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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 ! e( J; C- B: J4 O) s: B0 \! ?

    : h' F: ^$ K2 Z  G解释的不错" T" ]7 _2 @1 z+ s) }! r# U. C

    * h4 |+ a( V! e5 t4 n" |4 [5 C$ M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。
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    5 g2 o4 A0 M  D 关键要素( ?" E* O+ T) \3 W" Q2 D
    1. **基线条件(Base Case)**. c9 ~, k7 H% C' }) o4 j0 K; X
       - 递归终止的条件,防止无限循环3 ~( f* `7 Y, [. I: l2 d+ Q: L7 o
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    % `! N8 d( v& G# Z$ {: }2. **递归条件(Recursive Case)**
    " _8 @+ r( A, s6 j+ Z! d   - 将原问题分解为更小的子问题  n/ k0 Q) F% z( u0 _% S
       - 例如:n! = n × (n-1)!: P- v+ X9 c, x) E/ ?8 t& T
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    经典示例:计算阶乘$ R8 t, b0 i& ]
    python: s7 F# ^" R: B
    def factorial(n):! y  c6 q. e, ?! S# a1 z1 `" q
        if n == 0:        # 基线条件8 L0 s/ p2 x6 U& X0 n6 e& y# u' }* S
            return 1
    % r7 t1 W3 g( u- U; t" u    else:             # 递归条件
    0 [% s& W/ A( E        return n * factorial(n-1). `! N+ v" M- V; C# _5 G2 W
    执行过程(以计算 3! 为例):
    6 c, Z. k+ [0 j) {6 h, s  J: `* gfactorial(3)
      D; Q7 S" }/ N# [3 * factorial(2)
    . R1 L) b7 @2 ^) \) u* t5 h0 H3 * (2 * factorial(1))
    7 b) Q( T. f3 V7 r1 E4 j6 P6 Y3 * (2 * (1 * factorial(0)))
    - O' W' x4 l3 n( a) S" t3 * (2 * (1 * 1)) = 6
    : @; a& t! |' g" B) V# t
    : a' ?$ |6 h6 J5 r) e) @7 D 递归思维要点; i& G1 k2 I. [: h: L7 q
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑: X: F( M. z/ y4 d* A9 _0 l' w
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    1 o6 _  j9 f! m' z- G+ j3. **递推过程**:不断向下分解问题(递)( A5 \  c; g$ ?
    4. **回溯过程**:组合子问题结果返回(归)- K# j/ e2 H# z
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    注意事项1 X  b" q, M- I/ ]
    必须要有终止条件3 u( v  Z: t* r' }9 K2 j
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    1 `4 {4 K6 c# J* k: S! U4 f& I' I0 b: q某些问题用递归更直观(如树遍历),但效率可能不如迭代
    5 O$ f: Z: M. Q3 \$ l0 w尾递归优化可以提升效率(但Python不支持)/ ~& T* n0 K4 o1 M  @3 B* \" Z
    0 I) `  i' H- R8 r, a0 T* K2 t
    递归 vs 迭代
    5 d% N5 I0 B9 b3 v' {8 y  j|          | 递归                          | 迭代               |. d) S( N1 }+ Y4 R+ M( i. ?1 R5 A! w
    |----------|-----------------------------|------------------|
    % a5 g: W6 ~( ^| 实现方式    | 函数自调用                        | 循环结构            |  W3 }/ \: R: m6 u: P
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    6 L$ S7 k/ K2 |4 i5 p1 F| 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 o3 I  n/ j% G- M| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    6 o5 D; B# q" |8 [. b5 v& U) G4 ]
    经典递归应用场景
    6 s2 J4 _$ H! ]& P4 S  K6 ~  E1. 文件系统遍历(目录树结构)& u% H6 t1 ~4 H8 Z7 t
    2. 快速排序/归并排序算法
    , u* z+ e2 c% w4 P3. 汉诺塔问题
    & E* N7 Z* Z# f  U4. 二叉树遍历(前序/中序/后序)6 h% A% X4 S" p( A  Z8 H# x
    5. 生成所有可能的组合(回溯算法)& i5 I+ b) }! e2 W7 P  W
    0 y" C( A( n# V
    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 }( R6 K. ?3 K我推理机的核心算法应该是二叉树遍历的变种。
    & {3 T% `) {& t# E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:
    0 l  |4 H# @: W- z) j& ~! cKey Idea of Recursion
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    " r3 n" {* l, r) G9 V2 f) G* JA recursive function solves a problem by:: L2 d1 l7 x, j
    , B/ J6 {9 h: n0 _7 O9 N
        Breaking the problem into smaller instances of the same problem.
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    7 c/ m! Q+ b$ U) n) o6 J2 ^! Z    Solving the smallest instance directly (base case)., h( |* F1 p. L. v1 N& o) Y7 z# r

