t' T" G/ H! K, c9 d
    ! q0 Y6 [0 H% [8 Q* e& xAdvantages of Recursion
    : D) \$ z( X3 r
    8 ?1 [, r1 k( a    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).% F2 I/ l, C# ]! [
    9 w" z7 u* Q# D9 Q% h
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    , u2 P  L8 J: i0 X
    ' f, Z7 N1 l' N4 U: E4 L+ ?Disadvantages of Recursion
    2 Z: P7 J- C$ i, h/ y. I% g2 V' d) M& r5 o, L
        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.
    * N/ J1 d5 U% M7 x5 M
    6 r$ e8 F/ ?. |  g    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 E, k, t- D, M" k3 H
    $ N$ U5 v: s2 q2 C: I
    When to Use Recursion+ k( x' }. z0 d1 m+ u' p: l) ~

    # N9 g$ f5 x* ^' b" V* c    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ! M( Y* D+ C: Z( o- F9 ]
    3 D  ^8 d% I/ V) V9 C, i& c* r    Problems with a clear base case and recursive case.
    7 H' {: }) s5 w, ^% ~1 D7 }/ }: Y8 U+ n% T5 {
    Example: Fibonacci Sequence3 d$ H" P/ ]; Q: ~  y  y
    ' {0 Y7 S# W& V, _5 b! Q* X
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ' f6 D, ]* b$ F5 M7 G0 r0 X
    % p. {. K' a0 z) r9 x, D, m    Base case: fib(0) = 0, fib(1) = 1
    2 e: ?5 G8 D& _3 q9 \7 Y- U  L0 Q0 f( A) m1 L5 p
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    $ |! R: l$ y0 `. p8 i4 f2 h2 B1 J2 |" P1 |1 I" f0 c
    python8 _4 _( v1 ~7 R) }+ F4 ~. X" f. v3 _
    * Z& G3 c" i, E* T2 A

    - _6 J9 u2 c- W0 ?/ X2 }/ kdef fibonacci(n):
    & y) h3 n/ C9 T9 k" N    # Base cases
    . v% e& O8 r) U  @  {/ T    if n == 0:
    ; f5 \( s% }& M; T" t  R8 P        return 02 a$ i. {8 x* P9 |( |. e, M
        elif n == 1:$ {  x4 f! V+ A
            return 1; Y& r5 ~1 e, k$ N3 {  i
        # Recursive case1 z' D8 ]" {3 @3 H2 g
        else:- d# J7 _- Q; O+ i- b
            return fibonacci(n - 1) + fibonacci(n - 2)( X8 h) P& \! E7 J. {" g8 O0 i, J

    . q1 q+ B" S7 c+ F: b( q  T# Example usage  O3 W5 w+ j6 o" W6 H
    print(fibonacci(6))  # Output: 8
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    Tail Recursion) f& a3 h% A  e0 Y1 Q

    , ]6 _$ t: _3 c# c' }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).
    % b( x5 F# L  v2 d+ j$ K) h8 A2 N4 t+ L
    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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