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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
  • 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
    - l" h* u) ~2 o5 d+ Q2 r# A0 `3 m4 I
    解释的不错/ i/ N$ S0 ~7 K+ B4 }, t$ ^3 E

    / y# W5 {0 l5 Y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。
    5 g8 K6 F" Z& J) Q" R" p% a& Q4 r8 h6 S5 s$ ]2 n
    关键要素
    % v" Q' D0 b$ n* u5 _1. **基线条件(Base Case)**
    % X4 X) K  E& n2 X9 n8 P   - 递归终止的条件,防止无限循环8 M6 Q/ ^& r, v2 E
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 11 S3 f1 e+ d6 ~$ O# O
    ! D' x& n/ r  n
    2. **递归条件(Recursive Case)**  v8 k/ }5 O. O- n
       - 将原问题分解为更小的子问题: K* m$ g1 ~6 M2 A  b! D4 p. Q6 {  ?
       - 例如:n! = n × (n-1)!
    # z  J- K( ^  v6 C, l7 g) V
    + G- Q3 m7 C7 \9 Y$ `% _ 经典示例:计算阶乘
    ' Y: H# h( [2 _. u0 I: i, opython
    2 U8 O7 m" o+ jdef factorial(n):
    * i4 R8 {' z& k* s4 O    if n == 0:        # 基线条件
    ) j5 W- C* w5 [& o. i        return 1( n: `/ M: h) u
        else:             # 递归条件
    * I4 T  S+ B: }7 a  ^- r( y/ |! d        return n * factorial(n-1)/ l' q$ ?" \3 n  R9 m, R
    执行过程(以计算 3! 为例):
    ( I' }+ |2 B/ T+ |/ wfactorial(3)& N1 ]3 f2 F/ T& e; Y7 o" x
    3 * factorial(2)2 I: p- j" ^  N# q8 ^3 b
    3 * (2 * factorial(1))* A2 ^( D7 C7 o4 K( k1 z
    3 * (2 * (1 * factorial(0)))
    / p4 y' l6 C- I3 * (2 * (1 * 1)) = 6
    ' }7 p1 h# B7 V7 K; S* v9 B6 k
    9 N2 |: [6 b! a( y 递归思维要点
    : i( V$ |8 P. ]+ g$ [& B8 e1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    * p6 Z# _: u5 \' o/ ~* H2. **栈结构**:每次调用都会创建新的栈帧(内存空间)* V: q: A8 d0 X
    3. **递推过程**:不断向下分解问题(递)
    & e* L' c/ C) g  `4. **回溯过程**:组合子问题结果返回(归)
    ; n' u% X+ B# e5 \1 N
    4 Y+ P. B, F- P* Q 注意事项' Z9 |5 U* n2 ^3 Y
    必须要有终止条件
    / H9 ~- j# e6 w; W. {递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  {% \0 J  G, j) Y' u
    某些问题用递归更直观(如树遍历),但效率可能不如迭代( Q! z' |9 N1 c& Q
    尾递归优化可以提升效率(但Python不支持)3 M' R# l3 e0 b" x& G  @5 E; d; c# o
    4 ?! |6 N# [: k
    递归 vs 迭代( g. P0 p4 r  m9 m1 k5 c
    |          | 递归                          | 迭代               |( S, s3 W2 b3 K* x9 R9 ?
    |----------|-----------------------------|------------------|8 _1 k, Q  @' p& ^% |
    | 实现方式    | 函数自调用                        | 循环结构            |  ~9 Q! |$ P; }3 u; G: j
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |& D3 e! j( i3 I% r; Q
    | 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         |
    ; w, r" _! g. X( @# G4 G| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |/ |6 c% y9 f7 \4 R  Y- g

    " _) b8 q+ l4 H- v/ z 经典递归应用场景- }6 \- n( A" ^$ R) K, S. e3 j/ m
    1. 文件系统遍历(目录树结构)
    8 E! \4 Y+ h" c  y- @2. 快速排序/归并排序算法9 {: s6 j  P4 l; a* D; t3 O+ V4 u
    3. 汉诺塔问题  m  ]" H/ n# V8 |) E6 f
    4. 二叉树遍历(前序/中序/后序)
    6 v9 J5 a, `. S3 N9 G5. 生成所有可能的组合(回溯算法)
    , [) ^0 W3 A  v% E: R
    ' v# `& j: _5 H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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    昨天 12:17
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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,. S) M- S4 l* Z  ^1 ^
    我推理机的核心算法应该是二叉树遍历的变种。
    5 q: `$ S& B+ S" e5 H! o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:! L+ f& V4 h5 I. i! x/ b+ y
    Key Idea of Recursion& B& c- N+ Y& l' s& Y

