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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
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    - w" r1 s, }7 ?( `7 O9 b% V3 ?解释的不错) B3 D2 N+ G" u& y9 Y, t' o4 v

    # G( {, X# u9 E. D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。
    1 n& Y, U$ P( g* F1 g, N) h  [3 R$ o# y- H1 Q
    关键要素
    5 S6 C: c% p! M- ?* @1. **基线条件(Base Case)**
    % w7 G; W: k$ B  ^. J& S; ~/ `   - 递归终止的条件,防止无限循环
    , s5 A4 g, D& ^$ F  F0 K& V2 i& `* `  F   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    $ T" `; F8 L3 Z7 ]' K# F1 P/ q0 @9 q( `  O, S8 V4 w
    2. **递归条件(Recursive Case)**
    ( M1 F( o( L& ^$ D  w   - 将原问题分解为更小的子问题: @. K4 w! {$ o4 h) X
       - 例如:n! = n × (n-1)!- o- b$ a. z  n6 k

    , B0 r3 a( B5 i$ f5 B 经典示例:计算阶乘
    8 S6 ]. V- k  }- C! npython9 p& J4 R  f# t2 W! a
    def factorial(n):6 f3 j" I, F' _
        if n == 0:        # 基线条件
    % h: @( h1 d) O/ `0 S0 r4 y/ A6 x( ~/ _. g        return 1: {% w! h2 B$ n. E1 ]8 B/ j
        else:             # 递归条件
    / k4 w6 A/ J9 W* ~3 [% X, ?        return n * factorial(n-1)& v. m4 w: I; t
    执行过程(以计算 3! 为例):
    1 O  ?3 e  f" ~! G! @factorial(3)
    % h. @4 M5 E7 F' a: m3 * factorial(2)
    ) L: X4 l9 z4 i* @5 x3 * (2 * factorial(1))
    , H9 E5 A7 j: H  A  n# z( J0 V3 * (2 * (1 * factorial(0)))1 @, h1 w, K& p0 F) p6 ?
    3 * (2 * (1 * 1)) = 6
    7 W! Y6 o7 ]# J: T
    ' n; G+ y6 u! }# { 递归思维要点/ b- v$ z$ y9 c. j9 ?. i' Q
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 b; v* e7 b8 Y/ y, Q
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间): V! w) Q- C6 `: e; B
    3. **递推过程**:不断向下分解问题(递)
    . k- l% H' O! C* k4. **回溯过程**:组合子问题结果返回(归)
    ! l+ k0 b& U7 M+ e) d
    3 [& b+ A! z$ n8 L9 ~ 注意事项
    : \8 n+ I3 H/ e必须要有终止条件
    " w; l0 H( z5 p% A, P% Z" @! b# ]递归深度过大可能导致栈溢出(Python默认递归深度约1000层)$ N: y% B6 O. \1 L% K! t
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ( A! Q1 w! Z$ Q+ r+ Y: q, V尾递归优化可以提升效率(但Python不支持)
    9 d  X: h( n1 J. o
    / t9 x& N, b# z% K5 c! P- F 递归 vs 迭代! R/ ?  U4 @9 x8 @) n
    |          | 递归                          | 迭代               |
      {2 d  v' w9 a+ B5 z|----------|-----------------------------|------------------|/ @  X% S6 m% B" w9 K+ [
    | 实现方式    | 函数自调用                        | 循环结构            |; I; n! _) g9 R- O
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    # V) N; ?/ e0 l. B( k$ A| 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         |
    4 p! l; I3 }; H4 z0 L. X# q! o| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |- q: G3 Y+ i/ o" M

    4 G4 a# ?# y0 N  V. @( j 经典递归应用场景; v7 [- g8 `& A) S7 \
    1. 文件系统遍历(目录树结构)' r  o3 u; A5 w1 Z, o. b- P
    2. 快速排序/归并排序算法" f0 R5 B( ^0 s0 F6 m
    3. 汉诺塔问题
    3 _4 [4 n5 J& }& I2 o! ]4. 二叉树遍历(前序/中序/后序)
    + O; }; _. Y- U9 N" c& _5. 生成所有可能的组合(回溯算法)1 ]" C/ T0 `2 n. c

