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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| 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 3 I: k/ h( J9 x6 q% U1 }

    . b  [; o( F0 Q# |解释的不错7 [3 S; _8 `# L% j( b+ V8 |5 J. c

    2 S' ^& Y, d+ }$ d, |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。3 E3 y( p8 q7 Y+ x( _4 V
    ; q; M, X3 v3 \; u# L% i# A
    关键要素
    % ^% z) ~; Q  @1 F9 d3 f1. **基线条件(Base Case)**
    7 g, u7 x; r) w' }8 Z2 r   - 递归终止的条件,防止无限循环
    , N8 O) u- V& o- {   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    9 x; Q2 O+ x* f% B4 \+ @' U0 z
    0 ]+ U  D, x1 v, v, G+ D! F$ e2. **递归条件(Recursive Case)**
    ( h& Q/ R& w! G# t- d0 y& @   - 将原问题分解为更小的子问题
    ; }" K* C/ [' X/ {+ F$ t  [! _   - 例如:n! = n × (n-1)!
    ) T9 I2 `' i7 t/ B, q4 J2 q: [0 k2 ]7 d# I0 C7 U8 M
    经典示例:计算阶乘
      n$ v8 o4 s; D1 |) g9 A/ ]* s$ jpython
    6 ?4 I0 C( W, n0 X  n1 @* n8 }def factorial(n):; n& k; c4 g9 H9 M. H# J: i
        if n == 0:        # 基线条件4 b1 K9 P+ A4 q: e' ?: S1 u
            return 18 ^& ?1 A" ~5 X0 W) Y. b1 I2 h  \7 ^8 E
        else:             # 递归条件. T3 b1 I; s1 v" p/ Q
            return n * factorial(n-1). e" x# `+ H/ g, I6 f& b4 G% ]9 x
    执行过程(以计算 3! 为例):# j! }. C) T  f
    factorial(3)
    ; t) p% O7 g+ }4 a' W; \3 * factorial(2); b# ]2 ]! ]" _+ V; B
    3 * (2 * factorial(1))
    8 H1 s8 P* z& i, }. E! y9 h; R3 * (2 * (1 * factorial(0)))
    / T. O# Y3 n5 g  U# u+ t3 * (2 * (1 * 1)) = 68 ^: @% K2 g2 T0 U6 d; g

    ; l& V5 C+ b- p- D8 t9 q 递归思维要点4 Y$ G0 |) l( |2 X4 g2 k
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 M( a" d0 Y+ [4 ~& T
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' @3 t7 O: Z) U% K4 J- z2 B
    3. **递推过程**:不断向下分解问题(递)
    : f$ L& H- ^$ e4. **回溯过程**:组合子问题结果返回(归)6 a$ d9 N: ]7 K! \8 L- H
    - {% @' ^+ T! V# V
    注意事项, d# E3 g3 @- B9 U( W6 N
    必须要有终止条件1 |) j% j- p4 N" j
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 J1 G- i# k& ?, a6 I
    某些问题用递归更直观(如树遍历),但效率可能不如迭代, M, A1 l; p1 K0 d+ A1 o( Y0 \
    尾递归优化可以提升效率(但Python不支持)5 K4 u4 O8 |' z6 J! _

    ; [/ `  n5 c8 U6 S, v8 V, x& r 递归 vs 迭代! J, {, E$ i- q' \9 b3 s
    |          | 递归                          | 迭代               |
    & Y% ^& o! [1 S0 h; [6 B! `6 R, S|----------|-----------------------------|------------------|
    ! P( g& {  Q) @| 实现方式    | 函数自调用                        | 循环结构            |
    # X) w  p& ?# H1 d1 N% `| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |+ M6 x6 _+ T3 C% L5 [) a2 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         |( v3 j0 r: Q* J  s
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    # r" W. w( V; ]% g: w) D. t. \( N( p6 p+ _2 ]$ B( h
    经典递归应用场景, Q4 f+ \- i( ?; i# m
    1. 文件系统遍历(目录树结构)9 o8 E7 F" L$ V% p. _2 ?$ b
    2. 快速排序/归并排序算法
    ( M, e0 H+ ~' f5 t  ]3. 汉诺塔问题/ g0 P  m3 d& O+ m% ~( }3 [
    4. 二叉树遍历(前序/中序/后序)
    9 @8 N: V- x0 q* {5. 生成所有可能的组合(回溯算法)7 n7 a* ?! X! r' {. Z% {1 p4 g
    . {# f2 u3 `1 Y3 K4 [
    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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    奋斗
    昨天 07:11
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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,
    % H( o8 e5 A/ v* E, z7 M我推理机的核心算法应该是二叉树遍历的变种。
    8 ~, O6 b/ T8 j6 j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 }8 X# g# q5 L
    Key Idea of Recursion/ Q4 X9 R! m" ~" N) ~' d

