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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
    + g% _3 j& k' O1 p: z# s! o8 N$ k7 \8 m" N/ V
    解释的不错/ ]' F+ v& o5 _1 c& c' R4 g/ P+ M

    ! G. c4 A2 e, m, N* D3 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。
    ) ^( J5 T4 j( Z/ k' z% |8 j, B# ?' D1 E1 V' I
    关键要素
    # h& P: \( F1 d7 i' l% |+ R. z1. **基线条件(Base Case)**3 I; M  A( b3 I$ ~6 Q
       - 递归终止的条件,防止无限循环
    1 \& z9 K& M0 h& g7 j* e9 |- X+ s   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1) B0 w$ T$ b7 t

    ' K; t# K. T2 t5 G- s# ?1 y5 @2. **递归条件(Recursive Case)**) Q  l8 W0 Y) o: w$ o' j6 D
       - 将原问题分解为更小的子问题* u; U' \* b& A5 `% q7 [& b
       - 例如:n! = n × (n-1)!- f8 [! W# c6 b$ E: T5 Y. C

    # g3 J8 }/ \# i  s, r* Q9 c 经典示例:计算阶乘
    3 \& q3 k& t# {* B! qpython4 U. q/ j. ], V8 _
    def factorial(n):: E4 Q0 N4 ]. B  a4 s/ ?/ @
        if n == 0:        # 基线条件
    - h( ?; J% ]" R. N1 G/ e        return 19 O0 [; V. }' {0 e/ q' }
        else:             # 递归条件. V8 r  ]) j9 T% k; {
            return n * factorial(n-1)* U- ?  c9 _9 P2 _% @9 C6 w& z
    执行过程(以计算 3! 为例):' e0 u/ _* ^/ O: A1 v( q- b
    factorial(3)
    ) s7 E% V( v% E) |3 * factorial(2)
    ; O3 G* Y. J' ^7 u3 * (2 * factorial(1))* M6 [4 C5 n/ s7 G, Z$ B
    3 * (2 * (1 * factorial(0)))5 ^/ i% P6 f+ R/ w- u- @; R
    3 * (2 * (1 * 1)) = 6# o: `, l  l5 K; b# X$ w

    . K. A) M2 G9 s* ~' K 递归思维要点2 i; R2 f  V- a* t+ [9 s+ C
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
      j8 a7 U' P! G  G" v* l2. **栈结构**:每次调用都会创建新的栈帧(内存空间)5 M7 ~! s3 E6 R" T
    3. **递推过程**:不断向下分解问题(递)
    + H9 J  r# d8 e1 ~) `4. **回溯过程**:组合子问题结果返回(归)& R# e! t4 H9 @" v" u

    ! y$ P5 @- j" o6 p  E# y 注意事项; y1 _  M( C& W5 j; g
    必须要有终止条件$ M9 n/ N( G3 I: o' G0 Y' _" ]
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    / n" L. j6 _' d) A, k  y某些问题用递归更直观(如树遍历),但效率可能不如迭代1 }" O( Y$ [' `( H( \. ^: L
    尾递归优化可以提升效率(但Python不支持)
    $ `/ N' O! w# V" X8 w: R4 x9 h) e
    ! r0 Z6 j+ j% H" V9 a, r2 n 递归 vs 迭代
    ! F! i/ J8 r8 P, N|          | 递归                          | 迭代               |
    4 G% |  Q5 {/ Z|----------|-----------------------------|------------------|
    " {" V7 O* F3 k1 ]4 f$ M| 实现方式    | 函数自调用                        | 循环结构            |. g1 N: [- [/ }9 l: c
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |! X* u8 o$ C) d. r( t2 B3 Z
    | 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         |
    8 P- p5 M  e; B& g2 g* u0 C8 u' l| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |+ v8 o! h  t: v$ F& d2 b1 \
    + _. @. x& P4 D" }4 g2 i, X' _
    经典递归应用场景3 n7 v) Z2 e& m% N& A) h
    1. 文件系统遍历(目录树结构)
    - I/ P4 B$ J4 n) x2. 快速排序/归并排序算法
    $ M) t6 a' P$ ]) g: x. e4 w3. 汉诺塔问题& Y9 c& m5 \. f) t
    4. 二叉树遍历(前序/中序/后序)( ]& Y1 `# V8 i( H0 J  g9 `" r
    5. 生成所有可能的组合(回溯算法): m2 |5 |8 U# M9 {
    ! j! x3 T3 C2 Q8 a3 W6 i/ W* K5 ~
    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,# J7 A$ g1 y$ q$ \
    我推理机的核心算法应该是二叉树遍历的变种。
    6 G+ T) K7 F3 @3 T8 a; E9 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:( E4 {9 m$ i& q
    Key Idea of Recursion% b6 r+ j) |& |- j  X8 @2 o( S) |

