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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| 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   b" j) X/ M" m3 J7 v
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    解释的不错
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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。
    ! F. j- V( p( R8 h# c- [5 P
    8 y" x# C7 k- Q: A% V; \7 U 关键要素
    : U3 a9 V% f) z1. **基线条件(Base Case)**
    0 U7 w' R; k* _* r   - 递归终止的条件,防止无限循环
    6 G  a6 q; p3 ]  q: x- ?6 x) D( }$ ^   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 18 r" g) P0 a+ {  a+ {

    % L( o4 o6 I; E, x; d- l2. **递归条件(Recursive Case)**
    % _! x) k2 p0 Q4 y) c, G, e- q' o   - 将原问题分解为更小的子问题& Q: T; M) d; h$ h1 @) W# x9 B2 G
       - 例如:n! = n × (n-1)!
    . ]- k' t8 I8 X. m1 g- Y: F2 j
    3 V' [% d* B2 p" B% n 经典示例:计算阶乘
    , q0 S$ G! `( p, T" }) c: I- mpython
    7 Q. u2 v) J9 j9 c; bdef factorial(n):$ [  D& B3 t% p8 X, [0 j
        if n == 0:        # 基线条件1 m& D6 z/ G7 O' t) S$ @; R/ b- K
            return 18 B: U5 n8 g5 ~  f5 S4 p
        else:             # 递归条件
    " \7 p0 K( t8 g# ]* X        return n * factorial(n-1)" N7 k! v- @% G. `
    执行过程(以计算 3! 为例):$ W6 `' U. ]$ p
    factorial(3)
    ; Z- g8 d4 h6 \% l3 * factorial(2); |( l( `& m* J4 m9 I
    3 * (2 * factorial(1))
    " r( p3 C" N! l3 * (2 * (1 * factorial(0)))
    5 ^/ P! x. H+ {; F7 M1 ~3 x7 q  I+ B3 * (2 * (1 * 1)) = 6/ E) d& N. J% A2 z- m
    & a; U: X9 ?- E- Q8 d9 \
    递归思维要点
      Z3 C& h! {' j: C7 V8 u1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , Z. ^( s# p% }" j$ q. |2. **栈结构**:每次调用都会创建新的栈帧(内存空间)# K1 i! e5 k  Z$ P! e; C) N$ v0 I
    3. **递推过程**:不断向下分解问题(递)8 X& f  w8 D& _; v1 a
    4. **回溯过程**:组合子问题结果返回(归)3 ^. G/ d! N" ~/ R& Z2 _4 T* w
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    注意事项
    9 K9 c/ a; j( Q3 l# L' F必须要有终止条件# k4 q  Q" i' t( n0 {. V
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)6 S: L8 S# Y( b* s5 `
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ( S+ Q& r: q' M尾递归优化可以提升效率(但Python不支持)
    % y4 O& X9 n" O" O: f5 X  `# N6 K& B. K9 r7 ^! ?4 U8 @8 k  T% p
    递归 vs 迭代3 [4 p! M. t7 D) z7 r& ~( C2 |
    |          | 递归                          | 迭代               |6 y( I! }7 y; B
    |----------|-----------------------------|------------------|0 _) m. {" @; P4 m! {) [' t
    | 实现方式    | 函数自调用                        | 循环结构            |, e1 f# U. l8 ?+ }6 I& j
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    / P8 G$ c5 \& X$ j' |2 A' A| 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         |
    - j2 Z* ~5 q; H8 @6 `, {| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& s! s: J  w8 G9 h3 {
    , c: {7 w" t% X/ J
    经典递归应用场景
    1 I# F/ W; A1 U9 H$ U1. 文件系统遍历(目录树结构)
    ' ~$ t& ?# X& L( H3 Y; L2. 快速排序/归并排序算法. }( b+ m; M$ W4 d* ^
    3. 汉诺塔问题  _" a) b  ^  z3 E( }0 J$ g
    4. 二叉树遍历(前序/中序/后序)0 h  M+ {- F+ H
    5. 生成所有可能的组合(回溯算法)
    : A5 S3 @0 e# q6 P% h* k
    * U" B' w3 p; J! }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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    郁闷
    1 小时前
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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,) b* @% }  b. p8 m; S9 m# F* Y9 Y
    我推理机的核心算法应该是二叉树遍历的变种。
    $ }$ X" R: x! s& s- \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:9 V8 C, }9 V, s8 {
    Key Idea of Recursion
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    . F6 K1 Y7 R; I8 v  K; hA recursive function solves a problem by:
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      R3 f) }8 o) N, G& M8 u) I) p& r    Breaking the problem into smaller instances of the same problem." G+ V. ^7 J- W1 k9 x, t" I7 J

