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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
    7 W; D8 F: `& \: g4 V7 P$ E6 \( D) |0 D# ]: j3 n; W; R3 H
    解释的不错
    / C" \' l9 q  F3 i1 e4 E. X0 ~6 b: i4 Z
    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。
    ( Y5 x* J0 E* r7 C& b4 N- U: P8 n/ k# U* d
    关键要素/ F2 v" S* p: C. c( [3 {3 Q
    1. **基线条件(Base Case)**" Y& h# E; c5 `( @3 z
       - 递归终止的条件,防止无限循环& }2 {( f, X- L) Z" L
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1! j$ q/ D  v6 g( E. A& d- _

    ; {- D' M# o/ K* U3 ]( z+ v" a2. **递归条件(Recursive Case)**
    - O* i+ h$ e9 ~. F7 I7 V   - 将原问题分解为更小的子问题
    - H* P* R7 ?8 t5 b( n' j' r& ?0 `' M   - 例如:n! = n × (n-1)!
      A  Y8 D! s9 a  X/ l% n( ~" l, F5 h) b) {
    经典示例:计算阶乘& Y/ U& }6 D+ z: r" q
    python
    " x5 A7 L7 l2 ~8 W. ~/ o$ Idef factorial(n):0 S+ r! m9 ~1 {' F
        if n == 0:        # 基线条件
    ' U4 z6 L& ]" J; y# f        return 1# e" n3 Z) j  ^2 [! Y! Z
        else:             # 递归条件. [& ]/ k/ G' C1 s+ T  G1 [
            return n * factorial(n-1)
    2 P2 U- `! w( Q1 E* c" T执行过程(以计算 3! 为例):
    ; a) T* i0 E4 mfactorial(3)
    5 T$ z. V0 ?' ]. q4 U3 * factorial(2)$ r; W% ?4 \) D* ^4 c
    3 * (2 * factorial(1))
    3 N, H: E  P& q+ a7 e. a* M* r1 p3 * (2 * (1 * factorial(0)))
    + m; z  e) ]9 p0 j, x9 v3 * (2 * (1 * 1)) = 6" H$ k0 P- Q3 n0 {# H8 N% Z0 `/ |

    1 n# u) u( {* V, T 递归思维要点( u9 i1 m2 v( V8 u
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑# |8 X1 D* u& m  b# ?: ^+ [
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    8 s" |- n4 f$ W* |  H) u, J. e3. **递推过程**:不断向下分解问题(递): \' A. o4 p2 a/ p9 @
    4. **回溯过程**:组合子问题结果返回(归)  u% d6 N+ N: Z* N3 b

    2 k+ G6 N) z$ O9 B# G* ~. \ 注意事项! p8 n: y5 g, g$ R  ?
    必须要有终止条件9 f2 O4 k3 J1 Z  d/ s" v
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    8 t: B/ R. d, V. h某些问题用递归更直观(如树遍历),但效率可能不如迭代
    / m: K+ k' c; m0 N0 D* W4 l尾递归优化可以提升效率(但Python不支持)
    7 K8 L" ]# S9 e# _
    ! k9 q! I9 c! l- U 递归 vs 迭代* E8 j  {3 T: Y
    |          | 递归                          | 迭代               |) _3 r( {1 Y; A9 `. h& P2 D
    |----------|-----------------------------|------------------|
    " ~( m$ h$ C5 k1 i! o$ A( ?7 ^| 实现方式    | 函数自调用                        | 循环结构            |" S6 Z* I: L8 J8 u# ]! @# Q
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ! S) r4 h; C2 N$ }. a) i( T, `4 `| 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 R. u7 U( _# i
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    # m& L% z6 [. @1 e) [) @6 Z8 i0 I0 G7 o( o1 {4 _* r
    经典递归应用场景
    $ g5 V) B3 O0 f4 S1. 文件系统遍历(目录树结构)
    , j; N: l( D! a( Q! I2. 快速排序/归并排序算法
    & E! u2 M! `7 {3. 汉诺塔问题
    3 u( Z1 n4 n" |. S1 d. ]4. 二叉树遍历(前序/中序/后序)+ n/ {4 H( k) a
    5. 生成所有可能的组合(回溯算法)
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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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  • TA的每日心情
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    昨天 06:31
  • 签到天数: 3146 天