    + t/ j* {& b, M3 }, ?A recursive function solves a problem by:
    3 A, i/ g, U2 O4 J1 |8 J% u/ h6 s
    ) p! f; A2 l" F5 \& w    Breaking the problem into smaller instances of the same problem./ s# R" m. a& z) g  |

    - W: N1 U; m+ o; w0 L$ T" F    Solving the smallest instance directly (base case).
    6 J' p! Z8 {9 i# [3 X+ d9 `$ J: `/ J: x; C. x! \' w9 h
        Combining the results of smaller instances to solve the larger problem.
    5 ]5 W1 G" R# H& n# o7 _9 A' [1 [  _, o- X5 x+ P; i
    Components of a Recursive Function
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        Base Case:9 G. _: a+ a; O

    " G' e: S" q' A4 l" \( `1 ^        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    2 d9 C' ]+ h# P0 ~
    6 a; Z. d1 c& _& n+ M! c        It acts as the stopping condition to prevent infinite recursion.( u0 t9 L  U; s& L7 z7 x

    8 q, @* o$ P  D5 @        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    $ U- l, L. ]& J- e
    * Q( X* x$ ]+ M4 j    Recursive Case:0 |3 K! W0 R9 s3 ]* Z0 H
    6 w8 [/ [6 y8 ^8 S; v
            This is where the function calls itself with a smaller or simpler version of the problem.- n: m5 M! r, p7 q4 B0 E

    : ~5 o9 i8 F2 W' n0 m        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., u. S3 c  s" H4 \  n: F( a' h

    8 t0 T, @/ l! W4 T. oExample: Factorial Calculation
    ! r% P( ]2 i- d2 X; j6 y7 i: I
    ; [! J: z  v- {* Z: j/ Q2 oThe 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:' ~( c" {0 @' Y- V2 q# Z

    - u. I* d# O9 R) `" K+ T) U+ @2 [    Base case: 0! = 1
    1 l2 ]" J( ^4 W+ s% L: p
    4 G: Z6 [$ l2 f    Recursive case: n! = n * (n-1)!% a2 H+ S$ y9 M' O0 y% Q
    - O" [8 ]  s) ~0 I6 I7 M: y4 @
    Here’s how it looks in code (Python):
    . q, \9 ~6 W0 b9 m+ Hpython; j2 L: h' |- n9 U% [) {0 H  j
    " k0 U* a' z0 z' j& ]% f

    + t* a& R- _$ g  M1 L. g. Xdef factorial(n):
    % _$ z& p: i7 T" [    # Base case
    * A% h# l( e* V  m1 x3 x. |    if n == 0:' h6 L1 m) q& W1 `
            return 1: G% g" v) M/ B5 I4 f5 N, R
        # Recursive case  U' T* R0 T! k4 }1 H' v
        else:
    6 v/ S/ N  k" q% a  d! T        return n * factorial(n - 1)3 {3 J/ N& u/ p. {! N. ?
    4 L* ^) D" A, A3 j
    # Example usage
    ) Z3 b3 C9 I! A5 y8 zprint(factorial(5))  # Output: 120  K9 O* t5 X) l" e3 O

    5 {4 e0 s  P2 kHow Recursion Works
    ; K0 _$ |; |5 B6 i4 Y4 S
    6 q& W3 N$ k' t7 O4 v% ~: ^( U- I    The function keeps calling itself with smaller inputs until it reaches the base case.4 ]0 ]2 F6 P5 |- J1 T
      o( M! q, T2 H
        Once the base case is reached, the function starts returning values back up the call stack.
    % A  g9 h4 D! c. @0 k
    , p  U' J8 \, p1 o4 ^6 d1 v2 _4 a    These returned values are combined to produce the final result.
    , M* y$ k- u/ o3 ~( I
    6 V7 x4 B9 R2 N& TFor factorial(5):
    ; A0 w& i$ S% y7 |2 o- `5 A" X7 d5 C+ r) f. J