    0 @/ C6 ^9 P1 n  A0 c' {0 d5 ?    Combining the results of smaller instances to solve the larger problem.  m0 p3 J1 L# W5 t$ f  d3 {

    % {! D2 u* F8 kComponents of a Recursive Function: G/ S3 f4 [( S# a. k
    . B2 a4 ~6 L; h( b3 p
        Base Case:
    & q% N1 J8 p. @! s0 F0 h; z9 m  _8 J8 Y" p' ]8 K0 B7 C  G) N& t
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    . f3 s0 _: s* X, _
    . ]! F: F' {$ P8 N1 F7 f        It acts as the stopping condition to prevent infinite recursion.
    : g- C# Q0 h& x6 F& \- `" |8 `7 _1 y. G3 g
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( l! j% n4 `4 m

    1 g$ h0 V# Q' h1 {8 z    Recursive Case:# v9 K6 }& _0 c# K
    & N" x% ~( v4 ~
            This is where the function calls itself with a smaller or simpler version of the problem.
    . u, g* j3 r7 o' ]8 [7 k" M6 c1 T# d+ C. v
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 K1 Y+ _5 q4 G8 ]6 i' m

    7 x. I, |! Q4 V% |Example: Factorial Calculation
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    7 W/ K# _( Q. U  {- b  `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! h6 ?# ~3 a& F5 M$ W  Z) P# c4 I+ l* U* _( |& O4 |! G
        Base case: 0! = 1
    3 c2 d: F3 u7 d0 ?: x8 p. H& M& ^) `6 P0 R2 O5 F2 i4 I
        Recursive case: n! = n * (n-1)!* Q/ }# |" f) u9 E4 l8 F* @1 r
    : Y) P. e+ a. h; E7 u
    Here’s how it looks in code (Python):( j. ~7 U1 g. s* E3 h1 L( q7 q
    python9 C1 K* v7 n' _

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    7 K0 J/ _) l% `" j- y3 a* ^* udef factorial(n):
    8 ?3 Z" \/ a. r7 c: d' P9 G    # Base case: x. S* [& |' T4 X, v1 ?
        if n == 0:- r5 P* Z) q7 q- y
            return 1
    0 h9 J0 O2 J. a8 |# Q    # Recursive case# I: d4 h/ P. D  t: \# K% N- f' c! P
        else:5 K1 y4 W- k4 d2 d
            return n * factorial(n - 1)
    - ?, a( T5 {# U2 d8 H1 R6 c7 D2 F) d
    # Example usage
    , Q: k/ V8 q, u* q! xprint(factorial(5))  # Output: 120
    ! w9 t% n) z6 n) e, ^
    0 G0 o' b( J% E* ~# G; [+ WHow Recursion Works  [" T. r5 T. q) M) }
    9 q4 r3 L3 x9 ]' k- h- L/ ~: x
        The function keeps calling itself with smaller inputs until it reaches the base case.
    1 d  a% T# r# o4 _& A6 }8 w/ w2 A! r; v5 r* F0 j% }4 u
        Once the base case is reached, the function starts returning values back up the call stack.
    5 |% F& b& |+ e1 ]. t  |4 ?2 U- @5 v8 y! W' Y# J
        These returned values are combined to produce the final result.3 k& ?+ ^5 B  U9 o
      Z+ Y3 t* u9 {( ^
    For factorial(5):
    5 i. r% _( Y9 r; T. n+ u0 Z% h6 E8 G5 N' W9 r$ j; }
    % c. l' c, A7 f0 w  G. f) h. K
    factorial(5) = 5 * factorial(4)
    ' g  n, `1 u5 g) N5 B: H' bfactorial(4) = 4 * factorial(3)
    ' h( s' E, U$ u/ Dfactorial(3) = 3 * factorial(2)
    3 l6 t: D2 |! ufactorial(2) = 2 * factorial(1)* \& j- J1 }1 X! J" v
    factorial(1) = 1 * factorial(0)8 ]1 m- c: {! ?: }: t& @
    factorial(0) = 1  # Base case
    ' S+ H' ~( m& s9 V. T+ q8 Z& X  \5 f  V
    Then, the results are combined:
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    1 h) A8 H2 J  i& t  {* O& ^5 e9 b: {1 d. W" V9 d  v" W( ~: P0 P
    factorial(1) = 1 * 1 = 1
    / n( Y: Q1 K4 G6 Gfactorial(2) = 2 * 1 = 2
    ) `. [4 e1 X0 Sfactorial(3) = 3 * 2 = 6
    - d8 |& R3 C, q- s9 t; Pfactorial(4) = 4 * 6 = 24% F8 }$ C3 d" H: n3 g% Y9 Q! W- y
    factorial(5) = 5 * 24 = 120- `, S3 M: n4 j, W, j7 E

