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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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    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   }$ C8 J) E; ~/ }
    $ U$ F, i! Z6 g4 X! b4 W1 E
    解释的不错
    ; E% |6 Y5 y. e' b4 ?$ Y6 T9 c: Z3 D' z% j8 a
    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。9 o: y3 P7 n3 f( _3 ?- j: V- z9 M. l

    - Q/ ]  {# E% B* L4 a; u1 O' H) ~$ H 关键要素
    $ o  `4 l' Y) F$ `6 N7 r5 n1. **基线条件(Base Case)**
    5 C/ f: |0 C/ A3 u- U   - 递归终止的条件,防止无限循环& B9 P' B* U3 T+ g
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1  d% y9 u0 X/ ^. ^. K3 s, J- T$ G
    , m, o! Y5 a% y8 q! {+ x' A3 I* B7 S
    2. **递归条件(Recursive Case)**
    " c8 c. w" t+ \   - 将原问题分解为更小的子问题% M' E( Q* e' r
       - 例如:n! = n × (n-1)!
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    8 s1 B' L! w- f/ B8 o) w, I+ V 经典示例:计算阶乘% |! [/ D% b7 }
    python9 G( d  B( K7 \# g6 o
    def factorial(n):# J! @1 O# w4 G3 Q) Z9 g$ Y
        if n == 0:        # 基线条件& H. L+ c% U  T9 j
            return 1$ I# b) p& w! f9 `- g, h
        else:             # 递归条件
    ! t* A7 F1 I. K6 m! y- a        return n * factorial(n-1)
    2 X2 J) n1 t+ R; I执行过程(以计算 3! 为例):, Y2 {- J' K: h( r3 j0 K2 v
    factorial(3)1 v( {4 X2 }7 V8 F8 M
    3 * factorial(2)9 D$ j( R* m+ ~6 f
    3 * (2 * factorial(1))) s! [( k6 n" M! J2 U" {- X
    3 * (2 * (1 * factorial(0)))& N2 S% ]4 \" c& B$ Q" |
    3 * (2 * (1 * 1)) = 6
    6 U, X# U6 t& |% {, e' l. }8 i- Y. B0 l/ O) A
    递归思维要点- r- X* F* b3 I/ I. P
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , r! ?* I8 Z* U2 \3 f2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    - q; L0 c4 G# w1 W2 Q/ J5 Q3 d7 V3. **递推过程**:不断向下分解问题(递)
    $ Q$ w" t, `0 C, n" D8 m4. **回溯过程**:组合子问题结果返回(归)
    0 ]3 n$ P+ `+ I( [
    + ^4 c6 D: Q6 O8 z 注意事项9 E1 d7 Z: j5 N- Q* m" L* U
    必须要有终止条件
    + e; V* }* @, `8 Z* R. f: r. G$ N递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    + S1 w* n! j) T+ Y) W( l某些问题用递归更直观(如树遍历),但效率可能不如迭代+ S- h1 j# f8 R( q  `
    尾递归优化可以提升效率(但Python不支持); W+ a: v7 w1 p7 M: f4 g+ g

    7 j+ {, \: k4 u3 d7 @ 递归 vs 迭代
    ) |0 _- h; t! a$ \6 f|          | 递归                          | 迭代               |
    4 s3 _1 E2 Y8 o: F0 \/ S|----------|-----------------------------|------------------|
    . I! M( T/ @' }+ H| 实现方式    | 函数自调用                        | 循环结构            |
    5 K; w! z5 t/ s+ h) W# ^| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |  @0 b1 P( u! q' J6 g1 [
    | 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 w! E$ w$ q- I2 Q( p- U
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ) x  e9 E5 X' r4 a% L5 [8 E9 T+ g- h0 h+ D  J7 b
    经典递归应用场景
      z$ }$ d/ V/ ~( |% y) L6 ]1. 文件系统遍历(目录树结构)# g* l* t' x( \2 ~) Q+ l4 W  Z' z/ O7 A
    2. 快速排序/归并排序算法
    - t$ h* d4 f5 k- r7 ~3. 汉诺塔问题
    6 w$ X4 r, \1 l2 r1 z: M- \4. 二叉树遍历(前序/中序/后序)% @* h8 U1 z" Y, Z( |2 a! C3 S' x
    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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    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,
    1 s4 I  P* M3 W8 v" e: ]) J' P我推理机的核心算法应该是二叉树遍历的变种。( [; v- S, x, S& B) Q/ @# i
    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:4 x; X: t1 a, `8 g  x
    Key Idea of Recursion
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    ' Z. M+ ~& N5 A% HA recursive function solves a problem by:% g: J0 |. \, P: G+ T6 o/ j* P* E

