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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
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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
    , i( `$ s! n9 r4 R5 G9 ~# x0 D; \$ J1 ?3 q! r1 S+ z4 @7 y
    解释的不错
    8 Z) X0 T: l7 t! l0 A( M% S/ V+ j
    $ @* Y  h. \5 r3 P% R8 v. J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 m+ R/ j4 d% N8 f  w6 d$ [
    5 d8 F6 s( J5 v6 X4 W. x. u6 u, X
    关键要素. h  `/ n* C, ~. P3 d" p
    1. **基线条件(Base Case)**$ o, y5 D: t3 o: Z, u. s
       - 递归终止的条件,防止无限循环6 Y9 L" R, ^1 p3 }
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1, c: h' O6 e- M' p4 ]3 J/ x. _! o, e

    2 j8 |( r# i. M5 ]0 t2. **递归条件(Recursive Case)**
      _. G4 P  \. E% @# j   - 将原问题分解为更小的子问题. o+ ]5 i2 X4 ?9 x7 n& D
       - 例如:n! = n × (n-1)!8 T2 [" c3 S. b8 E- q
    ) T! K' O5 R) g
    经典示例:计算阶乘
    # c' ?) [" V$ ~3 `python
    / f3 J3 a0 i  `8 o# j. N# {$ bdef factorial(n):
    4 K+ r) u0 w$ a6 Y5 n    if n == 0:        # 基线条件8 J# |1 r# ?% e1 E8 I6 {$ h# r! n
            return 1* d1 I! D' `" m
        else:             # 递归条件6 @+ a7 w& s3 v, u9 O
            return n * factorial(n-1)
    : ]; k' b) D  _$ }执行过程(以计算 3! 为例):
    / Z7 p& A) y# W- e9 v9 mfactorial(3)
    / |# q. N7 C) l3 * factorial(2)
    * d! }$ a* v- {+ l6 O3 * (2 * factorial(1))
    7 `; j0 f. ?  W. g  q7 \3 * (2 * (1 * factorial(0)))* {- f- y  n7 _* z
    3 * (2 * (1 * 1)) = 6& O4 v5 o5 F( Y! [0 Q
    5 S( b6 k: X- P( P8 l. H4 K
    递归思维要点# }! K: \- e) ]
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    3 Z: u1 P( N( w% K; e2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    - S7 \' r2 n& }3. **递推过程**:不断向下分解问题(递)$ s% ]( \# u2 [1 E
    4. **回溯过程**:组合子问题结果返回(归)
    # m. b3 E. x* F
    : ^* `0 r4 {2 y+ D/ O1 H 注意事项
    7 U+ U# y3 o# T: K; U必须要有终止条件; @% I6 b& M2 c9 A8 D8 a
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层). t8 K8 C' s$ p6 [5 N5 x
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    1 A0 a3 H8 J* D7 x" ]尾递归优化可以提升效率(但Python不支持)7 E# B) k" I3 f+ ~+ e! J5 E; m4 F
    + H( u0 q- K5 q+ c
    递归 vs 迭代
    $ x9 q2 I4 X% [4 `7 y|          | 递归                          | 迭代               |3 I. \4 X2 t7 c
    |----------|-----------------------------|------------------|
    2 ]: \7 H+ x, ]8 `| 实现方式    | 函数自调用                        | 循环结构            |( K  Y5 G5 v( s5 E& k2 A, E1 F  c
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |( O4 M  S* K9 I) I1 h
    | 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         |, q3 Q6 z. x) q
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |/ E, b& z2 Z! b, @4 _

    # @0 F+ W: C: i. ? 经典递归应用场景
    4 g5 P, U7 c3 s# x1. 文件系统遍历(目录树结构)
    5 s# [) C6 t0 i% `5 u2. 快速排序/归并排序算法
    * G$ H7 Y: J8 j8 l) O+ T3. 汉诺塔问题/ z. h! ~& y! }
    4. 二叉树遍历(前序/中序/后序)
    / R$ V- p% m4 b& \  e4 W6 W5. 生成所有可能的组合(回溯算法)
    " c4 E8 x* l$ |5 Q) O& M1 @' e/ f3 d2 K- E/ u4 j
    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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    郁闷
    昨天 07:37
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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% ~1 L' _$ v' w我推理机的核心算法应该是二叉树遍历的变种。5 W0 U0 \. f. Q: Z  J/ K0 A2 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:* I* f* I* I/ |' z# a, I
    Key Idea of Recursion9 v/ |  l$ e4 C) m% ^
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    A recursive function solves a problem by:& I, o' {! p* ?8 Z* {8 g
    8 Q% o& y  q2 n! w) N
        Breaking the problem into smaller instances of the same problem.. U. _! f2 c1 _: _+ @

