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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
     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
    ( C2 \# j1 |3 Z9 Q7 ?  t/ v0 G* I$ k0 Y* {% p8 H- e. t
    解释的不错
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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。
    - @& [4 i* W. ?6 n$ I+ Q0 e" }
    % j/ _) J8 d( E+ D; {) l 关键要素' p9 Y. v% n# C9 y+ p- O4 Z+ G
    1. **基线条件(Base Case)**
    3 v3 w, a, u9 M. ^   - 递归终止的条件,防止无限循环& K& [4 |* H3 i& V( c
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      y* E/ s2 w0 p" V  k% i- p$ Y/ t% }" r$ i2 p3 z8 P7 ^5 s
    2. **递归条件(Recursive Case)**5 K: \" G5 F" a% g
       - 将原问题分解为更小的子问题
    / [1 s6 z7 e$ k4 @   - 例如:n! = n × (n-1)!% b0 i& K( I- T5 f  f2 w8 W

    6 V. _$ i" G; v9 I; i: K+ w0 Z! d. c 经典示例:计算阶乘3 ?/ @2 }' O/ k( ]+ D5 V. Q0 `% x
    python
    % |- Q: d) V7 m9 S  adef factorial(n):
    2 ?6 U! N( b  R' T    if n == 0:        # 基线条件
    ; V* D* M) \% K0 g5 A( U, K        return 1
    ; D7 ?, N! L# T" [    else:             # 递归条件
    * b) [& m2 o$ p$ ?  f  m! {/ u/ l& M* V        return n * factorial(n-1)
    4 u3 v8 B# H6 d/ D执行过程(以计算 3! 为例):
    1 C/ M1 f4 Q2 q8 T" y9 E1 U" f( Qfactorial(3)
    4 g5 s, }- \7 r& J; r3 * factorial(2)
    . q! b3 h+ i2 n4 D4 m, h3 * (2 * factorial(1))
    1 n8 R* R, l  ?) S% k3 * (2 * (1 * factorial(0)))  ^$ b/ f6 H/ r7 ?9 T4 Z
    3 * (2 * (1 * 1)) = 6$ e& t! ~6 }+ r7 Y3 X

    5 J  U! e  L8 M1 Z9 e- V  \/ q 递归思维要点
    1 p6 {7 P: a& m; Q  F& H1. **信任递归**:假设子问题已经解决,专注当前层逻辑
      h: r' s: L: o2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ) i/ y5 B# q$ A. `. K: \& f8 x3. **递推过程**:不断向下分解问题(递)$ q1 S5 p3 z5 N
    4. **回溯过程**:组合子问题结果返回(归)
    : ~/ D9 f' C  M$ S. ?& J5 Q& N8 K; ]& \
    注意事项: A9 ^% G" _* J5 `+ ?% ]# M4 Q$ `
    必须要有终止条件
    : P( N' {* D9 K5 V1 M递归深度过大可能导致栈溢出(Python默认递归深度约1000层)8 C8 P7 V' r; |9 e6 g
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    # C6 w9 V! f7 A) `尾递归优化可以提升效率(但Python不支持): Y: S6 P6 y3 c# a' [
    + h& k6 ]& f1 S; [
    递归 vs 迭代
    6 }9 X* U2 J* }( F7 D; c" j( `|          | 递归                          | 迭代               |
    0 K3 [( V7 t- R" X* F3 q; p|----------|-----------------------------|------------------|0 Y4 c& X; a4 S0 }) Z
    | 实现方式    | 函数自调用                        | 循环结构            |3 c' P% d- s% y# h5 a
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    2 I/ j, Y' E  E! D2 n| 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 u# g; G  L* X- u4 Y  t| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |/ L2 b5 y0 A6 G0 r5 ]8 m$ B( G
    9 }# s( w- C5 U( R1 q% r$ v
    经典递归应用场景/ ]' r# m" c' |" W" W" A2 c2 X
    1. 文件系统遍历(目录树结构), P" ]7 t# X6 i
    2. 快速排序/归并排序算法
    % |$ ?: ~1 E& c5 C$ p7 b3. 汉诺塔问题
    ( i" e4 e+ M/ T4 f4. 二叉树遍历(前序/中序/后序)' l( d6 ^( B# E( s$ j, p# Q
    5. 生成所有可能的组合(回溯算法). B* t# Y4 Q6 d7 V- K
    7 }9 I( x: F5 {+ i& S& ]
    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
    郁闷
    4 天前
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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,
    % B8 J/ _" {3 K我推理机的核心算法应该是二叉树遍历的变种。
    % ^2 B! J5 ^9 q  h2 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:6 X# K' J3 E" N2 Q
    Key Idea of Recursion; l; [& ]( o7 K: W
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    A recursive function solves a problem by:( D: w$ Z1 f  b" C

