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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
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    解释的不错
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    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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    3 K- e2 r1 f; p- v& t, C 关键要素, X" a+ E( f9 X/ k  f
    1. **基线条件(Base Case)**
    ( K% v4 X- a9 \& _0 \: e2 g   - 递归终止的条件,防止无限循环& q! o- U5 G1 I! q& H
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    2. **递归条件(Recursive Case)**; `( X# d1 H- S0 w
       - 将原问题分解为更小的子问题
    " A$ }; E3 j7 s- Z, _$ M2 k   - 例如:n! = n × (n-1)!, A* b! B9 F! U
    - w0 _2 [, V2 r6 I. j# Q& O7 ^" X& Y
    经典示例:计算阶乘! I$ F5 L2 t7 @0 Q6 X2 D: ~
    python/ X  `* j: T7 I0 r! R
    def factorial(n):& `8 u/ n3 ~, j6 @$ [7 C2 b
        if n == 0:        # 基线条件+ P; \- D& t+ n# l! A* L
            return 1
    : r7 T3 [0 _; i: D! I1 |" P    else:             # 递归条件0 t, b6 w4 c: ~
            return n * factorial(n-1): R# M/ X/ j' p' X( A
    执行过程(以计算 3! 为例):) \5 o; T  f# f7 u, Q; {( E9 k% e
    factorial(3)
    9 t; U- U6 u7 M0 n5 i( z" G3 * factorial(2)( L7 H/ e' X' N- Q8 @9 x% u/ r
    3 * (2 * factorial(1))
      }9 w* s+ s4 H% f3 * (2 * (1 * factorial(0)))
    % K7 i9 L" u/ W# d1 X3 * (2 * (1 * 1)) = 6
    9 \8 h% ]0 s, v# ?; i) @* y( }1 w( X  B. n: w
    递归思维要点# U- ^& {% v7 C
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 U2 @) L+ E/ ?7 |, u# v
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)$ }, V9 x8 p# ~+ \* C  }. _
    3. **递推过程**:不断向下分解问题(递), b2 O9 M$ i( g- e2 U1 v( p2 @
    4. **回溯过程**:组合子问题结果返回(归). W/ q6 O0 y7 H0 p4 K& N6 ?
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    注意事项
    ) R: ~* K* }+ h9 U必须要有终止条件
    6 z2 P" N  r* w. G, k递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ r* G0 u7 I9 [: [/ k: r/ ?
    某些问题用递归更直观(如树遍历),但效率可能不如迭代: b& ]/ n+ \% W) ^5 Y1 l
    尾递归优化可以提升效率(但Python不支持)% C& M' S( ?1 f  J: l( k5 f; x
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    递归 vs 迭代" n' F8 K( p1 p  Z! q
    |          | 递归                          | 迭代               |8 N+ E) W+ m7 a" ], {( M. ~
    |----------|-----------------------------|------------------|
    9 G$ g: ]5 B3 o5 p' f0 s. W| 实现方式    | 函数自调用                        | 循环结构            |
    / y* I4 [' C( K" w$ ^0 _" b7 Q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |  A8 _/ I  U  G9 H0 e. L$ X6 i  r& C
    | 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         |
    7 ~7 x  P1 {7 ]# R| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |! Q. `0 x' `/ e$ I# z  e

