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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
  • 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 1 {- ~3 `2 w) q; U  i  J

    - ^* Y" K6 j: S- L0 X6 q) V( f解释的不错/ w: z: I" l8 B+ B

    1 {; l1 ?( b9 m- w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。
    9 b# q* |/ c7 K% n
    * O$ J& h, z' i4 X 关键要素
    # \. F# w# N; f1 x- ~: u1. **基线条件(Base Case)**5 }0 }* h* H% Y; r6 _
       - 递归终止的条件,防止无限循环3 o3 r! F; [, k7 z- k
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 12 M/ ]0 Y" V- v/ v4 s5 g
    , F7 ?! K9 B' Y2 P$ k3 t4 b; B5 J
    2. **递归条件(Recursive Case)**  G5 Z. P- m- |& k
       - 将原问题分解为更小的子问题
    * [/ ?  B% ~' t7 a0 e   - 例如:n! = n × (n-1)!
    , r& T1 o  q0 ]! u: {' c
    8 P' V/ i0 d! f) Z 经典示例:计算阶乘; g" `6 n$ c; d# ~4 l
    python
    + {  X, i5 \0 G; \% w8 F" ^/ u" tdef factorial(n):9 Y9 m4 Z/ v9 i4 m- }0 s
        if n == 0:        # 基线条件7 l$ a: a9 C" U* v* j# J' b3 R
            return 18 z7 L) U, j& ^. o% k
        else:             # 递归条件
      b. L, E% C. }  T* X        return n * factorial(n-1)
    9 B$ R$ k: n' J, h# u* X, D执行过程(以计算 3! 为例):
    . Z! ^9 t( [6 ^. r/ A& \factorial(3)
    % L9 w$ t# }5 t! b3 * factorial(2)
    % s/ r$ n6 b$ ?& D0 ~; E# i7 E3 * (2 * factorial(1))1 |7 Y6 r. N5 I( l
    3 * (2 * (1 * factorial(0))): O5 P' \6 z# w2 Z- i
    3 * (2 * (1 * 1)) = 6' B; O* ^1 e* o

    : R1 Q! s' M5 N 递归思维要点
    : Y) s! g( w4 r! v1 R1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % z9 z4 Y& \) e7 ?, ?. `2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    : i& E6 g* a) b" i3. **递推过程**:不断向下分解问题(递)
      S) x! _1 J; t4. **回溯过程**:组合子问题结果返回(归)! p- [6 w) Q2 \7 D
      K; k1 d  U* A" B) \
    注意事项
    ! J, A! I0 k0 u$ @, H  x8 K必须要有终止条件! r/ V# ?! C( \6 w) c" Z$ P
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( w& O) e0 X: O5 Q. i! D  R* Q
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    2 w4 N5 C" q0 X0 y- c2 [% A尾递归优化可以提升效率(但Python不支持)
    4 x  b1 b' |8 c; X, S  F, B2 ]8 }$ G8 N0 e! g
    递归 vs 迭代
    ; b& \) N% N* z% Q6 t- f|          | 递归                          | 迭代               |
    ; B# ^- h& R) Q- r5 m; z& t# m: y|----------|-----------------------------|------------------|* H+ s; N. R  k% U6 w% Q! f: X
    | 实现方式    | 函数自调用                        | 循环结构            |
    + N1 S. M0 |" c( Q2 `| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |' C2 N$ r5 U, L& y* m6 R
    | 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         |
    5 q0 `# L: p- v2 v( O| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    9 |" l4 {! K7 E2 h1 E. p+ `% X; L5 Y; H8 P# T
    经典递归应用场景: n, ]4 k* V9 s
    1. 文件系统遍历(目录树结构)
    0 L- @$ ]1 z1 ?9 |& i! ]( v2. 快速排序/归并排序算法
    ( g  h; e! }  p8 A# a% c3. 汉诺塔问题
    ; Z9 ?' ~$ n0 c$ p1 a! c* ?4. 二叉树遍历(前序/中序/后序)0 {! [; B! K% z
    5. 生成所有可能的组合(回溯算法)2 D% O( h& J* d- Z  j9 D
    ' b- u! O7 N5 U- k
    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,8 d1 d; I5 b* j1 `' F2 I
    我推理机的核心算法应该是二叉树遍历的变种。+ p3 ]+ N  v3 C8 C" Y. M1 n
    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:
    " m2 U0 @; i4 _# U5 ^Key Idea of Recursion4 R: z# r9 p( u; w& g) P! [$ b7 n6 {
    4 o& n/ D2 Y9 d# w2 _- o5 t9 p- Q
    A recursive function solves a problem by:
    5 z; x$ f* v8 M; u9 v9 C7 B6 B! c3 P9 T4 N
        Breaking the problem into smaller instances of the same problem.3 [- h* O; ^2 b

