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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
  • 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 ; z+ f5 k0 ~; e# Q1 w6 A
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    解释的不错" Z) F: {+ R# O( ?' n

    # ?) v1 o* n3 n. u  o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。+ m' c4 U  \- E' q6 \1 ~
    / `( T5 q: M$ r& w# y9 f
    关键要素
    ' t7 T& ], }6 t) n* ]1. **基线条件(Base Case)**
    $ m9 r1 r+ q; b% O1 e6 W& R   - 递归终止的条件,防止无限循环- L- q! m' e/ X& `2 n* O! e( I
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    : L! g( Z0 V0 O3 A( }" _) V" O- h8 W, J7 G" g' D: [
    2. **递归条件(Recursive Case)**; R1 \; [% c1 N' j& }* i; Z5 @
       - 将原问题分解为更小的子问题
    ) @$ [8 F# N6 ^' n   - 例如:n! = n × (n-1)!0 a, h( s; A6 R6 A

    : C) L- J) c  Z$ b7 q" H, D7 @ 经典示例:计算阶乘% J( P% q. F. s: x' ~4 ?# R* r
    python
    5 m3 a0 I; ~& n& Y! Hdef factorial(n):
    ! }' J6 Z) `2 Y( ^3 C+ w    if n == 0:        # 基线条件
    ! I2 k' o  N" V+ v; m! p+ s; q        return 18 \( C$ B( P) @( `: ^
        else:             # 递归条件3 E4 `& @1 N" _
            return n * factorial(n-1)" x& P8 Z7 e! g/ K  D0 R4 t0 |0 p' |
    执行过程(以计算 3! 为例):
    + v5 B! L& N) b2 G- Mfactorial(3)
    ; \& Y8 ~6 @2 A; w3 * factorial(2)
    , I4 `: m) v; U: `7 ~! j; }' n8 D3 * (2 * factorial(1))
    4 A# h( g' R1 D7 {2 M3 * (2 * (1 * factorial(0)))
    9 Z) ^3 [6 R9 s' D3 * (2 * (1 * 1)) = 6
    # Y) m# h0 u! k; M" C; [; d( I+ z$ D: ]- S$ D
    递归思维要点+ N% h5 o6 g) c
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    . g) I- K5 Z# H6 i( g2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( q# o$ i0 L) n% J% D8 q
    3. **递推过程**:不断向下分解问题(递)
    $ T' W, o8 v% w& z! Q& a" R4 Y4. **回溯过程**:组合子问题结果返回(归)$ G1 F- r/ j4 |0 g/ |4 }, S) r

    ' J7 U8 ^. b, \  f' z 注意事项
    0 f* c2 ~) |7 q( M$ ]必须要有终止条件
    - e5 a, y2 U7 Z; U- \6 Z递归深度过大可能导致栈溢出(Python默认递归深度约1000层): _* R. E2 P: @' {
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    1 x! O8 i  @* w$ t7 ], t尾递归优化可以提升效率(但Python不支持)) @$ W* r% B% ]
    ( p: J4 u6 w' B' S$ P
    递归 vs 迭代8 N) M5 S3 y! [8 {6 a! z
    |          | 递归                          | 迭代               |& ^- c9 ]1 P2 q4 h& `! H
    |----------|-----------------------------|------------------|. h0 U" l/ y% {+ {0 Q
    | 实现方式    | 函数自调用                        | 循环结构            |5 V7 p6 p+ m. N2 U  [4 d: ]
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    4 Z" X  l4 _9 M* o4 P| 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         |
    ! N- l) j9 n0 ?1 e, {| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |# P) X3 j2 J# M0 p2 M9 a. ~; {% c% W
    1 l0 k/ H5 ~6 x' R- x+ X1 J/ l
    经典递归应用场景
    % W6 }; R$ B3 j3 W' _1. 文件系统遍历(目录树结构): i( U5 y1 M$ v+ p# P1 o
    2. 快速排序/归并排序算法7 W: }! K2 d( K+ z* A. j
    3. 汉诺塔问题
    7 n% {/ R" K5 l. ~4 L7 L" `/ O# @3 I4. 二叉树遍历(前序/中序/后序)
    5 C4 |! S) y* \0 F5. 生成所有可能的组合(回溯算法)
    % i) g. x* K; H# Z2 T8 E) Q
    : f# h- k4 _2 B4 c/ G  N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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    昨天 10:05
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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,
    ' g% }9 w5 m$ q5 w3 a我推理机的核心算法应该是二叉树遍历的变种。- j. U% ?8 e& k8 ?; b/ e# T9 _
    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:
    3 K' o) S, B6 M% Y! M7 s. XKey Idea of Recursion- T' K- ]; m8 N( _

