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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
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    % c+ z. D( U3 P4 Y6 T( u解释的不错* ~! }6 s, E9 q% {

    7 ]: j' }# K6 A. r" N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. _* X: W# v; |, u' N5 T9 b$ j0 f: J
    关键要素7 S* m$ b9 U5 b# m( Z3 e% n
    1. **基线条件(Base Case)**
    $ ], B5 a% \9 y2 R% Y; f' |   - 递归终止的条件,防止无限循环
    $ f5 ~% M) Z  E   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    4 [& l; S1 u% K! F, E: v2 Z" M9 _7 L4 |, U6 K; ?/ I0 n
    2. **递归条件(Recursive Case)**- b5 j* J1 z, C+ O
       - 将原问题分解为更小的子问题
    8 H$ T6 \6 Y9 p+ e" `3 u   - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘' ~( `% |. }; S
    python# d2 X8 e+ j, w5 x8 v2 x. |7 n
    def factorial(n):
    , J* x; _. ~" K1 U2 c5 W! S    if n == 0:        # 基线条件
    6 L% O9 k  W( K; s, _+ j        return 1
    ! j8 p$ c! q* O3 ~2 d4 e0 L    else:             # 递归条件- p' p7 e" v' e/ w; f" w
            return n * factorial(n-1)
    7 |" u# F# s4 Z0 [执行过程(以计算 3! 为例):
      n  C8 d! m' _! C1 }factorial(3)% x; r- N# k5 B1 P2 N
    3 * factorial(2)+ J' y8 a' I# l( j6 t+ L+ S2 `
    3 * (2 * factorial(1))+ `; u! o3 f: ~4 @0 t
    3 * (2 * (1 * factorial(0)))
    ( f! H. q1 e4 j4 k  A  h0 g' J0 \. T3 * (2 * (1 * 1)) = 6
    2 g5 Z" z2 S; C& ~
    " N" [$ B  |; W5 r" s 递归思维要点6 q( L& w$ n- l  ~
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
      T# |$ R9 ~  N2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    $ a9 v  Q1 O- a; `  |3. **递推过程**:不断向下分解问题(递)8 V9 U8 {8 g5 T8 R& C
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项
    8 |0 a, L! R' y  E必须要有终止条件
    * C% g# Q1 U- }: V' ]递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    2 z+ z& N7 E6 _, a某些问题用递归更直观(如树遍历),但效率可能不如迭代% d6 Q; c/ Q; i- V' a9 i
    尾递归优化可以提升效率(但Python不支持)1 o' A5 a( u" B) o# Q) ?8 C6 [; P( `

    0 v$ i) l6 H: I. V  _ 递归 vs 迭代: L5 Q( [0 J- |0 ^0 d
    |          | 递归                          | 迭代               |
      B& ~6 }( E) m& C6 A|----------|-----------------------------|------------------|* V$ @! L4 o4 I( k
    | 实现方式    | 函数自调用                        | 循环结构            |( B# B3 ]1 F  Q0 h; F
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |$ |9 v% w+ l2 m* l- h5 t  U! O7 [7 q; }
    | 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         |
    ; K1 m0 E( ?+ p. j5 F/ r  D5 U3 }  X& c| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    : U0 |% {& a. l' n, m: \; r, ?9 P3 ^9 l+ v8 _
    经典递归应用场景
    4 n  G4 f+ m$ r4 m+ u& d8 s1. 文件系统遍历(目录树结构)
    + P( u% `$ [( R  U1 o5 s2. 快速排序/归并排序算法
    * l& l/ I! L9 q( S% Z6 R3. 汉诺塔问题' i) P4 L- I( e* }- F. l
    4. 二叉树遍历(前序/中序/后序)2 ?, T2 q( \: N. s7 |; C
    5. 生成所有可能的组合(回溯算法): P9 K/ u  v8 Y, Q+ f* e
    - K( K9 g8 l# ]+ r5 h1 B
    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

    昨天 08:50
  • 签到天数: 3109 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,6 b% e: V' ?# k. x3 E% C6 c+ O! J
    我推理机的核心算法应该是二叉树遍历的变种。1 A; Q: @( k$ L4 |
    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:
    * N" B& \  e/ s3 H' }Key Idea of Recursion, H8 r0 u9 j& Q
    : @& f- e1 _% {% V( w4 S
    A recursive function solves a problem by:
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    % G8 L0 w2 X  u9 K2 M' t    Breaking the problem into smaller instances of the same problem.; \+ u' U3 Y+ ~- h3 S3 w# d

