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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
    9 L2 L3 R8 u) n! `9 M, U+ I4 Q$ c$ D  y+ r
    解释的不错
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    1 a$ ~: {9 n6 n. f5 N  m0 C/ n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。
    % J- z8 d* Y9 H1 U6 Z9 {7 V4 z9 H1 ~, [8 V& m+ E, K" h2 v! V, |8 N) p& w
    关键要素$ t, W% w7 C8 ]' A5 P) j+ b) z
    1. **基线条件(Base Case)**" Q* J: b' p7 D' Q- |5 ?9 a- x4 e6 U7 ]
       - 递归终止的条件,防止无限循环
    . u4 H# M( p) [8 C& h0 @1 H& M   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1; C" k& p0 `# L/ y8 \2 _* t

    8 _: @- S+ B2 N5 w2. **递归条件(Recursive Case)**
    5 L, u7 ?7 f6 e% s   - 将原问题分解为更小的子问题
      ]1 f8 E0 _3 p" ~   - 例如:n! = n × (n-1)!, R0 S0 d+ j1 p' B( F4 J
    - }& b) X5 ~4 s0 y
    经典示例:计算阶乘& N* u5 B4 g, ?
    python
    & M+ g, G6 G, q. T1 w* p2 V+ g& mdef factorial(n):
      w3 M3 f" U  b- `% S    if n == 0:        # 基线条件
    ( S+ w% k8 b+ a* {. \  D: Z2 Q        return 1, }( ]  }) z: N( f. L
        else:             # 递归条件, T- D" f: k# L+ G2 O' l* ^# J' p
            return n * factorial(n-1)% I# r2 ]' Y; c4 j3 A
    执行过程(以计算 3! 为例):
    0 v( c0 J# X$ l# `factorial(3)
    4 o/ h; z& {5 \7 [, E( z& X# o3 * factorial(2)
    7 p; w9 J4 O/ U, }% @# s3 * (2 * factorial(1))
    , R; X5 }) [3 k; v1 _8 @  H3 * (2 * (1 * factorial(0)))4 T4 e& ?  Y; X9 V7 T% n, W
    3 * (2 * (1 * 1)) = 6
    8 |+ ]& B# h" f; r* l4 M/ V% D8 T: N
    递归思维要点+ d! E6 K6 x8 c3 A$ r; d8 ?
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    4 y4 r+ ]& |% v' s$ H2. **栈结构**:每次调用都会创建新的栈帧(内存空间)! f& E8 v, q! Q) b3 F
    3. **递推过程**:不断向下分解问题(递)
    % r& Q( ?/ G5 Y6 N2 ?4. **回溯过程**:组合子问题结果返回(归)
    % X/ E8 g% C: @% S& s* F) E9 a0 M" b6 l1 U' K! ?* M0 C
    注意事项
    * G5 M; }! |1 X: ]8 Z, i' t( u必须要有终止条件
    % R. y* v' X; L3 i' n' ?递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
      h% f" E1 M9 ]某些问题用递归更直观(如树遍历),但效率可能不如迭代
    7 O. N5 a6 ]+ a' }1 t0 K: I尾递归优化可以提升效率(但Python不支持)
    6 C3 m8 `6 V3 i; w( G
    - `. G5 h/ X5 q; x 递归 vs 迭代
      c) `3 j2 q$ l0 T9 C$ Q  o|          | 递归                          | 迭代               |; S$ e0 E, D5 k  [! C8 t
    |----------|-----------------------------|------------------|* m. Q3 z; p7 n  b, S& d
    | 实现方式    | 函数自调用                        | 循环结构            |
    % d  s, T( _  R3 Q) Q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    7 ^3 g" }3 i/ g* d( `% z| 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         |( V6 |  a6 y- L' W
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    0 @, S0 I: `) ]' R# ?* [2 X  I. [9 k' N; v+ {
    经典递归应用场景
    : B7 M. q& B' R: `5 O$ w1. 文件系统遍历(目录树结构): K; l! f* Z# F3 k1 V! K7 p
    2. 快速排序/归并排序算法6 P, Q. f# W* r
    3. 汉诺塔问题  o1 b, F4 S+ n8 m" X. D
    4. 二叉树遍历(前序/中序/后序)
    % N1 \6 E. D& z+ w/ ^0 V" k5. 生成所有可能的组合(回溯算法)
    ) s4 g  W1 S" m( \+ \5 b8 h8 k) b9 ?8 `6 A% j6 o
    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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    郁闷
    6 小时前
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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,
    4 S/ X0 H; v6 ?0 w0 U, X4 ]9 @我推理机的核心算法应该是二叉树遍历的变种。5 x$ l/ L9 k% L4 _* F* o
    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:
      z+ D; Z$ \- Q, k+ GKey Idea of Recursion$ |( m+ X' x. J5 w