    % A/ v5 Z. e" @+ t    Breaking the problem into smaller instances of the same problem.
    9 n$ `$ V0 \  C: `9 _! y
    ' \8 t$ l% O! ^. \0 `2 u    Solving the smallest instance directly (base case).; v( z7 `7 V) a- ^/ ~- `
    9 M: E9 v; Y5 j  d& ]
        Combining the results of smaller instances to solve the larger problem.3 d8 U) ]; j4 A7 }% z, P2 R
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    Components of a Recursive Function
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        Base Case:0 w- F7 w  l% m, m# \5 I$ A
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. r1 Q9 I' h5 {& o$ |3 v

    1 ?( l! I) b* Y& a% I( k        It acts as the stopping condition to prevent infinite recursion.
      d' g" L- Q( i  d1 @1 z0 s, o+ M/ \. x0 d
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    2 M8 N  Y, l: `; \) u, m& K8 R- m( u; x8 X1 \8 D# d7 b
        Recursive Case:
    6 ~& ]5 \: `  q2 O7 |/ @
    ; l# V: W8 l& M+ U        This is where the function calls itself with a smaller or simpler version of the problem.
    9 d8 l. l% Q& o7 ~
    ) N$ e( t3 e3 B. R6 _3 C        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    / L# J# [. i" AExample: Factorial Calculation
    9 I# T5 b7 |/ S4 W% ]" y! t6 q$ e( N) ?9 j4 k
    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:. [* [2 N+ J2 z4 ^
    ) B% m: \& B, D$ w) L0 g
        Base case: 0! = 16 n& S, m& p" R" S  h

    ! ^6 t& O) ]' j9 f5 ]. U3 p% Y' y    Recursive case: n! = n * (n-1)!' B1 R" L: `! x2 L

    & d! }- R8 m- H& m7 ]5 w7 iHere’s how it looks in code (Python):+ m6 A8 L. M& s4 Y' [% t, j
    python* S3 g0 ^! }3 E$ I! U
    1 G) _" a! |' c# O" A: l

    ; [/ F" A( O  t# zdef factorial(n):
    , Y: ?# c3 q; l    # Base case
    5 B1 `4 E: B( y% Q" _    if n == 0:$ t: Y9 u# j! t' d5 i% _! O) |
            return 18 V" m5 o- M! r" N) q, K1 H
        # Recursive case$ b+ l+ y+ N4 I5 |
        else:% O# J9 H8 f3 s  G
            return n * factorial(n - 1)' @4 x9 h2 s9 K7 [1 n  e

    ; e" R" E# c' a$ [0 U! u# Example usage0 s3 P: Z! u1 A# d$ L+ e3 ~7 R8 S2 `
    print(factorial(5))  # Output: 120
    & Q* F& O; ]1 y7 Q8 O! j5 `
    5 U# U; m& ]5 y* bHow Recursion Works
    1 z2 O+ X* u4 i* g8 g0 @5 s: R1 u& p  ~# r. q$ v, O
        The function keeps calling itself with smaller inputs until it reaches the base case.
    2 a, J  J1 q8 Q/ z0 N) w
    # [1 D, ?* g# G2 \/ w6 @    Once the base case is reached, the function starts returning values back up the call stack.. b: {( s3 @9 N' ]# a7 r1 ?- }' b8 W) ?
    7 o+ \# E0 m4 x- J6 l' a5 w9 Y- I
        These returned values are combined to produce the final result.
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    For factorial(5):
    & U0 I0 C' K' r5 \
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    * S5 H: {3 w* l" |; Cfactorial(5) = 5 * factorial(4)
    , M/ ^* h8 E" x! N8 Qfactorial(4) = 4 * factorial(3)
      o! p. W" G4 L+ f& P1 X' K* Ffactorial(3) = 3 * factorial(2)
    ' f9 N. r& a  xfactorial(2) = 2 * factorial(1)
    ) f0 b6 e) P& Nfactorial(1) = 1 * factorial(0)
    + J0 y& P/ p7 X# O7 i1 A6 Ffactorial(0) = 1  # Base case8 G) _) z$ h4 ~( t3 g" {, X

