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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
    + V' b. W: S: {5 [" H: L
    ; I9 w  m# A1 p% q8 N7 C9 d9 b解释的不错
    ) ?* o- K: X% }. I+ {! H. F
    7 v" d2 q! S  `/ [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。
    - e# B9 V3 x0 ]# r; Y0 c) c3 J% e/ q* U5 C
    关键要素" e& I% v6 ?# b* V( [$ }
    1. **基线条件(Base Case)**
    ( ^1 C5 ~1 x, o" T( f, b( w: l7 d# x   - 递归终止的条件,防止无限循环
    2 I/ \+ i9 {' A8 }   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    6 s) A- w: b5 V* R* U  m) J2 ]. x/ z& F' m
    2. **递归条件(Recursive Case)**) r3 j8 g2 V1 Q: d7 g
       - 将原问题分解为更小的子问题
    + P- h+ ?9 J: B) x4 Q   - 例如:n! = n × (n-1)!
    0 R# k) N4 Y9 X9 n& k8 X& G) k5 K; ]! g, L, {; y- H' f( |
    经典示例:计算阶乘% M2 o- S  J/ f
    python  O6 Y8 h% o3 J! L4 a9 P
    def factorial(n):5 ~: q0 }! q0 T8 b9 @1 p
        if n == 0:        # 基线条件
    9 Q5 k2 G5 Z! a. J- i        return 1$ G: F; e/ b. j0 {# ~
        else:             # 递归条件
    ( S9 x- R& t( a$ |        return n * factorial(n-1)- {7 K& h3 x& ]! H* l
    执行过程(以计算 3! 为例):: v" a# x6 _" B/ y, u' E
    factorial(3)
    1 @! E( \. I' Q. E/ }6 }3 * factorial(2)& h, r  P5 D; m; g3 S6 B, d
    3 * (2 * factorial(1))
    4 v' p4 ]& F" \, A0 b: F  b3 * (2 * (1 * factorial(0)))
    % G; Q/ d( {& o/ R3 * (2 * (1 * 1)) = 6
    * m! M* J+ U0 d$ y$ B" R4 K8 \1 \0 J9 r' M* e
    递归思维要点% Q5 W6 _) ^& P6 p* H
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
      e( E5 v6 l& C2. **栈结构**:每次调用都会创建新的栈帧(内存空间)" ?% A7 l. X" A" r2 U/ V
    3. **递推过程**:不断向下分解问题(递)
    + |% E% m% k. w$ g5 t7 w4. **回溯过程**:组合子问题结果返回(归)
    " G1 \' X6 ]0 \8 J, C# F% s  _+ y& X3 P; H0 k4 @
    注意事项
    " j+ d/ V: l. P) p必须要有终止条件/ E0 e3 X6 m! h8 k
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    6 g; t0 e6 W- w4 k/ P) a某些问题用递归更直观(如树遍历),但效率可能不如迭代" J: X* D4 z  U* G/ Z
    尾递归优化可以提升效率(但Python不支持)& A  N( \) ^9 ?) B9 O. P$ u

    ! \9 u, m2 }5 v! D. e 递归 vs 迭代
    ( H- a( O2 j( v2 u2 A' _9 j3 @|          | 递归                          | 迭代               |
    # a  I6 e2 @  m' d|----------|-----------------------------|------------------|
    2 [2 q7 Y; r) n' O3 F& ]| 实现方式    | 函数自调用                        | 循环结构            |
    & D4 _" b8 V& T( ?| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 l( k1 I8 {2 p
    | 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 x! l2 L+ A/ N' ?( K3 \| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    $ k* n4 `5 y8 }& _; R) @' \$ q/ l6 f: n7 ~
    经典递归应用场景% o0 S8 Z/ x3 N) V9 `0 ~7 z7 V  }* ]
    1. 文件系统遍历(目录树结构)
    . F8 a& l- L3 Z2. 快速排序/归并排序算法
    " b: O0 l. J6 W! _* j8 C3. 汉诺塔问题
    4 Z$ Z6 a2 N( H$ v4. 二叉树遍历(前序/中序/后序)
    ' x6 V# H, N- L$ z5. 生成所有可能的组合(回溯算法)0 P) }! w8 K/ q9 f- g
    ' ^) m' ^* U* x+ i! c
    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,
    * t4 l) N* N& c  s我推理机的核心算法应该是二叉树遍历的变种。5 o7 S- y% y! V# E: |# x5 |+ z' Q
    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:# T' L2 e9 h3 b# T4 m# i( e& h
    Key Idea of Recursion; y( r% `1 w7 j  R! N% [  D% P