    # S. z; ]3 T9 p' w    Solving the smallest instance directly (base case).
    6 ^# n7 @* G, z7 N* }- X; m) r3 b5 d) }- N2 Y  @
        Combining the results of smaller instances to solve the larger problem.' I0 @4 P9 ]2 q6 H
    / M2 q2 ?$ l. o4 u: ^; `  b
    Components of a Recursive Function4 `& b/ J$ c2 c# F2 C

      b, k  h( ]4 M8 j8 a    Base Case:
    5 l; ]* f+ |6 \2 ?+ V
    9 ]# {1 k; @; ^6 Y+ r8 @: R        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. f+ A- b( F% o: \- c) w! G

    ( n5 E6 ^( w/ `+ J- N) q4 C# s        It acts as the stopping condition to prevent infinite recursion.. h! {1 _% z* r' P6 S2 Y/ n

    8 r. K+ z5 q6 X3 c        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    # H5 Q$ Q( |! K1 k3 u/ p* Z- e3 r: i# J6 a# M
        Recursive Case:* k' w8 ?) s# o- ]
    2 n" b4 P" C9 q  W7 l7 Q: q
            This is where the function calls itself with a smaller or simpler version of the problem.
    # D; r. W, X. n5 d- @9 m+ d; s4 t
    : D; o8 `: Y% Z: W; B* Z# _# D* S        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ! k- o: D( w/ j4 G2 p8 {
    8 A  b1 p0 v4 I, H  a5 l2 i7 KExample: Factorial Calculation
    " o( e( I+ h* H, U; ^! B  ]# ^' l: x' b
    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:6 _1 S) |1 e2 V% G  `! ?( S
    1 c9 n3 H2 @+ c( h* D: N. q
        Base case: 0! = 19 o9 p! a# M' C+ t, Z  \) v' o
    ; G' J8 g: o3 ~
        Recursive case: n! = n * (n-1)!
    7 \/ r" E4 `6 N& X! z. C2 p4 Q* f3 O: a$ D6 E9 B! o' {( }
    Here’s how it looks in code (Python):
    , t& o: M+ x6 ]+ A$ k! ~  E8 k' ]python
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    4 }9 z& b% v" k, V& {8 O, v
    def factorial(n):
    + |& \2 ]- H# b    # Base case
    0 E7 O9 Z7 x4 B    if n == 0:+ K1 n, u' l7 f
            return 1
    1 F# z7 R, v: {# c; Q; p  \" J    # Recursive case
    3 {0 j! R) G6 ]8 E. b! V    else:4 E( z) ?( [' u  D" t  A0 n9 ~
            return n * factorial(n - 1)
    7 O& ?$ U1 ^5 A! ^7 _( k7 v: t9 m9 k
    / d+ |( Z( K* K& s* ?# Example usage
    " E/ a4 P, {& R* T/ p2 Hprint(factorial(5))  # Output: 120/ s7 \' g% {4 i/ t

    ' ?6 p; \' |' R0 p  THow Recursion Works( X, T$ e/ }, a7 S* A# n

    * u. C  x; Q- Z" z( p: \" ?9 |    The function keeps calling itself with smaller inputs until it reaches the base case.5 N4 a- D* `0 A# j! `4 q' Q1 \

    $ x  \( I+ M( z9 C) [. B8 D    Once the base case is reached, the function starts returning values back up the call stack.: h+ o" z2 Y; a! D% U0 m1 p

