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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 & H) r* h4 I' |8 U' h& `! {1 F
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    解释的不错
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    - C0 D  c. X% K! t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。
    : \" Y' y. G0 s% ~' d
    & G1 `- i2 V2 ]: S" m0 ] 关键要素
    : y3 E; B3 e$ D1. **基线条件(Base Case)**8 V; k% V( S& y% Q
       - 递归终止的条件,防止无限循环
    3 d8 d% L7 s7 S. x   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1. z' ?# B6 M: S  ^& m2 w

    0 a; P6 M$ [: w! I( o2. **递归条件(Recursive Case)**
      r" U$ f. o9 m- n   - 将原问题分解为更小的子问题
    5 u; f, F% G/ S! r   - 例如:n! = n × (n-1)!9 k/ {, N3 \! z" G9 Q& L& R5 ^  M& s

    ) B2 |* g, n6 f6 T+ F7 G& } 经典示例:计算阶乘
    + m8 ^0 g6 H% T$ z% U7 S2 \python  s4 o) E% h/ E, Y- V- v
    def factorial(n):+ p5 g  J: y4 A% X0 e8 U  N
        if n == 0:        # 基线条件
    . h+ D) N* S; a# Z) G1 C. ]+ o, S        return 1) k( H' p+ n( Q1 ]' M' F+ {
        else:             # 递归条件
    4 t/ Y1 e5 [$ {- W% R5 c2 p        return n * factorial(n-1)
      d' I  r  M. A9 x5 N) d8 K8 ^执行过程(以计算 3! 为例):  v* o& G& u' s( A' t- x
    factorial(3)1 P8 t  r5 c+ a
    3 * factorial(2)
    ' g9 d& }, n3 m- U3 * (2 * factorial(1))
    1 e1 q3 L; j; L( O; B3 * (2 * (1 * factorial(0)))
    " C5 Q5 x, Y2 I4 h& T* m3 * (2 * (1 * 1)) = 6! s6 m2 K/ b) T2 c* Z

    + }8 G5 I1 t4 y$ r* y+ ~! k 递归思维要点3 Q& t5 p  p0 d* A* p/ `$ A* K
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ) R2 e# h' x# W$ y0 v2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    1 d" m9 {; @; u* l$ p3. **递推过程**:不断向下分解问题(递)0 @" p- d% C' {, m
    4. **回溯过程**:组合子问题结果返回(归)
    * N" i) Q; N2 M; b
    8 l9 ?. |' d; S% m7 Y+ ] 注意事项" `* I  y4 s1 m' m
    必须要有终止条件
    8 C' p" Y" \; I8 ]$ |: ~' ~递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    6 `4 D8 f" E: B) j- ]# T4 G) e某些问题用递归更直观(如树遍历),但效率可能不如迭代/ @' A9 t/ i- V& n/ s
    尾递归优化可以提升效率(但Python不支持)/ t: {1 @  }) H  j
    . U" }1 J( I7 ^/ \; G9 c6 ^
    递归 vs 迭代
    ! ]+ a8 h. T4 \$ m8 |1 k; T0 J|          | 递归                          | 迭代               |& r' i& u. `0 O. I
    |----------|-----------------------------|------------------|
    ) J8 V5 q0 N  F$ W| 实现方式    | 函数自调用                        | 循环结构            |# E3 K- p# k" Z5 x% I
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |% ^9 F+ ~9 K' d; u1 V
    | 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         |
    # y6 N& ~2 ]- S5 i| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    7 U$ H' t& C# j5 c* C" x6 R+ ^. C9 l" x. t
    经典递归应用场景
    / A* |( z2 l2 D! I/ ~7 H, |1. 文件系统遍历(目录树结构)
    * l. X" e, @: f+ s2 `& Q0 d2. 快速排序/归并排序算法) m4 L  m+ D7 t
    3. 汉诺塔问题
    # e: }1 a; U' ]; h3 S4. 二叉树遍历(前序/中序/后序)
    0 x/ P/ C8 l! u7 E5. 生成所有可能的组合(回溯算法)6 f6 ?- s8 Z% {; {, p0 K" }
      l+ V; a% O% G0 r7 G
    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 06:39
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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,7 ]1 k/ J3 O( V8 \( v' J4 Q
    我推理机的核心算法应该是二叉树遍历的变种。5 A/ {" }: Q: @, n# q, v% ~2 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:# x; X. k5 K6 v: C' g
    Key Idea of Recursion
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    A recursive function solves a problem by:
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    5 ~2 R7 j9 @, E/ b    Breaking the problem into smaller instances of the same problem.  S4 A2 N+ W& q( F