      [! D% @6 C& _- `+ A6 PAdvantages of Recursion
    , t/ Q/ P- ?: X) z; v( {/ X  D) E( |8 h: r) K, n% h9 k
        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).
    ; s! l& A! L& A, j& t; n6 F% H: M% V' @! k. t1 M& h2 P% u
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    0 w- R0 ?( \* y& T% x* b4 M, D, F/ ^0 ~2 y
    Disadvantages of Recursion
    7 v, x: t& E% @% Q  ]8 p2 {8 l6 F; q" L( L0 s0 c
        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." h$ b$ r" c+ {, Z+ E( g# N

    + t' a4 o( W! d9 m7 _    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    * H' e/ v/ S' B7 {& S' n& ~* [. ~3 {3 f6 X0 K8 @9 v% p. t, E
    When to Use Recursion0 T. Y  A# Z6 B2 M7 U

    / s. u5 G- @7 I* _/ f    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).) K0 K  G4 P7 r7 {) c8 g: z
    . U( c5 w+ E6 V% V4 F
        Problems with a clear base case and recursive case.. l. e. V7 }& b! F1 M3 b6 j

    , @6 h0 ?6 \1 GExample: Fibonacci Sequence
    0 B* q; ~% K- E& a0 c/ \0 W& ]( k9 q! g$ l% j
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    # b) ^' E# X/ Y8 q
    4 F- q; E$ I! W    Base case: fib(0) = 0, fib(1) = 10 p" x. b; q3 L( s' e$ M( e# D1 h

    7 M3 R6 `# h; [1 B    Recursive case: fib(n) = fib(n-1) + fib(n-2)1 C3 Z9 X# t7 g5 X( J5 v/ D/ U

    8 u. n- J8 D9 L3 @: }2 p# m& y  apython6 @) r9 q" B& J  [1 \

    * B  o3 q9 g& X: _/ d" v
    , r, i* m3 H# Sdef fibonacci(n):
    2 z7 m8 l* r, {3 t( S    # Base cases% ~, I) t+ {9 y% U) t
        if n == 0:  Y2 r+ o1 c, ^; o# D( S1 G
            return 0
    7 {- v1 e& M5 e    elif n == 1:
    7 D) _. w  A) i' r' p5 h! X5 Q        return 1
    % I/ i  s9 Q4 i3 k% x7 E    # Recursive case
    * o, E+ r) |1 I    else:
    , P5 x3 t' T) }  L1 h        return fibonacci(n - 1) + fibonacci(n - 2)3 Z9 H2 S  `" \  G7 N" x1 E+ {

    # Y* F9 W% }0 u5 Z, W# Example usage9 ?1 R) _" I* C
    print(fibonacci(6))  # Output: 8
    : P# D( ~# C. Q  i
    ) O% Q" k5 s5 \Tail Recursion! d( X8 _! a7 T$ N- Z2 X* J4 ~: A

    " B! Z0 |4 \" o# {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).
    7 W6 u0 }- o5 J' J8 Y7 z9 `) k9 M2 F- R1 b
    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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