    $ m+ w9 o4 D: ~/ ^/ _' Kfactorial(5) = 5 * factorial(4)& k: ?0 U6 s5 R7 Q& [3 q0 N- e4 v
    factorial(4) = 4 * factorial(3)" }; {2 k2 b% L. \! c
    factorial(3) = 3 * factorial(2); a$ k' l9 d& K# J
    factorial(2) = 2 * factorial(1)
    2 ?) X7 k# u# F* }9 [1 {3 y: u; A2 pfactorial(1) = 1 * factorial(0)( v; d. Z4 M1 X/ t2 z$ I! n
    factorial(0) = 1  # Base case
    # ?7 `8 Z6 K6 r$ Q/ p1 P8 t# @3 M! @, \) |
    Then, the results are combined:
    7 W: P2 O8 ?8 U+ l9 `
    - \# ?+ D, q! F7 p0 Z# s7 c6 U3 W( v
    4 P) Y; h& w+ }& ~- xfactorial(1) = 1 * 1 = 1
    ( S0 Y9 x% _* U# j" dfactorial(2) = 2 * 1 = 26 c  m2 r7 `, a5 ^6 U6 K% F
    factorial(3) = 3 * 2 = 6
    + x' w- @! Q; ^8 w7 o- yfactorial(4) = 4 * 6 = 240 H; G6 i3 `4 _+ d7 o" p
    factorial(5) = 5 * 24 = 120( F# D4 B9 ^2 H9 N; C4 c
    $ P: I8 M! L5 w( {1 x) D
    Advantages of Recursion% `) [) W5 w( r9 @0 q! |8 p9 Y! u

    2 `6 v! f6 v' V- A! 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).
    & X6 ~# v" d# e0 e8 N  @, U+ k  {2 R% l& D: h. g
        Readability: Recursive code can be more readable and concise compared to iterative solutions.. |3 h" D% H2 E& \2 m
    ' U5 o- @* T- p0 R/ Z" ?
    Disadvantages of Recursion
    - {% W' u9 m, q
    / o+ u. v/ q3 @3 S7 }+ V% r6 y    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 B8 n+ w  i3 p: N# y

    # o8 M" z% Z  }# \5 M$ }. u    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    5 I2 I6 V- ~( \! g' w6 l' i7 ^. o( }! L. ]( L, y* F
    When to Use Recursion* f* u- @) [5 T9 f

    2 m2 [$ A! h) R6 F4 `6 p4 G8 |    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).; t! Y) p' A1 Y: z
    0 C. D% [1 g  S6 M
        Problems with a clear base case and recursive case.; S7 Y; l1 D5 M# q; ?2 Y# j
    ! _! J5 b, {% Q& T9 x; P
    Example: Fibonacci Sequence5 g- [1 v" N0 U  g& M% F

    ) t% G2 S- k  \4 X: s5 e( \9 PThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:  {& C& F! f' G/ \  t, a" o
    1 H% h! n' v' L5 H" V! I7 O' _
        Base case: fib(0) = 0, fib(1) = 1/ K- {; u: I; K' u' V: v$ T: d

    / u: Y! G- {! q    Recursive case: fib(n) = fib(n-1) + fib(n-2)! @) j2 u) F+ M; W; }

    0 N- `8 J  g5 J- g- Q3 W, A( O7 Npython
    " Q2 z8 |- K& c9 V; `. d
    # h2 W8 M* ?$ y: c2 l! a& ~- n
    . n1 F! C9 H! ^' U; }2 A& v. S+ Fdef fibonacci(n):, R, h% M' u" a! z/ F! g$ E
        # Base cases
    6 {% d; I4 G( q& T( r  G    if n == 0:
    ' B6 B6 u- H$ t+ d0 ]" @: M; s        return 0
    # I' R% r& D, F( q, X) ]6 M5 t    elif n == 1:
    & p, ~/ I0 F  K  e        return 1
    ! p1 R9 Z+ t) {" f: M    # Recursive case
    + W* M9 [* k0 \; c    else:
    % A% {# g  r+ t; P5 u$ H        return fibonacci(n - 1) + fibonacci(n - 2)
    7 |! z  `2 ~( Y) M% h; R9 Y
    . Q2 [" c% L. ?5 A4 b# Example usage
    & o5 d7 [( m+ hprint(fibonacci(6))  # Output: 8/ s) x' C% j' L9 n

    ; I- E- l% A4 W4 D; lTail Recursion7 b" v" A  r4 l4 x. y/ H

    ! j6 j! c+ o0 [' e/ XTail 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).
    7 }+ ^+ q0 H2 @, q/ Z% `; _* C
    : k; s& ]. o) t3 ]7 G* pIn 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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