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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 0 k, _* y8 p( H2 r* e

    ( h3 ~0 p, U: ^: C! }1 K解释的不错
    9 A5 M( K+ K  [4 i: E) \8 X- w6 ]1 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。
    * s' j" K( f( m% P* f4 `. ^  F& u; v4 m
    关键要素
    7 L) ~1 q, t0 h1. **基线条件(Base Case)**
    - J8 ~$ i4 m. T2 c& z   - 递归终止的条件,防止无限循环
    $ c" z# H2 J! C" R1 p3 N   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    : C" {* c  F4 H: A- N! {+ n
    5 x9 s4 j! }0 @. e7 r9 q2. **递归条件(Recursive Case)**
    ; [* U! |& g# Z   - 将原问题分解为更小的子问题( y' s% U" J. t: U
       - 例如:n! = n × (n-1)!7 g' O* U% x- J/ W0 @. Y8 }
    : ]3 d/ s0 O* S' ^* o8 w( }6 ^
    经典示例:计算阶乘9 ?3 x1 m2 c+ E) e6 d: |, p
    python
    % p  D+ b6 A. l' p( J" g& ^5 u" y$ z4 L1 Adef factorial(n):7 [  M" ~1 ]5 K, X8 e
        if n == 0:        # 基线条件
    ) \( i1 ]2 s' }4 M7 C( y        return 1$ ?  n7 E' v! n# g* s( \( N
        else:             # 递归条件2 G8 j& N9 q# E7 \+ @& X; P
            return n * factorial(n-1)
    ! Z8 Z7 A" a& z8 u; i0 H7 P' Y执行过程(以计算 3! 为例):$ a% a# e& R- I& P( |" y
    factorial(3)
    0 y8 m) V$ H% H; u3 \% v" W- E8 {3 * factorial(2)- K6 J' ~  w% C4 m$ m$ d0 ^
    3 * (2 * factorial(1))3 a, U- x) ]4 l  ^8 o6 g; k
    3 * (2 * (1 * factorial(0)))
    * j8 n8 V; n( F$ q4 O3 e3 * (2 * (1 * 1)) = 6# n0 R  Y9 m0 C. _* Y! T! X% ^

    9 j+ y* x4 X( R( @% r, _7 K* S3 W 递归思维要点1 A* x- q6 O0 A  S9 E! |8 l( q4 A
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑* s) z  O. X5 c. D; ?: s# G
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    2 q3 E' L5 H  s- [( ?3. **递推过程**:不断向下分解问题(递)
    8 \: c0 Z- ~$ H/ b2 u4. **回溯过程**:组合子问题结果返回(归)* s9 O) s* L, ~2 R. M% F$ P2 F* z
    8 J, A3 y1 Y* i% s' G$ Z
    注意事项
    8 Q, Q3 D  M: Q) f( p, o必须要有终止条件/ V" g2 o8 v1 _! b1 u( l) S& i
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ Q4 \$ `9 J; ], k6 C
    某些问题用递归更直观(如树遍历),但效率可能不如迭代1 Q1 @  k( N/ f8 m- Y
    尾递归优化可以提升效率(但Python不支持)
    % Y& D$ }% i! \: p# E& l, F
    - s  s2 y' E9 q1 o/ J. Z6 M 递归 vs 迭代
    6 M; s6 K; B) @0 ]5 y0 q; f|          | 递归                          | 迭代               |1 d, H4 }( a$ B: [
    |----------|-----------------------------|------------------|
    1 x8 v1 B$ ?: Y+ J. \| 实现方式    | 函数自调用                        | 循环结构            |
    / o8 z: t1 v" r| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ; x( \. u; \1 ^7 ?| 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         |1 }0 a4 c8 R: ]: e
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ! J# Z4 S8 s/ d% C! \5 |4 z* V8 d: L
    经典递归应用场景
    5 `; I' c) |7 J, d7 L. c+ ]! f1. 文件系统遍历(目录树结构)$ R6 _, e' _" S& K1 p
    2. 快速排序/归并排序算法6 ?  X; e, _( _- ~2 R2 m
    3. 汉诺塔问题
    % G# I' i9 l2 r) K/ t4. 二叉树遍历(前序/中序/后序)4 `! I  H4 h' E/ _- e* o9 j
    5. 生成所有可能的组合(回溯算法): z- Z" u: _1 y# T" w6 q) M
      t5 y" s* Q$ H# j7 V$ C9 w% p
    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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  • TA的每日心情