    1 g% M, N; M& A7 b9 d7 \    Solving the smallest instance directly (base case).
    1 x$ ^: R7 z% s" J/ Q6 q, U' y/ v' Z, m0 O6 ~, x
        Combining the results of smaller instances to solve the larger problem.
    . D9 X8 o; u1 {& x  _
    & H( Z& Y. h+ `7 y; D9 v. JComponents of a Recursive Function! z9 ~) l; ^+ l

    2 k- Q+ n0 w# {1 |    Base Case:
    9 t7 I; ~9 z8 k' m8 T7 E
    + v7 S' o, ?1 B        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    8 E; i$ [$ V" u2 {! a' \" r" c! U1 I8 M) [7 Y) S2 h( f
            It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    ) T8 e6 N0 F) m( g" w    Recursive Case:
    / S" V& n. @5 [
    2 h' `- C. L, C, Q- a( U        This is where the function calls itself with a smaller or simpler version of the problem.
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    # s+ ~3 b8 c( {  \& b! n        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).: X5 R4 `& G# `
    % X2 j  f5 W% I' p: L, \
    Example: Factorial Calculation
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    ! h/ |7 d: r( hThe 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:
    5 d  j" h' g8 W  r5 y/ r+ k3 d+ {$ e$ [
        Base case: 0! = 1
    ' j: H9 d- _; [, S* ^" _  h
    8 S  |. F6 {# f. Z/ E$ }" g    Recursive case: n! = n * (n-1)!6 d; t% Z5 Y. Z/ ~3 j9 q
    4 b$ `. P; p3 T* G' m. L( S
    Here’s how it looks in code (Python):
    $ M' C7 w, S. t0 M) hpython
    % R7 \* G- K# ~8 E- D( K; C& N; `
    ! N6 ^. [$ Z+ d  y" x1 z
    def factorial(n):/ Z. p# d" G5 Q9 O
        # Base case1 n, y! r9 I1 X% p
        if n == 0:1 I9 I6 b: g: c  \3 A
            return 1+ j' |/ L2 @6 A) @. d
        # Recursive case- {& @/ [9 o% n! N( w$ l+ i8 |
        else:( J( w/ M: m. w- J& h4 Z; e
            return n * factorial(n - 1)
    + _' O6 F- e. E7 J* f$ C* d% W8 `- A
    # Example usage
    7 G, S8 E( u- l# b; p! vprint(factorial(5))  # Output: 120$ K; j9 B( v; F4 P
    5 s  _+ f" g. D6 _8 H4 P4 h
    How Recursion Works, y1 l  {# W: T! ~' O
    * [" m5 Z  i  \
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ' [9 d% l0 L1 p* q9 }: y: y1 L* w9 p/ b$ @! ~
        Once the base case is reached, the function starts returning values back up the call stack." ~9 y# I7 U5 a( L. s3 _' G' H2 J

    2 m+ o: i) i/ @' g/ T% @/ Z    These returned values are combined to produce the final result.' i7 f3 O( F" ~* `3 o! ?