    : S6 H' P0 q# H    Solving the smallest instance directly (base case).
    # S0 G- [6 i/ ^. Y4 K* t8 e1 B: M' I* p7 G4 L. T4 B
        Combining the results of smaller instances to solve the larger problem.# ]  {3 W$ X% H1 K% N' m
    3 K& P0 i& d' o
    Components of a Recursive Function, C# `6 _% C. `; Z- m3 p5 M3 G
    9 g: s; j$ h. N4 _" Q
        Base Case:. w; a5 Q& e8 U, @( Y
    ! `" C7 E7 m# h6 l: a
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
      v& O8 a  l- T; j! m. L' `2 d1 C* C+ G+ N& ]$ I2 k" E
            It acts as the stopping condition to prevent infinite recursion., T* L- H; P& j+ m: _/ \( @" c4 H( F0 v/ h
    . g! C6 O- a+ X6 r- x0 ?5 n
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( q( m6 W" P2 [8 L7 Z

    5 y+ @& t' L: I8 E    Recursive Case:
    : h% y  q- c6 N* d5 Q5 T$ G* I
    9 X* b, ]& }: l1 x        This is where the function calls itself with a smaller or simpler version of the problem.
    8 ]" Z. W0 t' _$ {$ g0 z" P) T! X* Q* T/ r
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ( y* u4 f9 [) c* W: G: `# W1 T7 N, O
    Example: Factorial Calculation
    & I; q/ b4 O+ m! ?( O0 {/ F5 ?+ T% V1 h7 i! ^
    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:7 f1 \. j6 n+ O. W( F- |8 z/ B

    1 U3 N# o+ R7 y( U% u, F    Base case: 0! = 1
    # V8 ]9 b% D1 r9 a) s! U4 w' E% ?( {/ i8 P* `3 r% w
        Recursive case: n! = n * (n-1)!
    : D  ]) C+ G- }2 P% v6 V8 _. ?6 @: u' ?% j, @
    Here’s how it looks in code (Python):
    & o. }$ x9 [4 lpython
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    8 ?* g4 h) v& [- D4 f3 F% w
    3 J# {% x3 I5 g5 [9 _1 `3 {  @# Edef factorial(n):) [6 J; I4 i  M5 S8 ^
        # Base case
    , P5 X& H; @0 K$ f6 R    if n == 0:
    ' Z' ?; w9 b& X3 h        return 13 \/ N6 \4 t' @, o' [3 K: u
        # Recursive case
    ) D$ b" Q$ I: I5 h5 l: R    else:/ ?; R% \4 c1 ]4 }: Y( O' {
            return n * factorial(n - 1)( ^+ u$ L) d2 s( ?: z4 t
    $ v; Q0 o% N, B; j3 m
    # Example usage
    . R- E' U* K! F9 w# V$ [print(factorial(5))  # Output: 1200 p# t  L1 W( Z9 N& H

    0 c# F5 f  e3 b, x( ]: FHow Recursion Works
    ( E) s4 K" X# m$ n5 X
    ) p9 t$ u& E. s9 L, U/ b    The function keeps calling itself with smaller inputs until it reaches the base case.  x' x1 S% u) M! M; R" h( h% `1 q0 K
    & R/ L7 J1 v5 ^6 K, G
        Once the base case is reached, the function starts returning values back up the call stack.
    / i7 Q( I3 T) @$ F! g/ z$ l; @( v+ |, o; [, m% j
        These returned values are combined to produce the final result.
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    * Q; I) V% [, {6 N3 X1 A' KFor factorial(5):( Y( H& [1 r4 F8 O; s
    + ~" v+ }( W1 G6 ?' L- W1 V; J