    3 M+ s  O- r% m3 p& Y& c! CA recursive function solves a problem by:7 e: t1 g4 W- a- |
    * v$ }6 \" h, p4 A$ B4 Q. N
        Breaking the problem into smaller instances of the same problem.
    ; ?# |! |( w  S$ E6 n) C, J3 E3 q6 K% |# M: t7 S5 {# ], k
        Solving the smallest instance directly (base case).; W; `3 k9 Q& {& L- F

    # U' e+ z/ u, W7 ]; Y    Combining the results of smaller instances to solve the larger problem.
    4 {$ g, i: k/ A5 u7 T6 `- t4 G# o) E5 Z: v- s: F- c4 L4 b! I& }
    Components of a Recursive Function) V. |* x  w2 l6 m# ]6 U
    5 E! o5 E9 P0 ]8 r& U  u% F7 ^1 R
        Base Case:
    ) _  B/ j6 C. v* M2 Q' G) d2 p9 x1 T( \7 {) Q
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    : i8 G7 ]& g  \. A% l
    9 G5 r; l( }3 R( ]4 L        It acts as the stopping condition to prevent infinite recursion.
    * P5 i6 R) K, M+ p# o4 K, I7 I5 V
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
      Q. `1 V- x% M# D- U, ]
    * A. @. z  ~& Y' d8 J) w$ p; ~    Recursive Case:
    2 X3 i# y8 i- @7 S  t7 r( {8 v) Q9 X- H4 l
            This is where the function calls itself with a smaller or simpler version of the problem.; ]2 z" Z# U. z8 i7 `

    0 p. ]- v/ Y# `; F        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    5 j; p" y% [% A& w9 i
    8 S+ X" h" z2 m- e1 A# a  sExample: Factorial Calculation
    7 K3 O/ l7 W( V+ H( }" @" T
    - X1 h: k7 i! \- O/ ]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:  v2 ^, u1 J) J% c) [3 b- J; j

    % p: k) E% M, M: ^9 S/ a) S8 R    Base case: 0! = 1
    8 Z5 C: \( h; m9 g  f3 k' {& H" i: `: q( x6 `# ~
        Recursive case: n! = n * (n-1)!
    2 p- N" n/ f" P" D) ]( B8 G% J# z* L) x9 y4 m. J2 X; ~/ X
    Here’s how it looks in code (Python):
    * \8 M& l3 c& C. |python
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    + A6 h$ e# ]- \* t9 I$ B! }4 S, B# A; g9 n. j1 V9 s
    def factorial(n):* d; j0 _, o3 ~+ h# n
        # Base case$ O5 l3 V; e6 ^) U4 o. c6 [
        if n == 0:& R# R1 F5 B+ W
            return 1
    ( g% [; g& A4 |$ R    # Recursive case
    & Z7 m, Y* q0 ?2 t    else:3 ^9 Z* e3 h, r  P* r2 G9 o
            return n * factorial(n - 1)
    0 ^7 _/ L( H$ q* Z* y! t9 f$ g' v0 C1 |% ?# K
    # Example usage# D9 q; n2 ?# o* m* N- |
    print(factorial(5))  # Output: 120# s4 @2 Z& M7 ]; ]( U
    $ y3 K. B) x4 X% n
    How Recursion Works
    ; s  D  y7 x& q+ w" d& q& q: _
    % t% k6 T  V% J2 ~    The function keeps calling itself with smaller inputs until it reaches the base case.
    * ]$ J* x4 E: S8 z) U1 C4 c: y* T
    4 A, r+ _$ ?( {) m    Once the base case is reached, the function starts returning values back up the call stack.: R6 o7 F( q# H. |6 o

    " Q, g1 F1 L0 ?# ^    These returned values are combined to produce the final result.
    * z9 Y* L9 u8 x/ F) O# p) ~. q
      s6 ?" p# D. b  P5 N! p. w* SFor factorial(5):( I5 X: X6 a0 s: \# Z6 M8 N