    6 P% F- [# U0 p* M/ lFor factorial(5):$ A$ V) r* ~' U: p

    ! w( L! O: O( J! S1 E& A5 C  K- x. V) {" A! Q% B3 N
    factorial(5) = 5 * factorial(4)" T# a8 u/ \8 l+ i. C
    factorial(4) = 4 * factorial(3)
    # r0 B/ ~! f6 Q) {factorial(3) = 3 * factorial(2)/ G% _% A7 u  \0 z3 Y
    factorial(2) = 2 * factorial(1)
    0 K1 W5 [  _& [+ l, K7 k( |) \  tfactorial(1) = 1 * factorial(0)
    ( j0 k" p6 j- Q$ i: y$ M9 Cfactorial(0) = 1  # Base case
    & w7 ~: v2 F; U1 ?
    2 z8 K6 |) |$ j/ W# aThen, the results are combined:
    4 ~, l- `) x7 H1 k/ t% |
    5 E; I8 ~' n6 p8 O2 w: W
    , r- y4 |# i9 p- c% d% W& A: efactorial(1) = 1 * 1 = 1
    5 a, O' {: ^1 v* o% f0 rfactorial(2) = 2 * 1 = 24 p9 R9 G$ f* L
    factorial(3) = 3 * 2 = 6- @. E" B! p. D: ]* ]5 m0 S
    factorial(4) = 4 * 6 = 24; a- a. j2 F) O: F4 ?& [$ t
    factorial(5) = 5 * 24 = 120/ y2 T) ~- U( S; S

    4 v. [! |% `% q8 X6 \& |- ~Advantages of Recursion
    4 n8 o. f0 W2 U; i" O
    * ]& {, d6 g; Z, C    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).! g8 w& J0 t4 O+ Z7 V

    7 B- `/ m2 N2 u6 G! W0 c9 K    Readability: Recursive code can be more readable and concise compared to iterative solutions.2 T( E( |& @' V. R: G0 G5 |5 b2 R
    ! e- h3 ^: ]7 ^) U6 s
    Disadvantages of Recursion/ b& a7 N4 m+ N4 }9 d

    0 H7 F2 n$ e) [2 O& D5 Y$ N. W    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.7 O% p, N  g( C# j4 Q
      k4 Q. g% F) S' ?
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    0 f7 B) [0 w* J
    , @# j( G* k* b% xWhen to Use Recursion  \4 d4 W" U' @* l0 F2 ?

    & z8 R, U8 M& ^* `5 m! U    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).9 q9 C# _6 m+ o, n8 O  \2 G

    + [; r8 Y- _! W, Y% p. |    Problems with a clear base case and recursive case.
    " i4 _8 y: k' M  |2 J1 |( m1 o- X! V* D. B9 O" y3 C
    Example: Fibonacci Sequence1 W, F( \% m. b- x4 g% I
    ( v! \( i" W* _' V5 Q9 s! g: A
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:. x- |4 c* Y5 o: e) J( |) m

    ) O9 \$ C# p3 B3 j) ~* Q7 z    Base case: fib(0) = 0, fib(1) = 1- V% q7 _/ X$ v; j9 l  [$ S4 m
    9 j$ P5 D- c# y/ a; S6 X
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    & j) |' U" ~0 S! H1 k3 j8 z6 B8 p1 B! p
    python# B7 S! x- [9 ~, m

    & l- R( e$ S) K3 X6 r  W2 e
    / r3 [7 L* r0 a! U- @/ M5 f) rdef fibonacci(n):
    9 x+ K+ v1 V+ p    # Base cases0 N: {* O1 O* ?3 G3 N: N3 [
        if n == 0:
    . Q/ y" u5 j1 T- {5 z# l        return 0
    7 M* X( |2 J. l% t5 L% x    elif n == 1:
    9 `6 V& {" {) o; ?5 L        return 18 ~0 p: O3 H, x4 N7 m  G
        # Recursive case
    ' B0 K/ H( U% ~4 G/ L    else:, A# r! n8 y! @* w
            return fibonacci(n - 1) + fibonacci(n - 2)! Q6 G6 z( t, _) n3 U* B

    5 h0 `9 v* s# z6 V" \1 H* ~5 R# Example usage
      Q2 b8 w& I2 @% X( ?! uprint(fibonacci(6))  # Output: 8
    4 U' {7 l& o1 g. C0 g: E, G) p% Y2 H- I, r3 A' K0 q& s
    Tail Recursion& l' c& M7 \' T
    / ?3 K5 U7 R4 {1 ?! U
    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).
    1 L$ {  {8 f; ?, t+ d3 b4 `7 }3 ^4 {9 K6 l, q; w
    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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