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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
  • 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 5 c( ^  T6 y; [# ^2 ^$ v

    ; A" x! I* K; P  c4 |+ E解释的不错" l$ d# n: N' e  y: e
    ) C/ z- p4 |+ T  @5 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。0 s4 y9 P) Z0 J1 q; F7 P2 N2 _- P. n
    + g' Z# B/ ^. e
    关键要素) [: l% S5 q7 B, f
    1. **基线条件(Base Case)**/ c$ A6 E2 K- f2 g$ T! |( c
       - 递归终止的条件,防止无限循环
    % |- R; I! L, n+ \   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1! |7 {( E. Z& W" A* c, A; _

    ' G" O" p4 _! d2. **递归条件(Recursive Case)**+ C7 {2 B4 C* B& A# B  c+ S* h
       - 将原问题分解为更小的子问题
    8 f/ }) [: j3 G8 G" ^" x' f   - 例如:n! = n × (n-1)!
    0 `! F# t! w+ f3 c# j5 K( m8 U4 G5 }& F8 @. o4 q2 e
    经典示例:计算阶乘
    5 n- Q8 v+ r2 Z- N& u! \python
    , a3 e" G1 z: Adef factorial(n):
    ' Z/ k0 K6 M. u2 n    if n == 0:        # 基线条件
    5 E( D4 j7 Q' d# |0 C7 W1 |; S        return 11 x8 [" ~% _( V: N% ~
        else:             # 递归条件: F; {: a  H  K3 ?; r4 l2 E( p5 ~
            return n * factorial(n-1)
    - q/ h' ?2 K& i0 {执行过程(以计算 3! 为例):
    8 J4 F+ D( \. gfactorial(3)
    4 z" A6 ~* }5 q$ [! \3 * factorial(2)2 ?% M. R/ U2 Y
    3 * (2 * factorial(1))
    % u2 ?2 w4 c0 c3 * (2 * (1 * factorial(0)))  h" Q! F4 A1 G4 A3 g' M9 D
    3 * (2 * (1 * 1)) = 6
    ' }5 E! t/ m! |. Z& r0 j# n- M' [! h5 D9 ]$ q5 _$ a: L9 }
    递归思维要点
    7 F* X1 Q9 s# q( {9 z7 r2 L1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    & {1 [' a/ M( Y4 S. c2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    6 W$ X, y- ]) J: F  ^3. **递推过程**:不断向下分解问题(递)
    8 n  W% t8 Q+ U6 v4. **回溯过程**:组合子问题结果返回(归)
    + @$ k* [5 G3 T1 y
    6 O( h! x4 Y& } 注意事项4 D& B" c3 ^0 _2 k9 ~, y
    必须要有终止条件8 b0 @$ E9 A( t
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)* I3 O2 y- }+ o$ [  c
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ( T9 m5 u  {/ j( l+ A% X1 s! \0 o尾递归优化可以提升效率(但Python不支持)
    - K+ s: O, P1 }- u/ C4 X) T
    1 g# `8 S/ Q; e 递归 vs 迭代
    # D6 Q! C1 a3 w5 V# `2 A|          | 递归                          | 迭代               |
    # Y5 G. ~; O+ s2 Q/ C9 q9 y* t# y4 g|----------|-----------------------------|------------------|: A9 [. y5 S! I& V- q/ r3 [
    | 实现方式    | 函数自调用                        | 循环结构            |
    / P+ |' ?. z. p4 p1 h1 `| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 O. {8 s6 V5 W, O0 I! {& q
    | 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         |
    ; o* z* F& U: [  v| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    5 d1 @; f# o  S# \4 G* h
    / I$ ?, V# S$ X. s$ @* T2 D" e 经典递归应用场景& O( a# s- i) p/ |! [
    1. 文件系统遍历(目录树结构)
    ! R$ h) p; W8 \: K& `6 b2. 快速排序/归并排序算法, i4 i( X1 b: D7 \7 v6 n
    3. 汉诺塔问题
    ' {. i$ I' Y+ o& K, F4. 二叉树遍历(前序/中序/后序)1 X; f5 `$ r" T$ ?: [, b7 z; a
    5. 生成所有可能的组合(回溯算法)0 k" P2 D1 h6 x! C7 S

