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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 ! j: I* g. i4 y
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    解释的不错
    ' h) ?) p! ~0 e, I+ ~* h$ C
    3 g$ U+ B  B( F  E* w; Y/ h/ j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 Y, r* V  ^) h5 ~/ s8 w
    2 X- S7 X- y" `' ~9 q6 Y
    关键要素2 u! G  S* b+ Z) J: [- |9 \
    1. **基线条件(Base Case)**5 L6 T! k- H0 p7 a, t
       - 递归终止的条件,防止无限循环
    7 m! }& W; L( ]3 h8 O# X; L   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# C2 x3 g9 ?' n7 v8 h2 I
    : C' C: @0 B3 Y7 u
    2. **递归条件(Recursive Case)**
      T; `" v- @3 T( ^! Y3 V' Y# [   - 将原问题分解为更小的子问题. `. f  ?+ E1 i% }- \
       - 例如:n! = n × (n-1)!- R$ T$ S+ ^2 b. R/ Y7 ~
    2 \' ^4 G7 N* S8 V+ |
    经典示例:计算阶乘% ^' A" N( Y3 e/ ?- C
    python7 l& \. Q7 E# ~- E1 h
    def factorial(n):# t* [+ F% K4 X7 P
        if n == 0:        # 基线条件
    6 k7 Y9 P+ `/ l! _        return 1
    + m/ B: @, m6 C3 E0 T    else:             # 递归条件
    ( B$ ]; r: F* k, l+ J1 N        return n * factorial(n-1)
    ' k! a4 _/ C; _. `+ P; w执行过程(以计算 3! 为例):8 _, N* O: F$ w
    factorial(3)
    : K- C# `; Z; s- V3 * factorial(2)
    2 O! m& P4 b& {; ^% L3 * (2 * factorial(1))3 R$ i2 b; g: Z7 U$ Y9 A
    3 * (2 * (1 * factorial(0)))
    7 O$ V* w; @  \3 * (2 * (1 * 1)) = 6
    : v& Y, j. l9 C! J  ]0 w. P& r7 X8 m. \
    递归思维要点
    ( i% C- G; j- i+ m/ N: \1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ' k* p5 A+ w1 t! E6 O! M2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ' ^9 \$ O6 C5 `4 j8 L! p: L3. **递推过程**:不断向下分解问题(递), V4 z9 u2 V, R/ z# [) |
    4. **回溯过程**:组合子问题结果返回(归)
      b. N4 N8 v- f2 u# H  Q; A& @7 k7 [! q- y1 b) ]
    注意事项
    * z% Y7 m6 M1 |' ]必须要有终止条件
    7 [9 |$ N* W( S2 t9 p递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    0 i3 k' A; j  @, n3 q6 T某些问题用递归更直观(如树遍历),但效率可能不如迭代
    8 M  j" I2 I# J$ _- Y尾递归优化可以提升效率(但Python不支持)
    - W/ B. s6 ~: A  C6 m; @
    " D7 t& ^$ K$ {" s$ Z! K; R 递归 vs 迭代1 b. B  H# W2 D$ U9 `6 J% \
    |          | 递归                          | 迭代               |' E3 E6 O. S# f. Z2 K/ G( T
    |----------|-----------------------------|------------------|
    / o$ y4 l) ?$ _/ l# x& {| 实现方式    | 函数自调用                        | 循环结构            |
    : b4 `& w7 H3 m, l- K| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 P2 @/ _( L% C7 A4 t1 B/ o/ R) 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         |
    8 S$ k# b) ~7 P5 `, s' s1 v| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& V5 {( C7 X- f) N, X# C
    " `4 ^5 I& ^9 v  P9 A5 ?* x  R1 X
    经典递归应用场景
    # ]- h5 s$ V+ L( s8 M: k9 s1. 文件系统遍历(目录树结构)- M& v8 L6 ~% t# s( ]3 ]# o
    2. 快速排序/归并排序算法2 @7 M. ~8 Q3 j; T
    3. 汉诺塔问题
    3 A! h! Q& [$ z) F0 E- v: S0 S4. 二叉树遍历(前序/中序/后序)- f8 \8 q3 p9 J3 x7 O, W
    5. 生成所有可能的组合(回溯算法)
    . g3 F' `. x' {3 q
    # _) f2 N5 V: X! A) q) [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,( U$ U3 @, h( E: l9 B- e
    我推理机的核心算法应该是二叉树遍历的变种。" E% `$ f5 E: }4 G- z; Z2 i% l
    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:
    : e) Z6 w! C# _Key Idea of Recursion+ A. B1 R: f: ]; h( G$ N' D