    ; z1 m# e( O" u9 z" b. S9 r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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    5 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,. W1 D' e3 `& ]" X8 Y+ _8 M7 r
    我推理机的核心算法应该是二叉树遍历的变种。$ m6 n* H6 Z/ s
    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:
    ' g' q* k0 f! |9 L1 kKey Idea of Recursion/ m8 N2 w* m! \$ H2 O$ }* X" I
    ; P% e) W0 }" J) c( n' ^, |3 D7 C
    A recursive function solves a problem by:
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    1 @) Z) E3 ~) S1 ~    Breaking the problem into smaller instances of the same problem.
    9 }6 Z" _4 Y" Q. K- M5 ]9 }. f3 g2 y" B! R, ?
        Solving the smallest instance directly (base case).
    6 M) H! F6 S% m: b% L
    9 J' b: g/ D- X. ?& h5 H* k    Combining the results of smaller instances to solve the larger problem.7 C4 q8 i9 e* H9 D

    & d' }- z8 S: _" S- pComponents of a Recursive Function
    8 ?/ l( }1 D8 a; `  G9 M. [
    7 _9 @( j2 f0 v4 z6 y+ F) M    Base Case:. _. N# B: o9 N. M4 M4 U( j; ~1 {

    " J' @! g0 R- C! \: u. i1 T% R        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ) [' p8 A" t1 s. ?7 _3 R9 d1 M: i4 Y1 j! s# G1 b
            It acts as the stopping condition to prevent infinite recursion.% W1 w- q3 Q" g; @4 n1 @

    4 x8 M+ @: ^- B: R- l: {" K* b        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:3 g* v( f6 \- F! J* b& }' u" p0 \. L

    . l; q3 ?( w; F0 r/ U        This is where the function calls itself with a smaller or simpler version of the problem.
    4 o! L$ t" f  M) r: u2 _! f: y" S+ j1 N7 g
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* S  N" D/ j( B+ V! y

    0 ?! [/ i) U; ]: m7 pExample: Factorial Calculation, s7 F8 q# I/ ]6 N& p& C6 s

    ' k6 q+ y# E8 ]5 o9 }* ?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:- \6 n% [& I8 {9 z: Z( u5 G0 d
    9 B7 ?+ h0 z2 P  @" T
        Base case: 0! = 1
      J4 R2 o4 _1 O
    * e" D% H$ s6 t0 `4 K0 s% u/ i    Recursive case: n! = n * (n-1)!3 D& U" H! }  R- R: z9 R

    ( |5 I( y& @& D0 s$ N" M; hHere’s how it looks in code (Python):
    2 B( U& b% S& ^5 B2 d& j% Hpython# N; ~9 J; X4 O& H; u: {8 ]
    ' D9 U8 U- c$ r- S
    ; l, u! Y  m1 r; K2 ?
    def factorial(n):
    5 V  o# m# _' y- d9 v/ l+ f9 S    # Base case+ I- V; }" `& L
        if n == 0:% _: ^3 D5 t8 j5 L' G( ~7 @* y
            return 1: n8 N6 v* P; f; `' v
        # Recursive case
    ' i  C$ g' o; ?! b) L' i/ M( L    else:
    # i  j: R8 X- P( X# w        return n * factorial(n - 1)# D" e; v% x0 B
    9 u, ~$ v, y9 D- r
    # Example usage' v& a" v+ B' Z- ^
    print(factorial(5))  # Output: 120
    / @  |  }# [: C3 b
    5 {  A! `: u" D+ ]How Recursion Works" Q. W1 c: v" h; T& w$ P( _

    # X5 e: C' d2 y! I3 Z; g) H    The function keeps calling itself with smaller inputs until it reaches the base case.
    - c6 @- i5 S) ?0 N5 S0 {% [5 H; f, X3 w8 u! O
        Once the base case is reached, the function starts returning values back up the call stack.2 V" E! y- ~% `" ~2 D8 m

    ' h3 H( }$ N2 T( f    These returned values are combined to produce the final result.; Z0 Y' k1 C0 e, |8 x2 V
    " ^  n5 L6 F4 q  Q
    For factorial(5):  s% |2 `# M( p3 K7 u
    5 p; Y. o6 x' s4 O7 |  [8 v
    1 |, v% u! n, g* Y2 S" l
    factorial(5) = 5 * factorial(4)' {7 L4 x7 G* ~0 q- ]
    factorial(4) = 4 * factorial(3)
    0 l; }4 a; q9 Z7 F! `factorial(3) = 3 * factorial(2), f" W* N8 N: K! X5 v" S
    factorial(2) = 2 * factorial(1)
    5 N4 d- C% r7 @  k" Z- Nfactorial(1) = 1 * factorial(0)
    * y+ A# `3 |7 F! y+ ?factorial(0) = 1  # Base case; ?& N) s+ t# n) Y

