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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 ! t" H' o- }/ t. g# @

    ; H- D, O2 T2 v/ I* N解释的不错
    , L) x% g/ Y4 e( j( r6 G0 e( i7 Z8 e1 u6 ^# h; y5 a, ~
    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。) n3 ?8 \; H, C

    3 i0 K- i- `6 q. p 关键要素9 w* [* W3 w% _! U; ]
    1. **基线条件(Base Case)**, O* J# \0 K+ P- c5 U8 E" n1 w
       - 递归终止的条件,防止无限循环
    7 H6 F6 B1 L; f0 C  o0 h8 w   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    6 r) R6 D5 x6 q, D
    2 `6 b) K" l2 H0 p/ i9 e2. **递归条件(Recursive Case)**
    ; N, ?$ `4 M% B   - 将原问题分解为更小的子问题0 _1 b) V& [* H
       - 例如:n! = n × (n-1)!
    ' h, }% n, M( ^+ n
    5 Z, A" c' N, w2 I+ w 经典示例:计算阶乘
    # t$ B  S! w4 a! A4 Jpython; X  \( ?; T( K, K
    def factorial(n):. E/ F, ?' b" M8 G( z0 ?
        if n == 0:        # 基线条件; N" P+ \: Z( c" I; M4 g7 S! W
            return 1
    + I  |1 S5 a: w9 c2 b    else:             # 递归条件$ r9 m6 u( m' c) H1 [
            return n * factorial(n-1)
    ) W$ m6 t7 X! |执行过程(以计算 3! 为例):; R- V! n! `" I! W* P
    factorial(3)" R. F5 M. z2 Q# p7 J, I) D8 Q7 [
    3 * factorial(2)3 d& e- {6 o' F# _0 L
    3 * (2 * factorial(1))5 M% g1 x  p! [: G2 _. h6 P& C
    3 * (2 * (1 * factorial(0)))4 l. R+ ^- [0 H4 A2 p+ Z6 }% }
    3 * (2 * (1 * 1)) = 64 Z; V9 D) d  i
    ' G  F" X# o; b+ W6 w4 c3 B# l
    递归思维要点( Q0 a8 h" i$ c1 a
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑% X0 e& J) `! f4 e, c2 @
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    1 r) V6 J# r6 b" Q4 p# u3. **递推过程**:不断向下分解问题(递)( W6 P* R8 N5 Z
    4. **回溯过程**:组合子问题结果返回(归)
    - Q: }" m, o6 r& h
    7 o& j7 e& A: d) P; L 注意事项) n: f( v5 ?% U9 s( T  f% w
    必须要有终止条件! g" @3 m6 D. w* m; [7 ^/ A
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)$ P* v7 z( b4 T' p: q
    某些问题用递归更直观(如树遍历),但效率可能不如迭代/ S, Q" U5 V3 A4 _' C
    尾递归优化可以提升效率(但Python不支持)
    & g3 T( v& B% Z+ [$ q* K* T& R/ y! F9 L" s, q9 [
    递归 vs 迭代+ m9 P" A' o5 Z5 _" t
    |          | 递归                          | 迭代               |
    . q# s7 {8 O' e( R|----------|-----------------------------|------------------|
    2 U, f3 d4 d2 C7 q; r2 [9 T# w) R| 实现方式    | 函数自调用                        | 循环结构            |- e0 s$ A8 z, y
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    0 O) Y: Y0 a. G( e$ a. N/ r| 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         |
    ! P1 U1 F2 P+ I8 @| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |1 e$ {) I0 {% r" D

