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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
    " @, j. R+ _; d  I/ n% x% F  {  M& ~$ ~% j( p9 |! t8 h
    解释的不错' ?! U. _( ~9 e& ?* Z
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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。! E* }% X9 {; T: Q* m: _& J+ _
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    关键要素: _7 N% y0 n1 u& @6 r
    1. **基线条件(Base Case)**/ t! \* w: P+ \4 N/ x
       - 递归终止的条件,防止无限循环
    # z7 N1 ]4 l( D( b   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    * G+ `; x* N2 N/ i" M' v& _! o8 K) s. i: D7 Z, ?! i
    2. **递归条件(Recursive Case)**0 f) s, O. L) A1 u0 {. u, G* |
       - 将原问题分解为更小的子问题
    2 z8 i$ j' {* P+ F, j) L   - 例如:n! = n × (n-1)!) n( Q8 c6 e1 r( C+ |6 D5 }2 n/ _
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    经典示例:计算阶乘
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    def factorial(n):( \3 i) }' Z& j
        if n == 0:        # 基线条件
    / ?7 _+ X2 D8 }, x9 t. R; J        return 1+ H1 C; T  o! {/ H. p7 j
        else:             # 递归条件
    , I/ ]8 I7 T1 ?4 U        return n * factorial(n-1)
    ; X6 p2 e. [" b: P( J4 l执行过程(以计算 3! 为例):8 L1 ~; h0 U/ ^, l. `+ Q4 x
    factorial(3). B# ^; g. d/ p& k; L
    3 * factorial(2)0 Z+ P6 ?- [  [- G% Q
    3 * (2 * factorial(1))
    9 l% [# K  t1 ?3 * (2 * (1 * factorial(0)))
      T6 g  L! q. a8 p2 z6 B& ~# y3 Q3 * (2 * (1 * 1)) = 6
    9 }! \+ N+ t9 K% z
    7 Y3 ]: r/ Z  S0 d 递归思维要点
    4 b% F9 R  u- r# M0 j4 |: h3 [1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    9 b- a5 Y0 u; R" N  z  U' f! {2. **栈结构**:每次调用都会创建新的栈帧(内存空间)7 ~5 a" Q1 b  I# N: c) k$ N( h
    3. **递推过程**:不断向下分解问题(递)8 y0 X3 ~3 C! d' ~% w( a. v- e6 b! {
    4. **回溯过程**:组合子问题结果返回(归)
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    & o- s! O8 O, w: ]1 @ 注意事项% E0 G6 [( V1 x
    必须要有终止条件
    ' `% p7 P8 U$ s7 x& b) O/ h1 _递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    : s- L- R7 \+ f) J某些问题用递归更直观(如树遍历),但效率可能不如迭代+ r0 {5 M2 e9 j9 o8 i
    尾递归优化可以提升效率(但Python不支持)
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    ; V2 K. Q. N, v5 ]1 _3 O 递归 vs 迭代
    $ V( i; q) w6 C4 j8 ~8 |% w|          | 递归                          | 迭代               |
    ' W7 \" E- t; J4 K( L|----------|-----------------------------|------------------|6 O# O& w1 u5 _! ]: q) M$ j
    | 实现方式    | 函数自调用                        | 循环结构            |
    3 |5 N, \2 ]5 s1 n| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & ^7 B- E$ o7 U4 R0 {3 f9 P9 o| 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         |
    - t7 e" Q$ b; p+ G| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ) p- Z9 e4 _3 X+ Q% c' @# E
    ! V1 w! J! W' J6 n; L. E7 ? 经典递归应用场景
    ( ?5 i  i( Y' ~$ B# g0 ~1. 文件系统遍历(目录树结构)
    ( q4 P4 H4 q8 j. f0 N* R. o2. 快速排序/归并排序算法3 l$ O5 `" H5 N8 X; p9 Q# _
    3. 汉诺塔问题$ _! C/ I$ L; r
    4. 二叉树遍历(前序/中序/后序)" \( u% p8 C& V' T- y
    5. 生成所有可能的组合(回溯算法): v6 h/ ~1 j; N
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    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,' L/ a, @1 K6 V  N- @, X- C
    我推理机的核心算法应该是二叉树遍历的变种。, t' ~. @. `3 q* M2 c2 Z7 X
    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:
    . E( [1 _: f5 [, iKey Idea of Recursion. e9 g: X: _  Y# D; u, u
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    A recursive function solves a problem by:
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    ! v' P7 W1 {, k7 ^5 ~" t    Breaking the problem into smaller instances of the same problem.
    * U* v7 w5 t! ]+ t- Z( V
    ' \0 J6 @# b1 T5 M    Solving the smallest instance directly (base case).
    % d( k' ~+ ?% `5 ^" k8 [! R8 r" n$ L, f* T7 o% N& P2 c4 B7 _# X& T  n
        Combining the results of smaller instances to solve the larger problem.
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    & y! U, r1 V$ S7 ]1 x8 A* FComponents of a Recursive Function
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    . G: C" B& |3 F/ Q    Base Case:
    " \9 V8 ^) [! Z: [0 `9 k& D8 l2 l7 n. j1 A9 @+ E6 Q
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    * ^  T7 ^7 U# G! K" H# A2 m        It acts as the stopping condition to prevent infinite recursion.
    ' g; ^7 H6 L; v: g: q/ @5 u& c
    % A9 k7 L+ B3 |/ Q# T        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; ]' l1 }4 p2 `- f0 ?4 J  @
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        Recursive Case:
    6 L7 ]8 R2 ]' X
    - j6 y: b# C7 _$ A8 V0 {        This is where the function calls itself with a smaller or simpler version of the problem.
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            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).  A( g- D$ @; v2 @! r