    . ], l( L: {+ v: IA recursive function solves a problem by:7 K% Y+ j& f( L: \+ h2 P+ M9 p# w

    % a% `' r4 `/ c  x    Breaking the problem into smaller instances of the same problem.0 l5 F2 H! {: T( w4 ^, K' T

    ! _6 [3 ?7 ]3 \9 }    Solving the smallest instance directly (base case)." W$ a$ B# q5 j, {7 P, O

    . K! q/ P1 X) E8 c    Combining the results of smaller instances to solve the larger problem.4 F4 E, \; n/ _

    # _+ T; p6 P' N. X6 lComponents of a Recursive Function
    + t2 G$ N. x: L" n/ K
    2 c! d- t9 n* w1 g* O    Base Case:
    - ~3 ]' E  _7 [& A/ M
    4 }$ L' H) o5 _) o        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    / ?/ M. [' i% ~0 R: g9 F0 z
    7 h, s1 s$ v7 B/ L' N2 R9 r, ^: M$ L6 g        It acts as the stopping condition to prevent infinite recursion.' `0 `, Z6 C9 a  {5 W

    % R& L9 A* K# B. E6 V        Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ {. o# i+ B/ |) @4 N5 Y( r- F4 S
    3 C% f" n1 O, Q1 M6 x6 ^# U
        Recursive Case:  R% C8 ?2 \# j

    ( ^! g8 T7 f9 t0 q- w) y        This is where the function calls itself with a smaller or simpler version of the problem.
    / U! O; i0 {9 [* W# [4 ]$ L& i2 Y% o" y% Z
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    & m1 G; A/ }2 }  y7 n! G2 {; Q1 f9 D0 {$ O0 C
    Example: Factorial Calculation
    " Q5 `6 E( ]: `4 l+ D0 S, N
    ' m+ s: ^9 [( ?6 Y1 q# Y7 BThe 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:
    , v$ _! \! }" ]: Z0 y7 a; Y
    + P8 }  g: q/ A- O" D/ Z    Base case: 0! = 1
    7 R3 L( d+ S# U, D+ y) ~3 @  E  r* S& C! W% W# O/ V0 _
        Recursive case: n! = n * (n-1)!
    2 V# r& H+ g% o* {$ o8 Q
    & z& f+ u2 e' sHere’s how it looks in code (Python):
    * W; p: T1 u) X5 M, u4 }" [python! w5 L7 C# f) j" j2 N

    / D. n" Z1 H( Z7 s2 I
    7 ^6 E; p. B+ j. e! {6 tdef factorial(n):7 \$ T; W7 B  O. z% I
        # Base case( M; K2 W$ ]# k! c. r5 P  M8 X
        if n == 0:, p! M0 f5 Y9 Y; L
            return 11 K& P, ?$ F: ?- q
        # Recursive case) @: Q- H2 ^- d$ I) E1 _# _
        else:
    ; g/ L7 I0 J; ^* q        return n * factorial(n - 1)5 J; ?: V6 v$ |7 `4 c0 X

    / ?$ n# n7 a* L# N1 w) C1 T8 i# Example usage8 r/ c! Q; @2 @
    print(factorial(5))  # Output: 120: @/ N# m9 ]* X: }# A' g
    8 u0 G3 o% R/ z( Y( ~' N
    How Recursion Works: Q8 A5 k  [0 l3 y; C3 }
    , g+ C$ t# Z6 k( B
        The function keeps calling itself with smaller inputs until it reaches the base case.1 b1 z' U' q3 K# x0 T

    ( L( ~! U; F6 W  m; D6 B    Once the base case is reached, the function starts returning values back up the call stack.5 k* C  {/ |! g  f$ m1 L" Y

    6 W3 N& x4 f. {2 K: C, H" f+ [    These returned values are combined to produce the final result.
    % u0 _; B# H' M! O% V/ _6 Q* M1 U/ o2 Z+ O% y$ a% P+ o9 f8 |
    For factorial(5):+ _# z% V+ W* v1 B0 ?: b
    ; f% A6 @7 i4 {' p# Q0 I