    0 Q" X2 t) P7 }- L+ ~4 N& G3 A 经典递归应用场景
    ) R$ R0 J' ~, \# M9 g) i  P1 I1. 文件系统遍历(目录树结构)
    : p9 f( d3 f, E4 [) t2. 快速排序/归并排序算法
    ( z7 U  E8 `$ z3 ^6 ^3. 汉诺塔问题
    : o2 c, p% c/ W7 ^4. 二叉树遍历(前序/中序/后序)) d' [. Q& {3 m5 C/ H7 H
    5. 生成所有可能的组合(回溯算法)
    : K1 V% u& g' k. B* ~. e4 ~6 M: s2 X  k
    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,
    ; A& l. J& m2 z& C& t0 ~5 ]我推理机的核心算法应该是二叉树遍历的变种。
    / P' |/ {' `) A4 E8 @, e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:1 [7 |' p' @0 H9 y: U
    Key Idea of Recursion
    ' M! \+ A* d( _0 f/ E
    & B& v, \( @2 R& D$ c: vA recursive function solves a problem by:
    9 i7 w+ x6 y6 o+ U' i0 u$ n9 }" P. X- `- Z1 i% K5 m
        Breaking the problem into smaller instances of the same problem.
    ! `) W: p9 i7 C0 p
    ' K# j% B! }) n" E    Solving the smallest instance directly (base case).
    ! e% j8 o9 T3 k( J
    0 j1 S; D5 q7 N* @, B    Combining the results of smaller instances to solve the larger problem.
    ( N4 S1 I* F% z8 J1 m; f, Z0 s! u8 a9 M- f0 ~% z( ~# q
    Components of a Recursive Function5 N1 Q% @( [$ `" n; |

    , o; u8 X) C( z4 Q' t5 p    Base Case:
    * Q4 c/ D6 h, V' _4 Z# ]% e- ^
    9 q8 i! [4 O9 G7 ?1 i: u" o        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    7 T7 d( p3 o* r6 Y+ m- p6 C8 T1 J  [
            It acts as the stopping condition to prevent infinite recursion.5 m" K" {8 }% `! ~

    / _: G6 N- p: f0 l        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 ]4 N  O2 y' x# ^  h
    0 w8 i+ w+ Q- W) ~3 J9 t1 Q
        Recursive Case:
    6 l" Z! d6 u& z3 {! j) u( z
    ! i7 o5 n% b# d* `" u        This is where the function calls itself with a smaller or simpler version of the problem.
    ; y9 V  `- I% n. X) h, s
    2 ^7 L) Q: z/ u# C6 n% F+ Y: c        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* c6 F/ K0 Z* r" k
    ' L$ ^  y+ _4 h3 m+ i9 o
    Example: Factorial Calculation
    / C" T& f0 ?  {' M
      r6 X) G& T! y0 `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:8 m; g, p$ M. X$ T2 d7 j
    & N. C" G1 S( \6 \4 x- B
        Base case: 0! = 12 Y7 f, W3 O2 I$ I" d+ k
    0 Y% `( R/ k. m. r3 t! ^
        Recursive case: n! = n * (n-1)!0 {2 a1 h. X& j5 H5 }. v% r, ?

    : t- \# p; I* }; RHere’s how it looks in code (Python):# W  Y1 L4 ?& K; R
    python  U8 q0 f2 f& h6 H. w1 V

    9 k  `& T7 H: B) X
    3 i0 O2 |# C" b: C/ Z: B- m3 ddef factorial(n):( g( s( c- [* r7 s  a* Q
        # Base case
    0 O' f0 Z# b# o: M' `9 w    if n == 0:
    + i$ v5 G( _5 U% g; O# A* f        return 1
    . ]& z5 {& h6 R- [    # Recursive case
    % R* u" |4 O" Q: j    else:
    - s7 b: T3 }0 N  y. \7 o* N        return n * factorial(n - 1)# c! _* W  V; [
    1 M# b. j7 u8 u2 t$ i' I
    # Example usage
    ; k; T5 T) l' eprint(factorial(5))  # Output: 120& `/ m5 r* n9 Y0 Z
    # L* ?  J3 p% L) D& i
    How Recursion Works" k6 B8 E0 x0 q& }
    2 T& e- `) Z8 G+ o3 L0 b
        The function keeps calling itself with smaller inputs until it reaches the base case.
    5 I! Y) b# u: e* m7 x7 n" N) r: o, P5 f% S: D- V. e
        Once the base case is reached, the function starts returning values back up the call stack.1 O* ]6 s6 N" E$ |2 Q% C, Y, a

    : x  a; \& E+ h: X/ w    These returned values are combined to produce the final result.. t7 I& U; M- V- ^2 s