    4 p' o8 J8 N- q/ JA recursive function solves a problem by:/ I; K7 b  q( v9 S$ F+ ]. _# R
    9 ?8 r3 G+ t* S/ R: Q/ G( ]6 I
        Breaking the problem into smaller instances of the same problem.
    0 ]( ^: a& \3 L: i% ]/ A) u% S- {, s! i  D7 Y- ]
        Solving the smallest instance directly (base case).; d0 [( i% X8 W. f( U& p

    $ j( n/ f# M9 u: \. K7 u    Combining the results of smaller instances to solve the larger problem." g' \9 H9 I; }# o
    + F' r# m3 R- _/ U
    Components of a Recursive Function, H9 Z' d8 I1 T6 B
    * D( c, f. Z% L1 o
        Base Case:' _/ T2 F6 i  G% [

    / N2 X; X+ P9 L; _2 d6 L        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.# v' }6 k7 b9 N' ]1 f) ~0 X
    3 Q$ r# O- Z- S. G% g* l) _& J% {
            It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.* M! `- i$ A9 f, i

    2 j" k' _2 G3 H9 d) l6 G% Q    Recursive Case:
      L# x" i+ w' y0 K$ q! p
    # t; U: D7 N- F- Q        This is where the function calls itself with a smaller or simpler version of the problem.
    1 U) j/ {* |, q, ^$ [( {) {# _" D6 _- A$ ^3 g
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 q+ K$ L5 B4 B
    + I# \# g' y6 d3 ~
    Example: Factorial Calculation
    + d* Q. R- p' [$ n
    - t9 X0 M4 j4 h  lThe 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:
    8 t6 |% V* ^" Y( `7 @# e
    / A. L* u) L5 Z$ j" A1 c    Base case: 0! = 1- q8 T3 }$ u$ I4 \- Q3 D, G
    : C0 k9 K+ l- Z
        Recursive case: n! = n * (n-1)!
    5 [; q: m: }; n- D. B0 x8 n3 V2 f
    - z+ l! {. u: ?: D2 bHere’s how it looks in code (Python):6 w' I+ B4 O$ `' f# r
    python
    # ?' J6 G/ H' S. A. ^
    " d7 c6 M* a5 }  e/ a; N' F0 U9 M, a- l( I
    def factorial(n):
      S. g0 Q3 f$ d+ P% U    # Base case
    1 v& g& n( @. w% p1 Y# U    if n == 0:- w1 w2 N% o: D6 z- u( g2 m* Z
            return 1
    % J+ @5 F* t* D% @8 s    # Recursive case
    2 K6 [! I* ^' Z9 x; S; K! F    else:/ X" n9 w+ m, n$ _: t; I. D
            return n * factorial(n - 1)+ a0 V' V! G& N) n

    & b: q0 ]+ u9 o: q: z7 B; X# Example usage1 d; d) A" v9 I" Z2 T) W
    print(factorial(5))  # Output: 120
      _/ }: P8 C7 z
    + Q* v! }4 X. U+ n; ZHow Recursion Works9 {4 e- l2 f" X' I5 n4 Q3 U

    . I, Z! e' v$ v1 R5 l7 U) r' S; i    The function keeps calling itself with smaller inputs until it reaches the base case.; {6 i' y% R/ G: M7 h

    ; O' @; }! k( D- u) L    Once the base case is reached, the function starts returning values back up the call stack./ R3 b! _' f6 S( K
    ' f$ j, O* W, w/ |2 G
        These returned values are combined to produce the final result.
    ' o6 K' ]: X: Q) K
    0 z% ?. C1 D2 W. m+ C8 aFor factorial(5):+ p  L8 O6 a$ W" Z

