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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| 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 0 @5 g$ d) n5 b* Y; ~

    / Y( J) C: ^  }% f0 E5 s* b解释的不错
    ( k  q- p$ `) i5 ~
    , a5 ^& W; Y/ Z, n  R, z, ~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。
    % U, k" |) E/ ]! A! S
    $ i2 X6 K$ ]" {. P5 m- R- \7 G 关键要素
    - g! u$ Y4 u" M1. **基线条件(Base Case)**
    3 f9 T8 U. J; f# [3 f   - 递归终止的条件,防止无限循环
    ' a- N8 T$ q0 q/ p' _   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1! {/ B* `9 E; L( Z0 @

    - M  N, f5 k$ [, n- e1 I8 c# q2. **递归条件(Recursive Case)**
    4 J) Z0 u2 k2 N% ^& X% z   - 将原问题分解为更小的子问题5 M& g  y& y# t$ Y" n3 C( O
       - 例如:n! = n × (n-1)!
    0 d% C; V+ {( C8 n' S" w8 T6 O8 F; j$ f7 {6 _, h
    经典示例:计算阶乘( d" U( K3 S7 H  N  u
    python9 k  F; h# X, a
    def factorial(n):; A. b' x# [. U1 v: N) l4 S
        if n == 0:        # 基线条件. A7 G' @. S& |9 d7 e
            return 17 }: o5 W3 M% O; }2 b
        else:             # 递归条件7 n' }  L8 _2 p) c9 D; l3 ?
            return n * factorial(n-1)
    2 x& H; W5 d3 L, w执行过程(以计算 3! 为例):1 M( P  u* }& U1 k6 X4 |- E
    factorial(3)5 d! l! h! K9 W: i1 L/ v0 w. m8 a
    3 * factorial(2)
    # m9 ^0 v; R0 c8 _+ O3 e' h3 * (2 * factorial(1))
    . g) L0 N* Z  @: y$ D7 f3 * (2 * (1 * factorial(0)))# \; g( k/ \3 R9 ~6 ~, f* K
    3 * (2 * (1 * 1)) = 6
    . M8 c5 K1 f  ], ?1 B7 P
    8 `# O0 ^7 {) P- _+ v7 E. o' ^( G- f 递归思维要点
    3 j/ N$ t. A# H: j8 w# `1. **信任递归**:假设子问题已经解决,专注当前层逻辑- E; ^$ E( n; `/ H6 C
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)# u" Y2 Z5 H7 C3 R; L9 f
    3. **递推过程**:不断向下分解问题(递)4 X7 E. a$ }8 e5 |/ E8 _
    4. **回溯过程**:组合子问题结果返回(归)6 {  l) f" H/ ?" y) _
    % Y* V/ F5 E% ?& s! L5 K
    注意事项
    - Z+ v& i8 l0 i4 e+ h必须要有终止条件0 F/ t, F# P7 Q7 G& J
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& ~) N2 g5 b* N% s, d) R. d
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    + ^3 i! d2 I1 x) \+ h尾递归优化可以提升效率(但Python不支持)
    8 {4 ?+ C- ~# m2 W% ^- K7 Z! {7 V( B
    ' Y4 h& _" [/ F$ c4 u+ W 递归 vs 迭代1 E6 w; X4 Z$ J' A) j
    |          | 递归                          | 迭代               |
    9 J& `3 A* l% E, i! w0 p|----------|-----------------------------|------------------|2 n0 W0 j# `/ e4 g0 O2 q: s* S
    | 实现方式    | 函数自调用                        | 循环结构            |9 j4 h1 F% W6 {. ~) {
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |2 `3 c+ p! P& s* w# A
    | 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         |$ g# K& e$ f4 V' G* p, K+ ~
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    * S$ t9 T  E, f* I- L1 v. ?
    ) f+ |1 _- y' i$ s8 L 经典递归应用场景& z3 m4 n/ U& T8 e# J+ h. }
    1. 文件系统遍历(目录树结构)* o  `: ]+ L: P2 N& F$ w! E) J) C# B
    2. 快速排序/归并排序算法' x' Q( M3 S! ~- K& n
    3. 汉诺塔问题) A- M. I! A$ X" Q( @: {
    4. 二叉树遍历(前序/中序/后序)
    ) _$ A+ \6 n6 b1 _( B+ g5. 生成所有可能的组合(回溯算法)
    ( U: R+ z1 c" Y% w& M0 z- ^6 B, _5 D
    2 m. I9 a) U5 R5 Q0 X5 l0 v6 _' L: z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,
    ( }( s9 L% ]7 c% Y( x我推理机的核心算法应该是二叉树遍历的变种。
    3 H' ?6 V) g" N" ]5 p$ e  y  _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:4 _0 G, Z) b1 v" ^7 B5 a4 y
    Key Idea of Recursion
    & d$ g! ^: j$ h; }' y* t5 y! ]7 m( @$ `
    A recursive function solves a problem by:8 ]7 M7 c0 j% A; D4 E

