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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 / C$ k8 k" U& b3 X1 X( B, Q& ]! n
    5 m8 v( J! L) j3 |4 v! R* Y
    解释的不错
    8 P) ]7 W8 p! h3 o% c! t- I8 }2 T, I/ {$ f. w$ M8 d& x9 n
    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/ [* d+ i* p! p# S2 I

    6 [; {0 m) u2 M! F& t- z$ Q 关键要素
    7 A! g( n# m1 E# g" B8 K1. **基线条件(Base Case)**
    ) P- T) F% x. p  H# l3 R* a   - 递归终止的条件,防止无限循环# C2 ~) ?0 p6 y3 \
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    3 i7 A) N" l! ~3 K0 T; B& L# w- I: L# ?
    2. **递归条件(Recursive Case)**8 A) Y9 ^! R+ j9 k: L7 e
       - 将原问题分解为更小的子问题
    * U% x4 u, S8 e8 L) ^- y6 i   - 例如:n! = n × (n-1)!% R$ |8 ~- s" \

    ' F, e: g7 v5 y  M  j! s 经典示例:计算阶乘) z7 o# T. [9 r
    python; L, Y# R" d0 K5 `0 t
    def factorial(n):
      l' ~* D" B2 d    if n == 0:        # 基线条件  y$ S; N$ X5 @& U, t
            return 1
    , o  H. P: ^# Q8 n- }. U" `    else:             # 递归条件
    . H: ^. b# f5 e3 {/ P        return n * factorial(n-1)
    ' i$ h- |7 s  T# c. d执行过程(以计算 3! 为例):
    & n2 ~4 \6 V/ J4 K- Cfactorial(3)
    1 c) ?3 W6 k9 o' i; j3 * factorial(2)
    . L, l1 b. p6 N  d, ?# o2 j+ w; }6 p/ H- z3 * (2 * factorial(1))
    % `- [/ u* d0 N% }) ]: R3 * (2 * (1 * factorial(0)))
      X  Q0 k/ B. ^, u3 * (2 * (1 * 1)) = 66 e1 t$ W6 L) f. C- p; T
    9 o4 N! X% R8 L* w+ Y+ B9 J
    递归思维要点
    8 B5 k. @% t7 Q4 n+ k4 D1. **信任递归**:假设子问题已经解决,专注当前层逻辑9 G' X/ v( b8 B! J7 v" A& {) A
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间). R! A$ K) Z8 L: f; b0 i4 C
    3. **递推过程**:不断向下分解问题(递)
    , {7 }$ X; d$ m% g4. **回溯过程**:组合子问题结果返回(归)
    6 @& W7 ^& p) j$ H2 Q7 d! ?5 c9 R% }: e% a/ q+ r, e( g
    注意事项) b$ C, z- ^+ k9 N6 b& A6 d2 ~
    必须要有终止条件
    # y9 k4 p5 J" T# v0 j: b6 U- D. ?递归深度过大可能导致栈溢出(Python默认递归深度约1000层), P, O& D  {6 {5 b* c8 ?% Y( l
    某些问题用递归更直观(如树遍历),但效率可能不如迭代- O. J: y% Z) z* n* G
    尾递归优化可以提升效率(但Python不支持)! E: Z) q2 T- z8 E& e; b( x
    1 z) o/ {& D/ _4 N- j# Q0 l, r
    递归 vs 迭代
      S% a$ y; W0 i' q* [|          | 递归                          | 迭代               |9 U* A4 H; ~* @( z4 [& }- R; }! r( I
    |----------|-----------------------------|------------------|
    ( ?( }* n' X# P, `7 a& e% f| 实现方式    | 函数自调用                        | 循环结构            |$ J+ Q& P& g3 X- y, Z6 l
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    , O. Y: S" Y# y& E# ]* V| 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         |0 Z# @0 A/ _' d. E0 o' F
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |! }8 p/ V9 T/ C( W1 X. A6 I9 L4 N

