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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
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    解释的不错4 u1 `: r  d- \
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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。
      p8 |, M9 x% W' r. M$ t* U# c  `  f4 k3 Y: d
    关键要素
    , _, L) m1 j& L8 H" h  D1. **基线条件(Base Case)**) y% `3 U3 u( _: N! Q6 c5 n
       - 递归终止的条件,防止无限循环' e, L3 @1 h/ i. |( C
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    , b  a# ?+ ]. c7 q; z
    ( Z3 {% c# j; ]2. **递归条件(Recursive Case)**+ }& a- J0 F5 \0 w
       - 将原问题分解为更小的子问题0 Q7 N' _* m, u/ d5 ?" U
       - 例如:n! = n × (n-1)!
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    ' G( z8 n* h7 t' S/ q 经典示例:计算阶乘8 }$ `5 h5 Q/ o8 q$ _5 R
    python
    + a, @% o6 I% u5 t5 rdef factorial(n):
    $ K- [9 R9 W/ `* A4 L# U! X! {4 |    if n == 0:        # 基线条件9 k9 q8 ]; s9 p0 m: W
            return 1
    , R, L+ ~8 W& k: \* T    else:             # 递归条件
    9 z3 w) k" a/ p' P        return n * factorial(n-1)2 T3 x& V' ~. Q' [
    执行过程(以计算 3! 为例):! q# ?$ y. X  B; W" @% U, l- M
    factorial(3)9 c" E0 v1 K5 R& j: R( s
    3 * factorial(2)
    4 a- i7 S, ]1 n1 Q" `3 * (2 * factorial(1))
    3 d; Y* f' H0 Q! F9 o3 M3 * (2 * (1 * factorial(0)))
    0 r0 [; Q' d8 Y& y" G. N" K8 v3 * (2 * (1 * 1)) = 65 f6 c* Q7 v3 X9 f5 r" ]7 d/ i" n
    7 N" U# d4 ?% p. k& t2 l
    递归思维要点2 {" ^; z1 n* h2 ?- ~. e3 T
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 q8 a: }+ \: r  z
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)& m' S% u# X" g' z7 K
    3. **递推过程**:不断向下分解问题(递)/ j% q! Q1 T& ~+ n& c* Y" U) n
    4. **回溯过程**:组合子问题结果返回(归)0 s7 ^% l0 c  T9 O

    : z  v% F5 B; Q8 A, B 注意事项
    - ?$ {2 z1 O1 s必须要有终止条件- U6 r* B% b0 n* Z* Q+ u% c+ L
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)' f3 Z& D; l/ n) v, G: N' V
    某些问题用递归更直观(如树遍历),但效率可能不如迭代; ^/ f! Z; D5 h7 G/ j  @
    尾递归优化可以提升效率(但Python不支持)$ o9 t( }/ O/ U2 B
      w6 A# l! {) l% T  A% }. X
    递归 vs 迭代
    0 k% B- m: I) ?|          | 递归                          | 迭代               |
    ! g9 h( M0 a% k, G+ m0 j|----------|-----------------------------|------------------|; s. ]$ \/ j! T1 l  T$ R
    | 实现方式    | 函数自调用                        | 循环结构            |
    0 G1 D& \" J% N9 b: e  j| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |1 w$ d7 `- I( R1 h" P
    | 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         |
    + }* i: p3 P; k9 u  ^* y) y& D0 d5 F| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ( Y* ^/ h$ \. p& l2 q# q- B8 Z( s+ o5 o9 A7 C9 q
    经典递归应用场景
      t7 J7 }7 Z: q2 d: O% P1. 文件系统遍历(目录树结构)0 L) t$ ~4 i! \9 {8 \& ~
    2. 快速排序/归并排序算法
    & f; f" S1 U, Z# m3. 汉诺塔问题
    ' L& v" j* |; c: W4. 二叉树遍历(前序/中序/后序)% ^; r" {- L; }. C- ^7 \
    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

    昨天 07:22
  • 签到天数: 3154 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,. N: T5 Z) D! q; V
    我推理机的核心算法应该是二叉树遍历的变种。7 t' B/ S5 i. ~4 R7 i+ h
    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:
    2 x' [, X$ s0 K- p/ _0 R; pKey Idea of Recursion
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    A recursive function solves a problem by:
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        Breaking the problem into smaller instances of the same problem.- i7 h5 T! S& J5 z+ {9 }$ c

    - [9 x, ~4 Z; X) D    Solving the smallest instance directly (base case).! ?$ g# q9 I  O* Y: w. y

    9 L5 q7 l& @* ~5 w* ^) m- \" S    Combining the results of smaller instances to solve the larger problem.
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    & J5 p0 Q9 r+ fComponents of a Recursive Function( P9 j& H( f: k9 c/ O( k

    4 b& a" E' F+ H" X! o. X    Base Case:
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    4 f* `# J* i9 G        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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    + N4 V& M5 N  Q, l# t$ N        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.$ j( {. ^  |" r7 a

    - V4 @. L8 _2 `    Recursive Case:2 r0 h+ G4 ]- g5 k
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            This is where the function calls itself with a smaller or simpler version of the problem.1 r+ r, q4 d0 H. }/ Q( I7 \) }
    : ~6 R& F( `5 O! b3 ?' \
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).! F% _% L: a5 I2 a
    $ L4 c: X+ Y$ c( V& h/ s
    Example: Factorial Calculation
    # r; u* C! G0 J8 J  s* y1 J& L3 Z: m3 ]( H
    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:9 Y* T, h9 A9 n, U2 u/ I

    ; }( ^9 ~, R" g/ N" f/ A0 e9 }" a    Base case: 0! = 1
    + v# `" n* ?0 V# F/ I! s: V+ C$ c& A
        Recursive case: n! = n * (n-1)!- z7 K( l% z. c  G" v

