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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
    9 ]/ Z7 v1 ^0 w9 e5 x9 p( G/ E7 H5 r* W8 D" m& H
    解释的不错# `, H. M+ l3 v

    / R0 P5 f$ P2 L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。
    & _. a' d) M& G* H
    5 _. `) G1 u: T; i7 s+ Y0 y0 b 关键要素
    9 W* ?/ h+ a' R( R: S1. **基线条件(Base Case)**
    - W# y9 Y9 C1 ?' G& l; F; B3 c   - 递归终止的条件,防止无限循环2 ]6 [2 s4 ]) |
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    % l4 u# [" G9 b$ b9 C' H5 l" U, O/ u' T. ]
    2. **递归条件(Recursive Case)**
    / d3 Y1 G0 W. P. c4 F; y   - 将原问题分解为更小的子问题
    / N7 y* u/ n) N- z8 `7 G* Y   - 例如:n! = n × (n-1)!8 f% X0 T! {6 }% G0 M" k
    # h& J2 h( Z! w* r
    经典示例:计算阶乘
    - o1 K2 H& S+ d( ?9 Qpython
    5 L, E  T) J7 e7 R* Y# w+ Jdef factorial(n):1 b& S4 m7 N3 `# X& r' t7 I
        if n == 0:        # 基线条件
    & i% ~' F/ ~4 N# k        return 1) r: O* W+ i$ [# f7 w: `( A3 A9 _
        else:             # 递归条件! b6 t3 E! Y3 Q, ^) d" T6 z3 \, B* S
            return n * factorial(n-1)$ Q! p' F$ [, g! r& D
    执行过程(以计算 3! 为例):
    - N1 k1 \# Z; b) ~5 C. x9 c% Yfactorial(3)
    0 W) s! ^5 L* X6 X' l3 * factorial(2)
    , X0 A5 x1 U% d3 * (2 * factorial(1))# j+ m9 i3 _8 K% b- p: p4 l
    3 * (2 * (1 * factorial(0)))
    ) T4 z6 M+ b- b; @3 * (2 * (1 * 1)) = 6
    / p7 l/ ]2 d2 K+ L3 p$ H% n
    & N  y% p0 q. L 递归思维要点  x+ V( O) Y( X% c$ u# V
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    5 z. ]4 D: E2 z! v9 J2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    : {" j8 a3 {# l0 K3. **递推过程**:不断向下分解问题(递)& C3 K) G, U$ J: N, c% s" i
    4. **回溯过程**:组合子问题结果返回(归)
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    6 Y$ H5 u8 b; u1 F  B/ `0 t 注意事项
    : |2 P% v6 R2 N9 `* m必须要有终止条件
    0 }! G; j' j  j7 ?1 g7 e递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    , d0 i$ Z+ f* J8 l某些问题用递归更直观(如树遍历),但效率可能不如迭代
    2 b" Y+ @, W  m: `* w: q尾递归优化可以提升效率(但Python不支持)* j0 ~( K9 S/ e0 b5 M

    0 R" f6 `6 t: a- d) _. [' f% h9 a 递归 vs 迭代; G  |/ l1 H6 d. w5 U$ U4 H
    |          | 递归                          | 迭代               |- ^7 z# u# q  _4 m2 d" c) y
    |----------|-----------------------------|------------------|
    ( h5 E+ d( @& w" {+ x| 实现方式    | 函数自调用                        | 循环结构            |- O, Y/ o6 j- J
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ; j" P# T8 j' x; d, f| 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 S' ^( X6 N/ u  z; I& u
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    4 b' ^, [" u$ H7 m0 \5 q& t# h5 L+ L; Q7 E. y
    经典递归应用场景1 s" x0 X, Q- j
    1. 文件系统遍历(目录树结构)
    + c' e4 x5 ~: A2. 快速排序/归并排序算法
    ! x9 ^/ e. U: c! N3. 汉诺塔问题# a# {. N2 C( T. l' l: _
    4. 二叉树遍历(前序/中序/后序)! \! X4 W: c* d$ z: ^
    5. 生成所有可能的组合(回溯算法)1 ?8 u: W; p0 q% u: F) B
    % S2 M2 x. h0 [  O5 r8 Y
    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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    昨天 08:28
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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,
      \8 y7 C$ G/ f7 [我推理机的核心算法应该是二叉树遍历的变种。
    - \" g. i1 {1 ?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:
    6 Z0 p% S8 o, W; y4 W! HKey Idea of Recursion
    . D3 l% j0 [9 P  \7 K: B7 w) T* t, {) G7 s
    A recursive function solves a problem by:
    8 Y" G- W6 `7 J& P3 R1 [6 F+ r; o: T
        Breaking the problem into smaller instances of the same problem.
    4 L( _4 _6 _. ~$ N: b6 Q* i# z4 q8 \7 |7 w6 Q' P& k6 o; P
        Solving the smallest instance directly (base case).
      c2 N$ Q3 i1 k4 b; z/ _7 V! |' U  A- N, ~; |) Q
        Combining the results of smaller instances to solve the larger problem.
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    1 q  d4 X' E8 c: {Components of a Recursive Function: s2 f" H' ?2 {3 B6 ^8 L
    : G& U& K3 |' h7 g/ R5 T: C, d9 O
        Base Case:% [1 @) o/ p) h! L! J