    , @+ F( b# a7 j5 a 经典递归应用场景* [$ z9 M, E( R9 |9 |3 y. T: U
    1. 文件系统遍历(目录树结构)
    - A* V$ f8 l6 E0 g* O$ Q2. 快速排序/归并排序算法/ V; b4 m9 G" g8 @6 Y
    3. 汉诺塔问题% Y( M# I3 M1 h- l( v# n$ m, N
    4. 二叉树遍历(前序/中序/后序)* h% P# G- o: d* S  u
    5. 生成所有可能的组合(回溯算法)
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    + E- P/ N4 K- d- P2 W" C4 g3 J1 g6 A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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    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,
    4 N/ u2 h1 ~9 E我推理机的核心算法应该是二叉树遍历的变种。8 j  e4 ~5 N4 U; R) i
    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 c- Q6 e  r* y! r4 G
    Key Idea of Recursion
    . p( g$ w- {+ W( C: G' N, i! g% V
    A recursive function solves a problem by:
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    # {  D0 o! ]2 a0 Q" K    Breaking the problem into smaller instances of the same problem.
    ; a: h; P( d  ]8 }1 }; a
    ' _0 E# N9 D8 }: h    Solving the smallest instance directly (base case).
    . I2 n& O1 ^! C0 n8 x8 L; j3 J% P% b1 f) I( H
        Combining the results of smaller instances to solve the larger problem.
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    ! L7 x) l& ]: c! I' n* @$ {Components of a Recursive Function) X# ]$ ^# N( R) U- x4 _4 m
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        Base Case:
    9 f  z7 D% y) q, x2 f
    9 r$ s9 s. Z+ y  ~: a4 ~# V        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    8 U3 H  |# Y5 B' c' U' A) w- u, Z0 p& h- S
            It acts as the stopping condition to prevent infinite recursion.. k; a/ r* K/ P0 {' ]4 K

    9 A- Q- Q; n: w* c% ^, Z        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    # M" V/ l& Y0 z+ G0 h# w$ }' ~# c5 W: ~1 I% l& m
        Recursive Case:3 `  G7 z" u+ u4 D
      s9 l0 ]2 k* x
            This is where the function calls itself with a smaller or simpler version of the problem.4 O1 z& k; L! b  j, W* w, q; L* w# {
    # b* {7 Y, Y0 y$ W2 p
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    & C4 F5 x8 `4 j; l. f: N6 R: |4 W; |" C
    Example: Factorial Calculation/ ~/ H( n  z, H8 }

    * W% N6 f" E# m$ rThe 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:# d1 B6 I/ N( Y6 N* h* l& q
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        Base case: 0! = 1, p* w2 w0 @; W# c5 _  U$ ?7 x, [+ C

    , r  W" }9 {- G6 L, b, i7 r    Recursive case: n! = n * (n-1)!) I3 y5 D5 w' x' k
    & {7 N) E8 i) k* e
    Here’s how it looks in code (Python):
    , o, e3 ^. x! D' ~  F# A) Upython
    & V* C$ d1 T3 F& l  |6 Z
    + R7 `. V* c6 K$ y5 d7 S( x1 Y0 H+ U4 ?) U1 }0 P5 U. C0 |/ M, ]
    def factorial(n):
    , B: d* K/ u5 x0 j& j    # Base case
    & J- o, g, s. f  l    if n == 0:0 q  @3 W$ n2 a" L; \0 j
            return 1
      C( i0 P) s1 c( a( x2 D    # Recursive case. G1 x; Q# f& F% l) e2 ~% ~
        else:
    . [4 ~0 |0 q* W6 h) v6 ]- }        return n * factorial(n - 1)% N% C4 p! u/ Z$ [( z0 z

    . r4 n+ o0 `+ E2 `1 B# Example usage, u: D. I9 r' a- s7 r! b7 h& ]
    print(factorial(5))  # Output: 120/ p& ]# H3 b& O8 p$ l, J7 N9 u4 g- p

    8 p3 h9 v# }6 t! yHow Recursion Works
    2 I) p& R3 T6 J  ~  d
    : L. M& t4 v8 J' Y# E) D    The function keeps calling itself with smaller inputs until it reaches the base case.
    ' T5 b3 _- i: \" `) |
    ! o7 P+ `) g, M    Once the base case is reached, the function starts returning values back up the call stack.
      t/ a2 r- U5 w( b9 d" o0 W6 }  H
    ; e) V$ m9 V& M4 S    These returned values are combined to produce the final result.
    6 N7 L% _5 p3 a1 K0 i0 u& W( `" A0 R0 W/ G
    For factorial(5):+ _' y; f* J4 ~% B  N/ H- \
    1 Z3 e: [+ B- X0 V9 r

