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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 2 H% I5 j0 H- g# W/ A

    ) [* o4 y8 M9 K! _解释的不错  n0 v4 c( b) J( N7 X

    4 O  C: b6 w% P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。
    & t3 b4 ^  Q7 E5 X
    % b: C+ t- ^( F+ R6 x+ e 关键要素/ C0 W$ L# o) h  E4 _9 q
    1. **基线条件(Base Case)**2 z( B- N0 j* m, p4 \
       - 递归终止的条件,防止无限循环
    : D& ^: w9 J% f1 q   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& t  X% f/ H/ g) a: x7 G: l
    . Y$ K0 {" u* t! H
    2. **递归条件(Recursive Case)**
    / \3 ^0 E: ?* Y$ ]5 h2 x   - 将原问题分解为更小的子问题
    9 a9 W% e' W: e- k1 Y   - 例如:n! = n × (n-1)!
    * [* U+ z8 s2 g; x7 y: S4 d# j2 L- y0 y1 ?( F3 y- h1 V* s7 V. J! K
    经典示例:计算阶乘
    $ U3 B2 O5 Y* \. y9 Mpython# t  v' H4 ?* f. y9 f
    def factorial(n):, v5 X" m0 j" o
        if n == 0:        # 基线条件* A$ h8 J6 E4 i& A) @
            return 15 I' ?& B" p% O2 {' ~( E
        else:             # 递归条件
    $ ?) G# H$ M4 V5 g9 O        return n * factorial(n-1)
    & e, w& J/ Y) y5 s" s. [执行过程(以计算 3! 为例):
    + B, i- S/ z! i$ M' _, k2 Afactorial(3)
    . L% V- [0 t; t8 e: S3 * factorial(2)
    / x2 T* X' h- ~! p: I; |$ J3 * (2 * factorial(1))# u" h3 S4 J+ q* r5 F
    3 * (2 * (1 * factorial(0)))4 u4 N. ~8 z# s8 D# R3 F. R4 v3 P
    3 * (2 * (1 * 1)) = 6
    - X, h8 C8 A/ e9 d1 i: v2 y  Q# |( r# U' ~5 ^" U
    递归思维要点) Z* Z( s1 T9 w8 T; @& i* ]8 S
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    . C0 X( S1 |, d9 q  M2. **栈结构**:每次调用都会创建新的栈帧(内存空间)9 P& f, _$ T- e  n: a; R9 ?
    3. **递推过程**:不断向下分解问题(递)
    - E+ L6 z" V: N& r  B4. **回溯过程**:组合子问题结果返回(归)
    $ R* I9 H! M7 a- m: k" }; w3 P. G0 l+ E- H
    注意事项
    0 `3 [' }: G: W0 E+ |必须要有终止条件
    ( R. @) k8 \- W9 |7 a递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    4 O7 i! J1 A1 v$ P1 i某些问题用递归更直观(如树遍历),但效率可能不如迭代
    7 D5 o& a3 Q  u9 J8 y尾递归优化可以提升效率(但Python不支持)
    + @: [2 a$ @1 b3 }1 I
    . F( [6 Q# u7 r+ g) z! R: W 递归 vs 迭代
    ; m( a# Q6 n8 k: I' [, V3 c  y! z! A|          | 递归                          | 迭代               |
    + [# @% w5 ]) [% [  a6 m2 R' t|----------|-----------------------------|------------------|
    # }; f5 S$ }' d$ B# A- u2 \| 实现方式    | 函数自调用                        | 循环结构            |
    / w* c* a3 D& z0 n! e! N, ~| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 \9 L; V* ]3 O( u4 Z
    | 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         |
    ! [6 Q/ p# U$ w| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |( c- ^- l, |. X+ B: H