    ; n) P' Z" V0 Q) X$ a: F2 ?A recursive function solves a problem by:
    4 ~7 g+ ], Q: s6 }$ D% H: B9 M( J/ ~5 a9 [) s
        Breaking the problem into smaller instances of the same problem.
    8 r) h+ T: X( n
    7 v4 J" b0 J0 a, A6 p: Z    Solving the smallest instance directly (base case).5 u5 V+ l+ a' q- S$ H, P8 ^
    2 d, u3 \6 a6 j: \" u+ W. s5 I0 B( h
        Combining the results of smaller instances to solve the larger problem.( `5 K: z0 e( }; K5 N) y& `
    ; N1 h* C+ R: l( G
    Components of a Recursive Function
    7 ?: a6 I4 w! N3 d
    ) @! P! I/ ~# Q5 n9 t    Base Case:; O. t8 R: x! V; Q7 S

    # O( f: D) w$ H% J7 ~        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ( c* Z5 K; o$ z
    9 j6 ?2 D+ o' M0 {        It acts as the stopping condition to prevent infinite recursion.4 p6 L9 m/ ]- U2 b( S* w( J- }
    9 o4 s% ?. l& g& |! N) R
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 _. K. R* G  C, G* F
    1 O" \  x& K1 U; D; P( h
        Recursive Case:
    # d+ ?, c5 h- d0 P
    , _+ K4 U! Q) @7 q' D( c9 c( Q) C" m: q        This is where the function calls itself with a smaller or simpler version of the problem.
    : J2 I; m% J& m" ^0 m% R; E) I
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).( L/ {# V5 C) }; w* E/ d- P
    : J1 E, u" J/ T. I
    Example: Factorial Calculation
    ( i, g8 ~" s) |' o# R+ a) f, X( `3 \, ~! }
    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:
    $ M# d& G, V2 i) S9 \: g; y& A5 ?# i
    / N" F: p) W' l, [0 s. I    Base case: 0! = 1
    # y& [6 Y- }* ]& ]* e% M& M7 F9 P+ u8 p- I
        Recursive case: n! = n * (n-1)!% e2 I: s; E% t0 H3 B
    8 S, u% o3 L0 k3 W* X
    Here’s how it looks in code (Python):3 V4 M( g2 G' M0 R& z
    python
    ' J* E0 F( x# w# [2 H/ f- L# M/ R. A1 O1 ~

    " J3 d1 c/ Y5 udef factorial(n):; p3 \2 j0 `5 Y, Y- y: ~
        # Base case2 f9 A. j, m) ]- I/ j' J& V9 b
        if n == 0:  _5 p! @9 l* A( T3 g
            return 1
    ' r6 l' \" Q2 G    # Recursive case1 r- y- K# b! n5 G# A: k
        else:( d; V; c  E7 k  `
            return n * factorial(n - 1)$ U* ^2 t2 I5 c& I
    $ v; j0 j+ I' r
    # Example usage
    9 A: l4 |, t; g% H7 G- `0 g. X! S# \print(factorial(5))  # Output: 1200 F& `/ N0 Q( W9 u! U

    , J6 X9 e' |7 m, g0 n- D2 f. n' YHow Recursion Works
    8 K$ K  E* d: m1 }- C1 u+ Q
    : a' o; U7 v" K* \8 Y    The function keeps calling itself with smaller inputs until it reaches the base case., M( t# Z9 j0 S1 I( n0 r

