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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 ) z5 y' N0 W9 m3 J$ x" @! K. }- e

    ; o5 j: n- T& I, S解释的不错
    ! B7 G& M9 I- c9 K$ m9 D7 f; g9 ?% T4 K1 j3 v
    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。
    - p2 Z3 O& e0 |2 i: g# G
    - Z% @, v& O/ z" V 关键要素
      f! v, |6 {1 S% Q1. **基线条件(Base Case)**2 e: z. N" B3 x6 R. O
       - 递归终止的条件,防止无限循环- W7 V% K3 D  l0 k; Y
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    1 j2 V. Q6 _- U9 h3 L! H3 Q* x/ m* _8 t
    2. **递归条件(Recursive Case)**8 \! g) ^# c9 Q% W9 O# u- L4 m6 P
       - 将原问题分解为更小的子问题
    2 M. ]1 x$ X$ Z/ \   - 例如:n! = n × (n-1)!
    : X* I% g$ T3 l
    ; A1 F; l' K* a- J& a 经典示例:计算阶乘% ~# F" C9 r. m; K
    python
    ! m5 \& ~( e' i+ F8 G. Z' rdef factorial(n):
    9 \/ U9 O$ X3 Y, A% R8 O    if n == 0:        # 基线条件2 A( D  @1 E' c: E" \0 R; G
            return 1
      Z: w; x9 t1 M" l0 E9 a, ]    else:             # 递归条件
    & D$ E$ \0 s. Y& L. {# ]2 C        return n * factorial(n-1)& V0 ^4 d# D, T9 v
    执行过程(以计算 3! 为例):
    ) X0 _  p0 T: C: G" lfactorial(3)
    ; R0 y3 E" F$ s; k+ ?; y$ C+ T) [2 K3 * factorial(2)5 [* ^, N( K( _# B/ s; ?4 e
    3 * (2 * factorial(1))  a& s4 L% `* i7 M2 Z& N) I
    3 * (2 * (1 * factorial(0)))
    5 ?- {  \! \% ~3 O# k2 r2 B3 * (2 * (1 * 1)) = 6$ d! a. R2 ]1 d1 b# w

      X. z2 Y) k1 r2 h" i0 |8 F 递归思维要点# h6 H8 \8 b" Z
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑9 ^! \8 ?) j7 R( U* y; ?/ a& {
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 m# L* z, {- j
    3. **递推过程**:不断向下分解问题(递)
    1 X7 U$ @- ~; Y- ?, S! O4. **回溯过程**:组合子问题结果返回(归)
    - O4 P% w" n# [8 T2 k/ ~
    % Q& I$ H8 I; `& t 注意事项' I* v* @  P7 _; o: i/ ]
    必须要有终止条件
    8 F" d/ e8 W7 f3 M6 @递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  S. k1 n" W" n; s1 P0 r+ N
    某些问题用递归更直观(如树遍历),但效率可能不如迭代. s) W/ C' H8 F
    尾递归优化可以提升效率(但Python不支持)
    $ G6 T9 w/ A$ B4 `3 N' F: n. f) t7 ^* b5 F+ g4 F, r1 ]+ y1 Z9 T
    递归 vs 迭代5 ]( b% B; E6 A
    |          | 递归                          | 迭代               |
    8 |+ S- x/ _9 ?; e$ ^|----------|-----------------------------|------------------|
    ! U0 |6 i8 N( R6 ?, c# A; R| 实现方式    | 函数自调用                        | 循环结构            |
    7 l, X5 I- e, M4 M" @2 N3 V6 i| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |6 e& g. k2 V5 Y9 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         |* E8 ?# j9 W* L
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |1 j2 U% M+ B0 H) {, W6 O

    ; G! J3 @( |! ^# T7 M 经典递归应用场景
    1 K- D7 {% ?3 @6 L1 h9 n- W1. 文件系统遍历(目录树结构)2 C  b& U- B  {# q4 W; W
    2. 快速排序/归并排序算法2 ?/ p( V# C- ], s% O; S
    3. 汉诺塔问题
    9 [5 M5 D* ^  I" p  h7 L& q) m4. 二叉树遍历(前序/中序/后序)
      [% k, R9 C3 D. }0 I$ z3 }+ u5 Z5. 生成所有可能的组合(回溯算法)0 ?2 I: W' [( C5 C, W- V; i; E! C