    2 Y% O2 Z' b6 ^0 h    These returned values are combined to produce the final result.
    * }( A8 q, S  }+ G- W. f
    . u; d3 ~4 J$ X2 Y+ S4 H0 s. F( XFor factorial(5):8 ]- r- g1 _, I, D# _! U) L: w$ g0 g
    3 ^4 V$ \/ z4 Q; T: B7 ~9 U
    # \$ U0 }& W; Y
    factorial(5) = 5 * factorial(4)1 z% W9 j1 y& V- g3 t0 A
    factorial(4) = 4 * factorial(3)
    3 B% l- N: p$ y  e: c3 p2 pfactorial(3) = 3 * factorial(2)& p! M3 E7 ^: W3 \) H0 o
    factorial(2) = 2 * factorial(1)7 `  J: {$ m3 @7 O6 U( H
    factorial(1) = 1 * factorial(0)
    - }3 y) f: k  h% g$ Ufactorial(0) = 1  # Base case
    " L8 j% w+ e* ?( x! s0 |
    * z) w. G: R# {; M$ r7 N5 a0 K: ]Then, the results are combined:
    ! |: j; [8 l9 c5 `6 T! d$ Q1 A3 {
    * D) h( d6 D5 _' o# s0 _& e" f
    factorial(1) = 1 * 1 = 1
    ! N  E8 a) C8 z- d2 Q. j& J+ mfactorial(2) = 2 * 1 = 2# i: c9 I8 R/ B9 S
    factorial(3) = 3 * 2 = 69 a% l( f5 A1 [9 N& k& E" U" {2 ]
    factorial(4) = 4 * 6 = 24
    " ~; s) o: V/ H5 h5 t4 n/ E5 a/ hfactorial(5) = 5 * 24 = 120' g1 B& j! O4 j; f, a" Z% y

      f; X  q8 [+ t0 g  ^) N3 }) IAdvantages of Recursion$ o( l6 k" M% R% h6 d& h4 R2 X
    4 L8 J/ o6 q' U7 V# U- @) e# t
        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).
    % n! \) b! h2 N; x2 {( X  W( N  ^5 M! G# f, D+ h
        Readability: Recursive code can be more readable and concise compared to iterative solutions.9 l; s, Z' f9 A
    : C! T- i- j& B0 P! f+ k8 C$ ^
    Disadvantages of Recursion6 Y5 O4 L0 Y- I& V5 @
    - h6 i2 r: b3 C
        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.1 _2 K2 ~. @* E  ]
    " n/ ?9 `3 B% n! Z
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).) L( s4 m! f$ i* T7 @
    ! C# l- G7 U9 [0 U2 @
    When to Use Recursion3 @8 m% `5 A2 I' @
    5 z' j; _( ?$ V4 x2 R( ?) K2 N# M+ W) k
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    : n3 p7 E. e. p) ~4 f6 l$ e
    # j8 v' X. A. @7 v8 s, F    Problems with a clear base case and recursive case.5 j: _; B3 B! E
    * F, b' h+ @6 I1 d
    Example: Fibonacci Sequence0 q9 G8 i4 x+ t) N

    ) z- B2 q- T6 ]) _The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:# s% |1 K+ q- q5 s" X  I

    & y8 C/ V; T! h. K  S" f- o  X    Base case: fib(0) = 0, fib(1) = 1$ A, e3 ?% n4 }  w& i

    . {/ V6 Q% E8 F9 I+ R4 G9 |    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    9 L8 @& F; a8 a' b5 n4 R$ i" w9 l# _
    0 p2 i8 J( ?# t4 }) [7 ~$ y6 ^python
    ( c% t5 n3 T5 Z* O( t1 d6 m/ L7 J' t5 r( \% N1 [* ?7 a3 H% u

    4 r8 M% }2 {% p* s- E9 F+ L; c( Fdef fibonacci(n):
    , e* G% \# N# b  o& I$ S, V8 m    # Base cases; ]8 x: y( e% G& W4 r
        if n == 0:
    ) x# I7 p' t; `' |        return 0# D/ f. d, t( J0 W! v
        elif n == 1:2 F2 A$ v3 o8 l* \4 v! `
            return 1) H0 k) I( [) X
        # Recursive case
    7 _+ W& ]+ x+ g: }/ e% x    else:
    + P! x' y1 A4 ]% G  Z+ P8 o        return fibonacci(n - 1) + fibonacci(n - 2)6 N3 c! k/ W1 N

    & Y: Y% s0 B0 C5 Q# j( i# Example usage
    & J# a6 [6 b  k2 _; Gprint(fibonacci(6))  # Output: 8
      ^2 i) I4 ?3 H- @+ p" c) t: j- d
    Tail Recursion
    . O8 R5 \1 V5 _; m% A5 B
    8 J& T$ p3 |8 o8 t- nTail 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).# w6 f2 u' H, O& [: A

    # Z$ |( @6 ~+ ?) l7 d+ F( JIn 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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