    1 }% I' ~) X) k9 T# @1 _    Solving the smallest instance directly (base case)., _" @' x& `7 e9 f

    ; J3 x$ Z0 I" U! f    Combining the results of smaller instances to solve the larger problem.
    2 E6 i! M! ]# ]" q$ j* Y2 |8 v. E9 w0 Z4 C% d+ E
    Components of a Recursive Function
    8 P! D5 j( e; R! E* P, d1 @" w* r8 J; _  k
        Base Case:1 ~7 R2 H5 H3 N3 W9 y( c) [" _
    2 Z' g5 Q+ G7 N7 H
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    : x$ y0 X4 H- B. Z6 a; F( u) }
            It acts as the stopping condition to prevent infinite recursion.! c1 L6 p9 q4 n0 _/ @' M; a4 G

    2 f% u* z( Y3 _+ H' p8 i* j        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 ^- h4 w! a2 X- ~
    % G2 ~! n0 M" C% l& U, S1 |# z
        Recursive Case:; B$ f& G# D- y2 P: G  C2 r8 i& H
    1 _  r) T% n' C/ q  }
            This is where the function calls itself with a smaller or simpler version of the problem.: z) ?  O. F, F" s1 s% _; t

    ! d; c1 x1 q- c        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* h6 p! m1 X, w* X# ?4 u6 b+ j

    4 G3 y1 L! h" m0 y, q# HExample: Factorial Calculation& c; K6 t6 Y- q) C

    / w& q( Z) }) EThe 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:7 `% L  R" w# c+ g  T

    # a4 Y% s: F( W# w2 g    Base case: 0! = 1% Y% @3 T; j. _5 K( R& }- t8 M
    1 p& f- t% @! L5 u! z3 K# j, Q) ^
        Recursive case: n! = n * (n-1)!
    / |7 n, l2 f3 N% U1 i; L4 O
    , ]% g9 }& U0 _! l0 v( S& PHere’s how it looks in code (Python):5 U; S( v3 n. l1 y' ?' y
    python
    + X. d* ^4 m0 }2 a& w, i4 |8 G! ^4 @3 E' X: o8 S* q. y

    & f. r9 y3 n; n1 B( L7 e2 R" z9 ]* o" Mdef factorial(n):  d2 Q% [2 s; W" U2 ^
        # Base case+ L2 O& b1 t3 w- k
        if n == 0:
    & S! v0 a  }5 |! o        return 1
    : s6 U) t8 M; l* F1 t% j$ J    # Recursive case
    . V3 k6 J  T& S2 ]    else:- V- z1 p1 P& ^6 w0 ]: k
            return n * factorial(n - 1)) p8 K  b1 z5 m9 Q/ V) o

    . s  v  K$ [- D+ b- I# ]# Example usage) F1 k' e( Q. {0 e( m3 a1 X
    print(factorial(5))  # Output: 120
    7 q- [. W! V1 v, [6 i
    2 _  ~# o. }9 rHow Recursion Works
    0 k$ Q3 ], m, L" s; A+ ]! e6 a3 v8 C5 s* W% v, Z/ v
        The function keeps calling itself with smaller inputs until it reaches the base case.
    * B3 P; w; {* ~8 h. P! G# k% w, Q3 [2 P: N9 N( z1 L
        Once the base case is reached, the function starts returning values back up the call stack.% L7 r8 @# e' T3 `
    ( ]5 h9 d% I6 u
        These returned values are combined to produce the final result.
    5 K% Y4 |( S, n! K: Q
    5 P: N( o) n: h% }. k3 p, I; @/ }For factorial(5):
    # b) Z* h  W! k) K$ Z% r. }% B+ u$ d& J9 e' f1 o