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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 x. U9 C/ {* y. a. J
    我推理机的核心算法应该是二叉树遍历的变种。* D7 k, W, V6 z  D/ m
    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:
    + ^! F/ z2 u' ?; WKey Idea of Recursion7 A9 N1 T+ S$ Q; W& u  A  a, _

    6 n  k4 {3 x  E3 E9 K( WA recursive function solves a problem by:  u4 I+ S) {6 T+ J

    0 U; I- ^# j9 v3 `. Y5 C    Breaking the problem into smaller instances of the same problem.
    8 K) X1 `, I$ U. v, c! ?! ?1 X/ U
    1 B# @* N$ C6 C1 @) a    Solving the smallest instance directly (base case).
    ' x: d( O  i* g: ~8 f
    * n, D& o  s9 C9 l3 ]    Combining the results of smaller instances to solve the larger problem.
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    7 u% u4 E8 X: C* \* @Components of a Recursive Function( A! V7 l: m' k4 n

    5 z" q! `  L6 r2 o- j3 J" r7 r( Y' ?    Base Case:( I3 e8 r# A9 y

    $ u$ N" I& t; ]        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.7 r  j+ v3 K1 R( J

    % _$ Q( t$ {1 d* ]        It acts as the stopping condition to prevent infinite recursion.
    . @* u8 s2 f7 \* I, }( x# N9 _! ^# q1 q' e4 D
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    2 R1 j/ m1 E) q6 C5 H7 k" v    Recursive Case:  v. U! r" t% B! |
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            This is where the function calls itself with a smaller or simpler version of the problem.$ B3 ^& z* G- ?& |* M

    3 D: _3 M- L  n% V. @# u        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).4 j1 }8 K* x( ?- Z) p
    9 [1 b+ ^7 R* R
    Example: Factorial Calculation  o- @% w) x/ r

    & T- F3 n8 m# @  d7 N6 ^8 JThe 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:9 x4 e' B' @! O0 J
    , Q9 O& s2 y. d4 R
        Base case: 0! = 1
    9 A- J+ x; k! |9 r
    + l% R% X- k1 ~: ]. ^+ h    Recursive case: n! = n * (n-1)!
    5 L  S  }9 N( w9 e/ p0 G; ]
    * {  Q8 C$ M/ m0 q6 u9 b1 CHere’s how it looks in code (Python):
    : M. y& k0 [+ H/ b! [* qpython; y8 R: Q' K( R1 |# X

    6 s' q8 V3 e# {' n, r+ _* ~6 W3 I9 G0 V8 L, z9 U# W7 v
    def factorial(n):
    4 r+ u9 K, q' P% B4 H    # Base case
    5 M8 ^  W; |/ _: _% R    if n == 0:1 j( K* ^# b( Q& R  e
            return 1
    # {' h( j6 W2 [9 h    # Recursive case0 L5 z& T7 U. V3 S% N" ]
        else:4 t& [1 M5 z  U& H
            return n * factorial(n - 1)
    8 E  |5 Y! I8 _
    0 J( L: D, O/ S/ B0 z+ o" ~* e# Example usage
    2 `' p! f; |# r+ qprint(factorial(5))  # Output: 120
    3 S) Y4 q1 q2 a4 u
    - K! o! s# Q  k) J+ N: HHow Recursion Works9 a) {: k* y7 O2 f, O
    : F5 m- h8 N$ ]: D1 y
        The function keeps calling itself with smaller inputs until it reaches the base case.
    2 h# u0 ~9 ]/ ~& |- F$ l  ~0 [4 ]; F/ V! O
        Once the base case is reached, the function starts returning values back up the call stack.9 {9 \& l$ ]6 Y) U
    4 b* c( ~% J1 b6 o( W
        These returned values are combined to produce the final result.% a; k( ^5 ~) |3 M) @
    9 W6 ]4 {' R; P4 n0 P
    For factorial(5):
    : w; M/ F" k  l( g
    . f* @6 J8 @, [; r2 G4 w* E1 H5 @' P5 _9 V. N4 r8 o
    factorial(5) = 5 * factorial(4)
    " o0 P  g7 V) `8 Wfactorial(4) = 4 * factorial(3)
    ) b: r1 \  }8 \factorial(3) = 3 * factorial(2)$ }; @9 }1 W( V: @) P
    factorial(2) = 2 * factorial(1)
    7 ^# P1 Z( s, |0 n9 m' P6 Sfactorial(1) = 1 * factorial(0)
    ( `9 D( l; g9 X5 U' ffactorial(0) = 1  # Base case0 Q0 L* _/ [7 v4 Z
    ( U+ Q$ x9 u. L% J8 z; ^
    Then, the results are combined:
    6 \) G4 \' o& S- m' W  T6 Q# S
    0 U, \9 f, i, l- F0 N9 r. U1 u5 `9 E6 ^
    factorial(1) = 1 * 1 = 1) F- [$ s) E, m. |3 ^$ J8 ]6 y- F
    factorial(2) = 2 * 1 = 2
    % U- v/ `8 j) C' S) |factorial(3) = 3 * 2 = 6
    ' e# {7 j; o$ e3 O) mfactorial(4) = 4 * 6 = 240 F% S) o0 d1 v6 Q5 D/ N; }1 a
    factorial(5) = 5 * 24 = 120
    % Q% q2 A" Z, l% K: b# m, [0 q' q6 p) m- y( _
    Advantages of Recursion
    $ f- k( |! \6 t+ k& T) p
    - {9 `3 B, F  x1 b) j2 @    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).# j0 |2 }: [6 W8 J% Q/ n: V
    2 e# r- G) D' g; [
        Readability: Recursive code can be more readable and concise compared to iterative solutions." v9 B/ Q  q/ ?. g+ A5 n. r

