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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
    % M6 z2 [* R3 B
    ' v4 w) b& N8 Y7 T0 J+ e解释的不错2 r  O- a# Q5 q5 t& k1 }
    6 Z. e- {3 _: }4 v6 S$ e0 B" f
    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。9 ?5 O: r+ h, R: ]1 y$ y

    3 e' C. B9 {: p& {) M 关键要素
    9 I5 B4 y* s2 h" t1. **基线条件(Base Case)**+ r4 u' G" N/ @
       - 递归终止的条件,防止无限循环
    * c, h- ?; s$ {  a. ^8 C   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: {& P8 M( B. g$ [
    7 H. }. B6 H) x, Z- K/ O$ b  i2 }1 ^
    2. **递归条件(Recursive Case)**
    0 N5 c; f5 k, z, |  L; T   - 将原问题分解为更小的子问题. R8 C( B/ H9 W; S
       - 例如:n! = n × (n-1)!6 a2 z+ ]% j- _. U
    7 Z& `5 o% ?& f
    经典示例:计算阶乘
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    def factorial(n):
    1 C) U8 ^. |3 k( H) R    if n == 0:        # 基线条件
    3 r! }9 S# Q0 T8 h9 k7 c4 K        return 1. `( h' }' j4 ^6 ~( U7 ]+ r
        else:             # 递归条件" `: i2 @' I+ T
            return n * factorial(n-1)
    . B% S' M6 a! j" w9 N5 `. _执行过程(以计算 3! 为例):
    7 ]; T7 V7 t3 ^! y2 q- `" wfactorial(3)
    / B( V  w9 t6 A3 * factorial(2)5 P3 \6 g2 `3 _1 N: r: y, p. `
    3 * (2 * factorial(1))8 x. V: [' p! v% }% x
    3 * (2 * (1 * factorial(0)))( ^+ d1 C/ |9 a( Q& q) q, c9 ]. c
    3 * (2 * (1 * 1)) = 6/ B4 ~8 U6 q! z: f1 |/ ^2 Q

    1 r) H5 t' ^9 q 递归思维要点# r: n3 u+ E. s% g
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    & J/ I. Z5 t3 G7 m2. **栈结构**:每次调用都会创建新的栈帧(内存空间)1 {: S. C" D7 ~5 R0 ~
    3. **递推过程**:不断向下分解问题(递)
    ( X, Z8 H% |% H4. **回溯过程**:组合子问题结果返回(归)
    1 y. B$ d5 k0 J( D$ {/ H# P3 E% a& x' O! d. _3 [
    注意事项
    0 S1 [/ c: r) b% a2 {必须要有终止条件/ Z( X" O/ D' G7 ]1 G/ O
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层); U+ r, ?3 A5 h! h4 N' N
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    8 W. h8 y  D. ~0 \尾递归优化可以提升效率(但Python不支持)6 y: A  I8 z+ K" \/ M" l: v! T

    % N. s! w' r( N* j 递归 vs 迭代
    . S; q- H  i, O|          | 递归                          | 迭代               |; _! E+ U4 }& j$ `
    |----------|-----------------------------|------------------|% b) D4 t% l$ f: A8 ~
    | 实现方式    | 函数自调用                        | 循环结构            |0 F: ]8 F  e- b/ o. B# v  {
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 M/ [: z/ 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         |& b& M- ]( [. Z  _  J
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |: T+ D- h7 i  t* `  k' T
      z  e- J& T3 r
    经典递归应用场景
    ' y$ V6 E5 z/ p9 `8 v1. 文件系统遍历(目录树结构)
    # p' p! E) w2 I2. 快速排序/归并排序算法
    $ f' S% V6 c0 z' J! p6 r3. 汉诺塔问题3 @$ a, s9 L/ d0 v
    4. 二叉树遍历(前序/中序/后序)
    1 u! R) M% K" g9 [6 m5. 生成所有可能的组合(回溯算法)
    " x2 _- T4 R1 P% ~& _
    3 S% L5 f, F" q% B8 S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