    . y" F: r9 e* O  R& i6 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

    3 天前
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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,# D- r$ b, k4 P
    我推理机的核心算法应该是二叉树遍历的变种。
    6 Y7 r; C& i3 j. N' o9 Y! g& w6 X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:2 N- I8 g/ X4 u
    Key Idea of Recursion: C+ _7 v0 X1 e- h" R0 v2 A5 h

    5 h+ K  a4 I; T  Z& \  A$ i9 I7 ?A recursive function solves a problem by:
    % X  l8 q) |  I% o3 M# p& l# c5 D4 V9 e/ X' k% ]3 |
        Breaking the problem into smaller instances of the same problem.# V. Z4 C" S! q2 ~* q/ F. p

    ) |# M* _" h2 W    Solving the smallest instance directly (base case).
    5 Y" I3 S0 s2 w! c4 U1 s) m' }, I5 D& h9 @! w8 `
        Combining the results of smaller instances to solve the larger problem.
    % W9 R9 W/ i" a0 }% d% b2 B3 T' {: p  X. u
    Components of a Recursive Function& G% v5 S" S8 e3 V
    + s& b) |% B$ p) P, a
        Base Case:
    8 G) I5 z* r! o+ t) p) K6 R' v* y/ R. ^; z) b2 ?" T
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.' g1 K& h2 [& E0 u" b6 p7 {

    0 |# V5 a- B7 v. B+ k        It acts as the stopping condition to prevent infinite recursion.
    9 T. U& B- W1 Y; ~6 P# R, h5 e1 Q& u% c+ Z% Y  T
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ y* B2 l5 o- Q, ^

    - c" I& _  e0 M! z4 B* H    Recursive Case:
    + ?8 W  N  L% w5 x" M
    ' X& p/ U: x- u; a3 }" V2 N: V        This is where the function calls itself with a smaller or simpler version of the problem.0 i2 L% \5 P  M, R$ X
    % \6 g1 E5 \+ T" s: z  r% n
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." M; u0 c' H! f; j3 y/ s

    ' \5 Q! c/ M" q: Z& y& b  {Example: Factorial Calculation
    & @8 X. Z5 q0 p8 \( S, T- @  m0 A& A" y' `! R8 n# i7 J! v1 T2 w. @; q
    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:4 c. w, h. y0 h  h$ Q8 T

    0 [: ?$ O. ?' z6 S" K9 y4 i% Y    Base case: 0! = 12 |7 F+ z7 |" o8 ]) Y# N5 N/ P/ E

    : N6 g7 M, [' U4 E    Recursive case: n! = n * (n-1)!$ h" A: q+ G$ P& N2 q# a, N& H
    + z' D: _; ]7 G' D' G) ^
    Here’s how it looks in code (Python):+ y% `2 e1 q* l
    python
    - g/ o# u$ v6 `- V7 h  ~& q, R- k% t! A. M
    2 L3 v4 v" l1 b4 u& {2 o6 a% T
    def factorial(n):* J* _; T1 n5 W* d5 h% H1 F9 ^
        # Base case
    ' C2 I5 O1 `! r3 ]( \) q    if n == 0:0 {7 x: W3 X, x$ K
            return 1# V5 ^5 E: }$ c4 {
        # Recursive case; l( `; E/ Z( T( p
        else:
    7 q& f( w* K8 T        return n * factorial(n - 1)
      n$ c  Q) |6 S! E% x3 g. w5 s) O. A5 k# P' _. b( ]
    # Example usage
    + Q& }& T4 J/ K  N8 T3 uprint(factorial(5))  # Output: 120
    9 W1 i6 A; t$ a$ [$ j* ?0 {; @" D/ o/ @: B, d% n
    How Recursion Works
    4 t% y" a& E3 Y* O: c) U5 ^7 L" o
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ! `$ O; F3 |7 U8 Q- F; [% K) i6 R' F
        Once the base case is reached, the function starts returning values back up the call stack.
      L* j0 j2 u1 G  R* e& [' P  k( z: Q
        These returned values are combined to produce the final result.2 _( @4 }6 E9 q5 g

