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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
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    楼主
     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
    ( I6 U; L2 |* {5 Z
    6 F- r1 A( @7 {0 T" B7 S' n解释的不错- A  ^2 O9 a; r8 l- r- {
    + s+ l( P' [) X
    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。0 ^' i# q' l& ]0 x  G" f/ s

    , L1 a" J8 N& [6 u! F 关键要素
    ; X  y" I' Q8 f6 m! r* S5 h; G1. **基线条件(Base Case)**
    6 D  Q) K( g# L: Y   - 递归终止的条件,防止无限循环
    1 M7 S: ?8 d2 k" I   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    5 K6 T9 j% Z9 ?: [7 }8 W4 z8 m2 S& U- h8 U
    2. **递归条件(Recursive Case)**" Z" E, c8 q6 j% l- F
       - 将原问题分解为更小的子问题0 ^- g# c% L0 Q# x6 u
       - 例如:n! = n × (n-1)!
    8 A  z  i8 d6 K% N3 Z4 w; o" Y, Y/ f7 o1 g% G9 A. v
    经典示例:计算阶乘
    8 X6 j+ `7 W( r) s/ cpython
    ; \  V7 L% ^# H5 \0 ndef factorial(n):6 A$ X8 E) l( g# I
        if n == 0:        # 基线条件
    7 _/ E: \, K3 y% s2 r( h        return 1
    # r  k) T) O8 h6 N+ G! O    else:             # 递归条件; Y3 v. L  r( L. v
            return n * factorial(n-1)
    7 K0 w2 l0 Q8 c" H# r  `6 c5 t执行过程(以计算 3! 为例):& u% A- G: H6 o$ @
    factorial(3)
    % D: l3 ~  u+ l0 L' Z: s  f! Y3 * factorial(2)
    0 s9 W, o- L& W& b3 * (2 * factorial(1))+ A! f3 e" q! y* q+ C# ~6 O
    3 * (2 * (1 * factorial(0)))
    + U; N( T! g& n" \2 k4 y! ]2 Q3 * (2 * (1 * 1)) = 6
    9 Y% V7 x2 b9 \! o9 I  D1 ^9 e2 E; e5 K2 w( q  p/ p* _: n3 }
    递归思维要点
    & t4 k! N, U) x, A! H1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    8 ^. i9 Y) S4 C2. **栈结构**:每次调用都会创建新的栈帧(内存空间). v7 Z: M/ B9 C3 R& w
    3. **递推过程**:不断向下分解问题(递)9 N6 d$ `9 U# [' }( w2 g
    4. **回溯过程**:组合子问题结果返回(归)9 `: @0 A/ I% ]7 x2 Z4 Y

    . p* S7 u2 S. ?' e3 O  {0 k& r" I2 R 注意事项4 `5 {* q* ?% j3 v1 H
    必须要有终止条件3 U; S# R! }3 c
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 @. y* `, U* s* [0 a  a
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ' C5 U* g0 R4 R/ f1 U3 \尾递归优化可以提升效率(但Python不支持)1 s% o: ~" d  _! y5 \+ q; }" C
    / q) h, Z: k: o6 p5 }
    递归 vs 迭代/ j" \: T: e* h7 G% H) C+ o+ L
    |          | 递归                          | 迭代               |! D. e( q2 I" g& u
    |----------|-----------------------------|------------------|
    3 q) X& j6 C0 I  ^7 a| 实现方式    | 函数自调用                        | 循环结构            |- @# e1 a; F! {. E6 _  C
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ) o2 Y$ ~2 N% b( U2 }2 C$ a| 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         |
    / c( q. l3 t. c| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |1 T5 G- L* c/ g2 x1 |* M2 }

    : v9 e4 q# O  N" P8 ]! R 经典递归应用场景
    0 v& T" P2 q' k" V1. 文件系统遍历(目录树结构), C9 `) l# r; X: z1 l0 n; t
    2. 快速排序/归并排序算法* {5 ?  o" l& M1 A6 L+ g% B' ]
    3. 汉诺塔问题
    - p- R; v1 }/ ?( G6 g4. 二叉树遍历(前序/中序/后序)7 j' ^$ x* M% s7 b. W
    5. 生成所有可能的组合(回溯算法), Z' }( s; v3 j

    - E4 A. k- b; s# {' p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,* d' Q3 J- m5 V4 g4 ]: B$ e$ B
    我推理机的核心算法应该是二叉树遍历的变种。
    2 ]5 y1 K9 p2 ]* p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:
    7 n# S6 k& o/ w% u1 VKey Idea of Recursion* z: S# E$ e4 e8 b! e# y