    1 _0 c( ?7 p& q2 X8 Q    Breaking the problem into smaller instances of the same problem.
    0 E" k2 u( _3 f' M9 Q; N3 v
    * A) g$ e1 V9 n" l# X    Solving the smallest instance directly (base case).
    $ \, J% H) k2 x( S, X: d0 K4 V: j# e8 p6 `3 n" v
        Combining the results of smaller instances to solve the larger problem.
    5 @5 z& _8 W4 }% R/ N
    3 o/ s/ H; k0 T9 Q  B0 OComponents of a Recursive Function9 p; F+ c/ k$ r8 E" \" a: U
    ! g0 N7 x5 n; F4 \
        Base Case:2 m" p; j0 x# x2 X! Y5 i1 H: g( D4 I
    + O5 n+ Y, {3 ^0 B
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; A4 T7 g8 R5 R$ s7 |% `, m" m7 U
            It acts as the stopping condition to prevent infinite recursion.
      _% S* M' {& Y, S4 E0 h% ?2 W/ A
    * m) P$ u+ j8 [( @        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    4 r# H* M4 @& s2 i8 P
    - H8 q' ?; ~* T/ V9 ^    Recursive Case:% {8 `3 g! [8 R; Y- l. s* z3 C
    : q& k* c- e( h' c8 n
            This is where the function calls itself with a smaller or simpler version of the problem.& r' y5 e7 f+ P: Y/ C( w
    8 Y% W* ^8 f4 j. o2 \# j
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., {+ `- {( a$ n! |; q: K7 Z

    " f' G7 I% ]2 Q; D1 ZExample: Factorial Calculation0 Y- y6 M. G' N1 ^4 v8 I2 t, g

    5 O: N( ?8 V- Y5 O2 dThe 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:* H/ w4 a, l* I

    # P& N2 x* {# Y; r0 H+ {, R) w    Base case: 0! = 12 E. A: E+ [7 \

    / [: @& z1 [3 q* ?7 g9 c* t    Recursive case: n! = n * (n-1)!
    4 K9 L9 N* p' T% i3 L6 |% ]
    * J& Z6 u6 \0 a  i( z3 `8 QHere’s how it looks in code (Python):' Q6 r4 @  ~: [( y4 m" k+ E
    python& Y7 f5 Q8 H9 d3 M* G2 N
    ! ]/ o# k4 @. a7 s5 S
    - `* ?; S% V% p, w* ~. ^& ?# r+ L
    def factorial(n):* g, e, m# ?2 S- H! o  p
        # Base case
    . x( G6 C2 A, D6 h" G0 Q; i    if n == 0:
    / J- A  V" q" i        return 1
    & T2 z9 V4 c' ~$ j# ?* Q# @    # Recursive case, U8 b( ]& J( c& h' f/ t
        else:/ n/ b2 w. f% n1 K; C& l
            return n * factorial(n - 1)
    $ G3 Z& H! E2 t, @' r& d7 F, B9 W. R+ _1 Z( U
    # Example usage% s' q' {( A) w$ [- y9 U
    print(factorial(5))  # Output: 120
    , V3 v! p0 C; ^5 I4 q5 Y* A' U7 u4 C: z) P7 J# I  a
    How Recursion Works2 \  Y3 y0 n8 L9 l
    1 R1 p9 }* `7 b6 m4 G
        The function keeps calling itself with smaller inputs until it reaches the base case." ?. [. b/ K9 s9 w1 u
    # `1 g* i* F. X5 V4 e
        Once the base case is reached, the function starts returning values back up the call stack.
    6 ?! V: h7 ]3 Z- x  D* E/ J$ ^! M0 v6 C% Y
        These returned values are combined to produce the final result.. r( Z; Y' e6 K4 V. n

    : D( G0 t! h9 D) @) F3 `For factorial(5):
    . [/ E1 C1 a6 Q: J6 {. i  |$ g4 n; ^* V" W

