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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
     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
    7 X8 E5 u/ K  N1 T8 r
    ; S: m3 d% J" v; U# y" A  y+ G解释的不错
    3 h3 V2 K5 x1 z& ]+ i
    ! b( F7 e7 K! k0 N/ @2 b0 c2 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。/ K. e1 @: w7 Y( H

    6 X" q! ^. y/ Z3 s" P  N9 | 关键要素9 [7 \$ s6 T8 X3 Z/ l
    1. **基线条件(Base Case)**
    - \7 }8 k0 p6 \% p+ U9 L! J$ Y% m   - 递归终止的条件,防止无限循环( h4 D# `4 K1 f; u* Q, X; L
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    * ^$ W9 w  \9 {- E8 `# l; U
    1 m' a1 o/ h1 a8 ]( b# T2. **递归条件(Recursive Case)**% }( G8 ?( q+ h
       - 将原问题分解为更小的子问题1 w- h( R6 R! v- E9 ?5 J
       - 例如:n! = n × (n-1)!
    6 J3 X% B& y& ~
    1 T! G9 }! U; y! f" y 经典示例:计算阶乘4 m$ I9 a! L8 h* P, n
    python
    9 N% y2 D- v, y( _def factorial(n):% P1 v! J) r+ ?- _
        if n == 0:        # 基线条件
    0 T# i8 b8 _, d3 G  Z        return 1
    1 q9 }$ u6 D! q( l9 `    else:             # 递归条件
    ! F; A6 B9 Q+ v# n( U        return n * factorial(n-1)- V! s7 ~& H; T- R3 [' z4 D
    执行过程(以计算 3! 为例):* t2 K: a  G0 h5 l% U
    factorial(3)
    - V- h' `7 W; T, D5 v& ]3 * factorial(2)! g6 P  _4 x4 _5 j
    3 * (2 * factorial(1))
    * P7 _4 Z, q0 k  d: @9 F3 * (2 * (1 * factorial(0)))1 N& N% S. M0 O$ j0 _: ^* o' |
    3 * (2 * (1 * 1)) = 69 @7 A- K$ u) h8 d% t

    ; C, |9 f( o; B9 F4 H7 l 递归思维要点3 t" t/ e5 a3 ]/ x& n9 o
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    * U0 K6 s$ V: Z" K( F# s6 R2. **栈结构**:每次调用都会创建新的栈帧(内存空间). r% i+ R' g4 {: g# k2 J0 N6 V
    3. **递推过程**:不断向下分解问题(递)0 W6 `% P( b, b& y
    4. **回溯过程**:组合子问题结果返回(归)
    7 W3 n+ G- o! Y2 C9 i& f
    6 T" l8 b6 T7 r- Q; x 注意事项2 ]3 B$ r! t. ^2 }0 X8 c7 |% d% z
    必须要有终止条件) z8 ?- _4 l7 m9 ], g
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    2 z3 u8 q2 {& e某些问题用递归更直观(如树遍历),但效率可能不如迭代
    # ~  |" s# X: `0 s( o7 ^7 }# E尾递归优化可以提升效率(但Python不支持)! P; P& g- `3 v
    8 E7 u9 d- n( U1 j4 U1 \
    递归 vs 迭代2 ~; y7 Y* @8 d$ i- P! T
    |          | 递归                          | 迭代               |2 M; K" ]. d0 @) o
    |----------|-----------------------------|------------------|
    9 B+ n" }* U& N! ~| 实现方式    | 函数自调用                        | 循环结构            |7 w) \- c& z6 m& r
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    . A$ z* _$ ^" D  L, H  V| 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         |
    ; g: u* j- o" E6 {* V| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |  f9 |$ |( _, l4 k

      a4 z5 X! C! a* i 经典递归应用场景
    ! a% M% N$ A  H$ j4 W1. 文件系统遍历(目录树结构)* @9 _) [# C+ j5 u+ f# e/ f. m0 }
    2. 快速排序/归并排序算法+ r0 d, E' `) V+ W6 x; G/ Q) I
    3. 汉诺塔问题
    1 M7 m  f$ c' e& U. i0 I) `# n4. 二叉树遍历(前序/中序/后序)+ J5 ^; `8 F. C6 H  Y. K% A
    5. 生成所有可能的组合(回溯算法)
    : C! y% \- d7 s/ x4 s! u' k* q; o! ]6 N
    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 07:10
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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 D8 T6 R9 a& u2 m, x9 F
    我推理机的核心算法应该是二叉树遍历的变种。
    1 U  E9 O" E" V  w2 @7 F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:
    ( M0 Y3 p" c! g( ?Key Idea of Recursion- S4 |/ V& G' J# p1 n