    8 a+ S. k$ H1 M1 o  wA recursive function solves a problem by:
    / F6 G; p- g  U' W2 p- T- P$ l6 z9 }& `3 M; i5 T$ Y
        Breaking the problem into smaller instances of the same problem.+ f$ ~" f* _5 @# @5 _

    & {! P3 d$ e* _8 W# r4 u    Solving the smallest instance directly (base case).' [% P3 v- H( H' y/ Q

    ) i! I! h$ Q( g    Combining the results of smaller instances to solve the larger problem.
    2 R! G8 m1 I' `0 @0 [3 q1 v( P$ d7 r1 X% ~& ~6 f( f7 N
    Components of a Recursive Function+ z# b( n, A) m* S
    9 v% F" o0 a" Y
        Base Case:8 Y; |2 j/ a% x3 ?7 q; Z+ I& e
    + V) ~0 b: d6 c+ |5 c- N
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    6 p5 F6 b7 l% M, ~" F  x# b" s& d4 J- [  d
            It acts as the stopping condition to prevent infinite recursion.
    ( e8 j4 G9 R* z" F- p$ Z& z! q; G, ]( o
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    5 W( R0 P# @; m% H  X7 A: ~
    - j- D4 ~' e2 ]2 @    Recursive Case:; \+ i0 b1 P6 g1 j0 e

    3 t5 @5 Y& S' q+ l        This is where the function calls itself with a smaller or simpler version of the problem.
    - o+ ^" B1 Y) Y% Z, B% q
    * E3 h! }5 F3 ], Q3 R$ H1 \        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 t; x9 ~/ @+ G6 @7 Q" m; C; G
    6 f' n3 m+ ?5 w) K7 @  o
    Example: Factorial Calculation/ V; S' T! G4 H: f' i. n
    " Q  x* r* \4 }$ |  E# ~
    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:
    1 f; }& x/ Q# H* H7 U7 \
    9 _( ?5 {/ x9 D1 z$ P    Base case: 0! = 1
    & x+ U8 X6 e9 s! @, J4 l# i) }; U$ C2 c1 X$ [& O9 z0 s/ W/ `. v
        Recursive case: n! = n * (n-1)!
    - h' N6 T- g; A- Y8 S$ {! z4 \: l6 h4 K
    Here’s how it looks in code (Python):
    ! @- M3 Q5 `$ [3 x1 x7 r- ipython
    / g( K- o7 E. u; X2 Q' D/ Q3 ~  h5 p9 u5 R" S

    8 S( ?5 M5 ^* D3 [/ V* ?" gdef factorial(n):1 l& M: _6 t, G8 m6 w* `/ ?
        # Base case3 t1 G3 s% B" v/ X1 A6 j8 N
        if n == 0:
    $ x: I/ n: Z& C" r        return 1
    ! x4 n6 Q) W( D3 u$ B. z0 w; y; |    # Recursive case
    3 m( a; }8 h  I6 t% {    else:9 {/ g  J5 X% e7 m  q! [. G
            return n * factorial(n - 1)
    5 w' J+ B/ G; J$ q/ [3 v1 s' A4 _" v5 H4 e9 v" a) Q/ b6 Y$ O. a
    # Example usage
    " I7 Z  @4 M3 U" x( D4 [" aprint(factorial(5))  # Output: 120
    " H+ F3 {9 N" \4 l6 A0 Y8 A$ F. w
    * j; S5 O# t; r$ p5 A) D7 w4 kHow Recursion Works3 ?& u- I% _7 T9 R* _; U  I- G

    7 V/ e& I2 o, _7 |    The function keeps calling itself with smaller inputs until it reaches the base case.
    $ b0 L; t5 j+ {: K& u$ z7 `( `3 N: y' p* F+ J0 P
        Once the base case is reached, the function starts returning values back up the call stack.3 K6 L) }( Y% q7 a' V" o5 e) B& m

      O! X+ s" n. t2 l% y( y' u    These returned values are combined to produce the final result.! d) }+ P1 c2 c3 v
    2 M1 E4 x% u( ]) ~
    For factorial(5):
    & D$ l5 _; Y' d1 b7 a- c  U, B6 [% _0 R3 |4 Z

