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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 - N* c# w( f% c$ [6 X: p; x
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    解释的不错! w- U+ s6 d& X' k' @+ v
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    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。  P" D& Y6 s& E5 c, T3 Q5 R) B0 u
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    关键要素0 P+ O( p: h( J) O
    1. **基线条件(Base Case)**
    5 M, ?# c/ }/ I( q7 e& v   - 递归终止的条件,防止无限循环$ E$ w# T3 O2 [
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    7 _& Z4 N/ t; ?: i" ^+ L0 K! h
    ( ?# C" ^4 b2 Y9 E, `. g) e: ]8 w2. **递归条件(Recursive Case)**
    7 [$ }3 e; x$ r. ^% @   - 将原问题分解为更小的子问题
    / }% u& n3 `, G- `5 e8 }   - 例如:n! = n × (n-1)!; z$ S. z8 w' ]8 Z3 [7 O- }* ?6 }7 g
    2 D  H6 k& }% N' a+ N( Q% q9 [) C
    经典示例:计算阶乘
    8 Q3 h$ Y7 y9 {) b, ipython
    + W' y% P" B, t* T+ Odef factorial(n):
      S+ ]# R3 Y/ ]    if n == 0:        # 基线条件7 m8 X: c+ P% j4 y6 J
            return 1
    - ]( e- u# W/ |/ e6 X( P    else:             # 递归条件
    % d/ @. x, `0 }( d2 D        return n * factorial(n-1)% t7 o9 q# H* G/ |) f: E
    执行过程(以计算 3! 为例):# M9 g( x- o# Y# N
    factorial(3)3 c# Y6 m7 M  z: ]& T9 p& O
    3 * factorial(2)2 u6 f* L9 I' @/ w. O2 X
    3 * (2 * factorial(1))0 i" B0 l# h1 o% K7 M9 b5 Q( w
    3 * (2 * (1 * factorial(0)))
    3 F# t: H9 \. Z. _1 K5 a3 * (2 * (1 * 1)) = 6
    8 M9 [# d6 M7 y' h9 _1 C7 {% @% _/ W) C: q! a  q5 G9 {
    递归思维要点8 a9 |1 s% [: Z: R
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    2 I4 a7 g, f7 F( p  h2. **栈结构**:每次调用都会创建新的栈帧(内存空间)& ?# u+ E0 i- Q& t. H
    3. **递推过程**:不断向下分解问题(递)
    . t  x6 n% F4 n4. **回溯过程**:组合子问题结果返回(归)* \. ~7 `, a) d/ X# h4 N
    7 V' @- }( f, X$ ~. G
    注意事项
    - r; }( W3 ^5 n. x9 l必须要有终止条件" G: Z% h5 A! U/ ^8 y' K
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    - }* W5 S1 `. f# `( H某些问题用递归更直观(如树遍历),但效率可能不如迭代) G9 {: t! ]  u: ^: u
    尾递归优化可以提升效率(但Python不支持)& T2 O: g7 a0 D% Q4 [" l
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    递归 vs 迭代
    2 V+ g7 n# }$ ?  ^8 f* A2 B|          | 递归                          | 迭代               |/ o) ?' O2 N3 f/ U/ ?  x9 T. q
    |----------|-----------------------------|------------------|
    ( Y4 q  Y- J8 t) \7 L' G# K| 实现方式    | 函数自调用                        | 循环结构            |' D9 }: C; L. i1 R7 j) m& w8 ?
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    7 }' h# F! p0 w| 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         |' y3 p# R3 H6 z5 `+ S: ?
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |) Q( [$ r8 i" {! Q" S* k- n2 b