    / Z* M- {2 f$ vExample: Factorial Calculation4 l; P$ W& n% E- @

    4 |9 U" y: G+ oThe 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:4 X$ G: t" B, q* \+ E: r

    8 J* ~, u, b- O: X% m2 d* ~    Base case: 0! = 1
    1 U( [4 k5 K2 _/ A
    ! p2 x" z# p: i3 x    Recursive case: n! = n * (n-1)!) e& f" j6 O! i8 z
    * p8 X; r% D; ]# V$ l
    Here’s how it looks in code (Python):8 E- s* b/ a& P1 O+ D! C" H& r
    python9 T' O, t, Q$ i5 N- k6 _
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    def factorial(n):
    2 K/ w% P3 j( g4 g; w0 Y6 ]    # Base case9 M7 g& U+ g- N) r& b4 B5 `% b2 S
        if n == 0:
    * N( e$ O  M) p  A: A$ ~        return 1$ n! M0 U5 h  i$ L7 U; o  N
        # Recursive case. b# q0 P7 n8 y7 w7 N: w( f5 x
        else:, h$ _  t- B4 K
            return n * factorial(n - 1)8 a7 R5 U0 J. F' p; j7 U

    8 o- V& M: W4 q; k# Example usage
    7 G7 b" i. ^0 l  M- O! ?: ^print(factorial(5))  # Output: 1209 {- `3 }5 W9 Z, x

    7 C, c' d! [& {/ P" RHow Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.# M, v- U6 o& c7 D: C$ i) i* w5 r

    , j: X# ]; e" J5 R, s9 J    Once the base case is reached, the function starts returning values back up the call stack.1 j$ |( i# b2 m4 [1 k- Z8 @
    + T) {/ d# u4 W5 |/ w) _
        These returned values are combined to produce the final result.
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    For factorial(5):. c) u2 n- A5 a* Y8 d8 A