    昨天 12:17
  • 签到天数: 3175 天

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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,, H3 c+ Q9 E  d) J! N
    我推理机的核心算法应该是二叉树遍历的变种。
    ) i* M% a+ [8 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:
    # v; ^' ^. d# BKey Idea of Recursion
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    ; S6 i. j+ P- f; u1 ]7 oA recursive function solves a problem by:4 c7 e' R/ Z3 X" F9 ?6 F. e% z4 J
    + Z* Y& i' H, x  A; H
        Breaking the problem into smaller instances of the same problem.
    ' T9 E6 p4 A1 r) A2 n; L
    $ [1 X" T/ {9 V    Solving the smallest instance directly (base case).
    , q) ]' ?  U, v$ t$ `" l; J: a6 x5 V0 p
        Combining the results of smaller instances to solve the larger problem.2 z2 v* \; q' a+ b2 z- h$ y
    0 i9 V  g3 B/ X+ c* n
    Components of a Recursive Function
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        Base Case:
    # g# ?# b1 x5 P9 J" I. e. A* Z9 U  W: F/ E. n0 p$ Y+ i1 V. `
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    7 ?4 h/ Y( J$ T- U  X( ?
    / k8 k+ J+ `. w/ q7 m& U' q( O        It acts as the stopping condition to prevent infinite recursion.
    , G8 ^2 e4 s' i
    . m" ^* ?4 Y/ R; Y3 a( k        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    2 B2 V7 X( n9 O$ c! ^$ Z; w
    : `1 |7 [- w) U& Q' i    Recursive Case:. p1 v" }7 {& e2 B3 w

    0 t- Y" \' S& g" Q2 w# y* J        This is where the function calls itself with a smaller or simpler version of the problem.
    * c- v4 Z" C# o! y% D0 y9 }5 O9 o$ `5 }
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    7 A/ c) A& F. B3 }. N
    - }! G) w$ i) J7 i8 SExample: Factorial Calculation
    - J+ B9 r8 f' V! U* s# {5 l/ j
      h5 J4 D8 e" n8 O. W# A7 _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:
    2 m; n: x  L$ _/ Z, Y
    ( P6 x* Z) `8 s% v) `    Base case: 0! = 12 D# A% O, c6 o0 |3 r* A
    % c- D" u6 ]  B4 N$ w* ^4 [
        Recursive case: n! = n * (n-1)!$ n# g, q2 U% A  l

    ' }8 o/ v* U6 r# @4 yHere’s how it looks in code (Python):
    , p! [8 B# _; l" o* ipython
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    0 j9 e$ t2 C% `) R! A# v  o+ z; n( D2 r/ f% R( ]0 n
    def factorial(n):
    9 L; G- E0 ~$ O% M    # Base case
    / j; z- G  R' a' K    if n == 0:! |, D+ r8 h5 K" @& X
            return 1
    , z' @0 f. s& H. W& f! t8 _, v% Y    # Recursive case$ C' g& {# F% n( u1 v9 w
        else:
    ! w+ Q. U0 O5 ~* K3 d$ ?        return n * factorial(n - 1)1 ?9 @" a8 U- X/ S8 p
    ! G( f  H4 G- a% P6 ~* J, \
    # Example usage
    , d; D# Y' Q% v, hprint(factorial(5))  # Output: 1209 V6 L. Y# e) d9 f  C

    # Y1 r. u+ T0 ?5 h0 r5 b- qHow Recursion Works
    & j, L7 |) [7 z
    + b+ D! F/ E' u. V    The function keeps calling itself with smaller inputs until it reaches the base case.
    - }2 n; b7 H% a- _  k) r+ _! `
    3 s" T! ^  I& V    Once the base case is reached, the function starts returning values back up the call stack.
    & t- o: V# |! |$ ?9 N% n* ?
    " b, B; k1 A  @/ L    These returned values are combined to produce the final result.
    6 Q0 P" m2 v! r* S' P& i5 C0 l. B3 B' o
    For factorial(5):
    ! A6 @" V4 B# P& W3 |7 `
      u/ J0 {! N$ h" k/ t6 r$ l1 p$ y  x
    factorial(5) = 5 * factorial(4)$ Q9 u' W. O3 Q: l( r4 D( A
    factorial(4) = 4 * factorial(3)% a3 K8 f1 ~5 l3 g' @( S
    factorial(3) = 3 * factorial(2)7 I2 ]4 O" C+ Y# W# s1 U& h
    factorial(2) = 2 * factorial(1)
      p' ?- M0 x9 x9 B0 \5 F. C5 s: S2 efactorial(1) = 1 * factorial(0)
    , K7 H0 B" E) M% G" `factorial(0) = 1  # Base case
    9 a/ J/ D( ^" c, h/ f8 R: T2 P. k4 u* t% T+ }7 Y
    Then, the results are combined:
    # K4 z; [5 V% k( m4 Z$ [/ q- y, @* v1 ]; m3 S8 r5 w4 s( X: H