    * Z, I$ A4 z0 L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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  • TA的每日心情
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    昨天 08:23
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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,
    9 v4 |" l$ b0 n7 R% O我推理机的核心算法应该是二叉树遍历的变种。
    6 ]8 t7 L- n6 n  u% E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:
    $ ~6 `! K6 w! K* {  \) OKey Idea of Recursion
    ! o; \  c" g0 }/ N; a+ W. V, H; E( W. I, N1 Q: o5 o
    A recursive function solves a problem by:
    4 M* l2 h0 P6 e$ k/ W' y' _9 ]' V6 i% j+ T5 m  Z! u# ~5 c1 T: F
        Breaking the problem into smaller instances of the same problem.# O& j, r  i+ O' ?, \2 ]3 m0 ]

    ' X1 c5 [* }' S6 y" ?    Solving the smallest instance directly (base case).
    & j$ a$ @1 N- H; q: F
    7 R6 t/ ^' S; P6 _$ w( ^    Combining the results of smaller instances to solve the larger problem./ w. V1 D5 D( e! V3 n: ?

    5 ]8 q% p+ F3 F! z# t# d$ P+ T  PComponents of a Recursive Function8 b( ?- w8 x) ^' I1 r

    2 M4 d* L" y  o& w! v    Base Case:! d$ j1 `4 i% [: U1 S
    * v5 t7 S6 D0 u4 \
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.5 w& P% T6 b- U8 W0 q" P

    ( K! T2 C; S% k3 }8 I5 h6 q: `/ p        It acts as the stopping condition to prevent infinite recursion.$ X# O! c* P/ [5 F# Z1 @% w

    9 v* d8 S+ f* K+ D) A* @' {        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; i( F; j* ~9 \# f; A- w

    ; [* R9 q3 s; y& {- f  M) Z* u    Recursive Case:: R, C/ c& w# r/ ?' ~; X
    % v* k0 c2 B: a7 g* a) A
            This is where the function calls itself with a smaller or simpler version of the problem.% k! y* D: O4 \' M0 W6 `+ C
    * K) x7 S) A/ I" w% F8 q
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).# [& N7 y6 F$ m, i) Y

    ' X2 ?7 @' V2 H/ \7 l) |Example: Factorial Calculation) I( u  N" T8 m( a4 B1 q

    ; j1 Z7 I. ^* y  A2 V0 h& yThe 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:
    & y% n+ r. F4 A) r; [% @: U: t( p
        Base case: 0! = 1
    # r  Y  M* y4 c/ b4 \. p1 P" O
      [4 p0 e, U: r: l: J    Recursive case: n! = n * (n-1)!
    - n0 b# M1 j& g: T
    + H6 `; b2 {! K- d' b2 m& e. N, X! iHere’s how it looks in code (Python):
    # l( Z) K2 V1 d3 ~' m" \7 ^python1 K4 P8 a" l& u
    ; g" ~0 R& t6 t4 `+ n( E

    - L3 O! B1 C, s" qdef factorial(n):
    7 _7 j& k: \5 D/ [7 {9 Z    # Base case, |7 |& C- _' L6 y! l3 k
        if n == 0:
    * s& L$ Z4 O/ @1 f) b        return 1
    # B9 K: T; `* p    # Recursive case0 y, i. A: r+ ^0 I, B4 _- Y& O
        else:6 d/ {% B' o+ b) Y- A
            return n * factorial(n - 1)
    & N3 E& l" L! k( i6 D( I' L6 ]
    9 v3 s: h- l4 q0 U: p9 P# Example usage
    " G  g* X6 W- c1 J( L" I/ Rprint(factorial(5))  # Output: 1203 O, w; R  L9 K% b3 D4 m
    4 {8 b/ I- D% n- h
    How Recursion Works1 Y' M) g3 O2 j: [: `" a% c- m
    $ z- Z0 T9 z+ z: ?  l# N2 F
        The function keeps calling itself with smaller inputs until it reaches the base case.
    0 @: T7 \% h7 X$ J& n
    * v7 A7 Z  K4 ^6 q5 `2 E    Once the base case is reached, the function starts returning values back up the call stack.- o3 q0 c* U  H; t3 g

    8 q- ]: G( J8 R8 w* q  Q4 }3 _    These returned values are combined to produce the final result.
    ( U3 ~8 r( D. b% }' b
    9 {; o. N. H; a7 dFor factorial(5):- A+ ?4 g3 y$ \: r

