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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| 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 8 [) e. P. |; Q# z
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    解释的不错8 ]2 P( T7 d% Q: ^; v6 b/ l

    $ `: t0 i! G1 [" a3 U1 ~+ L3 N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。
    3 w2 c! f- d" D0 E. a
    ' ~! N7 ~4 n' [7 ?7 }" @ 关键要素
    ! g* u" B5 }% q7 `& K2 |8 O$ S' O4 `1. **基线条件(Base Case)**
    " ~9 s" \" l% ~- [   - 递归终止的条件,防止无限循环0 o, j/ y" u2 S, H1 L- P, K: y
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    5 g8 a- P8 }4 E5 h/ s. P: @2. **递归条件(Recursive Case)**
    7 k% e: L) ~7 X9 f$ {, k5 |   - 将原问题分解为更小的子问题
    7 E# w, s! z6 L  J   - 例如:n! = n × (n-1)!
    6 y) `( T, T  e% ^. F
    2 A7 _6 M) g$ Y6 q& B 经典示例:计算阶乘3 g* q8 T6 a% t# ?4 m4 u- T
    python
    3 G/ k9 U0 s* m" P1 F( v5 H2 zdef factorial(n):7 p0 b" W& s' J3 N+ f: k1 R
        if n == 0:        # 基线条件
    " d8 i7 r/ a. p! S0 \7 o6 Q        return 1
    5 g: U: U0 G, M    else:             # 递归条件: N: y* \3 t+ H/ M6 T% C, N, T6 D- X2 v
            return n * factorial(n-1). m7 J5 b/ k, G" z+ E7 t
    执行过程(以计算 3! 为例):% j7 V7 M  B$ q
    factorial(3)* e& E- ]8 t- W2 X, p; i
    3 * factorial(2)
    . C; M: c% l3 D3 * (2 * factorial(1))
    7 J+ b. {3 U9 f9 q3 * (2 * (1 * factorial(0)))
    % r! b/ `3 v# u2 e3 T3 * (2 * (1 * 1)) = 6" o. e1 h4 O9 \0 E& v9 T6 k

    ; I/ H) Q" E5 ?2 Q+ ?. I; e 递归思维要点3 @1 ]6 N) a) s; @7 W
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑2 [0 x2 x1 T/ _1 y) d; P
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    2 k' |- y  Z2 o) H/ E, B3. **递推过程**:不断向下分解问题(递)
    ( s' q) D4 ]. k4. **回溯过程**:组合子问题结果返回(归)/ u6 J" |! {4 T' w- e( s
    & C7 g  @: Z9 x0 s% E
    注意事项4 R* A2 V. F$ t1 c+ _8 t8 Q3 d
    必须要有终止条件: Z# l# Z, n8 s1 R9 K* A; r
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ; e9 U$ w( X" N. ~! ^某些问题用递归更直观(如树遍历),但效率可能不如迭代
    & x+ p2 Q1 Z3 o) v$ l* l8 c尾递归优化可以提升效率(但Python不支持); g$ l3 U, W0 u  X  x" K% r

    # C1 j6 K1 m, c* G" \( ` 递归 vs 迭代
    5 S8 B; |5 S- P$ x& Z1 ~' j9 d4 {|          | 递归                          | 迭代               |
    5 j8 X* v. T9 i|----------|-----------------------------|------------------|  Q" ^, e2 l# [# l' I% Z  v
    | 实现方式    | 函数自调用                        | 循环结构            |
    6 V: J! q8 j4 ^2 q2 ?6 \9 z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & r! [' z8 G. r$ A) M4 m5 S# D| 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         |
    4 s7 m& B% u! j6 w| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |. \' h. R' j/ E# I5 Z* R, K& c# p

