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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
    + K9 y( q! e8 q9 k/ k: A
    % J$ p# ?" Y5 I& ^( C+ F& \1 _解释的不错% E- B4 E( L" o9 Q
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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。
    8 I, T4 W0 U" e- E2 t$ F4 P5 ~( d
    关键要素
    . B# ^/ e# ~+ M' N" A( c2 r* Q1. **基线条件(Base Case)**1 I, _4 m4 E2 @# d0 k
       - 递归终止的条件,防止无限循环1 _7 E/ Y# X: R  T9 ]2 t
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: f) Q3 w# v2 L4 [' Q
    # y9 s7 f, k% D& ?% @4 A0 Y2 ^
    2. **递归条件(Recursive Case)**
    ) u7 S7 s9 [9 q& U% y1 n   - 将原问题分解为更小的子问题
    4 _& w& }! k! ?( A5 v- E' {0 w+ q  o   - 例如:n! = n × (n-1)!
    : o3 F  {( [5 I1 ?& N! e7 W& j+ C( X/ q* V% _+ q$ p
    经典示例:计算阶乘" C4 E/ p& _* |5 Z$ s
    python! e) D, ^- d! g* @1 m
    def factorial(n):
    " T. f6 C, N6 D; ?2 B8 y    if n == 0:        # 基线条件5 p5 d# x% }2 F
            return 1
    , w' [: T/ J. n2 ~" ~    else:             # 递归条件
    4 d& `# ^3 ~$ z        return n * factorial(n-1)4 n6 D2 U) @- V, F3 M
    执行过程(以计算 3! 为例):1 V* V# c, B, |( G* }- ?
    factorial(3)
    4 D* ^+ a0 d4 I3 I6 b, q! b2 `3 * factorial(2)& [. t5 D! ^  x: ~( s1 u/ k
    3 * (2 * factorial(1))5 [7 P# Y/ ?2 B  g
    3 * (2 * (1 * factorial(0)))  W' j. G: L6 z) b
    3 * (2 * (1 * 1)) = 6) F2 ?6 V7 c' k! u: P8 p# t
    " F& ^; K1 f& S- M/ j, B" Q1 H/ P0 k" d8 N
    递归思维要点
    4 V  C- d( L9 `3 m6 O" C1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : h6 A) n, q! h5 ^8 l2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 ~. k5 I) p4 ~) i
    3. **递推过程**:不断向下分解问题(递)
    ! r- D2 [" t# ?$ t4. **回溯过程**:组合子问题结果返回(归)
    $ E8 J* k0 l+ \' y8 o+ p$ j( k5 @# w, M% i7 b6 ~7 b* P2 {- E
    注意事项
    2 r9 Q* K, ]1 q8 Q必须要有终止条件
    2 m/ Z+ Z+ k0 s. F( _) u递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    , x/ S; z2 P; ?- H4 ]) ~某些问题用递归更直观(如树遍历),但效率可能不如迭代
    + l' P; g( [' u4 p! K尾递归优化可以提升效率(但Python不支持)& a* M/ S1 d+ e7 A

    * ?6 E( N" `5 j/ r3 ] 递归 vs 迭代- k( N  q$ j( e5 [: h& E
    |          | 递归                          | 迭代               |+ ~/ o" w" c0 s; O  j
    |----------|-----------------------------|------------------|, b' e, ~$ A# y2 o& a- x
    | 实现方式    | 函数自调用                        | 循环结构            |
    . |! u# P- }) b2 `| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |  r5 |2 w0 ~+ u* _1 |! n
    | 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         |" Z4 ]0 V' l9 w" p( H# O
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |9 u- w9 |; x5 X
    , {6 h2 W! G7 O/ c% V3 v
    经典递归应用场景
    # G& Y6 G4 M1 v3 v* Y$ H1. 文件系统遍历(目录树结构)
    % [  h; m, u$ r* w2. 快速排序/归并排序算法
    - o$ a& l) D" M* o8 [3 k3 q2 |% e8 _% c3. 汉诺塔问题
    ) }  \5 Y4 B  k% ?5 M# @0 K: A  C4. 二叉树遍历(前序/中序/后序)$ ^; M' {# A" ?& z5 v$ f
    5. 生成所有可能的组合(回溯算法)
    1 ?5 i9 u( a% n/ l7 X; Z2 }1 o) o# O3 H; q: D# D3 `
    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
    郁闷
    3 天前
  • 签到天数: 3208 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,( q! {3 H" a& _, d+ O$ }: O" n. w% x
    我推理机的核心算法应该是二叉树遍历的变种。/ h) b. A# _1 ?3 y- |0 D5 K
    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 Q' E5 }: T+ q7 n1 @4 v
    Key Idea of Recursion0 s0 s! o# a5 W+ ^5 N" b
    1 @6 B: Y6 J/ D5 I
    A recursive function solves a problem by:
    + F% W- `( g% M3 u7 R1 R, m9 u0 _3 q/ W$ R% g* ?
        Breaking the problem into smaller instances of the same problem./ E# o  C8 [# E; X* x# o