    " |1 E" y- ~. _! `        This is the simplest, smallest instance of the problem that can be solved directly without further recursion./ d/ Q5 V5 {8 {6 ^" A+ `
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            It acts as the stopping condition to prevent infinite recursion.1 _, u. B5 ^* b. }
    4 c2 J: e6 L0 ^8 r- E2 X
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    # W9 U7 x, ]* x6 m2 Y8 v5 U* O8 [' p+ X& |/ {1 l
        Recursive Case:5 T0 x7 J4 t1 @1 P& q" ~

    ( O, E# g$ A5 F# ^3 ?6 H        This is where the function calls itself with a smaller or simpler version of the problem.7 u0 k4 X6 u" D6 h" T. O4 H0 o
    " W6 m4 I5 ]& P# J7 P/ q
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).8 j4 d4 b, u1 `7 n# g) L

    0 I2 P$ W9 y; Y0 g3 ^9 }Example: Factorial Calculation" [, K* R; F# y) p, K; D! g

    % f2 z0 x1 a) @$ i% c3 UThe 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:
    0 ~0 o, s9 B8 ~
    4 W. @; b# H+ i) [    Base case: 0! = 1
    * o3 I9 ?/ }- _7 W( Z& m
    ! M; h' q7 o# k4 `0 V# `    Recursive case: n! = n * (n-1)!8 o5 m' D' @$ O8 o  q8 O% R

    5 n/ E+ f1 B* Y% ~9 h1 NHere’s how it looks in code (Python):. C! _9 O6 K, g6 ^6 E9 K, E
    python
    : T1 K/ c$ O- A$ G; |
    + ?3 c5 A% V) f3 s1 R" _* q
    1 Q8 @7 B" }' \def factorial(n):1 t1 l2 J- X* m* m
        # Base case
    : K( J' K" t! H) J' \3 {# i+ W    if n == 0:
    , A) ^$ Q" z! q, E1 ~$ {        return 1
    " ^( ?# r: G: o* l1 y    # Recursive case: S4 q, n9 e6 l8 }5 Z  x
        else:) C- |( b5 h( e3 a, E4 L- I
            return n * factorial(n - 1)
    7 T+ Y( z+ o  e- i0 Q" f6 P. G! ?6 C. W1 {# G
    # Example usage
    0 S! c. U& H; Uprint(factorial(5))  # Output: 1203 C* G3 r* O/ X) P

    / Y+ w2 k) Q6 {4 E! w5 AHow Recursion Works
    9 x0 F& e2 V- A! p( l- Q
    8 |0 z8 V5 V: T/ x. f) ~8 p. c0 h5 p6 ~    The function keeps calling itself with smaller inputs until it reaches the base case.
    8 p  U: I  e# D! n! u
    ' X/ [1 c! V' g) z8 w2 }1 p    Once the base case is reached, the function starts returning values back up the call stack.  }) |2 g) A+ r6 F- q- @
    4 `- Y, F2 i* ]
        These returned values are combined to produce the final result.+ t& {: u  a$ u
    ' v1 `0 q1 {6 y8 @7 A
    For factorial(5):
    9 r6 _+ Q$ N" l  E- n4 O, S5 ?" I% d) w- l' [5 s( O) q