    / l& @9 U8 G+ G% B7 j8 ZThen, the results are combined:
    3 \; W  z8 q# V# e9 C$ p! b) N; ]- f/ K) I( T
    / S7 v. d2 P$ P7 z  S1 A
    factorial(1) = 1 * 1 = 1, ~4 F  t. C% ^" V# j$ E& _" S
    factorial(2) = 2 * 1 = 2# o3 c( H- [: W0 p! M
    factorial(3) = 3 * 2 = 6. {& ~# S1 H  T( l9 O, ^
    factorial(4) = 4 * 6 = 24
    / s" Z( l$ |' U$ Efactorial(5) = 5 * 24 = 120
    ! J- w8 o5 a! |. |" c3 e1 [1 O7 n7 n  Z
    Advantages of Recursion
    / Y9 N9 D6 o4 A) l( ?
    2 k& e" }. D3 D2 n7 Q6 l8 D    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).
    # ?" b0 I( W- J' v) R
    * Z! N% Z8 @3 O- c% f3 o" }# d    Readability: Recursive code can be more readable and concise compared to iterative solutions.8 u/ h8 e' m, n+ n9 r4 S7 M$ P
    & E3 u1 m9 a0 h/ D* Z
    Disadvantages of Recursion% `6 L% T% P4 O' w, ?: O
    9 i+ B4 k& k2 a8 ]+ ~8 }
        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.% @# w+ Y& Q6 \4 h- j, V, @4 k
    ' C) z. |) f' c3 p# Q. d! W  j# X- T
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).. J; c) O" E1 a3 B
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    When to Use Recursion; C4 y- M5 b7 ~' i  a2 Z
    $ q. u+ C* P% K3 P' Z
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ! B$ E4 j( z3 j, L+ e5 K
    & q9 }% n+ F/ W% I2 R6 I    Problems with a clear base case and recursive case.
    4 s$ v) E! r" _5 V: O1 P) b2 H6 p5 {2 W5 D
    Example: Fibonacci Sequence
    6 I$ \6 H: f( h9 D  [- d9 _3 _' K4 f9 s& V/ e& d+ s  k1 Q
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    6 n* \* ~% f& L; F8 j) u. M; z( ~6 T( q8 x; H( {2 g# X+ S' W" \
        Base case: fib(0) = 0, fib(1) = 1
    - X' j9 B' _$ a6 U! L( E  _
    ! c3 ~# ~6 v/ e( m8 t    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    + \+ g, X, s( H  A* b. ?5 r+ u( C
    ( X  O+ J. {' b6 z! ?/ q5 Opython) Q& l6 h5 Z4 S5 }& T' l( u

    % R& ?* C- L' y
    * l5 V; P/ k" g" H4 b1 Udef fibonacci(n):
    ) O& x/ @% Y8 F, R1 C    # Base cases  x, C+ d( g5 q5 `) \5 x9 ?
        if n == 0:5 q" w$ T' |9 A5 o: `# H
            return 08 R& I) ?" Y4 @# k$ W3 T
        elif n == 1:; C, _5 h1 c; t
            return 1
    & M' z: z* h3 G) {1 M- J  X6 G0 e5 k    # Recursive case
    ! a8 h5 I4 Z( A0 }. R    else:3 _5 n7 b% Y' d4 U3 S6 l
            return fibonacci(n - 1) + fibonacci(n - 2)
    # j9 r8 E8 I) c; h* x" b& g" Q
    8 M$ K: v& s6 J4 D' d; L6 K6 s2 ?# Example usage0 ^. Z/ M6 S" R( _, d
    print(fibonacci(6))  # Output: 8
    7 K/ W5 u7 e$ Y8 H! F% l
    3 r" }: s% w! ?3 X- KTail Recursion% ]* `( ]1 ?. k1 M2 \
    $ g) T- }9 t6 r9 w( a! T
    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).
    : v3 n$ ^9 c$ l
    / D3 o4 b/ [( l* p# CIn 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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