    5 Z3 K% Z* y/ ~" v' Tfactorial(5) = 5 * factorial(4)
    4 g5 J2 @! R* o  Y* Cfactorial(4) = 4 * factorial(3)
    7 ^5 g0 J8 n; M- x' x, q7 zfactorial(3) = 3 * factorial(2)* r: E, P3 g- E! Z$ |% @* ?3 _
    factorial(2) = 2 * factorial(1)
    + s3 a: h0 q+ p" s- Q/ Ufactorial(1) = 1 * factorial(0)' f3 F: n, Y6 Y2 w1 _' m0 [
    factorial(0) = 1  # Base case/ D, r* `7 B9 S& L$ K4 ]% N

    $ R5 g) h" Y0 A7 NThen, the results are combined:6 ~) q6 R( {- J' U) s
    3 k! ]6 e0 D1 w! W0 ~7 o

    , B" S# ?/ K7 K5 @8 i1 Nfactorial(1) = 1 * 1 = 1
    - T3 Q, g- N- Jfactorial(2) = 2 * 1 = 2
    3 _5 l  J/ W* c: `- y# `8 U  ^factorial(3) = 3 * 2 = 6
    + s9 W; o# t) R! ?+ V3 n0 Ofactorial(4) = 4 * 6 = 24: G: c; u1 c6 T2 s- B
    factorial(5) = 5 * 24 = 120
    0 _5 ^2 V6 _3 g4 O$ B9 y% p1 f) \2 P. |: c9 J* l* ^4 Q  z
    Advantages of Recursion/ {3 |1 m2 T( I  N  @; `- C0 t- y
    / [$ p4 |" t" _' t% n  ^0 z$ s6 y; V
        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& K( h# O1 _; f" `. e
    # d" @) s9 w6 Z7 S, d8 t    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    8 P. }& l+ F! Q6 c. Q8 t5 R; \; S# n' J+ {
    Disadvantages of Recursion$ N# d* G, I4 G4 z% K2 B/ ?% G/ L
    $ k2 g. u6 X7 R9 r+ P) U/ g
        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.
    7 j0 n4 h0 Z: q3 o. j2 E" V2 s7 s/ {: Y* i0 z6 f
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    - e, F( A% `5 H% {4 F) N
    & h8 B; P; [0 A& DWhen to Use Recursion
    , D5 o0 f, ?* C+ i- V% S
    6 j4 p  Q/ z4 Q+ {4 v8 f) A& x    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)., l9 v. [% J) P0 r

    ! W, `" A/ i  S) h1 D( `" |4 A0 S    Problems with a clear base case and recursive case.
    ! r5 u& N- ?4 H* I+ k( e) T
    + s7 z7 z9 {! p# \& UExample: Fibonacci Sequence' d. c0 Z3 w8 }( A/ _8 a$ K6 _$ F

    9 L2 o/ D7 w# ~  X  b: V+ A5 wThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 A1 X+ F  B: B$ m! g' [$ O

    / j4 p. @' f* {7 M    Base case: fib(0) = 0, fib(1) = 1
    5 Y$ T( E4 W0 x* f1 A- Z. f! t/ A/ `! w  C& L& {1 _  j
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    $ K1 u7 t, F( g4 p' `
    4 @1 H; V) [( L4 i0 u4 s7 \. i* npython! \1 J) t7 c3 D' j7 O9 G

      d5 }. ~9 L3 B" H9 E& [' q/ T" I# R# Y" l% d9 T: m' o8 h
    def fibonacci(n):
    . C2 `! J. o% f6 T3 x- c    # Base cases  `6 F: h& u" T; d) a& @
        if n == 0:; N) g/ `! b+ g+ W7 q* A
            return 01 B4 x' c; ^6 J; q( }
        elif n == 1:4 T* k9 W- K' _2 s
            return 1
    ) S# f1 j; k" ~7 [    # Recursive case
    # N9 a. o% {; r( \. H    else:* E! A" p1 J7 m* W4 G
            return fibonacci(n - 1) + fibonacci(n - 2)
    3 |, T8 q# i- \8 f- Z' k
    # o# J% v+ y6 D+ M# Example usage
    0 M* t- r) z" sprint(fibonacci(6))  # Output: 8$ M0 y, f1 [% x0 [2 C- a

    & |) G' r2 @. T0 }Tail Recursion
    4 R  n0 F: V3 u4 G
    $ |, i/ @7 ]) Z/ K. e4 t. uTail 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).
    6 ~- n' N- |: l* s7 u' K6 M/ @4 ]. ~5 i, x. e0 d- j- r# ]" U
    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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