      k. j6 W( [6 ?2 h( Z) Afactorial(5) = 5 * factorial(4)
    7 K+ a2 b+ h& G" ufactorial(4) = 4 * factorial(3)
    % q8 a" S4 B3 Z& u. f+ q$ jfactorial(3) = 3 * factorial(2)
    ) |! R( W0 }; [7 w4 D# Jfactorial(2) = 2 * factorial(1)
    $ V" ]* Y& R  }factorial(1) = 1 * factorial(0)9 Z9 M/ a6 y; N
    factorial(0) = 1  # Base case8 `3 n8 ?1 S, p
      O9 V/ L5 ^2 ^, L  N$ W
    Then, the results are combined:
    & ]3 C* R) @0 I; [) S; A, _# W! ^: g& t
    2 e' E3 P) w3 \
    factorial(1) = 1 * 1 = 13 c: E' k" y  a/ T
    factorial(2) = 2 * 1 = 2* X6 N) ^3 g% x- h! T& ~
    factorial(3) = 3 * 2 = 6
    ; J8 e  m+ Q1 e! e) yfactorial(4) = 4 * 6 = 24
    % n* F: Y, X& o8 u; f' J0 d8 Zfactorial(5) = 5 * 24 = 120
    8 r3 K: R+ Y% }. F$ U- n. A9 P" G$ l" M
    Advantages of Recursion
      v7 T- j) }1 U( [1 s+ F
    , E6 Y+ r$ H  h/ O  F9 `6 y; Q4 I5 E    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).
    ' K& N% U1 D( ?  S: k5 m; y5 V/ Z+ K3 r6 W- T, x$ a3 n/ y
        Readability: Recursive code can be more readable and concise compared to iterative solutions.  j+ Y6 s2 C  q$ ?) B+ A+ q: t
    ; Z4 u7 D; a9 [% R
    Disadvantages of Recursion4 d! t# J& b/ o7 Y0 [

      q3 ^7 B+ l; U  |% b    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.8 G# d. s% {6 {; u# z* [
    $ h; d7 Z% |3 G1 R
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    & s1 `  J0 r7 P
    & y  ^- {: k* u/ yWhen to Use Recursion$ H5 U% O$ P9 z7 h5 K0 B
    $ z; ^/ q& {4 }& p+ h3 Y
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    4 y( t. `( C/ f- ^) T& a4 Y# @' N% c2 B
        Problems with a clear base case and recursive case.
    7 N- ?2 v# ?" b5 y# z7 S( s6 s  o- y* W8 U5 x
    Example: Fibonacci Sequence0 e, s6 v0 k9 |  ~/ k+ B8 A
    ( ^9 o- u/ V) r, E6 x
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    5 u2 \  Y  s9 o. ?/ `& z* Q5 q8 U$ e
        Base case: fib(0) = 0, fib(1) = 18 `7 o2 C& f0 G! Y; @% a

    ) }$ t0 j$ y9 B- ]4 H6 C    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # j0 ~1 }0 H" {  m+ }* N- ^2 b& s: B
    python
    ( q  P1 x) U% ~9 [: z
    ! z8 U6 o- j, N- B: J  `
    ( J  v. u9 g" I+ u* L$ o* sdef fibonacci(n):3 |3 _! h+ ~4 [' W% c
        # Base cases0 J8 C; Z, R, E  v5 m
        if n == 0:
    ' b, z" [0 {6 K( D' Z, Z        return 0  o& ^7 ~# N" p$ e
        elif n == 1:
    : w7 D7 K7 i% A+ _, u( C( m" _  L" \        return 1
    7 L+ O' f6 b8 N4 p    # Recursive case6 e5 d* ]* I. I# ~2 Q# J
        else:
    . Q6 G1 [6 s% v/ I/ {        return fibonacci(n - 1) + fibonacci(n - 2): N# W% L$ E  r; g7 Y

    ; f/ j2 Q  B' l( g2 N' N# Example usage: K: k% |! y8 C2 ?+ E1 x' s  j! ]
    print(fibonacci(6))  # Output: 8
    ' ^" E" P" K( S1 e
    : Z3 M3 E& D+ k( m; M6 ~Tail Recursion
    / e, j# i3 A$ j$ z' C7 O& m. V  D7 y8 g
    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).% b4 p+ F  |7 ]( p7 I. U
    ' X% U! t2 t  u1 K$ E+ _; {1 q/ K
    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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