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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
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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
    ; i5 n7 m2 D3 z1 @* E  q/ F( A9 x7 G5 v' b9 W8 L
    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。; @$ M9 _$ Z) N6 r* K
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    关键要素
    2 }. r, E7 @$ ?) Z1 J" ^& n1. **基线条件(Base Case)*** R; m& s0 y! [1 a- W
       - 递归终止的条件,防止无限循环
    ( A$ _+ G" q6 ^  D, g   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ; I. J, j( X5 g% _
    + \6 T# @% f- z' E2. **递归条件(Recursive Case)**+ {# M+ G. B: L; l  e
       - 将原问题分解为更小的子问题9 O* \8 p+ G  Q$ e( _
       - 例如:n! = n × (n-1)!! F; {* f; U) i' Q5 L

    5 r0 z$ n3 e: |  N' f# p& |. G 经典示例:计算阶乘
    6 U: C5 r* }' q! i) E0 Jpython
    7 ~, W+ i9 F  sdef factorial(n):9 M+ l  ^7 |) H, R
        if n == 0:        # 基线条件! g! }/ L  G" w7 n, q& y
            return 1
    & N6 Q' }9 q9 ~, C9 }, T: _& [    else:             # 递归条件3 r( x9 T$ y! D* O4 j  |
            return n * factorial(n-1). Q5 f/ K2 H- n. ]$ C
    执行过程(以计算 3! 为例):/ A4 u3 Z% x/ y- j
    factorial(3). g7 T/ W7 o7 d4 v$ G
    3 * factorial(2)
      t7 x' _4 e9 e3 * (2 * factorial(1))
    " r+ a) e5 p. x6 V) M3 * (2 * (1 * factorial(0)))
    8 |; L! x; }# B) q6 g: M3 * (2 * (1 * 1)) = 6
    + e3 D! Y/ }( n" ^' Y( M9 ^
    5 i( t- Z0 q) k& V 递归思维要点' N7 |# k* f3 X8 X. K
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑6 ^' x0 d8 z7 i3 A
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)1 A: K2 f5 w) D" t6 w" O
    3. **递推过程**:不断向下分解问题(递)
    , Y3 w6 ?) t9 u/ w8 `4 m% S# w2 x4. **回溯过程**:组合子问题结果返回(归)0 y) T2 X: d% |! ~$ C- g

    . o6 B) }# D0 C) E 注意事项! r( E# ?3 x0 {" {
    必须要有终止条件
    5 \+ D8 ]. K8 L4 u; K! k: j递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    2 y3 Z* ~+ o$ s* X* S% L某些问题用递归更直观(如树遍历),但效率可能不如迭代- G) a4 h( Z: I) p8 @+ K; D0 R
    尾递归优化可以提升效率(但Python不支持)
    7 J8 W6 v" I2 O- j9 d& ?; Q* E$ {& a/ d$ G
    递归 vs 迭代
    8 G( H6 N' t' U2 K: L& i|          | 递归                          | 迭代               |
    & O- j9 X) i2 n9 S5 P2 }|----------|-----------------------------|------------------|  e" i# I4 j7 b
    | 实现方式    | 函数自调用                        | 循环结构            |3 S) `0 S3 U* ^7 O
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    $ F* n0 ]$ {/ N  }3 \4 K. D) o| 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 s+ L2 v2 U8 @) }3 P| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |4 Z+ w4 `) O/ e3 U4 ]7 l
    # f; T0 [  v+ H( k
    经典递归应用场景
    9 t! g- k5 u" [1. 文件系统遍历(目录树结构)
    & n* K9 u" {% c- L3 J; h8 c" r% `2. 快速排序/归并排序算法
    6 t' L# l. C* H8 `) @) ]3. 汉诺塔问题
    8 a! r* b, @# ?4 _3 c9 `4. 二叉树遍历(前序/中序/后序)8 P2 c) r& J) y/ u
    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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    2 小时前
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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,
    . I, q8 K  {% a( W3 a  _我推理机的核心算法应该是二叉树遍历的变种。& O7 \# O" g, @/ O. F7 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:; _& b0 u& C9 ^4 y  n6 M0 p! @
    Key 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.6 L" s9 W5 [3 Q) _0 a' `% V) u

    ; M; _+ H# X$ j    Solving the smallest instance directly (base case).: F: A2 K  K$ I3 a6 d4 i! j

    8 P8 n0 r: U! c2 b' U, k    Combining the results of smaller instances to solve the larger problem.' {% c" L( b* O% z0 }/ P
    # `. L6 h1 ^2 j$ ~5 O$ c
    Components of a Recursive Function2 N2 q( g5 K, x6 a- i

    ! a" z3 B9 j7 \! S& o    Base Case:' |* Z. T4 B) m) F# r5 D# Q$ i

    0 u7 r- U& G* A2 @* `) I( t0 t9 A        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.+ H. g/ E) L9 _# o9 y' e5 @7 [
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            It acts as the stopping condition to prevent infinite recursion.0 n8 x  w& N3 Y# Y5 J8 y
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; I6 C! \, F( M) G9 `/ h
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        Recursive Case:: n6 U8 z+ Y: l  M$ Q
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            This is where the function calls itself with a smaller or simpler version of the problem.
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    ! v% W, S7 I! K% V1 P        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' P  x4 _' |% ~( _7 u* Y  a- k& R2 ]
    , @0 s8 H) k. ]
    Example: Factorial Calculation
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    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:' w, L/ a2 ~( x1 z8 ?3 s