    # p# c5 M/ k, y/ m& T1 w; A( n* hfactorial(5) = 5 * factorial(4)
    ( d; B; b3 N5 v& Y) _- l4 Cfactorial(4) = 4 * factorial(3)/ q6 a# _/ p* k* O. B# [1 g1 f
    factorial(3) = 3 * factorial(2)/ f$ V! P9 F% I- d. m% l
    factorial(2) = 2 * factorial(1)0 z/ `. @/ O6 z0 e  U% ~, x
    factorial(1) = 1 * factorial(0)
    & y7 Y$ T) z/ f( l1 \# cfactorial(0) = 1  # Base case. a, S8 n' O' W2 k

    ! b3 u4 e/ M9 \. D, N- t  }Then, the results are combined:+ ^7 P0 d+ V6 Z) Y' R
    . z3 A7 H0 Z1 O( B  p& p$ _! f
    9 X+ L, I- G6 \' U1 @9 c2 Y
    factorial(1) = 1 * 1 = 1
    ' [5 E( b5 k* H5 Ofactorial(2) = 2 * 1 = 2! |9 H1 N5 e9 _4 h6 W9 I
    factorial(3) = 3 * 2 = 63 E5 e. R/ b% L" |4 q! B
    factorial(4) = 4 * 6 = 24) s' z7 ?6 n2 D  f# R
    factorial(5) = 5 * 24 = 1200 s3 A  ?9 y6 g3 ?
    % ^7 E( m, U; Q# \2 {' _' \+ O9 }
    Advantages of Recursion7 w4 f4 I. |6 \5 x) @2 u

    # E- c9 [) ~2 [( O0 r, b' Z    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).& \& y; }/ j, l$ f

    7 [7 d8 Z. Y( P4 z: j& U    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    " H0 W9 h" H& C9 `, A/ k1 `" X* i$ }3 A9 ]) L' U9 Y* {0 |* |
    Disadvantages of Recursion; |0 I& u$ B0 {/ r

    & V5 \; s2 h* w% k5 M7 l    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.' n- Y& p5 J: E

    : }9 b8 `7 J1 s$ H' l% U    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ! B) s9 X* f; N" @1 g
    / R+ |' I4 q! ~$ w! J& M; X# S) x7 AWhen to Use Recursion
    5 U- V* F# u6 a/ ]; r. @8 F% F- T. N3 n4 j: \, Z3 [6 N( f; @
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).  Z5 V* j. i0 ]6 X) J
    8 G8 p, q# ]4 f( E- D
        Problems with a clear base case and recursive case.! t- ^% t" ?5 u

    8 i# r6 k( I; [' jExample: Fibonacci Sequence8 J) s9 g& \4 E- i4 l
    / D3 [8 ?3 V6 q0 r" d0 O
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    2 T' p1 ^# N, |6 T8 ~1 _( B- q, [$ ~4 m9 z  v) X5 ~
        Base case: fib(0) = 0, fib(1) = 1
    5 y/ e9 Y8 Y% h+ u' C5 L
    ) J! ^+ }+ g: I! b' G    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # E, P: [" W, Y# P1 F* b# p: f8 ?; E4 `; ~+ _( x9 Q2 {
    python
    0 J, @# M+ }' g1 V8 d0 t; s- n: f
    ( x7 |: ^% N- V2 d
    % Q! Q  m0 E7 ]1 ~6 ^% ^def fibonacci(n):
    $ ?. `: I; j8 @    # Base cases
    . a) ?1 \7 E0 d1 \    if n == 0:
    9 A* P: t. ]( R! t        return 0
    " [* z6 c, f6 e" ?5 ]% x    elif n == 1:
    0 o+ A& ?, Q- G        return 1
      k2 c1 I; X, E' q. c    # Recursive case
    3 ?7 R: S3 r; c+ h" r- G: I: o1 ?    else:
    $ R, q" v' R! g# J: U        return fibonacci(n - 1) + fibonacci(n - 2)
    * @5 Y0 Y& [6 W- ^+ Z, Q: E  `
    1 s$ _+ @# T* K/ y+ F; @( B# Example usage
    $ G% \/ a) ~0 t3 I& t( Uprint(fibonacci(6))  # Output: 8
    ' v, U4 d7 C. ?6 K& l$ b; k# Q6 u* E; }/ G( b8 W6 f) m
    Tail Recursion: x. p/ t4 A* N
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    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)." x2 d  [9 i. C- U" n7 J

    + H" T7 u. R% w: nIn 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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