    * i% Y( I  V' }; J
    - t1 w5 Z$ }8 F, ?: Cfactorial(5) = 5 * factorial(4)$ e$ w# H0 `$ m6 N+ C: F/ J
    factorial(4) = 4 * factorial(3)( Q/ ^* @& e; a0 {- H
    factorial(3) = 3 * factorial(2)
    & E& Y! |* h; zfactorial(2) = 2 * factorial(1)
    9 U: ~2 E0 s, {$ L& sfactorial(1) = 1 * factorial(0)8 _1 _6 H: {4 V1 h% H8 e# I
    factorial(0) = 1  # Base case
    6 N8 ]  o9 H8 W- ]% u$ P" F! b1 |. R0 c4 h- q% j3 l, j% i5 c* S% N
    Then, the results are combined:+ I) H) i: j" j  K5 M* Y

      w/ v* W+ E' q6 |  `
    8 c) Z0 i5 @) Sfactorial(1) = 1 * 1 = 1
    # U! C% ]  C/ u9 ^6 M0 {2 qfactorial(2) = 2 * 1 = 2( q  r+ _. P+ ], ~( U$ u
    factorial(3) = 3 * 2 = 6: Y- E: s8 {) h' z
    factorial(4) = 4 * 6 = 24* A- e; A* O4 _9 [$ j( Q
    factorial(5) = 5 * 24 = 120
    7 x/ K7 w! ^  E8 }  Y
    . u, t5 k9 Y6 ?: _, j  t& V$ P! tAdvantages of Recursion
    ) @7 x5 r) g( T: ]1 a% G
    , T3 A# j* j0 g* N2 Q, k( m7 J& g( L    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).) Q/ O9 [4 E5 G# ^# W! D

    ' e1 z5 f5 Y; S    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    $ x% I- M; b0 y# |# _& G5 `1 E* o# m' B& a
    Disadvantages of Recursion
    / c) @1 y) Z, n1 H4 V0 E8 j9 X$ n- A) k% 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.: G8 I" g- L! l' ]: n% B! T
    8 J4 d7 {/ v! n6 S8 v' B
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 y' }, k7 l! T) M2 W( q. F% ^- R

    3 g* E' F3 V& F* z. d1 EWhen to Use Recursion7 _9 x7 T) y1 X

    ' E6 k6 g- J0 U: |- O) `# d    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    0 t3 A& ?/ C$ V8 K* O! ]: q: A8 m$ m* o* n/ p* F; O
        Problems with a clear base case and recursive case.
    . q4 @6 f# k0 [+ G) d5 Z) L  z. c6 J
    - Z$ {, |6 H" g$ w/ H" v0 ^5 MExample: Fibonacci Sequence2 q' x6 \- c1 v0 {6 w3 O  d) Q

    ( w& y! t; s2 s1 I: ?' d/ RThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    2 k( M  |0 \" Z2 y7 N
    3 J; y7 b1 [0 F$ ?. p    Base case: fib(0) = 0, fib(1) = 19 v: X! X" {! G6 o- |$ v4 J

    % q" G: U$ W+ X! l    Recursive case: fib(n) = fib(n-1) + fib(n-2)- d: {4 M, H; H
    ' n5 m1 r, P$ `* D$ D* a
    python
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    9 t/ N+ Z7 R) z3 z: v' udef fibonacci(n):
    ' p$ Z, C2 E0 {0 O    # Base cases
    $ [% |" a( e+ p) y1 g2 E) ]$ L    if n == 0:+ r1 L) ~% [3 h+ Y- t9 z
            return 00 _- O7 d/ r: n
        elif n == 1:
    & {5 }+ E& _* T; ?7 ?        return 1" A6 n; T) o- A
        # Recursive case7 G% k. r, H4 Q
        else:
    ; E) [# v( d. d6 L& O        return fibonacci(n - 1) + fibonacci(n - 2)
    * H7 x! g+ w. [3 A$ e
    4 R) A, [$ _. J- z: f; Y; R# Example usage
    + \, t3 y8 c* Q' z5 Q- lprint(fibonacci(6))  # Output: 8
    , W" O" n3 f/ O
    % z* t7 N* M, Q3 m0 i' `! v3 r/ Y  JTail Recursion: {* }$ F9 g/ I% N+ w
    ) }: @+ G. b; s: ^% S9 \& n. E5 x
    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).! D# [3 S1 I4 T5 c

    % T, Z  Q& \" RIn 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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