    - D! F7 f$ o3 p" a) { 经典递归应用场景6 g! ^$ ^8 M; V( t
    1. 文件系统遍历(目录树结构)
    ' v9 \" A+ T) Q, D, n2. 快速排序/归并排序算法
    - U. f" y4 O9 ~. B+ o# U3 a3. 汉诺塔问题8 |+ }4 P+ z9 i$ Y" W: r
    4. 二叉树遍历(前序/中序/后序)
    ' G4 b$ f9 Q6 j* ^! h$ d% x' E5. 生成所有可能的组合(回溯算法)
    ; d* R7 @+ G4 k/ z1 f  I
    ; ~& n5 ^7 x# U5 T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,5 d+ Q# ~9 V4 b
    我推理机的核心算法应该是二叉树遍历的变种。% y  m: l! f2 n) c3 c
    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:
    : q8 u# Z; e, a0 F9 o0 j  q, S$ x5 H) \$ NKey Idea of Recursion
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    7 F6 T7 B9 `8 c; j  s' r( X9 w2 AA recursive function solves a problem by:3 d' C) |; @( u; r: p) i# T: b; m
    7 O3 ~% }' H3 w4 O
        Breaking the problem into smaller instances of the same problem.
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        Solving the smallest instance directly (base case).
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        Combining the results of smaller instances to solve the larger problem.
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    3 w' Q5 i: z) s, w! x7 _Components of a Recursive Function+ y2 B% b; n4 \+ M$ d

    7 `1 |' f" u" X  [    Base Case:2 K$ C2 \) L/ T9 M0 t
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.- y- v0 o" J& c8 h

    / x8 |) I! I+ C4 l9 e        It acts as the stopping condition to prevent infinite recursion.
    8 ?- n+ r7 s; D
    9 f0 z9 X% i+ u# e2 }        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( H& {/ i7 |& N3 @6 t, s# R& f  M8 Y/ ?

    1 t( `2 G! @' ~! d1 I1 E# F# t/ W    Recursive Case:
    : F% F6 O* {. e/ E% u% g) T- ?
    , Y6 D/ Y6 @* P  T        This is where the function calls itself with a smaller or simpler version of the problem.
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            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    0 q. P0 f5 V# h% r0 lExample: Factorial Calculation1 b! J! h. }/ j# S

    6 ~: S/ k' g$ H+ I$ x0 CThe 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:
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    + j* c2 V" [  }7 K# J% u: M    Base case: 0! = 1/ n" `5 F3 g" @/ q5 F" l2 Z
    2 p+ W" ^0 k5 _. U: i1 h$ i
        Recursive case: n! = n * (n-1)!; a: m9 R2 D4 A4 _) k
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    Here’s how it looks in code (Python):
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    " u- f- P; E- h* v+ [- b

    . I$ e& v  F1 J  }- Hdef factorial(n):
    $ t8 j2 v6 v4 s& K5 x    # Base case' y8 }- w; U7 o5 I( d; V3 K5 m
        if n == 0:
    # R( Z! K  D3 [" t        return 1$ w6 T# A& P: j1 c. `
        # Recursive case% T' Z- n, \0 e( T
        else:$ M: o5 A% O& @! q
            return n * factorial(n - 1)
    8 E. `( E# K( _0 U
    # y. P5 A7 c4 q& H# y# Example usage/ B9 ?0 _  `  r! Q, B. s
    print(factorial(5))  # Output: 120
    1 r# t* G/ D- D( L2 B
    + Y; i& t" b3 i* F: YHow Recursion Works
    , M4 j  a, f( V7 F. Z1 Q( G/ p: V
    / ^7 i$ Z+ r3 y9 q+ f. G( n  W* b    The function keeps calling itself with smaller inputs until it reaches the base case.
    / S' L0 U5 [& |& [
    3 T) d0 u$ }: E, J7 I    Once the base case is reached, the function starts returning values back up the call stack.6 p6 s! _1 z  K. v& }

    , |+ Q: P) i  s) W5 i& U    These returned values are combined to produce the final result.4 \* E. U: j8 S0 P4 n2 m5 V