    ; _" f9 h7 q8 c* P$ cHere’s how it looks in code (Python):
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      e, w: @$ W- k; p; k4 G

    ( a7 s; z* _' c, ?9 Edef factorial(n):5 G# ~0 S1 P) V8 ]( ~
        # Base case
    / ~# R4 l. `. U2 C$ S( l: G    if n == 0:
    ( E& i; c1 ~3 C$ r/ n        return 1
    3 @: I! \+ M* O# U- {. h" P. d; |    # Recursive case+ f" O' n# s, @( G
        else:8 K2 e" |0 F8 Q
            return n * factorial(n - 1)+ b# e4 g; v- e+ v# b

    - g+ i2 b# h# {# Example usage6 V0 Y0 A- P' U% B- F/ u5 x
    print(factorial(5))  # Output: 120
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    How Recursion Works; H' W- \4 T9 E8 i5 T
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        The function keeps calling itself with smaller inputs until it reaches the base case.
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        Once the base case is reached, the function starts returning values back up the call stack." l( }9 m7 N0 }
    ; ?. P6 s8 D7 O
        These returned values are combined to produce the final result.
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    For factorial(5):
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    " [# A' I; \4 D4 q( M  j0 x- }7 F$ j5 H
    factorial(5) = 5 * factorial(4)
    1 f0 b& X5 Y6 {1 D( t8 {factorial(4) = 4 * factorial(3)
    ' E' L: q+ d& d" W+ p: Cfactorial(3) = 3 * factorial(2)( D' N0 D) e, ]3 Y, f
    factorial(2) = 2 * factorial(1)
    % V% u) n) C1 ]8 x2 j' d! wfactorial(1) = 1 * factorial(0)+ s+ W7 V  ?8 C1 B$ F
    factorial(0) = 1  # Base case$ r  w% V9 \3 |4 P7 n

    : {5 b1 K# r" g* N$ kThen, the results are combined:
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    % E& M. u1 f- {8 g
    5 X$ Q* ?4 `/ w( Y2 Q  [1 H# \5 [factorial(1) = 1 * 1 = 1% M. K2 C* P3 J6 S. m% I
    factorial(2) = 2 * 1 = 2
    ( B% M. V$ z" k% k( y7 W2 hfactorial(3) = 3 * 2 = 6
    ( d1 d4 j; C7 y. B5 h$ ufactorial(4) = 4 * 6 = 24+ j* Z" K  ?- m5 v5 v# h
    factorial(5) = 5 * 24 = 1207 ?! H8 F2 z* \

    " g* ^3 q- W) M; a8 ?1 Z/ b1 rAdvantages of Recursion, l0 y: ~  h0 O0 W; H8 N) b
    * g$ |& n4 [7 M& h9 p) P  H
        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).
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        Readability: Recursive code can be more readable and concise compared to iterative solutions.# \# F1 r5 m+ M* M3 D
    ! j0 J+ I7 u: b3 O! m5 ]
    Disadvantages of Recursion  [  J# a! ]4 p1 E6 V& K: a! q

    . S! z$ ~9 P( d5 W    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.* \0 W' N0 Z& P& m( m" l: g
    8 P2 J9 U* F/ Z' |" Y' h$ v9 J
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)." k( h" e7 Z$ u/ D8 i# N' L

    " `8 h; d2 ?1 IWhen to Use Recursion
    2 ?) r. Q5 o5 \) f# z2 h- [1 ]
    ' F; e' D  O- Z" C  Q5 x! A& m$ t    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    % i- s5 S3 w( o5 W. Y5 r1 x# F  [6 z0 g8 K
    $ T, z2 C7 }3 q1 Y8 y$ U    Problems with a clear base case and recursive case.$ Z1 j2 v$ l5 n8 a/ }
    & a0 ?) Y$ e; h0 h$ q9 T
    Example: Fibonacci Sequence2 g. j/ V; i( P" m/ }5 d

    * c1 ^, `( `5 U" Z5 P/ |# B+ aThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:- ^1 @1 O7 B: l; o
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        Base case: fib(0) = 0, fib(1) = 1
    8 Q7 N9 B$ ]- c# @  Y9 {8 o
    - _/ S9 e1 p/ O    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    , L! Y& D& U: h- Y# a: Zpython
    , X+ V2 G/ e) {4 W, R9 h  Q6 S
    8 ^2 b; W6 v9 F% ]2 U( e2 c1 b8 P
    1 Z" R. m- S* L; cdef fibonacci(n):
    / K9 u. ~0 N$ e  |) r" F$ {: h' ~    # Base cases& G8 g) b$ d, G; F
        if n == 0:
    # t5 I* B- n) m5 n- ~2 Q2 O+ Y        return 0
    & }3 o" N% t4 A) W0 P3 m    elif n == 1:" l6 W; I, Z+ W0 A* H
            return 1
    : n! \6 [+ y- W" C" [, o. q0 a    # Recursive case9 Y# o! q8 }" d: h- P
        else:; |9 i  |6 A' ?* L
            return fibonacci(n - 1) + fibonacci(n - 2)
    1 ~7 O! @9 ~9 v# F, W; a8 g- |% Z, w# M6 a$ r9 H  i# ~6 }
    # Example usage. v) B  w$ V# c) b8 A
    print(fibonacci(6))  # Output: 8& ^" m# b" ?8 N8 w
    , g4 Y& ]' `9 u: T8 Z
    Tail Recursion
    ; e' Q9 R' @3 m/ w5 }% ?8 u! ?" w
    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).& H, L% |8 B$ a" @2 d
    ! j2 e. m* v! o* D7 L/ v+ X9 p
    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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