    / s6 W. G5 p. f$ ^" Cfactorial(5) = 5 * factorial(4)' X' o+ I" {- N. ~! k, f; \7 W$ [/ i
    factorial(4) = 4 * factorial(3)
    7 h+ f7 l# Y6 v) Qfactorial(3) = 3 * factorial(2)1 {4 ^5 j8 u. W
    factorial(2) = 2 * factorial(1)
    / a* Y* L7 @; bfactorial(1) = 1 * factorial(0)
    ' s9 l4 F( o7 N( S* P1 ]factorial(0) = 1  # Base case7 y/ S* Z9 G! i1 Z8 l
    2 N0 j7 T4 g3 s3 I7 n3 V# Y
    Then, the results are combined:
    3 n8 u3 D8 g- D7 X8 m1 g- n( E+ j3 o7 t9 l
    : E, c8 I, U) i% Z( A2 h2 i
    factorial(1) = 1 * 1 = 1" f9 _+ V  y  }; D: O
    factorial(2) = 2 * 1 = 2
    - a$ k( A. f3 x) K! ~8 Bfactorial(3) = 3 * 2 = 6
    & W2 `" l0 }6 Ifactorial(4) = 4 * 6 = 24: h5 R- O1 V, t* ?  ~4 B
    factorial(5) = 5 * 24 = 120
    " j6 [  A& y/ a2 R7 M
    $ M# U* O# k5 n$ d* }8 jAdvantages of Recursion- U5 [+ x7 u8 M  h, K8 [

    - G# [- j1 n% t2 h" L7 X    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).
    ; g# d4 x: x, t4 o$ S
    ! x1 ^, d8 I! l( U    Readability: Recursive code can be more readable and concise compared to iterative solutions.' D  m4 @- Q2 B
    ! C5 q9 \  E- h& r$ j7 L
    Disadvantages of Recursion+ ~3 ]4 N, r: S8 n6 q6 d4 k) s7 A

    ' M9 ?9 L) ]+ t+ k1 L( y9 s2 N    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.
      }- A' x& @2 c" `4 o
      ^% G( _; h% J  u+ c: l    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 x8 }# {* X# y6 {
    7 `  M: g, Q, a2 Q$ F
    When to Use Recursion
    . l9 ^" l8 M; k+ [& t2 R9 [, Y" B5 F
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).$ k2 W( r, E0 l9 H) C& h8 h

    . M* f& |8 \# \. Y7 x    Problems with a clear base case and recursive case./ L& E; m; H% s! P, n
    8 u6 T* X  R) T* q) i
    Example: Fibonacci Sequence
    # o3 A4 W' c; R0 ~
      P( N1 o* p# D& R( i3 E/ t- qThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:, E: _& _# l* @5 M
      m( }( ?1 f, W3 ~# |; t) ]
        Base case: fib(0) = 0, fib(1) = 1- P: s! d; g$ y1 r! @' z) o" Q

    & u: L; F3 c$ k  }& d! y    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    * ^/ c% p! T( E' p
    - p  E# C- V6 d, u. d9 ^0 }. t& B' Epython
    / C& K  F& R2 \# a- C
    - x  T4 G& G  j: @8 s" I8 w- u" Y- J3 Z+ ?: n
    def fibonacci(n):: w( @3 D. e/ D" a, j
        # Base cases3 b5 A: q; ]1 T9 K
        if n == 0:( n# s8 z' S1 y/ A. M
            return 0
    2 c! N1 O- N3 m- I  B' {* n    elif n == 1:
    + M! W) r: f$ y% y* M# a" \( w5 I        return 1
    3 u# h2 `; ]3 Y0 u. w4 {    # Recursive case) q1 F4 v; X7 O( d' h4 i' r8 T
        else:
    0 m9 Y# j+ s9 l- F0 H7 e        return fibonacci(n - 1) + fibonacci(n - 2)1 q) \+ F0 }$ a. v. `/ M$ _
    $ |5 g4 m( P* P4 `1 ?
    # Example usage* C- N4 s' H, H
    print(fibonacci(6))  # Output: 82 h" d6 p- o1 F2 M5 ^: M1 o* o

    2 ^& m, a; {1 o8 n. _6 U* z# zTail Recursion5 I, H) B. Q6 x* O* e

    , X2 A( y- i" o2 ~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).9 O, t, G. _3 g1 t8 c5 x& ?" Z

    8 U8 v% {; Q' C' q, Q; [. dIn 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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