    $ H! b, u& V" w1 g" _1 SA recursive function solves a problem by:
    # s+ O$ W) u5 O. k) i5 m5 Q2 h
    $ z. Z/ O& |3 y! n* t) ?/ C& U    Breaking the problem into smaller instances of the same problem.4 L7 u/ f+ O: O/ a( l# H" C

    5 {2 R/ D7 ^: ?- b    Solving the smallest instance directly (base case).3 N# R8 G7 C3 [8 ^2 [# \

    2 n' c% |$ O: C! B: U, K    Combining the results of smaller instances to solve the larger problem.8 J; T$ u5 U8 s2 U; m

    8 q6 y6 s+ s8 y6 i/ P+ T7 PComponents of a Recursive Function
    : \' {- J! L" G$ y1 |5 \
    ) L& g! Y- j0 i* q& r4 g5 O    Base Case:
    : G( U1 Z6 B) w; O' c8 O, }! b+ T
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    . ^5 [1 d, G/ d7 s
    $ I2 i$ ]5 J' S/ D# g        It acts as the stopping condition to prevent infinite recursion.' e7 W: o/ M, \5 f2 Z4 r
    & k  w0 P; E% n$ i. @
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.. p! G. H0 U) p0 i* G

    ! J. W/ v) b0 L1 t. ~    Recursive Case:5 @! X! v. Z: ?$ F, z# x7 \' H/ E

    $ ^1 g  v* [" W        This is where the function calls itself with a smaller or simpler version of the problem.$ x% C4 O9 ^7 E

    3 \  Q+ h# Z# t, s        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    % @$ a9 U) v: T' D* @7 Y5 Y9 D/ Q2 Z( X. H; b
    Example: Factorial Calculation6 z% K4 [2 M8 F/ A
    7 }$ e2 ~; J& ^1 y  F. x
    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:
    ! v' R0 z1 d: h5 [% E
    5 `6 V. R6 v" H' H7 }    Base case: 0! = 1; b0 y3 \* }. g7 r+ ?* _
    3 \: Y% t2 s* {
        Recursive case: n! = n * (n-1)!( J; U# v9 d7 @4 S' _: q3 {

    ! A6 w$ p  t' ^/ J. G, THere’s how it looks in code (Python):
    + j) J  E' ^9 K6 ^; O) Q/ ]. N8 epython- O6 {( ~/ o; A$ e3 N

    ! x, M& b4 Q) ]) p& C/ ~! F/ K. `- D* A- O
    def factorial(n):+ T. f2 r% W8 k( L, q
        # Base case
    0 g* C5 X; l( w) c    if n == 0:
    : D4 s' ]  c! C% d9 ^8 ^7 r        return 1
    ; _( ~5 W6 |# @4 H    # Recursive case" m1 v5 `7 {- v: ?. w
        else:5 {& d8 K4 A' ?" |" U3 M+ J
            return n * factorial(n - 1)
    ! ^( E+ k3 V/ S2 g8 x
    8 C6 a) x$ I" X7 w% F# Example usage
    3 u- C& q$ U% L3 l4 A) z9 [print(factorial(5))  # Output: 120
    0 D/ ]5 F1 q7 z7 i* q; Y$ s) {  S* N) [" X
    How Recursion Works0 [- A; E5 a. n, C: b
    ; W! M( f7 M  V* [; d8 u5 c/ `, m
        The function keeps calling itself with smaller inputs until it reaches the base case.( z$ f% \0 d# P3 {# M( ^9 v5 I
    + I, y( p% t) r0 V# A8 I
        Once the base case is reached, the function starts returning values back up the call stack.* M' g  Z, q: E* [
    ) a2 R4 H4 ?) X9 M0 g
        These returned values are combined to produce the final result.
    # o! U) t* t0 \& y( e6 H" A7 Y2 j  ~% X9 U8 i* b
    For factorial(5):
    : b6 C3 R% V  O% o
      I, n% G4 k6 s
    3 q. n- `, O) n9 ^! Dfactorial(5) = 5 * factorial(4)
    2 p7 V! X: j+ I' vfactorial(4) = 4 * factorial(3); M/ b6 ]: l* c+ k. H
    factorial(3) = 3 * factorial(2); _! S; R) e1 k
    factorial(2) = 2 * factorial(1)" P% G. H; L4 d
    factorial(1) = 1 * factorial(0)
    " S9 h: r) s( N2 }3 P5 p6 a7 Dfactorial(0) = 1  # Base case
    ) B  Y) h* C6 w2 k
    * j0 z6 d1 A0 L  v8 {Then, the results are combined:
    3 a: Y3 {  s% c3 m" i/ ]; r9 a" j4 S( s' }