    ( n7 z& s/ F7 a' }2 g 经典递归应用场景" q2 m0 u) c9 u9 s$ \! U) c0 \6 d0 p
    1. 文件系统遍历(目录树结构)
    ( q1 K6 F( m/ H% s7 q2. 快速排序/归并排序算法
    ' g0 [0 k; z+ Q# X, Y$ j3. 汉诺塔问题
    ( ?" e$ ?( q7 g6 z; F4. 二叉树遍历(前序/中序/后序)
    9 V4 X1 G6 B. O9 X5. 生成所有可能的组合(回溯算法)
    9 Q. G( v8 g9 m1 j7 R& i
    ! r7 c2 o0 c) }! H3 @0 a) 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,
    ' U) u" H, r1 Z9 Y/ R8 b我推理机的核心算法应该是二叉树遍历的变种。" K. H+ J" k# f4 [
    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:( h% v+ d# u# v8 I6 z& O4 h
    Key Idea of Recursion8 |: _1 C0 Y: J
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    A recursive function solves a problem by:* Z- M. f5 R$ q* [% A* |; J/ u
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        Breaking the problem into smaller instances of the same problem.
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        Solving the smallest instance directly (base case).2 {9 D3 F% z* P8 e& O* W
      |& \  j/ {6 R- K7 M6 I8 C
        Combining the results of smaller instances to solve the larger problem.) R( \0 i5 I9 m

    : `! S# L; I6 }' H$ X; AComponents of a Recursive Function/ V0 ~! Z; L5 H2 H* c7 A  ?! ^

    " U% I  i! R4 K) W! c& I5 x2 \# C    Base Case:/ B: R( \- W% |$ e) v* c) F0 H
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* [; G, E/ ~. D# Q" N+ J9 [
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            It acts as the stopping condition to prevent infinite recursion.
    , s9 d6 j- L1 v0 U; ?3 `% A1 I- c# H/ \; ^4 e$ _- C
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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      o# }% I3 f* B4 ~2 W    Recursive Case:, q4 S( i; A0 R0 s' q& q; t
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            This is where the function calls itself with a smaller or simpler version of the problem.
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    # X) q' ]8 X# M$ z# C& Z& I- \6 X. s( w) H        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).. x! D2 z* q9 Q  k0 w: Z. e

    $ _1 Y5 H9 z% \Example: Factorial Calculation, ^# K4 V3 ~7 e. q/ v
    4 N5 m( `  }" H; U1 V# n+ }
    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:- Z6 Q  p) b8 i. s
    2 a; e, g% ?: E. Z$ H! Q: d
        Base case: 0! = 11 y, J0 n; }2 q( d6 s

    & |% n2 o4 K# A, ^# H8 j" Y5 y    Recursive case: n! = n * (n-1)!
    ! y4 f& l! i& ?+ n6 J5 p# ~- H/ E0 W( m
    Here’s how it looks in code (Python):( H+ V: Z( l/ B& F& ~* z# V5 g
    python  a$ I0 f( n+ u* h
    ' w  {: k: w# h5 `% Q4 ^4 y

    , Q" {- X  v4 J) T9 T2 zdef factorial(n):- ]- T, f  Z( s: `& z/ v8 p1 `
        # Base case2 h# L4 C7 O; ^" k# q! m
        if n == 0:  a0 O# t& W: y8 d# ^9 |" P
            return 12 A! ^3 [+ ]) \9 ?& u
        # Recursive case
    8 r- k( o8 ?4 A0 D" \, T    else:# Y9 F& F3 o, f" c: F
            return n * factorial(n - 1)( k3 [% p) }4 Z" U

    & R0 Y4 ~6 F/ S: y2 R8 ^0 g# Example usage
    4 Y: F& O! Z0 I2 U2 u0 Pprint(factorial(5))  # Output: 120
    / e$ V8 `. N) k$ j5 l0 X
    1 m; c* W4 L9 Y) XHow Recursion Works  C. Y, B) h- p

    - F% o0 D$ H6 O+ E" S9 Z    The function keeps calling itself with smaller inputs until it reaches the base case.
    , y/ t4 T5 p4 X  T; Q6 ~; c9 K: w; D% d# a
        Once the base case is reached, the function starts returning values back up the call stack.
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    & `# h8 W+ \* x4 m, x3 K/ a    These returned values are combined to produce the final result.
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    For factorial(5):
    6 J$ p/ S# R. c+ i, n& v& [
    ; |6 V1 W6 H# H( l
      K$ f$ R$ J9 lfactorial(5) = 5 * factorial(4)
    - X* N/ n4 H3 hfactorial(4) = 4 * factorial(3)# s% h* c* Q8 l% g% {7 z; i2 v
    factorial(3) = 3 * factorial(2)! l$ u* c+ w: h* _8 S; O
    factorial(2) = 2 * factorial(1)
    1 ^; r) M% v8 _4 H% w/ Q, Mfactorial(1) = 1 * factorial(0)
      Y4 s3 Y" I& Y& n4 \. mfactorial(0) = 1  # Base case
    2 K  A& r) b4 c) g$ ]$ Q3 @$ p2 ^: @: T
      q3 w9 M- |7 o7 ~- vThen, the results are combined:/ V, m( W" k; U  J  ]  F+ [