    / k3 |* ?  z) B5 o) B    Once the base case is reached, the function starts returning values back up the call stack.
    % ?* X* c" R. i3 r  A
    ! |8 Z7 N  }! [" D2 r+ ~' G    These returned values are combined to produce the final result.
    1 b0 c1 W* L% N4 x" t: g$ [, ]1 |+ e8 n
    For factorial(5):
    , p/ f& v1 c  r/ n: O; }
    ; q# ^* {" P0 ^4 J6 j; y$ h: s
    5 {1 z+ O- u  j: K+ T. i" m% Ffactorial(5) = 5 * factorial(4)- o5 X5 L* r" C+ x' b% Z( B
    factorial(4) = 4 * factorial(3)! i; ]1 e/ y$ @2 r  K  o' s
    factorial(3) = 3 * factorial(2)6 V  P  F1 d  s2 E( B7 f# T
    factorial(2) = 2 * factorial(1)
    5 a+ i9 j- N9 Kfactorial(1) = 1 * factorial(0)
    ' F9 i( ]2 n. Q; K" k2 Y, Cfactorial(0) = 1  # Base case
    1 ?' u1 V- A: x, u# p" A1 `3 F, F+ }, V( c/ z/ X% a
    Then, the results are combined:
    - O8 R2 U  ?) U  W3 Q
    & a% v( e9 ^( M9 g- r# g: S6 W( `. c2 m. V9 Y
    factorial(1) = 1 * 1 = 1
    - i" x# L; d! R' u+ H/ p+ ofactorial(2) = 2 * 1 = 2
    , r0 T5 e; s1 G. ^: \factorial(3) = 3 * 2 = 6$ s+ ^# c4 z6 b% a. B
    factorial(4) = 4 * 6 = 247 H7 Q& l0 l+ a7 t
    factorial(5) = 5 * 24 = 120
    : x0 I" j" h8 l. `+ \6 y4 ^0 I! E' E, I
    Advantages of Recursion
    6 [" [- x# H5 I) [( `& L, ]) A- R, N) V7 m+ `, S8 Z
        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).
    9 L. l. O, [0 E3 ?, z; w/ j. W2 `$ g9 c1 M' n$ o% W
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
      g6 t0 U9 o- \" f1 X. d/ G9 O) [- k. o
    Disadvantages of Recursion
    2 X6 b' B' A" d( _. f% R$ y- M- w" C0 M# \8 K) U7 K
        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.
    " S" Z. i% H( ~0 a3 h% f3 O! F# r
    % a4 w. v0 R7 K    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    . p* a. f* R! W1 R
    9 _3 a/ U+ C3 [' J8 g  KWhen to Use Recursion
    ) n4 W$ k' r) L7 u* C
    . A, H+ V& {! p* i    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).* ~) Z' }7 X  b% v0 u: ~( N' W

    ; N0 m: J0 y$ I$ M    Problems with a clear base case and recursive case.
    2 j: c! J- k/ |0 C! H: w  I9 n- b
    ' T9 A, d; W  h2 eExample: Fibonacci Sequence
    9 j, G0 z' L0 _% N1 s; d/ \
      y: B8 o- F$ }1 W: m  R% EThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:# u4 R$ A6 K% I/ U# j6 I

    6 `1 G9 G* F% `' l- D1 l  m  J1 m" S    Base case: fib(0) = 0, fib(1) = 16 e" p+ m" C  |+ X) B4 I9 K. X/ ~# F

    . w: r; l  C; f: [5 u; K    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    % k8 U4 X1 i2 J. w- D2 U( U& k
    + i& A# B6 b0 V8 H4 N3 `python$ g- ?3 p& P3 n; k* Z

    + p; W2 o, I* m0 r5 y9 v. y' |1 l5 a
    def fibonacci(n):
    - U* o4 E, o! V6 ?) x0 D" V( ^    # Base cases& `  k; D) u7 W  A, w. b
        if n == 0:! F: Q4 P7 o$ _9 {
            return 09 N$ t, Z' j) i& P
        elif n == 1:
    % v. N' [1 ~. ^  B' e( r1 U: D        return 1
    9 k' P* ?' n' V$ |  b3 B    # Recursive case
    * i1 z6 _% S1 e, t7 ~3 O" R5 Z    else:( M2 U1 D+ y# h) T9 ]$ M
            return fibonacci(n - 1) + fibonacci(n - 2)0 K* I8 ~6 I$ `
    ! [/ N' ]+ k3 I% M
    # Example usage
    8 ^7 R* e- C+ E3 {. A6 Oprint(fibonacci(6))  # Output: 80 F/ j3 Z' q5 |8 k# S$ ]
    , z2 \' f5 G6 E$ E/ l5 x1 Y& o
    Tail Recursion* j  F* ~; {% Q
    8 `/ f, }0 K. x
    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).
    # j- k7 w0 y" l9 y, J  s6 d2 z' N( d: B2 y) J
    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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