    2 U/ l5 `( u1 Z. S' n$ ]  mfactorial(5) = 5 * factorial(4)9 F6 y7 ^2 ~7 w! M5 Z
    factorial(4) = 4 * factorial(3); M, U5 _9 k6 Q9 a7 M1 v
    factorial(3) = 3 * factorial(2)
    9 v0 K0 S- F( M1 Mfactorial(2) = 2 * factorial(1)" M% L. J& U- @# U7 K6 u
    factorial(1) = 1 * factorial(0)
      \3 r/ e/ D& L5 \- q! }5 _factorial(0) = 1  # Base case* b/ m2 N1 ?4 H3 E6 Z2 T# U! c
    6 k4 ^1 A2 Y+ U( u/ g
    Then, the results are combined:
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    " h" L6 R. [9 Q7 `9 B8 b
    factorial(1) = 1 * 1 = 1
    ! S) L2 p$ J5 T5 {4 c0 mfactorial(2) = 2 * 1 = 2
    ( J- U% _1 |/ Qfactorial(3) = 3 * 2 = 6" N" r' K- n5 p# |  m
    factorial(4) = 4 * 6 = 24
    6 O' D! W8 p* g  Jfactorial(5) = 5 * 24 = 120
    : |" y. r3 G4 F* u: |- ~0 @( j1 f2 o- I5 [
    Advantages of Recursion, ?: R: K% r, @0 H

    7 @; ^! r/ {- H6 T0 v+ ~    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).
    4 a$ i$ A) x& R6 l7 Y9 O
    " B# R: v% e8 D) h' H+ f$ h; z    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    # i# |, Y* c$ o3 ?
    ) `- b- M2 }$ v3 u8 K/ k; R$ qDisadvantages of Recursion" _" O. Q4 [0 y/ n0 L6 i

    - N2 l% ~3 F5 |6 ^    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.( l  Z4 v5 ]; o8 Q8 U8 a% u

    & v. d7 q6 m# v$ l1 J- P7 p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).: F  F* U' K$ |/ T
    , X, h1 B7 c2 |9 ]7 }5 f4 l2 G
    When to Use Recursion
    3 D" r7 E$ R8 H3 ~4 \7 n4 G7 D+ @& @# i2 ?" j( m6 l
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    % {! c% A: [1 d+ s) E: S4 D: d7 `1 h" y2 Y1 i$ z2 }
        Problems with a clear base case and recursive case.  \+ z8 l, }- R5 s
    % e% b& i0 W: a, L- W  Y
    Example: Fibonacci Sequence
    / I$ B) `" z) z  o2 Z1 u1 X3 z% y# }
    . h+ }: S- O! A+ J" x+ R( J5 H+ GThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ) N7 i2 D2 _7 w, P9 o& G1 K7 n0 v* }' s. Q2 ~. P+ k
        Base case: fib(0) = 0, fib(1) = 1
    6 T" ]6 F8 J) r2 j' U( \. V# V% ]# X. k" E- Z* N& }. S
        Recursive case: fib(n) = fib(n-1) + fib(n-2)& N# X# I* j; D; C
    . J6 ~: X% C" ?! d" [: W! ~  r
    python
    # E, J$ x, p' A# [$ m
    " J% ?# i4 q$ D2 n. _9 a3 [' }
    + C% h# h2 x1 u, Q; i- Kdef fibonacci(n):3 c! S- K' w( s4 `1 l" {6 h" W
        # Base cases
    5 Y% |' ^6 j% z    if n == 0:( R& L/ d8 E" x; x- V
            return 03 R2 o. B5 |! d" T2 u/ ?3 y5 G
        elif n == 1:
    % f1 ~/ K2 f- M2 T" v+ \2 G        return 1
    7 e! D5 @' o& N0 `! ^    # Recursive case
    5 x' F# Y- _) \3 x3 `' t. D    else:
    5 c& @: ]1 c1 F6 T' N        return fibonacci(n - 1) + fibonacci(n - 2)
    ; D# x; P( R, {/ F
    ( V* |5 r! Q& O+ k5 Y8 g# Example usage7 \+ M) S& ]: U! [" Q
    print(fibonacci(6))  # Output: 87 q0 e! W! D0 W6 x' L# ?9 S
    : C/ J) x5 m1 K. r; C
    Tail Recursion8 Z$ d7 i5 Y, @2 G

    % Y/ M+ `7 ?2 Y3 O6 W" e6 ZTail 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).* \' j" Q- F  i
    ) I- i% b& ^- l
    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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