    / j+ [) L& w7 U6 _5 CFor factorial(5):
    " @5 ~% f7 a% V9 ], }8 d1 P! c
    9 b2 x3 A: l. s8 {8 l. q6 {* R, i1 t
    factorial(5) = 5 * factorial(4)  l! W% @! `' G/ n3 T& L; f
    factorial(4) = 4 * factorial(3)
    % l  z/ f. n5 D" Ifactorial(3) = 3 * factorial(2)
    ! S8 b2 s% [! K* h- U7 Afactorial(2) = 2 * factorial(1)  Q+ p' b! O% k
    factorial(1) = 1 * factorial(0)' |) A& ]  r5 ^% c: S
    factorial(0) = 1  # Base case
    " b4 B8 I! o& q) q" a6 Q" V4 s2 U
    Then, the results are combined:
    * q% G( i2 N5 y9 U+ v
    , z( l$ S- N9 [) |3 r. b( ]
    # {7 p1 f: Y! D: B$ S9 H+ x: b3 bfactorial(1) = 1 * 1 = 1- \6 }4 e+ l. W4 ~. Y5 m  }
    factorial(2) = 2 * 1 = 2
    * h; V, B  ^3 Cfactorial(3) = 3 * 2 = 6
    6 O# G9 W; m  X! ~# ~factorial(4) = 4 * 6 = 24
      [4 t2 n1 l! C6 \' t# Ofactorial(5) = 5 * 24 = 120
    % e% j5 H5 ?* {( C- u4 j# H) O
    / F" `. `  ~8 Y7 q+ e; H4 ]! AAdvantages of Recursion3 a; n. a! S6 m- \: N! Z- R0 c
    , `* ?5 n2 T. c2 Z8 w6 [7 O
        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).) I0 s" q8 e7 U7 T, `

    4 Z8 H4 }3 F. t    Readability: Recursive code can be more readable and concise compared to iterative solutions.; ]7 G4 d/ d+ y

    $ x  a* l, p$ L' DDisadvantages of Recursion6 T. e$ B: Q" g9 p
    8 ?. B$ [1 O" [2 P8 q/ D! w) v
        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.
    ( N8 I) ]2 V! V& ^. n# K5 [8 ?3 ]' y- X7 Q
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
      Z* j+ `& E1 M) v3 ^
    ) X. f  P6 L) ~. G) U4 QWhen to Use Recursion
    , x7 Q- @/ t8 I' f
    6 X% v/ ^& e  }$ M7 ?    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    1 w; D% ?8 m1 \$ z* k& ?. W! h  x; M# |
        Problems with a clear base case and recursive case., |# B9 W% ^& m  ?; O) ]
    1 l9 A: m) C! A$ f4 y
    Example: Fibonacci Sequence5 N! w% I' D/ L/ d$ V
    * a5 j  l0 a5 O
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:* N' Y( E1 B, I' N% K
    5 ^& n' f- U. c4 v
        Base case: fib(0) = 0, fib(1) = 1
    # d0 f  e1 B7 j& Y
    " C' {+ m7 l6 n5 A8 I  Q' N    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    $ R0 C7 N' Y+ X& g" A7 B/ x( @; F& b) @+ g# Y2 A) Y
    python
    5 v) f' j& r& P7 {( B9 y- M4 c$ N5 r6 _1 G" W1 R# O
    ) B7 o6 R/ u+ H4 U9 S1 P! D6 A
    def fibonacci(n):) X/ G0 d( b9 ]1 C; R! p
        # Base cases
    . o5 x8 h' R$ O4 b5 V    if n == 0:
    - F- ^1 g; Q+ h9 K        return 0' L. I* r  y  Q- E
        elif n == 1:5 }' P' o6 O' D0 @4 J6 U. L9 S
            return 1
    ! z; R+ k# Y6 l    # Recursive case
    $ a" o* |  V' ]    else:
    * a5 }2 m( a  [1 U        return fibonacci(n - 1) + fibonacci(n - 2)
    " m  ?  k9 K. f  u3 ^4 W9 ~) U6 T! B0 s! _
    # Example usage
    2 |: k) C8 Z) ?/ U3 Mprint(fibonacci(6))  # Output: 8* }* ~9 p! h  S7 L
    2 F$ j! N9 j. K# p5 y
    Tail Recursion# S' k. `4 X0 |

    + M6 h8 ?; ^% Q( JTail 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).' B$ `( O6 j9 C* G' }  [2 l" u
    ( J% X- N/ I4 B4 U- J
    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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