    ' `6 s4 r  C# T* s' [
    ( h% R# v( k0 W4 {* W$ Dfactorial(5) = 5 * factorial(4)
    8 C0 P5 V9 y1 O3 }* ~% j; i9 @factorial(4) = 4 * factorial(3)
    ' V4 g4 h! x$ Ifactorial(3) = 3 * factorial(2)7 q! J7 Y  A8 m1 T" U- [" o
    factorial(2) = 2 * factorial(1); H4 q: `: O# b% i0 n5 E9 ^, h
    factorial(1) = 1 * factorial(0)
    # Y2 T- ^3 [% @: pfactorial(0) = 1  # Base case
    ) e  G7 u+ t% R; d' t7 c+ ]2 r' W) X7 u0 d% ?
    Then, the results are combined:2 W7 p- |( N! x
    ' h8 @) F% o$ Z# H# ~/ O- R

    " w% ^8 j$ `/ I4 j9 e" [0 Gfactorial(1) = 1 * 1 = 1& \% U$ N. f; C  o! h2 b$ z) I! V$ u
    factorial(2) = 2 * 1 = 2
    / n- p, s! o4 o+ [factorial(3) = 3 * 2 = 6
    6 H: S+ N8 b- n- @$ J( y9 Rfactorial(4) = 4 * 6 = 24
    / n  z4 r5 W9 h. @: m' r7 tfactorial(5) = 5 * 24 = 120
    & Y1 p: k# i2 A: g, Z' P. T
    ! q  W* d8 N( `2 wAdvantages of Recursion
    + m2 P: G5 y4 G' Z9 H( O# S. u; Z* I: y  s8 q3 G
        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).
    % u, R! O/ g1 w% Q9 t4 e4 S2 Q( j4 K& R' s6 y
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    3 s3 Y  L: T$ M* `; I. s: s7 ]4 N! A
    Disadvantages of Recursion9 r$ s; l/ h' W7 I' f
    ; J5 I0 x1 u, V1 \! e% M! H$ m
        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.+ ?" f8 J4 f+ n) x

    $ L9 Q/ ?' L5 \6 H7 D- r, z1 E    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    + x& O' V  p2 h/ B) i2 f( z7 k" k1 i
    7 _* d0 \2 z" W# i  `5 `When to Use Recursion; Y0 F6 N% E" c3 l# u. M

    ( z' ^2 s2 Z( _) y' O4 p+ J) ^    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ) o* s7 X& w. G8 K0 D
    + Y: T1 f: O- {& P- r7 k7 q    Problems with a clear base case and recursive case.
    1 s' C; e. ^0 V6 k2 j& s. ^  G) m* c- O4 q' ?4 d4 f
    Example: Fibonacci Sequence8 X% r$ H' b. @
    5 P- T# O8 O7 C: r  K
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:8 d0 \5 X' ~* f% o; |3 h$ w

    * J$ l1 G7 e' c' }$ a    Base case: fib(0) = 0, fib(1) = 1
    : J2 v. F- y% W- O1 ?8 c9 h  w# @/ \  M
        Recursive case: fib(n) = fib(n-1) + fib(n-2)+ \& ^6 l. v* E5 K* v
    3 U1 i- B+ u; M& u
    python* N) n4 k  a$ k& Z

    ; W% ?: v, [4 d0 o9 I3 H
      x8 j- P$ ~2 K. K4 l- D1 @4 X* _( {def fibonacci(n):
    ) ~: b7 ?! F0 V- d( B% \/ S    # Base cases
    . [4 v; ]2 o) ]6 c    if n == 0:5 p5 p: _. ?! y/ v  {: c
            return 0* X7 g3 \8 U# c* Q; y
        elif n == 1:5 k$ n5 c6 x+ V3 s
            return 1( V4 g  O* L" i2 t
        # Recursive case
    & \$ h! ~' Y2 V4 o    else:" D5 I% m% B  O: s
            return fibonacci(n - 1) + fibonacci(n - 2)
    * U" w( l9 p$ t* n1 V: q5 y- w1 }3 O6 o- r  y9 l
    # Example usage- A3 S* g/ x3 f0 A6 _4 ?
    print(fibonacci(6))  # Output: 8
    1 Q$ k6 ]8 U* g( F- L  o3 k# h' c# [" I
    Tail Recursion
      {8 O+ f9 V* g0 D$ y1 g0 z( C
    3 p# e9 t" D3 R# E% C5 hTail 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).) l. G0 P4 j5 u
    * @! I3 t0 H- Z. G. p
    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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