    2 \" r% ~& `$ Y0 q: C. CA recursive function solves a problem by:7 f$ D4 p- K; d

    8 }0 h' }9 b, g8 U4 F. N7 h( v. n    Breaking the problem into smaller instances of the same problem.
    : u+ ~( R0 M- k% U$ h0 _; l4 \
    : p( w8 R9 W/ z3 @: u, g    Solving the smallest instance directly (base case).7 a: [; j) G5 Q7 v0 x

    ( F' K0 o7 I" v# ?/ Q" j7 b( k    Combining the results of smaller instances to solve the larger problem.& K3 L3 Y' A1 ~! U$ {

    7 l# x1 S* g% C! E; X- bComponents of a Recursive Function4 t- H. H! W/ W- H- @
    & @/ l# p# U( N7 _) q, w1 e/ U
        Base Case:8 e) w% J* H6 i2 t9 |

    9 k- K; G$ N4 s; ^& M        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.4 G+ b0 M% S0 j2 N6 K$ k

    + {7 e8 w7 W9 ^( q8 c$ i/ q  S5 U$ K        It acts as the stopping condition to prevent infinite recursion.$ L, g2 h0 j  y5 p
    % D, N# J# F$ u# n
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.& N2 G7 x( O* B- `# T+ l: v
    7 B7 l% K1 _: I2 d
        Recursive Case:
    ; [- H* J0 ~1 a- a. Z8 v+ F; v; N# m" K8 F* _
            This is where the function calls itself with a smaller or simpler version of the problem.# u, W$ r; Z& v

    4 {; P; x/ a) W  F        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    5 c" J3 m. T; P& D7 _
    ( ~8 I: K2 X' W8 I* o$ |" i' iExample: Factorial Calculation
    0 l8 Z# J' T" u# s3 b: s
    % o8 h, u0 s: u0 @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:) l# N' W6 i" ~
    5 L: A: v& O6 p: c0 ^* Z
        Base case: 0! = 1
    * Q6 k" X# W- r$ \: M: Y% J& n- i+ L9 g0 M. Z
        Recursive case: n! = n * (n-1)!* W" m8 ~+ R3 N; I/ a% C$ k

    + |3 s! h. i- V) [3 u" `Here’s how it looks in code (Python):- }7 k( x) }. E/ ^# f/ H( U* V
    python/ G- Q# `$ N0 i
    , z3 W$ Y. J. w' Z( t! s; Z
    # D- H  T- a! y% }: p4 B5 Q
    def factorial(n):
    / R) U+ S+ {+ E9 J    # Base case
    : T. P: {+ J2 W& v1 A  V    if n == 0:, x# X( H7 J# G% T: Y  `5 q
            return 1
    & H  W1 B7 C+ Z! c    # Recursive case8 y$ W2 U2 c( s4 j# _  M3 {1 Y
        else:4 _# T+ w  k2 \6 n' `9 F
            return n * factorial(n - 1)+ l8 ~* G$ V+ x, K

    & p9 t# T1 y1 d3 G% y1 k# Example usage
    : h: Z5 `3 M! e2 @' q  h& n) O/ P9 Pprint(factorial(5))  # Output: 120% u0 Y7 Z. t/ I

    ) J$ o( N1 c* M3 I4 R' u7 l" y" BHow Recursion Works% r+ P. F3 b4 q

    7 E+ v  l  I) N7 e  y( }  D    The function keeps calling itself with smaller inputs until it reaches the base case.
    . z2 J: m+ N; D6 ^3 r" D* B
    5 U+ g) z' v+ _) R8 \    Once the base case is reached, the function starts returning values back up the call stack.
    / t( t# P* I  L  U: O; B8 S) W1 ~" Y- V- W" c2 p7 Y
        These returned values are combined to produce the final result.2 G, n7 R  x' N+ H) \