    ( C* W' ^  N; v/ LA recursive function solves a problem by:
    . |1 x+ j! Z2 c0 Z8 L8 X! `. l( l! l; ^% y) l
        Breaking the problem into smaller instances of the same problem.
    8 @+ Z; w9 ]+ z& X& G# B# Y. ]+ h: F% C8 R
        Solving the smallest instance directly (base case).  Y, ^' a" w; _  `" L, \) t5 l

    ) J) ]0 d4 f: L: {    Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function
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    ) i; v" S: n6 M5 t1 d9 Q    Base Case:$ n6 n6 n9 s3 L0 ^
    9 [. N! z% a, o3 q
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    0 }' q5 }. ^4 w9 F# b
    - h* f9 A3 f+ \: p; e) ~2 M        It acts as the stopping condition to prevent infinite recursion.
    ) k  }$ \4 j/ }5 T, q2 Q
    5 K: |- I; Y$ H  ?. q- O# L- x7 |        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 Q+ `! J; s  m1 a6 ?( u0 s
    . @& g0 g7 _8 a- y9 e8 Q. }  E
        Recursive Case:* C9 X& P: z, s: t! I1 |# w% T9 |

    , a6 e) E  K6 m, R$ [; W2 @- G        This is where the function calls itself with a smaller or simpler version of the problem.
      Z+ c* ^9 H; k, O; F  {8 _
    9 E1 g: N+ T. m  G        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).& G1 _, d! j1 M0 v5 R
    0 x/ c% X' O$ h9 O" z
    Example: Factorial Calculation- A2 S! u9 q4 N

    . V$ B( b- a# w3 N, `) n/ f" tThe 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:: Q4 W+ j4 }, Q' H: B3 {

    9 u3 L0 Z2 A+ k0 X    Base case: 0! = 11 M" b0 J. ?. M4 u

    6 {$ B/ N& I9 K6 [7 G    Recursive case: n! = n * (n-1)!
    + w  Z8 n9 k: S( l$ _: D, e. a8 b! v5 y8 U4 a
    Here’s how it looks in code (Python):2 B5 v7 t* u, ]! p/ L
    python
    , ]: {/ E0 s" k7 Z( s& |! |& f6 c4 b: }+ z
      k) U: e" y4 \* H4 x
    def factorial(n):. X/ ~) }1 F7 V
        # Base case
      z; `- V- ~6 K    if n == 0:% d# f8 |% v  ?% I8 w
            return 1
    : x# e* E, a$ x$ |2 f/ g    # Recursive case
    $ K* {! x( i. q$ t. j. X/ F9 ~    else:+ |# C' c6 E3 O; [# R
            return n * factorial(n - 1)
    9 `0 t, ]" }8 S+ g4 i1 c+ D
    * M8 ~0 Y# \( n+ {. R# Example usage
    " A+ f3 h6 g" m# [1 r/ pprint(factorial(5))  # Output: 120; s! A2 J: q& u; t1 `$ ]" t7 q
    8 O( R$ x# R; U7 Q, _
    How Recursion Works! d7 b' j1 z2 D0 V) k1 h% R) p
    : ?" x. b" u  A$ R% x. w
        The function keeps calling itself with smaller inputs until it reaches the base case.6 q5 G/ `" J& j! L" n, \+ C1 D/ ^

    & U" O& M8 Y! j    Once the base case is reached, the function starts returning values back up the call stack.1 G( f# _4 O  ?$ \9 X  b
    / o6 Z1 U- m4 V& b9 O8 C
        These returned values are combined to produce the final result.
    0 |; k( p- b' J2 @3 j
    ) f( {+ V0 t8 Z& dFor factorial(5):0 G. h" M1 `! n2 U# ^
    # Q3 n1 }- j) X+ `9 L

