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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
    8 j! K, O; @% i7 r& R' d1 B% ?
    1 s/ h+ k# h5 F! G1 f# m3 v; @解释的不错3 c: D. p2 e- E6 @: w4 G& k1 C8 P
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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。
    ! l, A$ L' J8 M) d2 J  z  |
    4 D$ ]1 P& j2 q5 L9 I3 }. ] 关键要素
      ]4 F* ~7 r; e: }; _1. **基线条件(Base Case)**5 y/ F/ |& b, M
       - 递归终止的条件,防止无限循环
    , C8 `# _6 ~$ {   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    + {0 s7 Q( G6 U$ ^. T) j/ H! N/ M' Q0 c2 _7 i* S- @- @
    2. **递归条件(Recursive Case)**# j9 p+ O( E% s! \
       - 将原问题分解为更小的子问题" v7 E- ]- U+ j9 f4 {- t& v
       - 例如:n! = n × (n-1)!1 Z" m; O7 b$ M
    ( w/ h- m) N8 N7 w: n
    经典示例:计算阶乘- ~6 [( L' H8 z" s: ^
    python
    6 ]! n1 f1 w+ J! U* \( Gdef factorial(n):+ B0 z' y6 e% a$ j) b* m
        if n == 0:        # 基线条件
    : ^6 N1 r! Y' {' w* R4 S" a0 x/ l        return 1
    ! ~. t8 y; U$ D& {  ?. T    else:             # 递归条件
    * i' k. @  A; L        return n * factorial(n-1)6 g' D) ?1 T- j; H: X. p: l
    执行过程(以计算 3! 为例):0 F% o, P( `+ c! o
    factorial(3)1 t' H- Z- W* L  s9 {% D9 F- A
    3 * factorial(2)( V- I: X& |* K& Q0 D
    3 * (2 * factorial(1))
    & C3 i. b5 Z+ E5 W( |3 * (2 * (1 * factorial(0)))) k8 ]1 p8 r8 v. k
    3 * (2 * (1 * 1)) = 6
    8 `" p$ u4 ?' P! t/ j1 F
    4 _3 n/ f. x6 `7 S, N2 X 递归思维要点
    + B; K) J0 S9 O1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    / L2 B$ E6 R+ N, y2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    # a' T# H4 `- k7 n3. **递推过程**:不断向下分解问题(递)
    1 `6 j/ ], n, O4. **回溯过程**:组合子问题结果返回(归)4 ^8 t1 i" y/ ~. {
    0 t3 |$ g% h8 u. \, Q/ h3 [4 U  g
    注意事项
    1 V9 n- z# w) h3 f/ H必须要有终止条件
    ) s! i9 r7 `; S- T% ]递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  i$ W. z. Q0 M  `9 m4 n
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ) M1 ?  L: |: M. J1 S6 ?7 G: S) y尾递归优化可以提升效率(但Python不支持)9 e4 D8 ~  f6 W, t

    ) u- a9 P& o+ z" t0 ~3 T 递归 vs 迭代
    . E, S" `4 P) y- n|          | 递归                          | 迭代               |6 A# R3 c1 }3 N% J/ P* }
    |----------|-----------------------------|------------------|6 J$ M7 c$ |0 R+ Z
    | 实现方式    | 函数自调用                        | 循环结构            |, \& z; R" c; O! G
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    - k- v) D" R! v# ~  s% w| 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         |
    1 n. d. \; S6 H: i/ Q| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    7 w) ~7 x- H' ^  A9 f, r9 o- @( q5 H2 S. b$ x, I+ K
    经典递归应用场景9 l; t* i0 I5 ?( U2 h
    1. 文件系统遍历(目录树结构)
    2 o2 q0 n, D4 U; ]1 D' a2. 快速排序/归并排序算法/ W. j. Y3 y: b5 k  C7 W
    3. 汉诺塔问题
    * {/ `9 k7 E% _2 q4. 二叉树遍历(前序/中序/后序)/ z8 S4 r4 q! w- j& N- ]
    5. 生成所有可能的组合(回溯算法)
    : o4 ^4 t% W( \) e
    - f6 I% m! T4 ^( Y% 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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    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,
    5 K! ?' u7 z9 \4 E8 C我推理机的核心算法应该是二叉树遍历的变种。
    ) L2 d# T6 ~) B; x+ l, J7 f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:3 @7 d3 t6 G8 @6 ~
    Key Idea of Recursion$ P( ]  M; z! U- \. _) j2 v0 M
    4 n2 S) F" Y. R3 M# o  w% G
    A recursive function solves a problem by:3 T2 _- Y; n, [2 o9 R: J7 F