    7 D- u) W4 I2 O; x# q2 Q; q    Base case: 0! = 1
    # L# A, v& K9 B3 z( I9 o% Q/ R5 u
        Recursive case: n! = n * (n-1)!
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    ; u+ v3 u; h5 w# o1 a+ pHere’s how it looks in code (Python):
    2 W1 t. Q9 @' h! }: e* S; t" Tpython) Z7 k; a2 S+ Z( l- R/ c
    ; m$ G) r0 G' w. x2 L

    3 C8 A; |+ f' H$ i; b5 idef factorial(n):, b/ ^" X% p: C
        # Base case
    , {& H. [: |  x9 q) w    if n == 0:
    0 n. ~$ q* I2 l( D- ^6 m        return 1" _5 h% H% q+ x$ [/ J& U
        # Recursive case. t. ^* q9 @6 v0 f* ]
        else:3 Y! O8 \, d' `* c- K
            return n * factorial(n - 1)
    7 M3 |+ G: M! p3 T* u7 }6 W+ \, H0 o+ h8 ?/ O1 H
    # Example usage
    : a5 |9 t, @& P0 C% Aprint(factorial(5))  # Output: 1205 b. y! R2 z4 |

    ( R7 w( g# A+ H8 R! T" C) Q6 HHow Recursion Works1 @$ T" `# o) O4 z
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        The function keeps calling itself with smaller inputs until it reaches the base case.: u7 a9 \3 @* n; x0 p
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        Once the base case is reached, the function starts returning values back up the call stack.$ u7 o/ T" F5 i4 B5 m
    + S, O$ C( s/ x, }" Z/ r
        These returned values are combined to produce the final result.
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    For factorial(5):: U8 b. s! G! `& U+ S
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    factorial(5) = 5 * factorial(4)
    % [; Z+ i4 P, t0 G% [/ Ifactorial(4) = 4 * factorial(3)
    ! L, p3 O6 `: o! {. m( ^0 tfactorial(3) = 3 * factorial(2)
    - f5 c8 S1 S, X% [factorial(2) = 2 * factorial(1)# s+ W2 L/ e) G, j
    factorial(1) = 1 * factorial(0)
    9 H, \% R8 K5 z% E4 T2 L+ |factorial(0) = 1  # Base case& ^; ]" \/ k( J/ n3 I

    ! l( B4 y- I0 k4 U4 aThen, the results are combined:( \4 s7 S& S' I6 V: l) {: ?: C* `  z
    & ~4 F6 j" w% }: Y* E9 G4 Q

    % v. o3 u# G& Xfactorial(1) = 1 * 1 = 1/ j7 q6 L1 Q0 b9 K/ ?! d/ v
    factorial(2) = 2 * 1 = 2
      F2 V- A3 r% v' C/ r& I& Tfactorial(3) = 3 * 2 = 68 s3 s& H8 ~% ]
    factorial(4) = 4 * 6 = 244 \5 Z6 i" M( g( i  X
    factorial(5) = 5 * 24 = 1205 D7 {8 i/ [# z% F) z) G

    . c( m  h: Q, e6 ~2 s( mAdvantages of Recursion
    & Q" W% z' ?# v: m! |7 P) p* f7 r8 C1 t2 }
        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).3 y0 h1 k; C! C( }$ f; f
    3 L7 S9 g+ ~5 Z5 g2 t
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    6 ~& ~2 r  A; s7 c, eDisadvantages of Recursion
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        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.; ^- A( U/ F0 |- H- K6 C) t
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        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    $ L  i# O& p* A- [  e# F) C( W* X/ F" u8 T* t% b3 Y- @. V/ |' H" Q
    When to Use Recursion* t# M4 [: k1 d' q

    9 X: M1 l3 T' w9 s, L    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 A; h0 Z. Z+ H* P$ h

    : @% b$ k$ F* k, }5 \5 a+ K2 s3 H5 u    Problems with a clear base case and recursive case.
    * R5 n# r; N! N2 @0 X# @  Q1 G
    ; k9 R$ A5 @% S5 eExample: Fibonacci Sequence
    & p8 R$ C; \: O1 j# G$ m
    / I0 g2 A1 e4 G, h2 ^The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:' B5 _) W% v: {5 L9 ~0 w
    ' Q$ Z$ L! D/ r1 O6 U4 h
        Base case: fib(0) = 0, fib(1) = 11 b3 x+ N$ S( h/ p0 N; l" J& _

    : W+ m& t9 \! `! H: _( U, z    Recursive case: fib(n) = fib(n-1) + fib(n-2). _: g. L- B) V! F6 ?& @

    ' T' |+ f" @+ F  W! l! T; e4 hpython7 C; O9 W' ]4 V4 Q) N2 @

    1 t5 s% j- ?/ a" v9 U, k( ]" R: B, p6 Z
    def fibonacci(n):
    4 q2 i% K/ ~. h) X    # Base cases6 J  `8 A0 T4 \: Z  m- @
        if n == 0:
    " N% X* v6 g1 t8 ^% f        return 0+ E. O8 u) S% f7 B& ]
        elif n == 1:: ~3 n" M+ H( C4 {5 U
            return 1. n9 X0 R; U* K6 g( y) Q) C1 h, E
        # Recursive case
    ! h" n" G! r2 S! A: t1 p$ B8 L    else:3 g1 j5 J+ A- v( e4 y% z( V
            return fibonacci(n - 1) + fibonacci(n - 2)+ G$ m) `- f8 L1 ~5 n6 y

    + e& C0 P$ |0 H/ W3 g4 E0 k* c# Example usage7 }( X& ~; R+ C4 T9 \- i; U
    print(fibonacci(6))  # Output: 8( I; P0 j/ P! @+ h1 ]

    ! y$ e7 ~" z  g' K# ZTail Recursion, _; W, c( M3 e6 S
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    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).
    4 ]0 ]( E' v7 _( [- t* d4 d; b. ?0 x9 [0 H1 G
    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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