    / c) X7 m! t& h3 X( C% z' A. dThen, the results are combined:1 d2 E/ z3 P9 E( [; z
    3 l+ a( M( c+ v
    # l1 S5 q5 Y% d$ p
    factorial(1) = 1 * 1 = 13 t# @7 }: E$ U! Z! q
    factorial(2) = 2 * 1 = 2
    9 w! Q& m$ ~. B- m  sfactorial(3) = 3 * 2 = 6
    1 o4 i$ K" s4 N  p2 @factorial(4) = 4 * 6 = 24
    ( t3 U1 V: E0 |' q: ^1 O" X/ Efactorial(5) = 5 * 24 = 120
    9 G9 }# }' Y* e& L" F5 B
    ( `+ o2 ]5 ~) Z5 f: p3 Q; m8 YAdvantages of Recursion) r8 Y9 p5 X; I( X. ~8 j  a1 a
    0 s0 s9 c7 i7 L! 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).# w8 A8 K) K9 N1 q4 B7 W4 X

    0 {$ f4 Z; g3 h4 [$ [6 ~    Readability: Recursive code can be more readable and concise compared to iterative solutions.- U  d7 {' v( Y( c$ [" s" K( P
    ; N6 M3 p, a; W5 r, t& I
    Disadvantages of Recursion" q8 ?8 D  B, U7 P) `4 j+ U

    / K7 M4 q$ ?! b! \- W2 ~' D; s0 z1 v    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.
    / J! Y- T4 q5 K8 j1 t& M
    . m% a- ?) N! _' R9 r    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    2 j/ g1 d# n  ]
    4 v& x" V+ q+ @When to Use Recursion! U( V! h  }' J$ g

    3 e1 u/ b& x) D( k- A    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).4 L5 B* k: U5 Q5 f- r; t2 q7 x. m

    4 I; Z- D5 }# ]# P" `    Problems with a clear base case and recursive case.& b# v, k! J& ^

    . u$ c" A5 [1 h% XExample: Fibonacci Sequence+ L8 F- @. l6 M. [' o, q

    % h5 ~: U% n. F: q* n1 MThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:* Q% ?  n4 h6 z" m/ }2 g
    4 M) x0 c$ I. B7 u( ]( N. b9 w
        Base case: fib(0) = 0, fib(1) = 1
    , s) h5 v! \- h  U. b( \6 _2 O7 c5 H9 b1 H" H
        Recursive case: fib(n) = fib(n-1) + fib(n-2)! H- \1 _3 J$ W! B

    # _& Q: I0 t+ z, t3 h) [$ a& N' xpython
    3 D* c( T  J! i7 I, H& s8 [( s3 N" e. l+ u7 C2 c
    " E0 Q$ @0 }# A' J3 {
    def fibonacci(n):9 Y7 D' D/ i( ?4 F2 w% s7 F3 j6 c
        # Base cases/ M/ h, H- b* L6 c/ ?1 {0 r
        if n == 0:. U9 r- }0 r$ Q0 q4 @/ U
            return 01 t' E$ c7 V* B0 {# J  x
        elif n == 1:% ]) \- c- ^6 t0 G# r0 u& S, Y
            return 1
    " Y& Z, l+ o& v' f% W  X; ]3 ]/ t    # Recursive case
    5 R. D! H' O5 q9 G    else:  N# j5 z6 t! j, ~/ }! K# W  [
            return fibonacci(n - 1) + fibonacci(n - 2); _, f+ m; p. B. K' K
    ( W0 K: R1 D- E
    # Example usage
    # w2 v! C. {2 }: k' i. sprint(fibonacci(6))  # Output: 8! y8 i/ z% z# l4 [# d7 x

    2 i. c/ D# t6 W0 @, lTail Recursion
    ! X, b7 Y; n7 F
    6 ^2 M! F% V2 E* y) p( bTail 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).
    / f! f& P8 y$ r- K! e  a: M
    6 J& C& l7 V2 NIn 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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