    % H& k% l5 N) o- Z7 J$ O: ]9 Qfactorial(1) = 1 * 1 = 1
    3 M4 v. v; L# ?$ g% n( u5 a, Y9 C5 yfactorial(2) = 2 * 1 = 2
    , o' H, s4 r( B3 L% e" O7 H. Lfactorial(3) = 3 * 2 = 60 v2 t* N' N4 v. ]0 {
    factorial(4) = 4 * 6 = 24
    ' C8 T5 u6 m. tfactorial(5) = 5 * 24 = 120( g( L/ D0 S# L+ X
    1 o# M1 ~, N' d+ G  ?
    Advantages of Recursion
    + i1 y2 P+ M# N" m1 a3 e) |( G! T) d3 i
        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).2 i3 ~! r4 \$ h) ~1 E' J
    ! w9 ?. A' `9 X+ g: L3 c' F# s
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    6 M5 u; X) k/ `8 b. r7 k  o) J
    / K. F! s* U1 u, sDisadvantages of Recursion
    - P. l$ _' |# B3 ~* Z! \; Y1 Z9 m( A% ]/ o) [
        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.( I" I+ c& F6 g

    2 F4 ^/ f5 Q3 p+ Y4 T4 m/ c! l    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).9 A4 X( o5 j9 u5 a) }8 |# e+ U

    / C/ F; v/ ^; ~When to Use Recursion$ t' C" u6 C2 ~% p% j' l, Z

    ( u1 ?6 _: f$ k# }1 D; F; [    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 t2 O4 c: ~' M4 Z6 y8 Q! }6 k7 N7 l
    , n3 f$ o4 O( V( S4 _5 R
        Problems with a clear base case and recursive case./ I% l. M! m0 b, Z. s8 T. l( t& y1 U

    . }* Q& t$ B3 Z" T+ \. BExample: Fibonacci Sequence, `" e5 V) D* S9 {& s

    ' @" A7 v) Q4 Q' C2 X, NThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:! b: S+ v7 ^+ O7 I" |
    4 v6 R5 }6 f( J9 j& ^/ A6 }
        Base case: fib(0) = 0, fib(1) = 1
    % c. W' H4 E( n$ D0 c, p* Z! q( [4 Q, V
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # j( \' |1 u8 {
    3 r3 [. \& m& I* I  O/ ypython2 I% ~* n' x& m8 P3 @9 O* T5 P% z
    % p9 W) ^1 U, G5 `

    * J( G) G% t! ^" Udef fibonacci(n):% G, o4 ]$ {& X! m) G9 O) D
        # Base cases7 _( `2 J. x0 |+ u
        if n == 0:, W& m6 Y/ W! c% r
            return 0% u& F- d( |( |' p4 V: F2 v# J7 |& u  X
        elif n == 1:
    ( I, W) c% Q. Y1 Y4 S& C        return 1
    # J/ }6 M) k7 r' y+ f    # Recursive case+ W* A0 j! N! _0 G: G% `$ i
        else:
    " b* P' f  [6 w* e        return fibonacci(n - 1) + fibonacci(n - 2)
    # V" B2 g  A& g; q6 G4 Q+ e: i5 A  t( ~2 J& n4 _) P8 i, n
    # Example usage, E# E3 a' h! M
    print(fibonacci(6))  # Output: 8) R- W$ L2 ~  O1 o( }. Z0 f- I

    % r2 U# N' S) P5 F$ E8 L% _Tail Recursion) w  b8 j4 q, i- s$ ]9 ]  m

      d; o  b. g* o9 }& WTail 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).' ^6 B4 {& _' f: R

    2 B8 j5 B( N/ \$ g; f' A4 VIn 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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