    / P2 N) [+ U7 H2 J' A    Breaking the problem into smaller instances of the same problem.0 ?' `( O( }) i1 C. \3 d

    ( F  X- t. O4 |( X5 v; G6 d( u    Solving the smallest instance directly (base case).& a& ]2 f$ B! t! P. n! Q/ ^

    1 T, t. @$ D% j1 K- ?' d9 B3 |    Combining the results of smaller instances to solve the larger problem.
    1 t0 ~4 _: n; m1 l3 Y* Y2 c: E  g/ N7 s
    Components of a Recursive Function
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        Base Case:" ?' O# W, g/ k: s' s/ [; S
    1 p% N/ C! Q. y2 l
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    - ?& m$ O6 E/ z- {. V+ i
    / C: R; `% m, s  T        It acts as the stopping condition to prevent infinite recursion.4 }+ S% E6 P* s6 v' d8 ]

    5 k; o" t) E, n1 x2 I# D* f/ r3 j! ]        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
      Y% e' O' ~; g+ \
    ) ], r" f4 @. s7 w# ~  z0 j& G    Recursive Case:% S$ z' I$ ~/ D6 M  X7 q8 g# f7 R, N
    9 U5 A5 H4 Z& b5 y5 v) v. {$ ~
            This is where the function calls itself with a smaller or simpler version of the problem.
    " B+ K$ ^5 A/ v$ Y* E7 G. {! }2 b6 R9 Y& G* X" F3 Y" P- ]" v+ |( ]5 k% l
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ' S4 u" @  d" \, `* Y3 V2 S: e0 t. n
    Example: Factorial Calculation
    6 i& Y7 C% o* O0 ^
    ' ~# C( x- q- ~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:
    ! l" z2 T' Y8 t5 k
    $ E9 c& I4 A/ ?' }* \    Base case: 0! = 1
    . `( m7 i$ d% q2 ]
    4 f. E( z' E9 m' `9 w    Recursive case: n! = n * (n-1)!
    : _2 p% n; u% t' w2 h  I$ i2 \: J) [! x3 [/ f' _$ i% ?( |6 y. f
    Here’s how it looks in code (Python):9 @# I& K6 u8 f- H# j  @
    python- M, {( {4 `: d/ e
    $ i( Q3 I& b' e

    ! J% E4 w9 @9 \: p3 O' Kdef factorial(n):
    7 T7 t% g7 x) o/ C0 t9 _9 O  t8 N    # Base case
    : y2 N$ d' W0 B2 H    if n == 0:
    4 _8 Q# ]% v4 s9 ?1 l' w        return 1) ?9 W5 Q' f! `# n& v9 @
        # Recursive case: k8 e6 `; f0 \" G
        else:. f0 {3 M$ n1 h
            return n * factorial(n - 1)
    ) _3 \% t! P/ R) r" |2 ]- r1 ^. n% U; V- F/ p% d/ Q6 k
    # Example usage
    $ y% x" g4 I) r7 h: [1 H; J8 R2 Dprint(factorial(5))  # Output: 120
    ; `! K3 X, n0 E: ]8 E7 Y! R# J
    How Recursion Works
    ( b2 x; [, s- Y& u. A! d! Z" P* s) ]4 C% C2 P
        The function keeps calling itself with smaller inputs until it reaches the base case.
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    . `3 p& Y. T0 s    Once the base case is reached, the function starts returning values back up the call stack.
    # g* H* E9 r& N2 S$ E4 i; B1 r" K/ X4 X+ g0 r  x
        These returned values are combined to produce the final result.
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    For factorial(5):. n' @% {* U9 Q1 i# E% t8 M" }, W- _
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    % j- z! g. d# a! d, E0 ^: U
    factorial(5) = 5 * factorial(4)) l5 P. u8 ^/ F4 h8 K! K4 l; W
    factorial(4) = 4 * factorial(3)- Y6 S! T1 L: b/ k7 G% j1 z
    factorial(3) = 3 * factorial(2)
    / D1 G; O7 X: Y# H' e) m5 ufactorial(2) = 2 * factorial(1)! ^* }8 @! K) K& g$ \! F, l
    factorial(1) = 1 * factorial(0)
    4 ~( Z5 a  \) B3 ?1 \6 _factorial(0) = 1  # Base case
    6 u0 c, W& ?  P9 B0 ^; E! d6 r5 a; x' H2 T) f. f; L4 h
    Then, the results are combined:0 w# L( Y& D" S: U