    ) |4 a) H; F$ s- J% f7 O0 f5 a6 |: i" a& s% }1 H  h' r1 Y
    factorial(1) = 1 * 1 = 1. d- n7 J1 H3 R. M" n1 w" u
    factorial(2) = 2 * 1 = 2
    2 d& w1 y7 x  O  R  C, v. e* r# ofactorial(3) = 3 * 2 = 6
    . H$ B/ A8 O/ S! L- |factorial(4) = 4 * 6 = 24  j& L% |1 @8 C: Q9 `/ r
    factorial(5) = 5 * 24 = 120
    $ d, ^" V4 a* L) V9 ]. `1 k! S  a
    0 d- {2 p  |' \( [; F! w5 `+ @Advantages of Recursion
    , f; U  G8 W- H& v, E& r) A4 M) ?" O8 }0 l5 h/ D# u3 _
        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).
    : Y: X: ~+ \1 T( o$ z1 N/ r4 b- W6 n; `! Q0 |# v
        Readability: Recursive code can be more readable and concise compared to iterative solutions.$ Q  i7 R5 _' L6 [9 W/ @; O
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    Disadvantages of Recursion4 F& G, ], Q8 }/ Q1 w! O6 [

    $ L) r/ Q2 J* h! L8 Z3 i- T% n    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.$ m) X! Z( e7 b  W9 S9 B
    5 G: S& l% w: X8 z
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 T( g3 Y! l  J1 n5 ^
    8 z# C8 p* {* M, |+ U" @. L" s4 X
    When to Use Recursion
    0 W; D. S0 Y4 U- m: ]! l" y1 {5 V, k% m
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ! u9 {; `9 Q* i& A6 ]% c( ~$ z5 l/ h+ x6 i! O! u) z
        Problems with a clear base case and recursive case.. \- w1 P' q: O& t! _( g  ?6 l
    . B2 g  F3 n3 k7 F
    Example: Fibonacci Sequence& y' f$ ^" D6 H4 w* Y9 w

    9 N9 {2 {( s: h4 w& `The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:/ S7 n: Q+ v; b0 l

    & Q# O2 K  W( J* V4 x$ h! s+ O    Base case: fib(0) = 0, fib(1) = 1
    / H( n% |- o! `1 H5 J8 w- ]# H9 }8 U4 ]* S7 Y7 B
        Recursive case: fib(n) = fib(n-1) + fib(n-2)7 I; s' {3 v, h6 O* b( F

    6 H; w# K. e* Z8 ]/ N( v( Apython
    ' p( C- n( r9 X" G( l% }! o6 [" x) a- [/ G
    - i: u, i: e2 ^6 n2 }6 J
    def fibonacci(n):
    5 ]6 l3 V6 {! K4 k. Z& e" y    # Base cases
    . b( m, [+ J6 V    if n == 0:; D0 C* |2 U9 V
            return 0
    ( U: ~- c8 _, P( A8 V  n    elif n == 1:
    9 P9 S9 a5 y4 f0 z9 g3 W        return 1' d+ [9 _# W( t3 i2 l
        # Recursive case% x! i% J% C) G2 t; w+ Q2 r
        else:
    . f( g6 E' f' G/ J4 `        return fibonacci(n - 1) + fibonacci(n - 2)4 T3 a6 M* A* p5 v

    8 d! k* Z- v; a# Example usage! |, X% L" m5 b& U& d  q
    print(fibonacci(6))  # Output: 8
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    ( e) B2 _0 D' k7 O+ GTail Recursion$ J: w0 w% k3 W" T7 G
    " {2 e( p+ M( f6 o  `) l' F
    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).
    , b4 ]9 m  a4 n- F7 s! B$ |
    ; X5 D9 A/ J1 n( 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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