    ! w: Z) `$ M! F6 Z$ a+ w; hDisadvantages of Recursion
    3 F; s$ r8 |" p: J1 a' i. e0 {8 e* ?0 T& z: S% W$ F$ I
        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.8 |1 D0 ]# m* L. R# K+ E

    ( g9 H* i$ _9 P: H4 p9 f    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).& _9 e  ]/ t3 t( f" F) o
    1 b2 G( R' X0 S
    When to Use Recursion1 i1 {7 t! J1 p$ A4 C, o
    - c: _, }: d1 H. D; q
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).. a4 s8 p0 B3 I. q) {6 @

    5 c' Z% ^9 e* T    Problems with a clear base case and recursive case.: X! v: |  b4 v1 G7 D# w: V

    6 b* e7 e% \: P$ Q, X% _Example: Fibonacci Sequence2 X2 B% w$ [; j. K4 i
    . u4 ^: Q" G; |' N; I0 k1 h: u
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:% O# p+ e+ `$ [8 \/ r% \1 L' r; T8 r
    1 J% X0 _9 v  d
        Base case: fib(0) = 0, fib(1) = 1
    . ^4 i+ k7 B8 U0 x) S2 D. h3 T$ W
    : Z3 D& }) W, P4 ?2 l    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ) e3 T# n7 ]% ?' ~
    9 w. N/ ~  ~' ^( f) @python9 a. D" W- Y* y

    - K( _8 o* L- `* v& C8 h2 [( \3 r2 Y! \
    def fibonacci(n):# }9 W( Z! I1 O  @# E8 M
        # Base cases7 O9 L% @! Y! i! s
        if n == 0:
    # R1 C5 K3 @# h8 R        return 0
    - I6 ?: W+ B( L. f- I1 ~    elif n == 1:( @/ P- _& L; l% n/ S; T
            return 13 c% T( ?+ \6 X
        # Recursive case6 U, {5 d0 G9 D' L) k$ j% \( S) G
        else:$ p! r5 r% _; j2 t/ |% E
            return fibonacci(n - 1) + fibonacci(n - 2)
    ; a: L* M4 G: k! f7 [9 ~- r5 g2 K# m4 b: v, K  H2 P
    # Example usage
    # o- X2 b, i* K3 Uprint(fibonacci(6))  # Output: 8
    6 `. f, q- D/ k% S3 N( r% Y
    , Y7 }4 u* L. f$ _+ e0 N4 kTail Recursion/ y; m: d/ `$ N1 b' ]- @

    ! X0 m1 O. K! f  W6 z$ PTail 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)./ S7 C% ~, Z- N$ y. m' j

    , W' P3 M, N0 n. G* ^1 B8 y( ]  ~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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