      j- K7 @: j9 l7 L: G    Solving the smallest instance directly (base case).
    ! K" Q+ W8 G/ ]+ N. \8 [  ~0 y
    " d$ n- K- k0 p+ s1 W7 w4 _/ \    Combining the results of smaller instances to solve the larger problem.
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    : L/ z( d5 z) k: VComponents of a Recursive Function
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    6 ~/ z. O+ B2 J    Base Case:
    ( Z% P3 W% U& e- W' b8 j6 e3 F0 t. F3 y; \7 V2 E) `
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    9 ]7 G' Y/ M' s+ ?3 J; Z2 N! G3 A, I
            It acts as the stopping condition to prevent infinite recursion.. m0 }! C. e! Y) l$ W. e: o

    / v4 ~: u- k/ B& n. D        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:
    4 O" x! \; X/ n6 C
    ) e  d- g1 G! E; h        This is where the function calls itself with a smaller or simpler version of the problem.
    * K5 F1 M% |: v1 Z/ q
    ! N$ L8 e1 D8 H2 k, k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).8 k4 k+ n: k, S0 p# Q8 ]

    0 P, ^; n' o, L9 cExample: Factorial Calculation
    # _& _0 M; ^3 K5 |1 d
    ' }1 l# d8 d1 r. hThe 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:: ~3 R7 n3 }& C* @

    + M% b( p2 y; z! Q3 h1 M) C    Base case: 0! = 1
    # z# u! d5 c+ b: _
    8 \  l' q- J) k9 Y+ W+ \) C) ?    Recursive case: n! = n * (n-1)!
    . `- g8 L. ]* Z/ d: i6 s3 e/ s% x0 j  f, X
    Here’s how it looks in code (Python):. h$ @; R, o( Y+ W% D
    python
    $ G" t$ i) w0 h4 W8 t: i. e! z! j- h2 y4 m9 y+ K8 s

    7 D7 x2 i% Q% F. D  ~, ddef factorial(n):+ b' m. V, E% V& h: y- U/ L
        # Base case
    ' p/ ]' ?( o$ x  J# `7 v7 x    if n == 0:
    ) a" s; ?* T3 d) F. \        return 1
    4 Y, I0 L4 q4 S" t4 k6 {+ r1 E    # Recursive case: |0 r+ A  _1 O2 b
        else:" a  U% S3 D) A5 `  ~# N6 u' \
            return n * factorial(n - 1)1 F1 n( U) M/ U* o6 a
    & f! z% |% X2 z1 G7 {2 A& ]
    # Example usage
    1 s* U6 p- j6 D3 ]) ~print(factorial(5))  # Output: 120
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    How Recursion Works
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    $ q; r( K: t$ K% ?4 ~' p% v    The function keeps calling itself with smaller inputs until it reaches the base case.
    5 w- V( Z4 R4 u4 l$ M% d
    2 H& [* V  c( G    Once the base case is reached, the function starts returning values back up the call stack." Y6 |2 ?8 _) g" w& s& m' K9 P

    ! E% w5 n3 m9 A/ n& |    These returned values are combined to produce the final result.
    % m# X7 d+ k  b1 l+ M3 K
    7 y8 Q% a9 w& y/ _- I  hFor factorial(5):
    5 m  \! y' p1 ?$ N* @4 K3 W9 u3 k" Z4 J$ Z