    * ?/ m. g' y  [; l 经典递归应用场景
    ) Q" F$ B4 y) W: P" O2 w1. 文件系统遍历(目录树结构)' C, U% _, s9 y3 B, a0 u8 s
    2. 快速排序/归并排序算法  J) I# Z; q" F; X
    3. 汉诺塔问题
    2 {4 n3 j/ Y* G8 a4. 二叉树遍历(前序/中序/后序)$ _" M. u: C+ h7 u* d8 T0 q  F
    5. 生成所有可能的组合(回溯算法)
    3 n# P* G) J; P- y! H' {3 j# }, q; D( ]1 B  g. k9 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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    慵懒
    昨天 10:05
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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,
    , x9 ^3 S& O: R( a我推理机的核心算法应该是二叉树遍历的变种。
    " V0 |+ |, t# B4 ~- g8 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:; g3 E2 O- _1 [' L- C0 v. ^
    Key Idea of Recursion
    / O6 N) d- s  G( |, p* l2 s6 ?
    7 u6 `$ V1 E5 r0 s* KA recursive function solves a problem by:' Z$ S" ^# ^2 r! g
    / O' m* J% c$ Z5 X
        Breaking the problem into smaller instances of the same problem.
    . N9 m8 C+ q: g3 j% L
    - A$ n1 |: V8 Y$ V    Solving the smallest instance directly (base case).
    4 H, K! ~" E& B* ?! c8 a. ?( F( G& A
    ( d+ ?9 V) j# V/ w- t# B# u    Combining the results of smaller instances to solve the larger problem.8 \1 \# ^5 Q2 q/ J6 m# W

    + L9 U/ [, N5 Q( x( sComponents of a Recursive Function5 |1 ^# L2 K! O9 T! n
    , j; ]: l. s2 m. S
        Base Case:
    4 T/ M# l% \+ g0 e4 w
    4 N' u/ @! C# z+ p        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ! L% e8 `! t# B9 y$ I8 J3 W. H% q/ w: @4 ^
            It acts as the stopping condition to prevent infinite recursion.
    + ]0 _  O. }8 V* A* k% B5 z% G" h
    , S+ f+ ^5 h2 B0 \- f$ M% H        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    1 y- y* [0 x+ {4 @8 x3 _
    ' i$ m7 v) ~/ ?4 M! p9 N    Recursive Case:
    5 F# h. O2 [+ E) x# A/ N* o
    + B0 e6 O2 ?3 b0 ~' K' O5 C        This is where the function calls itself with a smaller or simpler version of the problem.
    . w$ U+ W( S8 A
      }; c/ Q/ ]7 U. F        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% M8 x9 d. k! A. y# c- f+ |
    : L6 E, h& }4 ?  C$ Z
    Example: Factorial Calculation" k3 ]% t$ T' q/ D2 s

    7 H  W8 N; T% s: x. ^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:5 O5 [" `' B8 a7 }$ x

    5 E$ I9 L2 n9 Q, |    Base case: 0! = 1
    5 S  ]5 G" h# A: r
    9 t7 j' p; q. B8 C8 L! G. U/ y    Recursive case: n! = n * (n-1)!/ z4 T. ~. h8 p& q& I' K
    ; v; }: n" N# x1 x7 D
    Here’s how it looks in code (Python):
    ) @6 s' e2 s6 i- f4 \! tpython  X; X# f: Z9 W" ?& f
    ! U  {2 g; S2 W+ D& a0 }0 E

    $ ~' M% e1 @" Q# \6 pdef factorial(n):) Z9 }" X( X3 ?* H
        # Base case
    5 z9 B; _  \: W9 \- k, W1 r5 g    if n == 0:  @% _2 o3 P2 F" \
            return 1" S; v& @- ^9 Y1 ~8 o5 v2 x1 ~
        # Recursive case+ d) K* I. t; T9 c
        else:
    9 }1 z7 \/ [7 j        return n * factorial(n - 1)" ?1 X. k% G: v- e3 e' `
      t6 N, S7 h! e2 v
    # Example usage1 ], e9 ?' _7 v* U! D
    print(factorial(5))  # Output: 120
    . I1 e7 w7 z4 q( l: L! w2 u
      K5 j! n; |; Z; o$ K9 O, A" G0 S9 bHow Recursion Works
    * S. x! q: S/ L7 d5 e( h; x* M) Q$ S6 T. J' r/ u* ^, o! d: n
        The function keeps calling itself with smaller inputs until it reaches the base case.
    3 j7 {5 Y# Y8 z0 ~+ ^6 [& p+ c/ v! a5 I( ]4 S' B- H, I2 V# W- H8 _
        Once the base case is reached, the function starts returning values back up the call stack.; w$ T7 S7 H/ c# ^% W3 e! y

    8 ]2 x: {) X* _8 \  e) U# T    These returned values are combined to produce the final result.* ]% E2 k/ S; B- u! i7 j% k8 z

    ' r8 O) W' c) f7 `/ Y0 J9 ?' }9 rFor factorial(5):+ q. k3 S6 F* Q  X
    9 |; `) i8 i  {2 J