    + C1 [5 G, X( L% y    Solving the smallest instance directly (base case).1 e- E+ A; q: H9 v

    ! S5 w7 I3 u) d1 C7 ^' `    Combining the results of smaller instances to solve the larger problem.1 S- G. b& g1 C5 h
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    Components of a Recursive Function  m( {% N  K; U7 s9 r5 x

    " g' ], P2 `) _; @- n7 V) U    Base Case:1 i( K4 }4 a  O0 y" D

    / ?& A9 H: J0 f7 @. @( R        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    / f/ Z) l- `/ [5 w5 `- ^: Z' z; R1 q4 |  e& w( L3 N) D4 U
            It acts as the stopping condition to prevent infinite recursion.
    , D! q- z# Z* |9 X: p, D+ U1 D2 ^
    % z! T( x4 P6 U4 h: H9 ~" }/ \# M( Z        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    / v9 b1 H) `6 _
    : t* S+ u$ m' o% D5 P9 w5 ^    Recursive Case:; b. l" E5 V; M1 _5 J9 H1 y6 h( F5 S
    # ^0 j* F( C/ y& v
            This is where the function calls itself with a smaller or simpler version of the problem.
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    ' ]& K$ {5 y( g  m0 P        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation
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    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:0 ~; L1 D* k) X; K+ U! Q9 D+ N
    9 h& z5 r2 I1 q2 i, Q
        Base case: 0! = 1( Y- E. D2 @$ |0 D1 i

    ) J; N" G) ~, D; i. l    Recursive case: n! = n * (n-1)!& Q7 e. h( _/ I* s+ S" i
    + W  w% k3 y. y4 u
    Here’s how it looks in code (Python):8 }! G7 ?0 A% @* B# f
    python* N* P2 y4 N1 _: ?0 y

    * s8 Q) g! E) ^9 ?) k, I
    1 l! W! s+ @9 a& c6 P$ q$ T% H# ydef factorial(n):, ]4 m1 t' ~: d
        # Base case, @3 J- o3 E8 H7 X6 g5 o' Z- Q3 {
        if n == 0:
    6 o: R& W- ~4 F9 L& \        return 1  T& A4 X) ?) Z% K4 B) z8 e
        # Recursive case
    2 ]* Q, Z) T* O  r2 d    else:
    + _' c6 k4 \" R, \* t        return n * factorial(n - 1); S( f3 L9 G' u6 R; U+ i3 d* z
    ; W$ H; c# D3 p. I: X
    # Example usage
    ! t% B, z7 z4 O5 U* z! sprint(factorial(5))  # Output: 120
    : Z/ O, o3 _/ g3 m" K* J/ y# h+ J! ~3 C6 n2 F, D" P" }9 l. k! _
    How Recursion Works9 G1 U/ J! `0 \  s5 Y- i  Z9 @
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        The function keeps calling itself with smaller inputs until it reaches the base case.6 D' g1 x! W! |# Q/ c
    + i3 J) J, K1 `$ n3 c6 q+ {9 \
        Once the base case is reached, the function starts returning values back up the call stack.
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    # M/ x4 Z1 `2 }! P6 C/ O' [    These returned values are combined to produce the final result.
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    For factorial(5):* Q9 S/ ?+ g; l' C& ~' C" V