    ( o' h! M7 ?' }; M 经典递归应用场景( m0 m( {1 K( Y& [
    1. 文件系统遍历(目录树结构)% L- \& b$ N4 x& K: B
    2. 快速排序/归并排序算法
    : ~+ P% U# T1 p4 t3. 汉诺塔问题  m3 H( g7 E$ v$ |+ }$ [6 U
    4. 二叉树遍历(前序/中序/后序)
    % {  k7 b) b3 b# G5. 生成所有可能的组合(回溯算法)
    ! e8 d6 v2 J4 W% K
    9 I' H/ b" m) D7 g5 S: Q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,; w6 k8 |! r* F0 l" a, k& T
    我推理机的核心算法应该是二叉树遍历的变种。
    " a/ ~$ M6 K4 v& 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:% g; n7 e6 j7 S) P) s: A" X
    Key Idea of Recursion9 {3 ^& q9 k  V3 F6 L/ Q6 o
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    A recursive function solves a problem by:1 y1 H( U/ G" O

    4 }! w  @. {% @+ M9 t, d    Breaking the problem into smaller instances of the same problem.
    0 c$ B3 W- z8 N
    ' j! c/ m  C1 k    Solving the smallest instance directly (base case).' `9 w$ X# v; r1 F/ j7 ^  N

    + e$ N1 p2 [7 a# s- |5 D/ O/ v    Combining the results of smaller instances to solve the larger problem.) n- \! f+ Q# t: [
    ; ^' _2 `" L* C5 ?7 J# a; t
    Components of a Recursive Function
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        Base Case:
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      f2 C: \; l5 [8 r0 C        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.& H  j4 Y) ~; t8 E% G

    " t1 P/ x- ^' V/ o/ t2 _        It acts as the stopping condition to prevent infinite recursion.
    - c/ Y$ V8 g0 G  |: |- A0 `6 u6 d4 k7 l
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.) h! z8 O: E8 C/ b
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        Recursive Case:
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            This is where the function calls itself with a smaller or simpler version of the problem.5 r/ q9 K7 g$ R& ?6 r) G- j
    5 q; X5 Z* p7 J: Q, Y
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).( M6 E; Z/ ?' O+ U4 _

    8 Y: ~6 N' |/ P* n3 UExample: Factorial Calculation; f4 ?& K6 O  o: l( a$ X
    ' _* G1 h: y4 S, \, I, s6 Y- L
    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:; r' w4 i" T3 X1 S1 J, O7 G

    7 ?% |9 m; H$ r/ ~) ~& C    Base case: 0! = 1. V$ X9 J4 \. |) O: N

    : F$ Z3 N! A: V4 r7 J; c    Recursive case: n! = n * (n-1)!! o7 @9 ^# r9 `8 M0 |

    % K- L( Z  X/ \9 P+ L8 EHere’s how it looks in code (Python):
    / M1 u* y2 O3 U! ?6 W# h. O3 Ypython
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    def factorial(n):- x, B: s, I* x* b, E7 z' ?
        # Base case
    2 C. Q$ _" I% P3 J" ^# v; ~    if n == 0:% g& O; r, h7 J# t
            return 1. w0 R( y& M" _$ M" j
        # Recursive case
    ( a" p: w! j4 q( m+ l    else:/ ?- d) p4 L- G: ?2 N
            return n * factorial(n - 1)
    ( e/ P4 i; Q* ^& |# w( r6 ~1 G4 A1 S) q7 ^
    # Example usage
    1 k) C9 X7 |( e. w* X0 _print(factorial(5))  # Output: 120  T4 ]6 r" C5 y  S: }
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    How Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.
    $ p6 l& _# E" Z
    4 P9 ~+ F& D3 Z: U+ |    Once the base case is reached, the function starts returning values back up the call stack.$ b1 K- X% `5 `2 ?

    ' z$ G- I1 ?2 Q! m1 ], s    These returned values are combined to produce the final result.: m- {$ i5 Z$ k$ x' x
    & r2 C: z. E+ b, p
    For factorial(5):" S; q+ Y' I6 X) h9 r

    6 y5 \. j, Q$ {9 {; c; L% V: N3 b
    3 p- x9 E( w( `factorial(5) = 5 * factorial(4)" e4 ]9 a' S1 z% N+ q/ F
    factorial(4) = 4 * factorial(3)7 K2 {" k0 X5 U1 u+ H
    factorial(3) = 3 * factorial(2)
    9 [9 p$ W3 G& bfactorial(2) = 2 * factorial(1)# s9 y/ o' j; @, `6 b5 U
    factorial(1) = 1 * factorial(0)& ]# l0 r. f" F* f/ N
    factorial(0) = 1  # Base case& \) {" Z2 e# Z/ U