    5 E7 |" W% u8 Z; B8 T    Breaking the problem into smaller instances of the same problem.
    / D9 g0 Q6 X: a& g/ b) b0 d& Y/ f1 `' k" ~6 c- ]
        Solving the smallest instance directly (base case).
    3 M* @* C  G- j" s* ]0 m
    ! W8 g: d: e' N5 q9 b7 S" \# H9 X5 Z    Combining the results of smaller instances to solve the larger problem.
    ; X7 d$ `3 H0 i( _+ S" J0 G; Z! z
    ( R: r7 p6 X9 l1 P. a: p0 dComponents of a Recursive Function
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        Base Case:* a4 t. `& G2 k% v4 j4 Z6 [: Y
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            It acts as the stopping condition to prevent infinite recursion.
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    9 u8 j; j0 O0 U$ ~8 Z* c! j        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.  v; b4 L5 t) d5 k2 O2 F$ |

    $ b1 c9 P! O& L* E6 d    Recursive Case:6 m* Z) j/ {/ E$ j4 |1 d

    0 m6 {+ R  v+ Y( k8 L! ^        This is where the function calls itself with a smaller or simpler version of the problem.7 A- v. ?# c/ O" l( _1 ^4 s

    - n% r2 W4 a! I* j        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    , n' K' `- d  o' c/ T# t  z+ `4 I% o: j0 |( f: X+ L% f) F* I) T
    Example: Factorial Calculation
    " `9 E6 Q! L- f3 }( ?2 n! m  D
    * h0 j* x: p$ f9 \7 iThe 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:' r6 W' S7 A0 c$ w# m

    3 @: u7 Y5 w7 _: J$ f    Base case: 0! = 1& A+ J7 ~; V6 N: v3 R

    ' [. |" s/ w5 A. a    Recursive case: n! = n * (n-1)!
    2 y' E% l) s* _. D  H2 v2 g3 d* a' r9 H
    Here’s how it looks in code (Python):/ V$ Y% ?9 E$ A
    python- e. p4 A, j4 C  u& [
    - j* ?% Z* b, ~& m' |9 x

    $ L8 ?) X( w9 D9 |; Pdef factorial(n):
    , ]" w3 ^) h. A; i  E! o5 I* w( ?% Z& ~    # Base case' D6 r$ ]  p* V; }  N. n* ?2 t
        if n == 0:2 h6 R0 O2 i% c: A4 c9 x; J
            return 11 t2 m# n5 @" B5 b$ k
        # Recursive case
    & u/ H' f) W% j0 d3 p2 z/ P    else:
    8 N' `9 a! `; L, F8 l        return n * factorial(n - 1). c# J/ E( ?6 |. E7 v  d
    , Z: j/ P- S% U0 q2 |
    # Example usage
    ' q( Y$ o& R4 T' G& j8 D) x# uprint(factorial(5))  # Output: 120
    4 |" k) ]0 C6 ]& O7 o* \" |
    % p- R  \2 W3 QHow Recursion Works* z- A! O0 A: m8 s
    " ~9 ^$ j$ r$ E  y
        The function keeps calling itself with smaller inputs until it reaches the base case.
    1 I% P! Q/ `  x  N* `+ {$ K) T
    9 [) T1 m7 {8 k0 f& k8 f( y    Once the base case is reached, the function starts returning values back up the call stack.# [+ f. H9 V# |( \

    ' ^3 ^. j2 ~' i6 H, J2 @9 H    These returned values are combined to produce the final result.8 I; W+ ^& k9 z

    . ]$ e$ W) e- Q* rFor factorial(5):! s4 P3 D: m1 e* v4 w* w0 H& d$ S
    ' v* [3 x9 j- K, `1 W& z- i7 f- h