    ) S- {$ i  Z. i. |% \4 vfactorial(5) = 5 * factorial(4)
    + t+ l7 K' ?/ o+ afactorial(4) = 4 * factorial(3)' g' t5 o  M% r0 r0 p# [
    factorial(3) = 3 * factorial(2)2 w* n4 C- w' N" j2 x+ v
    factorial(2) = 2 * factorial(1)  ]( Y* z0 D: M7 h, d1 X
    factorial(1) = 1 * factorial(0)
    $ _" M" D2 f: O; X# c# z! `factorial(0) = 1  # Base case4 {* A$ d' ~5 F4 v( f3 w5 _

    + r- K8 Q/ J3 W. [8 A$ EThen, the results are combined:' f* [' T" C* \" Z9 E* I

    & `4 i; w! Y( w8 H: d2 V5 X8 z6 s: s; E0 L" e& H3 c* l+ t
    factorial(1) = 1 * 1 = 1
    3 d7 B: x% T1 b# p( rfactorial(2) = 2 * 1 = 2
    - x# w  [) v9 `factorial(3) = 3 * 2 = 6
    * i& V6 x; e7 L+ D2 C4 P# Pfactorial(4) = 4 * 6 = 24
    * K- t8 j- k' Q% v* H  |3 Zfactorial(5) = 5 * 24 = 1203 a: ]+ }% y1 s

    9 u& F. Z7 }3 A" V/ ~Advantages of Recursion0 `/ N) c# v' D0 Z$ v- @) w' S$ h

    ) v; C8 m" l' ^6 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).% X9 n% E7 Z* O& H: q" {
    5 Q( a, R& C' X  b6 p6 F# Z
        Readability: Recursive code can be more readable and concise compared to iterative solutions.3 Q0 V% N- D: Z5 r( b/ K" n

    4 o0 c6 n. J) |' LDisadvantages of Recursion1 @" L5 @6 t& `3 [

    ) v( L$ S, ~1 \- ^: l# v    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.
    . B* ]: R# ~, b4 i& E% |9 A! H" u7 g, r4 }% _: T
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    3 z: ^: @9 T& P+ z/ P: N* O* @2 Y
    ( o6 [" M, U( JWhen to Use Recursion
    ) I, X6 z1 Y; J7 O7 t
    " x7 R% K' U+ Y2 c% c    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ; ]8 l0 Q! G- [" s
      _  `3 y1 X- D* H. A+ X    Problems with a clear base case and recursive case.
    % n8 \% B) u* O. r* t2 d
    3 G; n4 ?5 \) `9 L) uExample: Fibonacci Sequence0 y& Q6 `- ^0 t8 E

    $ x1 D6 {( K: n* F) qThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
      K# t* f& M7 a, o# [( U1 I, f* _9 D- P) s1 I) Y2 _- V0 R
        Base case: fib(0) = 0, fib(1) = 1
    / i$ V2 ^& V! J( X+ n1 D9 L9 t5 i( D+ X6 o  P
        Recursive case: fib(n) = fib(n-1) + fib(n-2)2 n& R2 Z' p, d6 ~& R& `) x3 P, b
    8 k9 E# R# l( H; C
    python: I+ i7 Y& v8 l) M: }
      N9 B$ r8 W6 D, S/ J" P- n
    ! L8 h, V7 m2 r' ~7 h, k
    def fibonacci(n):
    . R( I% l% F5 U3 x) N    # Base cases
    3 v0 t5 l7 _* u8 m8 v    if n == 0:9 M' N. J; {8 @% E8 B  ?: k: V7 }& u( J
            return 0  L' \$ A& M6 L* S3 L. `6 Z: a, R) L2 W
        elif n == 1:
    4 |  N0 Z9 u! S        return 1# \. \% v1 L8 q* g
        # Recursive case
    5 H3 u+ }) l% s2 P8 ~  z    else:
    3 X0 s$ T4 w1 I' o3 N/ N. Y        return fibonacci(n - 1) + fibonacci(n - 2)
    ) i  E) x9 i$ \7 D6 Y1 {7 H& }7 m) A, {
    # Example usage2 j) p- Q8 S& l) ]
    print(fibonacci(6))  # Output: 8
    , ?6 p* g7 ^; P' v7 E8 y
    7 u7 O4 ~( t8 G' K5 ^" M% zTail Recursion
    5 u1 x" B% _. l* z) y8 ~, `0 H+ @& F5 l& l( W( {# o5 w7 Q1 M
    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).
    0 J: {& P  \2 O) }) I( i% R2 |
    4 _% T/ |" d- `. o- ?- L5 K% aIn 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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