      {$ _8 }3 _" e7 S# j) i3 M1 n" hFor factorial(5):
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    + x: k. V& j' H' [: Z' z* O8 s# }: y% _7 F( f- t4 x$ N' _1 [
    factorial(5) = 5 * factorial(4)
    $ n4 D9 D. V& P- m: [% Hfactorial(4) = 4 * factorial(3)
    + F7 p! q: d3 }$ Z2 Ofactorial(3) = 3 * factorial(2)
      ~, Z' F4 P/ wfactorial(2) = 2 * factorial(1)
    , N7 @5 Y5 }0 p/ `/ V" Afactorial(1) = 1 * factorial(0)
    0 }! G/ [+ u0 E, H; V9 |factorial(0) = 1  # Base case
    2 o2 g7 M, [, Y6 R5 z: p- E0 a8 ^1 i" O5 u- Z
    Then, the results are combined:1 n- R" A& k' p, o) K
    5 ~9 H) N% ]' T# P

    9 p+ E) B% U9 P  @: P* jfactorial(1) = 1 * 1 = 1
    9 z3 t5 m4 k# ~( _6 F6 Cfactorial(2) = 2 * 1 = 28 T3 }- A4 k: A4 e7 V
    factorial(3) = 3 * 2 = 6
    & @  S" b$ `# t1 Dfactorial(4) = 4 * 6 = 24
    2 o) g6 s) K. ^( M5 w4 `factorial(5) = 5 * 24 = 120
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    Advantages of Recursion* f$ _/ ]+ E3 \; S
    + P+ o, d) H1 t& m  ]
        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).
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    1 T7 {. ^! ]( \$ ^2 y* b, f    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion
    ) ^7 Y# U- A5 l) R5 ?0 y3 r& @# i. r) w/ D% R
        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.
    ! ]# M1 O5 V6 D" h, t% z# w# U1 c, Z* Y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    1 R8 r7 I' t6 K2 f8 \. j
    & H4 s, s2 J: d# V* F; T4 E# }When to Use Recursion
    . D8 s6 a) g. j% E- r
    7 R9 w6 j9 J' y& `2 Z9 x; h/ u7 d    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    1 \& d9 W0 l- W% ^$ }% y6 x
    . G  M, ~7 Q, \$ t/ i- _1 ?    Problems with a clear base case and recursive case.
    & M! i% o0 L" [, z% v# B  q( h1 K
    " y3 B- X4 v, {  s3 t) x& NExample: Fibonacci Sequence
    3 |+ U$ L- Q/ m6 @
    - v8 v9 P; R( v9 \! U: N8 o% EThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    3 @/ [7 k  {) |% |* \! P/ p7 P& l2 P
        Base case: fib(0) = 0, fib(1) = 1) a0 i$ Y1 K# O5 h9 n
    4 I9 S* G) a* Y8 F2 m# ~
        Recursive case: fib(n) = fib(n-1) + fib(n-2)+ y; Y" U7 C. S$ B6 y  U0 `
    ( f3 J! c7 ^' f+ ~/ P% S# O
    python
    , T, n) Y3 D$ x& x2 K' S8 K- H; x0 n% T* {* ]4 r

    9 ~( J1 L% N) W+ T5 T% i( Idef fibonacci(n):
    : R  [) N5 @: m1 O( J+ _; Y) g    # Base cases0 y) U% h+ c' E: u) A/ t
        if n == 0:
    + z, o. Z- Z9 v6 [0 j* d        return 06 |$ X9 ~% B3 j! F+ I" ^7 M
        elif n == 1:$ Y; t7 a  J0 W% }) X( P
            return 1
    " I& {4 c! Y& N- P' I    # Recursive case( n$ a2 Q  x8 u& _
        else:; U$ g/ Y. j9 _
            return fibonacci(n - 1) + fibonacci(n - 2)+ C5 q# i0 n  }$ q
    7 X3 Q) @+ R$ U$ c& j
    # Example usage
    4 Q, Y9 A: A& K% f8 x; n8 iprint(fibonacci(6))  # Output: 8
    0 T: S( [& [* z& ~! ]% I$ K: x9 g, G# g6 f3 `( m
    Tail Recursion
    + m$ D7 M9 K- \0 M( v' u5 a! B  O' h% b. j# s- V. O3 P
    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).% n+ B$ y8 ~8 x: x: p

      A% |. F! B" K. O  q: G6 sIn 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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