    " F7 @! H1 a) m, ]" ?+ Z9 E: B! n' R4 K/ v8 n/ }2 Y
    factorial(5) = 5 * factorial(4): O5 ^; h+ l  i1 m# s% b
    factorial(4) = 4 * factorial(3): ~0 u) i3 P$ e1 ^
    factorial(3) = 3 * factorial(2)
    ' f- P' J: [) hfactorial(2) = 2 * factorial(1)
    2 r3 G* D' _2 ~factorial(1) = 1 * factorial(0)& j, X  C: S+ |6 n! \! u4 ~
    factorial(0) = 1  # Base case
    5 w$ [6 `  A) @( P3 B# F' D. G4 L8 R8 @' S) t# n9 b3 ~
    Then, the results are combined:; m1 r# l' M( f' |0 l3 N% e8 t& l
    . ^+ i. _: E0 U$ A
    # J) A1 e6 X) \: x+ [( z' r$ W
    factorial(1) = 1 * 1 = 1
    ) X" `+ a2 B2 V- ^; X+ [8 Afactorial(2) = 2 * 1 = 2/ y; Y8 r9 o. q% g5 x
    factorial(3) = 3 * 2 = 6* H% T5 J. i0 Z, W3 j' B
    factorial(4) = 4 * 6 = 24: l* s# L" f# A2 U& d- _/ u7 p
    factorial(5) = 5 * 24 = 1201 N+ E! ~9 P7 L
    $ s- d% M4 w3 H4 a" U. u
    Advantages of Recursion
    $ Z' o2 e# n9 n' ]$ M* T6 Z9 x0 e; f8 _* l5 z- 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).2 y# S1 z' N! S1 B: p% ~8 q6 K
    - s& e; N/ ?+ B9 I9 @& b& V
        Readability: Recursive code can be more readable and concise compared to iterative solutions.! A3 w* ]- V! o, k' E7 ~+ ?: M

    ) G1 p+ |7 K" x9 aDisadvantages of Recursion- _9 x' w+ E2 k; U+ T; {% l
    $ M9 [. v# ^6 ^  N' I  a0 }
        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.6 }0 o* N3 f5 p& A- Y
    - o/ Z( E: _; y' v* p3 j0 A
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).! H; L- ?* w6 d4 V
    # o9 N6 Q5 W! \5 _
    When to Use Recursion
    ) Z  d* r% i& U7 K3 T7 }. D
    ' ^" z! Q( L2 ~% C5 W9 c    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# G+ u% M9 y' u/ ~5 }
      X0 r) D- }9 L0 l3 y3 h7 u
        Problems with a clear base case and recursive case.
    + p) x' Z" l+ V3 F: }( E; H& o; \# V; {' X3 D8 e
    Example: Fibonacci Sequence' ^3 D- d- Y3 v' Y" c
    0 J' _! W( i; P* G- m0 ]4 b% B, G
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 \, f3 g  }; T- W9 |+ ?8 C% a
    ' [  ?% `. B6 G$ _* `0 _. c
        Base case: fib(0) = 0, fib(1) = 11 d9 ]& P: o: L. G, v

      \6 a, w( }# V1 W4 S7 c' I' d, ?2 p    Recursive case: fib(n) = fib(n-1) + fib(n-2)4 {' Y/ x, U6 y& m
    % j- G" T" g7 \3 z3 }7 k. v5 W
    python
    & ?& N6 M2 S5 t  H# R# a5 X, W& V3 j" S+ U2 A0 O2 m6 g
    $ y  K2 }) J) R# e/ d& {4 s
    def fibonacci(n):
    5 e8 A9 d0 F9 f. }    # Base cases7 n( E# r( z: Q0 b9 L6 x/ u
        if n == 0:
    5 h( Y6 G8 O* C! H6 o4 V" ?: f1 ~/ @/ Y        return 0
    , U# D' ?" k- f: e' @    elif n == 1:
    3 R8 J+ G% |# C$ C$ B) ~        return 1) A/ B+ e7 ~( J* M# A9 p5 J& q
        # Recursive case
    ! R( P! t# w: R; d( t" W    else:
    7 P$ T; D0 O/ p        return fibonacci(n - 1) + fibonacci(n - 2)4 r: j" H- j6 D$ J$ U

    , P, `8 `- _( i0 ^" G% Q9 y# Example usage  [* R3 v" i- z4 M
    print(fibonacci(6))  # Output: 88 A1 |8 d4 o# Q

    ) L0 @: @1 z. Q& v7 QTail Recursion; D8 J! `4 U# S3 k+ M

    1 I6 ?! A. \) s9 mTail 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).
    , z" x& |( y) F8 |4 p6 e! r
    ' M! z. k: K+ t) ~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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