    9 P  V5 t% ~+ h, P+ F6 Y6 a3 Afactorial(5) = 5 * factorial(4)
    & h0 a5 Z& a, i! h8 _: ]factorial(4) = 4 * factorial(3)
    ) [% H! Z- L0 X9 `1 [factorial(3) = 3 * factorial(2)
    4 ?7 \; d. ], `/ u" V8 z- Kfactorial(2) = 2 * factorial(1)* X* L& `$ f- r
    factorial(1) = 1 * factorial(0)
    ! A8 J- t! g9 m2 h! T8 Nfactorial(0) = 1  # Base case
    ) T, S: K/ E: P! W& K. m7 {' w: v) X& E3 r( g& Z
    Then, the results are combined:
    4 ?! O( c3 E- M" e1 L7 B7 r; G9 d1 V4 }
    4 p( f# _: `; X# b6 f
    factorial(1) = 1 * 1 = 1: @8 {5 D" ], C- h: n' r- f. l
    factorial(2) = 2 * 1 = 2
    . S7 Y% M4 f3 c7 Ifactorial(3) = 3 * 2 = 6+ {3 a/ R) e' x+ O# E3 H
    factorial(4) = 4 * 6 = 246 M# ?! @# K' e$ o6 ?7 L( t7 _
    factorial(5) = 5 * 24 = 120" F4 `. O, J0 m2 |* c) {
    9 {) X# e3 M  F+ x4 ^3 j& \
    Advantages of Recursion
    - P$ o/ R. j4 q1 m; v  O! o3 L0 D& L5 B7 ?! ?3 P- R) N
        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).' N! D  _, m, \/ n
    : h# A- g& Q8 `- R5 J: Y
        Readability: Recursive code can be more readable and concise compared to iterative solutions.. T( q* d6 B; e; i+ O2 N( w2 a8 A

    ; u9 R0 y. t$ R  k6 l& iDisadvantages of Recursion
    5 `# l- J. ~4 D. @; }0 R0 \" G# N  K. x6 Z2 u* N
        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.& a4 r; I; J# z6 u# C! O6 g
    9 x2 h0 i; w# c; a- x
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).+ ?6 c- n& H5 h3 c* R

    9 |. {' x3 ]/ }: D. K, o  tWhen to Use Recursion9 E- O2 Q; ^2 o1 g

    ; b# i7 l! K$ g    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).  Z$ q# q+ D8 x4 {# K

    , [. Y( p! q: M    Problems with a clear base case and recursive case.' Y0 d  M: I9 J% T/ Q) ^
    0 O$ {4 X: B: V) q% b$ h9 b
    Example: Fibonacci Sequence
    & W5 I) t4 \, ]1 k1 ~* f% L5 s; h6 s! F
    ) {: H7 P7 `! \: z' K8 vThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:' |$ s/ X8 v) u9 o8 v6 {

    # T. I, ^2 ?/ |% V- W* \    Base case: fib(0) = 0, fib(1) = 1& Z9 I+ o# k& ^# ^" c' M! s7 j

    ; ~5 c. d1 |2 J7 U& v) }    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    3 K! ~3 F1 N3 V0 f' o- v6 m0 m# Z: G) _
    python
    3 B. E% v& k; z  W1 ^/ S1 s/ N, l
    6 X% z$ E( }9 q7 {, i5 G! ]# g( K9 n$ ~
    def fibonacci(n):
    4 _% i" W2 D/ v* L0 L2 ?    # Base cases
    + i: a4 K& \  q# \: e( {    if n == 0:
    % _* F' _( o9 ~# s2 G        return 0/ S( y5 L- `& w
        elif n == 1:
    ! S5 P( q. E4 W* o7 ?9 s        return 19 Y, V8 g; C6 Z0 b9 x! u
        # Recursive case* F. r( g% s5 B7 `8 S
        else:* A2 B$ u6 r# Q) l: F
            return fibonacci(n - 1) + fibonacci(n - 2)% o# Y$ B6 Z4 t7 ]2 c5 Y" l

    4 r8 s  N0 j, Q2 @5 Q# Example usage
    - E/ R# X; F% _) e% e( M" h1 R- p9 ^  pprint(fibonacci(6))  # Output: 8
    ) T. G/ X2 z2 H( P
    5 T) }: u" Y! D! l7 o% k2 WTail Recursion
    8 t5 o; K2 T+ O3 n. @$ o$ q
      E+ C+ G# L4 v* g  qTail 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 |% x2 x. q* c  ^% V
    ( l0 I) ~. C* C7 v
    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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