    " b- [+ }, _2 P3 Mfactorial(5) = 5 * factorial(4)
    . A- I. C" r# U% Y# cfactorial(4) = 4 * factorial(3)/ Q& G# B0 U' S6 m% K/ Y# `
    factorial(3) = 3 * factorial(2)  K2 I& \- E  w! W. R# x
    factorial(2) = 2 * factorial(1)
    , g# W9 ^, T2 x# Hfactorial(1) = 1 * factorial(0)
    , c& A/ D7 r9 V1 Ufactorial(0) = 1  # Base case  H8 e  s! g) q7 @8 _! o2 G+ ^

    - I; N+ A+ F! d4 G, Q# n" `Then, the results are combined:; v2 }4 y! n: w. v$ M
    7 Z0 V9 i3 Z5 |
    2 j5 h6 M- ~% u+ l- |4 B' m
    factorial(1) = 1 * 1 = 1- S1 A! A* z- E7 _( ]/ `, Z  s1 i
    factorial(2) = 2 * 1 = 2
    . {! a9 ~- K( v7 z  x' Pfactorial(3) = 3 * 2 = 6
    ' S- @" r9 h9 r8 |3 Qfactorial(4) = 4 * 6 = 24
    6 M0 V; N5 l- L1 o0 L2 Gfactorial(5) = 5 * 24 = 120
    ( ~$ p+ S. i. K8 a+ I1 `. E* o7 C: I4 h% f
    Advantages of Recursion+ L" v6 Z) D0 G5 I$ |$ l

    ' [7 X# \# P4 ?! 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).# `6 x2 p2 _8 X9 F2 _
    5 i6 m$ K7 v+ O: j3 D( X. `6 \
        Readability: Recursive code can be more readable and concise compared to iterative solutions.* C3 q( Q/ h8 y. n
    ( Q9 _7 A3 z8 X: h+ f
    Disadvantages of Recursion
    0 C) `1 \) S8 A. G  j, a* d4 ~- p* B( A: h* |. 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.9 }% p! S; L' K* n. [2 }+ r

    8 H2 m4 h, Q8 x4 U    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    : A9 L+ k, i, i; }) R, T& O/ i7 Z" F0 s2 D5 ^# ~; L
    When to Use Recursion
    " D7 J# p8 A1 y" X" a1 }, D* [6 ~% M: ^& e
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).7 d2 R: n* O* a( S6 x

    + ^* C3 I4 m7 C; M3 y4 Z    Problems with a clear base case and recursive case.
    0 Y% v1 D' m/ H3 N3 ^+ a3 O- F6 H
    ; O" ]# c/ l6 C) [Example: Fibonacci Sequence& f3 _3 _/ U% D; n. Y; n6 V# z. A

    $ Y8 F, v$ V/ U# I( `1 pThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:+ L/ T) {! |8 X: u* X( v

    ! ?. I+ ~+ ^  R; D4 s8 V2 L    Base case: fib(0) = 0, fib(1) = 1
    / H% T$ c4 ~# b! a+ B. x
    ( {8 b5 K. A+ j6 S' x    Recursive case: fib(n) = fib(n-1) + fib(n-2)/ u" H' A/ v0 E  _& O. R
    " y: ]0 m6 w  i" X3 Y
    python% ]' c+ a1 N5 J. J0 @. s/ g) }& k

    ! `# w6 a8 Z( f* a; q. R! p
    & _1 \2 z/ Y% x: qdef fibonacci(n):1 L5 W9 Z$ [; Q; l7 j/ Z" ?
        # Base cases
    3 c$ Z# I0 t; W. D7 V' ~' o: i    if n == 0:
    5 c; u: r! h' @. o  D, k        return 0. h* m- w2 {. F
        elif n == 1:/ g. T+ f! _$ x: F& R: G* b, n
            return 1! r- a+ @5 S9 s
        # Recursive case
    8 ^7 l% i6 u' K& i3 }6 E& H  _    else:
    : y) v+ t1 h& D& a+ C( |: d        return fibonacci(n - 1) + fibonacci(n - 2)
    ) p" }* ]9 T5 ]
    3 {0 X2 |- D" b4 v/ E# Example usage' q% R" ^4 P& y
    print(fibonacci(6))  # Output: 8: f2 i8 I' o0 c5 J+ ^
    : R0 v2 `8 @/ p
    Tail Recursion' V( x3 k4 @& v* G' B
    7 c3 x: ?0 \& Z' S; n6 i
    Tail 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).
    * f' Y7 B, B0 P. V7 S; ~2 j3 F6 a- {0 n
    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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