    3 ^  i$ ]$ ]  R& t( y# Kfactorial(5) = 5 * factorial(4)! B4 v' ^9 e: E* L
    factorial(4) = 4 * factorial(3)
    ! U$ T  a; s. {factorial(3) = 3 * factorial(2)
    & k1 x9 Y, q+ Q9 p. W/ ufactorial(2) = 2 * factorial(1)' K2 L3 {5 [0 W
    factorial(1) = 1 * factorial(0)
    / d: i3 C5 D# L8 ?factorial(0) = 1  # Base case
      R8 ^8 f& |, y8 y# Y" m5 X7 I. @" R% }1 d% a3 y- z8 Y
    Then, the results are combined:
    / h( d6 }  I$ d: {- y8 E2 J% G' h5 r9 o7 D2 I8 \
    ) w& Y! J# c0 G( t& A6 i, h# B
    factorial(1) = 1 * 1 = 1
    4 g; y2 w# U+ E% l; m6 pfactorial(2) = 2 * 1 = 2* A$ U' H8 `2 D; x. V
    factorial(3) = 3 * 2 = 6- C0 p4 V+ V4 C5 ~9 `: b# q; r
    factorial(4) = 4 * 6 = 24
    , f* j1 v. K8 s  z: Yfactorial(5) = 5 * 24 = 1204 [* d! I* ~0 n" k# _* o2 I" f
    + m0 |/ i' T* i- c5 L5 D4 ?/ r  E. Q
    Advantages of Recursion" w+ _  {: X$ b  ]4 S1 l

    , V& M& l3 B% @! i1 |8 ]: d. M8 T    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).# N3 f, V4 m3 K3 i* N9 C
    5 q: ~/ K! ?8 z) g. F9 b
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    6 p6 g0 W/ R, K  l: J1 Q! m
    : n3 t' S% _* nDisadvantages of Recursion" ~+ l# g( W# A0 [# Q+ O3 X" `$ P
    . l9 B' ?$ ^6 |& a  s
        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., T6 _# W! }/ n; P
    ( o& K' n8 i% O. Q
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).' z. _, ~$ ?0 e
    9 S  T: y$ B% U6 D0 e# r9 D
    When to Use Recursion
    ' p: p" g6 R) X& X& k) B3 \8 m+ l: f" s- [' y8 G$ J
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)." D( D- o' g5 |( Z" U

    & B# c$ n0 V5 `+ S/ m/ [    Problems with a clear base case and recursive case.2 [$ R1 R4 F# _* E

    9 D  C) E6 S7 G/ ?! B% ~; FExample: Fibonacci Sequence9 I4 e6 y/ r3 x% k7 G4 j# f
    : p$ ]  |* [: y0 V6 i4 p
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:: @# n; ^3 {4 E8 a5 M/ \, @6 i

    # [8 F' D( |7 i' w* [    Base case: fib(0) = 0, fib(1) = 1
    # H1 g4 ~  k6 s) C, {. ~9 [+ G, K- g, ]0 n* M4 z; b
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! |5 ~" Z  c7 i9 n  ^0 H7 K
    % e! u( }7 |! E' Qpython
    1 B) N+ |3 N1 y" \: S! O  f% r% D* _: i$ J0 _+ f
    ( Z9 @+ L) R& @
    def fibonacci(n):
    0 C3 T; r+ _% T; n& X" M    # Base cases
    2 n: a5 @# w% b    if n == 0:
    6 E( W. W& z  {6 N. [2 H        return 0/ P( U  U3 y1 b  }
        elif n == 1:
    ! \& }$ l& f& l4 h+ y5 O; c        return 1
    , N; ?7 w  [, R! j3 m; K* ~/ B    # Recursive case0 p' `+ Z. T6 Q- E( j, x
        else:
    - O4 d) d: a6 I  W6 A" i- `        return fibonacci(n - 1) + fibonacci(n - 2)' G+ ]. Y4 m  [

    1 d2 Y+ \9 \# a( D- t6 K  X5 ], U# Example usage3 y/ v8 r# G$ D/ y4 r5 k
    print(fibonacci(6))  # Output: 8
    # F; ^. [  c/ ~' t/ o6 o3 n- C9 m4 v$ Y* b6 G6 Q! I) G
    Tail Recursion
    ' e- k9 g  T+ J( d6 x. x9 `5 _; D$ D% K" k7 d4 O
    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)." x( n5 c) k5 M! U
    ) _- {' Z9 [& ]* \7 Z& s- }- i
    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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