    8 G% K. u# Y/ i. E9 l- ?5 Y1 lFor factorial(5):, n9 E& S( m4 Z
    8 G8 s" b6 i# V; {

    ; I  x4 I2 K& S! q/ M/ V0 Ufactorial(5) = 5 * factorial(4)4 w, E9 d% h& l8 A  J1 C* L
    factorial(4) = 4 * factorial(3)% q# T  P1 [, X, p( ]
    factorial(3) = 3 * factorial(2)
    - N: I0 W$ m) ]$ P  Zfactorial(2) = 2 * factorial(1); l9 V/ \9 {0 [/ c) ]6 S5 V. o2 F
    factorial(1) = 1 * factorial(0)
    : S1 \0 R( w3 T; O, @0 ]factorial(0) = 1  # Base case  V4 Y% J% G. N( ]8 Q7 d! U2 h" L
    8 X) F' g& c& t1 p! [! }
    Then, the results are combined:
    6 x% Z3 d' P8 u# ?' }- |5 k1 d; ?- K" ?/ l5 i
    % f4 D3 f6 ~- d7 c: ]
    factorial(1) = 1 * 1 = 1+ A. b# a6 ?3 i; v
    factorial(2) = 2 * 1 = 26 L% i  D$ Z( {& I4 H/ a) d0 R6 V! c
    factorial(3) = 3 * 2 = 60 s  b0 C0 `' A
    factorial(4) = 4 * 6 = 249 O+ o& q: b  X
    factorial(5) = 5 * 24 = 120/ ?, D( a5 l9 M& l- F* }
    * W. y. K1 q9 Z+ K0 |+ L$ j& O
    Advantages of Recursion
    2 ?2 j6 R, i) X' E& p: b0 h6 `$ C' S  n* @" O9 J
        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).7 v% B  i; S9 `1 c; V, F) d
    6 L, P4 @; V+ q+ N
        Readability: Recursive code can be more readable and concise compared to iterative solutions./ O- `+ B7 @& ?( v. |, F

    * |, W$ ~' |/ V! k9 }Disadvantages of Recursion
    ; P/ F. e* ]( ~- i" I$ ]7 m$ ^2 b  n- S  j; F; C+ G
        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.4 A7 \4 e4 k4 H7 n0 L: m7 s- r
    & L" b& K: P- y! p
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 m4 y% e+ e- i4 N3 q
    - x7 X, ~) }: a  k3 w$ @# @
    When to Use Recursion) G% A& o/ L6 L# t
    4 U$ @2 F9 _0 k2 d$ K
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).5 f, z8 b, E5 a5 I) E
    1 {, x( ]$ w8 O/ g. b3 G
        Problems with a clear base case and recursive case.
    & O- W4 Y7 I2 r' Y  C7 l9 x* C4 R5 \
    Example: Fibonacci Sequence% [, e3 R* r& p5 k. q6 O

    / n# |+ B5 e% O! k9 K- |The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:! g2 |( Z9 e6 H- j+ j7 D
    6 F% |/ V: n3 N! a
        Base case: fib(0) = 0, fib(1) = 1+ p7 Y6 \" M9 I7 B; B

    # f& `4 ?7 F( h  Y( {0 u# h    Recursive case: fib(n) = fib(n-1) + fib(n-2)9 {5 |) x8 U$ [
    3 y9 t8 Z+ j$ w2 ?! C
    python
    , Y$ W( e8 d% L0 E  x. P/ T  ^. S1 y6 T( x/ f

    ' t: y6 v2 e# A4 l% n, r4 S' fdef fibonacci(n):
    + r+ I% |. m1 w0 C    # Base cases
    , D- D5 x7 s( n7 v# t    if n == 0:) _7 a' M' C9 H
            return 0
    1 f& T3 a2 a7 c" E" l6 l    elif n == 1:; b# u5 [5 h9 p( E+ S+ w) z
            return 1% d4 X3 y, R4 {& }( \: [; D
        # Recursive case/ n. n+ v; c+ U: }
        else:- o& {, O8 [9 \. A- w. R2 M
            return fibonacci(n - 1) + fibonacci(n - 2)
    $ }7 m  |5 D/ ^+ d1 f3 @' X5 l9 w# ~. Z8 j+ s  |2 W9 {' [. w2 J
    # Example usage- Z7 y! f* G1 o0 s9 ]
    print(fibonacci(6))  # Output: 80 e# z. {4 I  b! F6 s) {

    / k2 R+ f$ R) Z6 u* z9 wTail Recursion
    , L( b9 j- M; |. M, T  E
    - E) U: p3 s6 W# LTail 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).8 B5 u% y# z+ i: p

    ( Z6 O0 j- S1 r6 D4 \" `3 \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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