    . l/ D5 u; L' W' t5 Tfactorial(5) = 5 * factorial(4)* c2 r. v5 k. `
    factorial(4) = 4 * factorial(3), Z3 {, o( f0 m2 i
    factorial(3) = 3 * factorial(2)1 C6 J8 m5 v) R$ n# z. S' B% o1 I% O
    factorial(2) = 2 * factorial(1)
    . Q, D" M% G" }" E4 v' [1 Vfactorial(1) = 1 * factorial(0)
    8 h9 F* a. L/ x8 S) v% c& P8 Afactorial(0) = 1  # Base case
    - Z+ Y( ^$ h# }& Y; Q+ g  e8 i# o" s6 v
    Then, the results are combined:3 t9 _, e. G, ^  ^! L' T3 B0 u

    % M& D: P) C: }
    & b6 I$ n, S& E) J- E  h+ I' x7 pfactorial(1) = 1 * 1 = 1: [6 _! ^( n  b. z& A
    factorial(2) = 2 * 1 = 2
    ' S9 _4 N2 J9 Pfactorial(3) = 3 * 2 = 6
      M! R5 Y" {8 ?9 N4 ]factorial(4) = 4 * 6 = 24
    + l8 p* U- }& H: ]7 R6 ~1 ifactorial(5) = 5 * 24 = 1202 e9 j4 `5 l" a- ]

    8 S$ a( n9 e; \# Y! b& g+ B7 `Advantages of Recursion
    ; ~% y6 h% S7 W5 j6 m6 {' m: Z6 m9 I  `: c) Y0 ^
        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)./ b, i5 I  W: L% t- e: _

    ' R, ?, L. x4 q    Readability: Recursive code can be more readable and concise compared to iterative solutions." S; Q( H. c0 j( H4 c# i
    ' |) B% Y1 _8 I  |8 r
    Disadvantages of Recursion3 o! \! S) f6 F* e/ n4 S
      h. D+ E, h1 O( j. f. H
        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.+ P- s+ W% G) o4 B+ Q

    ( d& u( R  |9 M& h7 ~6 E) `' _7 W7 X  K    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)." [+ v( t8 ~# g8 }/ P, y  n- i" P; c3 \( O
    4 S* B7 ]2 G0 v2 L
    When to Use Recursion. v7 r) ~) W( S/ ^0 z0 N0 W" i9 E7 |; C

    6 R; e# P7 r& U1 u* S    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 L  }) @! p# w7 P; `" c
    ! f# H5 y' w) F
        Problems with a clear base case and recursive case.+ _, `  C9 G1 j* U6 [  J6 Z5 Y
      H& C$ h; O- P2 ~8 e
    Example: Fibonacci Sequence/ Y4 C0 c# D: d  H: p9 C

    ) _$ w0 |4 g0 yThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 i% ]) X8 ]" B8 a6 H
    8 L2 V2 ]( G$ Y6 ^" U: w
        Base case: fib(0) = 0, fib(1) = 1$ x/ ^3 Y% I4 s8 a$ V0 J$ Y; z
    ( g7 @0 ^; r2 ?1 f# @4 O0 i- F
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    & @4 K# a7 t$ t$ [$ A3 b0 g4 a1 @; x& b, S+ f4 \4 u4 V( y: d
    python7 @, I9 J3 M) H/ u" y

    , h. i6 M4 q0 c
    ; W6 m% F7 C* Z& }( Bdef fibonacci(n):
    # {' m$ C8 H! s3 J    # Base cases3 ~0 z, l" v, _' ~/ `
        if n == 0:
    " d  `" a; K7 u        return 01 Q5 G# Q+ J( ~9 O% v* K! u
        elif n == 1:
    4 T* N8 n9 I7 C2 A6 F        return 19 p) T( u9 g; Z! R7 O
        # Recursive case- X0 f* S/ H0 Z- E- K' h- v' p
        else:
    5 T8 A; L. y3 G' t( L, L/ _: D8 S        return fibonacci(n - 1) + fibonacci(n - 2)( H; Y( m2 _& u) J0 I

    ( U- p1 e  Z- m/ D" X  _% F# Example usage
    : Q* A4 E5 r$ z1 J1 Bprint(fibonacci(6))  # Output: 8, B# W1 K8 r, [. `; G5 u

    - L. Q" I8 B% L/ fTail Recursion2 T& |. z( V, Z) ~' G

    : @/ O, D2 m- u  @& s0 U% |6 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).
    8 }+ A+ J' f# i# n9 i
    % G8 x* Y1 ?/ W- AIn 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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