    & s* g% N& `4 g, A. ?2 F) m$ iFor factorial(5):1 m3 M1 \4 M# p9 _* y
    4 @8 w+ e7 p" U! B7 r3 p
    0 y# Q4 _9 w$ J7 m
    factorial(5) = 5 * factorial(4)  p) A) ^5 H* v4 K; e
    factorial(4) = 4 * factorial(3)
    2 W. u! W0 [. X4 B" Z1 _5 nfactorial(3) = 3 * factorial(2)
    $ X3 ?7 t* z( E! I& \0 Rfactorial(2) = 2 * factorial(1)
    3 I: J" Z, ?2 Ofactorial(1) = 1 * factorial(0)) D2 V% e2 e2 |) U* Q9 O! f
    factorial(0) = 1  # Base case
    7 c4 x- B- ?1 Z) p4 S0 `/ ]
    6 n" g5 T8 d6 [& bThen, the results are combined:) O2 F2 m4 E% F' n' v4 Y9 E

    / Z3 C- o9 Y0 u5 S" J9 f4 u( z
    ( t- V+ @  u- t. V" T; lfactorial(1) = 1 * 1 = 1
    * P, e" u: V( l2 o2 g4 L; J/ N3 Ifactorial(2) = 2 * 1 = 2: L3 D( l  m- N4 o% m2 P/ M
    factorial(3) = 3 * 2 = 6
    ! I$ M7 F# T6 S3 Wfactorial(4) = 4 * 6 = 24
    " u# \* p( u3 [( c. v. w0 ~factorial(5) = 5 * 24 = 120  F- @; j; k( b* @8 Y
    * n0 B( R3 T  _! Y. r* b
    Advantages of Recursion0 q* j5 L% q, [9 J

    % a+ Q; \3 \$ v+ K5 t    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).& W+ {/ |% u) Q

    $ L  M* ?& q1 Q- K* b    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    * |! r% M- I* J3 e; K% ^: \: \) F+ ?* r+ h. `, T4 o* H9 }( C
    Disadvantages of Recursion: x( g. \$ B- a& P

    * ~- U7 W0 W+ q4 Y' s' S' ]5 P1 i) h0 g8 l    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.
    6 J% y' x# M: z7 U* O( i6 x: H/ V1 e: {( J8 m! x( r
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).- {9 G1 O. K0 [
    ' t4 x) X5 J3 e5 l9 X6 m- Q
    When to Use Recursion
    6 R& y; j' e) r. d+ {+ s$ }7 M1 X* P. z  D" h% r4 l0 f
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).5 o8 A; X6 R7 i/ q* P

    . s0 A+ V$ C" `" q    Problems with a clear base case and recursive case.+ j0 D& Y5 L. ~; |; w
    : q& Z; A+ c5 E& |0 o6 m( M  Y" w
    Example: Fibonacci Sequence
    8 f8 A3 C. J) j5 z+ v1 c) {; h5 K, r4 l" A' y4 l  q5 X  F- H- ^
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    8 J1 S- i( ]- `" J! ]
    5 u: E) Y) |& x$ M( s) l, u3 |& b1 i" B    Base case: fib(0) = 0, fib(1) = 11 T2 x8 F$ l. \9 a% E* W
    1 H; K" j; z0 e! [+ Q1 m3 X
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    . c; a  @! V! o: x$ U/ O8 u" C! D8 O( e
    python
    ' d+ Q. ~  K/ i6 i9 i% V4 L! v7 r6 u
      T$ ~! O8 ^/ W0 s9 o8 O7 @8 _
    def fibonacci(n):
    $ k' t4 x' E' t6 N* }" {& H/ G) w    # Base cases
    + t* G% c& V* w" ^    if n == 0:
    3 K2 y, f* C* c: A6 }+ p7 k+ c        return 0
    8 h; J+ C3 Q; \* |    elif n == 1:
    6 w( u$ t0 p, K( a# \0 W. C* _        return 1
    ( }1 R5 @- D& p" p  [    # Recursive case
    0 k( [3 O; F3 M4 P. M    else:% N& t* `* e2 J5 U! \
            return fibonacci(n - 1) + fibonacci(n - 2)
    0 o0 ?- x; C3 m- x  g& t6 q5 G  _
    : P7 K* N9 O# o; e  T4 ~( p% K0 J# Example usage' c7 p& i" N  K; B% e( G/ t( d, {
    print(fibonacci(6))  # Output: 8
    ( G6 y! ?/ {) x. _8 W- g: z
    2 E3 ^, l& H% i# I* Z' xTail Recursion
    9 }( m) n3 Z# ?+ D% o
    & V) x6 v! w: p9 M& VTail 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).- h5 r9 I3 I$ |1 B' S8 d1 P

    1 _  K# N9 ?  h9 t& o) ?" o# BIn 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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