    / z- F! ^$ s9 G; M) J( W+ N4 l( P" h, j+ k7 t2 R# E( f  I
    factorial(5) = 5 * factorial(4); a: ~- R' E6 D# m6 V
    factorial(4) = 4 * factorial(3)
    * C/ X- E3 n# z: L! _factorial(3) = 3 * factorial(2)& k7 d8 d/ ]. s" f
    factorial(2) = 2 * factorial(1)
    . c" L* }; ]# F' `factorial(1) = 1 * factorial(0)
    ) ]: D, h8 _# Z! Q, {. o6 Tfactorial(0) = 1  # Base case. I. B  @" x' W3 ?* J

    " [* ]) U' K9 q; d0 TThen, the results are combined:0 X& ~1 ^6 v( M6 `
    ! ]& z0 p1 B  U

    $ b, U  U) H0 P- {. d  o# Dfactorial(1) = 1 * 1 = 1
    6 t: E; {4 ^9 O2 |factorial(2) = 2 * 1 = 20 H+ T  O& i& [( w
    factorial(3) = 3 * 2 = 6
      |, n2 N& s3 h6 U6 c& F8 ]  Ofactorial(4) = 4 * 6 = 24
    & \0 ?! b4 J7 p1 gfactorial(5) = 5 * 24 = 120
    3 G- H1 ~4 s  t. w& Y7 @) D3 S( J5 J# c5 P8 }$ L
    Advantages of Recursion
    : K* n% f  |9 J3 P; K/ v. j/ i" d
        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).
    3 D1 E: T9 J8 [
    " \/ I1 ]# z  i4 Q' J1 a# D- J4 D* @    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    / q# b7 G& a3 ~6 ^" V. Q  @" }& A3 a, ~
    Disadvantages of Recursion
    2 |3 Y6 V) y& M& i% g* s8 G0 E6 X. b9 M- q, L  K" e$ b5 n. {
        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.: @: K9 n! |' _

    " i- }6 j; s3 M" [0 V6 a. q    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 Z0 E) |, k" u; d' r

    3 c  p& x5 t7 j7 P" mWhen to Use Recursion
    ( q. P* R  H; [" z( o. ~! J- [2 N" X5 O
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    5 w- _! P1 [: Y1 U& q$ ?9 k
    3 Q, F3 U. K. b: d    Problems with a clear base case and recursive case.
    4 n/ r5 d# S$ n# ^; ?" x/ O& d4 Y4 M2 w8 {0 f5 a# ~% k& F
    Example: Fibonacci Sequence: m" U6 {- N- k% {2 p* t5 P+ |

    / @; q+ U, T6 w# E& ^6 S" e+ pThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:* V- S1 _: p: e8 `4 _4 Q5 H

    . `+ E6 T1 R$ ?( Q. ?6 `    Base case: fib(0) = 0, fib(1) = 17 L" Y* l& U, W1 E8 b0 E! [4 k! _9 t9 q
    * S( n& o. g$ y, D8 @) E( a. q$ s2 v6 K
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
      X( L6 j& j( L5 M& P
    9 K. W/ p7 b- R+ y7 ]python9 g5 Y; K5 K! K( _; ?9 E' [+ u: f

    % a& K5 o6 I8 h: O/ O* k7 z2 s; ~- q% m' f  i9 }& y; J/ X
    def fibonacci(n):
    ) K; \. z1 T; F" K6 B! W. P3 t$ P# f    # Base cases/ _8 _# S# {0 i4 }. H' f) y0 a/ M
        if n == 0:
    & e% q5 n2 \8 i& Z+ }$ S/ z+ q5 l! u        return 0
    4 k: |# M6 K7 h1 I5 v9 f. d; g    elif n == 1:! \& J6 `* Y/ _8 J- `$ b) R
            return 1' [) m' \" {) Y1 P  u! S; \) [
        # Recursive case- |9 {, B( _; A, ?! {
        else:0 P# o' A" v; l% ]4 C% n3 M, u
            return fibonacci(n - 1) + fibonacci(n - 2)$ s% {9 s8 e8 y5 u- `

    6 }) r# `* p# Q& l# Example usage% u5 @! E! z2 f' c
    print(fibonacci(6))  # Output: 8
    4 Q: c) Y* r# b3 {" U% W1 o
    # f' K, d6 x9 `" L6 uTail Recursion2 c( Q2 M1 i( g2 t: h

    , u5 t6 P% n2 H0 q2 kTail 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).$ `; G5 \, i4 q& f

    , m& |, e' `2 Y' v. B3 MIn 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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