    " e! F$ S3 r3 G4 ]
    , n8 C6 T5 B( Tfactorial(5) = 5 * factorial(4)
    ( `7 I" O; O7 L. F% l3 `factorial(4) = 4 * factorial(3)1 \% V& i: t& u' J" X+ }
    factorial(3) = 3 * factorial(2)
    & g: t2 U2 p! y- U' G4 lfactorial(2) = 2 * factorial(1)
    & _% j5 I! k3 U, n1 qfactorial(1) = 1 * factorial(0)/ d; \( e; }( r9 b* L  w1 R
    factorial(0) = 1  # Base case
    . W0 ?7 \9 w) {5 ]. f# S, M  Q, V6 n. z
    4 D1 U3 i. P+ u# P, }Then, the results are combined:) L& ], o1 ^* g0 \
    % c. Z; t7 j9 d% m+ T& m8 W+ t
    - ]  C/ ~. [# A6 m7 {# ?5 U4 A8 y8 T
    factorial(1) = 1 * 1 = 1
    + ~7 C) \; i4 P4 S$ d" L) v' Efactorial(2) = 2 * 1 = 2
    ' p  P/ M5 O* ~% gfactorial(3) = 3 * 2 = 6
    ' ^9 P1 w- f' d* wfactorial(4) = 4 * 6 = 24
    1 P9 d1 B4 O' \- i/ g" Cfactorial(5) = 5 * 24 = 1200 ^5 ~( ]: ^' o- ^, a" {  _

    ! K( W' o" J4 Y3 KAdvantages of Recursion7 f) }* T4 u. M/ m1 G5 x* v
    4 \9 {* c2 }( p' {" `4 {! E
        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).  X. q" M- y% t( `& v" E" A

    ; ~4 o/ i+ W' _- X* v. p" P    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    $ U7 G0 g' V/ \% O: u, ^$ @6 h, R6 Y6 ^& A* _) ]
    Disadvantages of Recursion+ e. k0 c" E! M/ i

    " f& _9 G: t; q% Q+ u) J5 b    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.! Y. P+ i$ q" J6 u0 q
    ' J$ n$ j: c* g, A+ o# i! B: r1 ]
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).- A7 y+ Y1 j8 ~# z

    1 I' ]7 J2 t- Q! E  l# {( ?When to Use Recursion4 f5 b  U2 u0 ]" V
    4 q6 c: x8 n! \) `: B( @# O- m$ w
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    - Y3 Q2 f, p6 A4 x5 l& o
    " M2 W! I( W6 a0 {: m3 \; h+ `5 I: [2 f    Problems with a clear base case and recursive case.
    * W  \. [1 v& X, n  i
    " D+ h( K8 a& B8 H4 dExample: Fibonacci Sequence
    2 I, W" \/ `: p- B. H8 p- [
    9 q8 C& V( d0 o, s1 SThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . C' |3 z$ f7 B4 l  h" N/ n& @
    % d# M5 k5 m% q. p- b9 E    Base case: fib(0) = 0, fib(1) = 1
    " H) z; f8 W6 G2 M3 V8 P
    # c9 ~" {0 D6 s! d9 ?2 q1 j    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python5 C' w1 q1 ]0 H. D/ U

    ) S+ j2 D: |/ a: J$ a7 J6 {5 p# ?) i- A0 D0 L# O% B5 F( z- n
    def fibonacci(n):
    * W( {6 v. R) g! g6 F0 I" A) M  F    # Base cases; s' Q+ R1 r, m  q+ E5 M
        if n == 0:5 S1 k% }% h& M2 W- T
            return 0
    * v/ k+ n+ I# ~- j+ L    elif n == 1:
    ) @! M; J3 _2 D! }$ |        return 1) {' |& N0 N( e6 L  [) a
        # Recursive case) Q$ o  _) _: B3 V- p  i, D1 S4 Z4 |; z
        else:
    . V2 w6 ~% z% b, g# a0 D2 ]        return fibonacci(n - 1) + fibonacci(n - 2)
    & w& p* K- [7 b* M
    " Z9 ~' x3 N8 U) I/ Q+ q! ]* z# Example usage6 i) Y+ K- k1 C6 }! t' B) M
    print(fibonacci(6))  # Output: 8
    ( M" |# p/ u% `* L1 [
    / Z4 k- [7 G3 l6 d+ `1 u% i# ]- TTail Recursion  S' d9 l  K! f- Q3 w

    8 u; j5 |, X3 t7 KTail 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)., G2 }. s, {. V
    + T7 p! r8 _, w  b, s+ ~
    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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