    4 w$ _" M3 L1 b6 W1 Tfactorial(5) = 5 * factorial(4)
    * p% P2 H9 a9 V" Ofactorial(4) = 4 * factorial(3)
    - A! I2 R8 ]* \factorial(3) = 3 * factorial(2)5 F0 l* U) _: [/ q
    factorial(2) = 2 * factorial(1)
    8 v5 t/ Q7 J& ffactorial(1) = 1 * factorial(0)( [+ I9 v' K8 W3 C% z* s
    factorial(0) = 1  # Base case2 t8 ?; z+ r  \- p6 G
    2 ?8 h7 s8 w; j
    Then, the results are combined:7 m: r4 G& C+ K

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    3 c5 Z8 a  O( Y9 ~8 G% \3 e' @( Efactorial(1) = 1 * 1 = 1
    ' ?( }9 N' s  w# n! a3 P: Z" ~+ I" ufactorial(2) = 2 * 1 = 2
    ) W! C, t  w; v0 z4 B" F1 U" ffactorial(3) = 3 * 2 = 6$ Y, a& u# D/ h/ S; X* |8 g
    factorial(4) = 4 * 6 = 24) ~3 r; A# l0 n+ g  E) b6 h9 g
    factorial(5) = 5 * 24 = 120  e4 L6 w  F* Y5 M" i* I
    6 ?& `, E! P4 y0 X3 T' _! Z1 {* U
    Advantages of Recursion/ ]" m% |& |! G/ V- g, \

    , y2 n  w6 q& A4 I( a" ~; Z7 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).
    " a. R+ F1 K7 h# c/ w( G
    + y% m6 m, X+ w1 M. k    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    4 c" ^2 y( D1 W9 T& w- g) k2 r1 m: M- B9 e8 E* n  j, Z1 Y
    Disadvantages of Recursion3 c- {- P) {" N9 N9 `
    ; X. }" X5 s1 i5 ^
        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.
    $ v9 p: n' a+ X# |, {( ^, g9 j5 Y' R6 h; s! Z
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 v, s& [$ e/ D- }" r+ ]

    # {2 `% z: |" ]2 Q  gWhen to Use Recursion( c, g+ k, L% a. }+ G# n

    4 P6 Q3 c! G( Q! F5 B9 i7 y1 Z5 `    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).3 S  Y8 e" f+ ?3 _2 B  S9 g
    : i, z/ J. f$ W/ H
        Problems with a clear base case and recursive case.! N6 H' }. P8 x+ A: _0 A, F
    % V0 |1 \2 @! \; u
    Example: Fibonacci Sequence
      i7 f; k4 M4 J" s( r" d4 s
    # T; f1 S4 W2 V3 yThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:% y8 C% K, G4 x2 [

    / ]* V) Q4 j6 {) [    Base case: fib(0) = 0, fib(1) = 1! i1 e  H7 y' A+ Z9 t- a

    5 j7 |4 ^2 ~# G! d: ^    Recursive case: fib(n) = fib(n-1) + fib(n-2)6 k( P) e+ K! ^& e6 _# N" C

    2 U6 a. f, Q: ?. ?! l- K& Hpython
    / R8 Q9 ?' U/ Q4 \0 I  o
    1 A' k3 R# n6 @8 \& ?  ?" D( J/ m' c8 u. M" s9 k1 d0 }+ C9 ?
    def fibonacci(n):* e8 `: ^4 E1 I5 t* T! k
        # Base cases: \+ `, U( _& {' S4 U, ~, q
        if n == 0:
    . v# G! a* b1 V: K. O3 u" s        return 0$ t5 g0 `- g2 n0 u
        elif n == 1:, W: ~* v* H& b
            return 1
    ; A5 q5 }8 Y% s! s    # Recursive case
    2 m6 W) p$ S6 I0 u1 X$ Z7 O0 E) }! g    else:
    + [; a  R+ O& e/ P& i2 V  B( U        return fibonacci(n - 1) + fibonacci(n - 2)9 y# `* a" U7 l9 A. M  n
    , r- u- i! ?9 t+ {
    # Example usage
    : t& H0 G9 ~7 oprint(fibonacci(6))  # Output: 8% t8 C) y- t% d4 }3 k8 \
    4 A/ t, b# g" n5 y/ l
    Tail Recursion' r, o8 l0 F3 P9 Y: {) U
    % V0 [! ^5 H; \: H& U; E
    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).
      u3 f$ l$ O! e+ ]
    : P$ C- j: w+ T/ U: 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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