    * V) O7 V/ _" F, B! H$ ?. n9 b+ q$ i0 c2 C4 Y0 `
    factorial(1) = 1 * 1 = 1
    ( V- i' D1 j7 C9 ]& Pfactorial(2) = 2 * 1 = 2* M7 p# q% J# u2 i1 V+ B# _
    factorial(3) = 3 * 2 = 6* v0 T* y8 z2 o) m" j+ f& |7 w
    factorial(4) = 4 * 6 = 24
    ' v$ g' m0 a- P. Z8 L& dfactorial(5) = 5 * 24 = 120
    * @$ u; n) t8 K6 Z  e. j9 R: `" V% s
    Advantages of Recursion3 }  C, s) Q& f& q/ Z1 w

    " N; |0 m( h2 Y) Y1 n7 ]1 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)., M  a( M. `- {/ L3 M

    : e& x- m0 W8 b$ w& Y; P    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ( v6 v( m$ u  T' z1 _# n& ]# @. T: Y: O7 |$ q. b3 i
    Disadvantages of Recursion
    ( n8 y. }+ I2 I8 a, S2 _( u* p$ S2 P: C0 r
        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.
    ! h; i4 W' z+ G$ I, i. B( Z0 Q5 x! Z- \  F
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    1 q( ]! ]: i/ H% [! F
    7 B, w: w8 x) ^8 |6 n: BWhen to Use Recursion
    4 D. S- k( d- B
    , D& }3 C- c' v. c0 K    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    " }: e6 P/ V8 q5 g4 [8 w6 W' m1 q
        Problems with a clear base case and recursive case.# i  t/ D) r+ X8 }0 p, G( ]

    2 o. k( e, ?& vExample: Fibonacci Sequence% P+ A4 [" L- e* R3 L' @

    ) v# L* g& H" x: rThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:: K0 X4 `/ T' o: A
    ) \; g. b% V; V' R  E0 Q$ [( A
        Base case: fib(0) = 0, fib(1) = 1
    , f9 y' x" S& A5 t" V
    % P4 V6 K; c! O' @1 G. H$ g+ f4 e# H) u    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! [$ L- j" R0 n' [" ^% q6 W! f5 t& m& F( c- I) q" @  x5 E! u" V
    python
    # T- b: \+ c9 T$ f) ?+ J. y5 J, c# m: F3 U2 }0 A
    7 k) v! o# m% t/ |5 P2 l- \4 R  U. n
    def fibonacci(n):0 @3 g# k  t/ ^' X9 s! a! U6 V( A$ }
        # Base cases5 A3 B+ K- I) V5 m
        if n == 0:
    : l! L* N6 x- n0 b        return 0
    2 S! h9 b& @- U2 L6 a0 K    elif n == 1:
      F$ @& |* ^8 z: F7 p        return 1
    . T: |$ p# l4 B3 T8 u    # Recursive case/ _, V* V1 P, o4 M. P
        else:  [: F* R2 K$ S6 I; x) j# s
            return fibonacci(n - 1) + fibonacci(n - 2)
    . J; C( o0 W% P) O/ V+ b5 ]& h+ z- Y) b. [7 Y. x3 F3 s8 b
    # Example usage
    ! m& c2 }; A, `( G: U/ X$ R3 nprint(fibonacci(6))  # Output: 8
    ! m1 X- i7 l9 S" F5 m+ V7 a! Y( D& P
    % p7 ^. h8 `) E0 q2 J% TTail Recursion
    8 N. h% d: f  X- X" y  J  e+ [6 v( s
    9 p% z0 Z2 b' e% u' W# k$ 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)., i, B# h+ o$ t: C# @* x; E! s

    3 T) a3 Z4 v6 y+ i2 X% lIn 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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