    ) |, d( N1 D8 h4 X. j: _Then, the results are combined:9 a( d( \6 V8 I5 \

    $ H% a1 z+ K  Y2 [& n8 K3 r8 B0 V+ a' ]
    factorial(1) = 1 * 1 = 1; }  e( v' n* P' e7 H4 j: ?
    factorial(2) = 2 * 1 = 2/ z, |+ p( N1 i9 E, A, t( u% [$ [% z
    factorial(3) = 3 * 2 = 6# C0 S* H# T' D
    factorial(4) = 4 * 6 = 24$ J% |) q+ k4 ^* x! `4 R
    factorial(5) = 5 * 24 = 120; h; |% g3 @6 _6 [2 X+ f+ @  g5 [
    6 x% }' R2 E* I# K- U6 K
    Advantages of Recursion
    3 y9 u5 ~, f# ~0 ~9 t. D
    2 h( K3 ]! f- l; B- e1 i" I: I$ D8 P    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).# u0 f. I; y( Y) \
    6 W; O2 U, N5 K: t
        Readability: Recursive code can be more readable and concise compared to iterative solutions./ y5 P; r; G1 }

    9 w( L( D+ u- P) r1 [Disadvantages of Recursion
    + l: ^+ P! ^2 b% E+ r
    3 N5 ?2 U  v: e+ j; q8 z2 W    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.
    + A$ H, ^5 X9 v6 Q+ a) i, Y! M" F
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).) J2 l1 L. p0 h+ L. J0 C
    8 R0 H- w' X2 k
    When to Use Recursion/ S4 z# _' s2 u+ _  A; U8 r

    7 a- ~& q) @0 E  V9 {    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' ?& U3 y/ b& Z- z- }) T
    ) W" C/ a7 _; c! i
        Problems with a clear base case and recursive case.) L! ?: Q, W. s! [

    % V! c' L) l9 u7 B  b5 lExample: Fibonacci Sequence1 b/ D- c$ o4 o7 \7 N* f

    ; ~! }: v0 T5 ?2 K# S0 dThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    5 f8 B% ^$ X& r! u
    ! e6 b1 a7 W$ G1 u5 o  _3 _/ U) q    Base case: fib(0) = 0, fib(1) = 1
    ! ~, ]( c0 [" ~5 O2 a2 v# u$ W2 e4 V8 t% E5 t7 N
        Recursive case: fib(n) = fib(n-1) + fib(n-2)5 U8 e0 x$ J. e1 B: a2 i
    5 W5 |2 U9 Y( _# d
    python
    ( e0 r4 C- v8 c1 C2 a8 {: l
    " U8 z- ]" v. N; B5 `" q7 J5 Z1 l$ I! s5 H% Y
    def fibonacci(n):
    $ v7 p  i& U% q& N; k" w/ N    # Base cases" q4 V5 `7 t, p; f2 E& P, V
        if n == 0:) E4 X* Y! d4 Z7 H
            return 0" K; L% e6 k. K* w( E) S0 \4 X
        elif n == 1:
    + l' T* j+ \8 e4 z        return 1
    ! u  F# A! k3 q, A% u6 i  ]$ g6 T    # Recursive case
    6 p% M5 y6 U- u9 U$ X6 I1 a0 w: V    else:2 o) q. {9 U2 i9 |; C( m
            return fibonacci(n - 1) + fibonacci(n - 2)- ]' e$ V) Z4 a. n

    ( E, H$ B2 R# {- E# Example usage
    # ?) W( G. v# d- ?5 [- o" rprint(fibonacci(6))  # Output: 8. R2 m7 F. m  |6 f' S/ [

    6 G2 a/ x: @( U7 d7 z  h5 STail Recursion
    ' G: u+ b: I% J3 q  F: L) s+ a  [# \; \
    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)./ _5 ~* ?7 ~# e1 U, ]- O& C. m7 x' E
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    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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