    0 R* d3 Q, z' {' f" q. Ufactorial(1) = 1 * 1 = 1" d0 ^; [3 D/ s6 W: ~! C. l& ]8 t
    factorial(2) = 2 * 1 = 2& ~" ~" ^/ `& F. p7 s
    factorial(3) = 3 * 2 = 6
    / h, J" d2 k, Z, b3 hfactorial(4) = 4 * 6 = 240 A. L' `& b% [4 `* k
    factorial(5) = 5 * 24 = 120
    0 H" v  M4 s4 s8 |' O4 q' h) {+ ^
    & T6 ]; q) T, c: n0 GAdvantages of Recursion
    : i; R4 p( S8 F& r' M  X$ Z' t
    ( }' d' H8 G: h3 V+ l3 P/ x, l8 S$ t4 g: S    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).9 Y, m3 [4 c* f' O
    ( [. `9 E. P3 ?3 Y& M8 |+ T+ T  r2 ?
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    % }: r2 R1 W$ z& |- W1 @9 i3 w( X  m! O, k; h9 v& Z
    Disadvantages of Recursion" F! k$ n0 K7 L& b4 U
    1 x; X0 L* [) g1 j
        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.& E2 A6 k; F0 X* e. n6 o5 [. X

    * ^. }* e3 x4 o    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).2 F9 d1 b) ~  F1 Y: G
    / D' U$ E1 E  A5 C, q
    When to Use Recursion
    5 \! f6 O- J% y5 ~% X0 B2 O0 e+ u9 h! F: _2 D. d  y
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).( O9 h( Q8 y4 ^4 O
    9 {* P* o" W/ B% A6 k
        Problems with a clear base case and recursive case.
    - Y- O/ W9 z7 D/ C; d' Q
    , p1 ~4 k  k6 IExample: Fibonacci Sequence: Y* h' D+ ~& ]1 a1 i

    : s+ x, _7 G+ U  QThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:/ q, S2 l/ m, s3 o  H
    9 n( v  ?8 W3 e6 q* i: \0 G
        Base case: fib(0) = 0, fib(1) = 1
      _* d( e5 v# X3 Z% n' |
    ) W- f9 U7 Z3 Y. t  i    Recursive case: fib(n) = fib(n-1) + fib(n-2)8 Y+ ^6 e: z+ K! Y2 `4 T- I& w

    4 B) F5 [" A9 V8 }python% J5 \  Q3 I6 r0 L& E5 U
    ( x8 M5 t6 S! Q% S8 P
    / T* [6 r8 t. i' h4 {
    def fibonacci(n):
    2 z: M: ~. ?9 T# _* b5 G% S    # Base cases3 z- v$ b9 N! r
        if n == 0:$ J) y& `4 }1 ~: L- J8 M, _
            return 0; ?7 x) v6 L  F! Y- d( ~' E
        elif n == 1:
    * D6 Q3 c0 A. U8 l        return 1  _$ z( B4 Y. h. ^! W8 n3 a  ?
        # Recursive case
    ' x/ }9 `  I$ T/ h0 j4 @    else:
    1 `. f+ m( `4 V3 t/ W/ a4 E        return fibonacci(n - 1) + fibonacci(n - 2)
      D4 Q3 t1 ~8 W; g% ~6 t& e7 ?2 E& l& S& G5 S0 S: ]
    # Example usage9 J$ A2 f# t$ j9 g4 Y- }* X
    print(fibonacci(6))  # Output: 8( f& D& X+ J- p# r/ t2 m7 f

    : Y7 G2 @) T8 n/ ?0 w( k* H& l0 I, a8 BTail Recursion
    1 A6 c. f+ {8 d: k! s, q+ C. m! x+ J* {! o
    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).1 d0 U4 h, |) W! r" d6 R' M
    ( c( {& w. p* @+ j' W' U8 ^& w
    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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