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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,
    9 z9 v+ |7 }" }- d# I/ D我推理机的核心算法应该是二叉树遍历的变种。8 Q7 Y6 `# w7 q. W
    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:6 u% n# U) ~: b
    Key Idea of Recursion
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    A recursive function solves a problem by:- P0 f4 `; [* \
    " _$ r6 i: p2 b9 E: i8 y
        Breaking the problem into smaller instances of the same problem.+ t' i5 y. V7 R* }+ Z
      s: s2 w  m4 X( m  \9 Q
        Solving the smallest instance directly (base case).& k2 Z- t- P, S% O, b
    : W& J$ M3 R7 S0 R, g  E" w8 p1 ^0 j  m
        Combining the results of smaller instances to solve the larger problem.
    5 k9 b4 T4 N6 T1 W( S4 _6 E
    8 L* F( U  {/ F4 sComponents of a Recursive Function4 h! I1 Z( v( c5 E

    4 Y: F4 I; x* K+ E- o' d) x    Base Case:) y- @5 a( U" q8 q+ E
    ' B; p# F  m: ^5 }
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    3 v0 c8 ^$ Q/ o% o6 W' q8 E! l$ w! P9 {1 ?+ c) a. _$ A
            It acts as the stopping condition to prevent infinite recursion.5 c" x! S; r1 |$ C2 Z: ?9 b
    . A0 _" \# s9 A5 D( e& f6 _* h0 s
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    " a/ M' O$ T4 n. N3 c' }
    ) t/ o6 y. |% {1 a* C" c% d    Recursive Case:% H0 n' \# O( r9 f0 g% q
    * I  ?" F+ g: o) H9 m5 _" Z  E
            This is where the function calls itself with a smaller or simpler version of the problem.1 [) ~' O' x$ n0 }

    - K4 a7 k+ Q" c7 f        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    . C4 V# y7 v/ D, {2 D. q  }2 U; P: `; D
    Example: Factorial Calculation
    ' K: l. t2 e% {3 A
    ) T1 e6 ?$ M* [: o2 {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 r" ?+ `; P1 T* S; G" M2 O* L8 w; J. q
        Base case: 0! = 1' x/ z2 w* E: @: n

    : O" S! j% x. `0 W! @5 C. J% ^/ }    Recursive case: n! = n * (n-1)!. }6 N7 S( x! O, s" H1 B
    , n$ b8 }5 b7 K4 u$ \
    Here’s how it looks in code (Python):) H8 _$ I- N. j& S& n
    python% f) i$ I, \7 Y8 O1 \  G( I
    8 Q' |8 ~4 O6 @- ?( C

    , r7 u* K# k# @* S* }def factorial(n):* [5 q8 S9 X" Q; H$ q! C
        # Base case: l. X& M# Z2 n
        if n == 0:5 g4 I8 V; x5 }6 X2 m
            return 1
    ; w& n1 K4 v6 p9 l$ ^! G; J4 o: l    # Recursive case7 Q6 z$ u5 Y7 Y  A
        else:7 a/ ~' Y6 _5 V" e
            return n * factorial(n - 1)
    . p0 B; ]& Z: X( u) A1 [
    . e' K& c7 T7 p4 S# Example usage
    2 B: B4 ^8 `& ]) v8 sprint(factorial(5))  # Output: 120
    " s$ ]. L- d! Y/ T7 g
    9 \: q$ z6 G% C- [8 qHow Recursion Works0 w# j1 S$ f# j* k# O' S* Z
    # z2 ~* x; ?( M+ c% ~8 G
        The function keeps calling itself with smaller inputs until it reaches the base case./ f$ M  `7 w( T3 M4 U
    0 f$ Q; V1 z% P7 q- D
        Once the base case is reached, the function starts returning values back up the call stack.
    7 S2 @" \* Z) ~+ T# k' G
    - _' j/ V. v' ^9 }# i6 a    These returned values are combined to produce the final result.  }  Z4 D0 ~' u# E" H2 N
    ' ~1 \: g: o- R' E2 G
    For factorial(5):
    * e, d+ g: b$ \2 `8 e0 S/ |. t1 R0 e! o* q3 O/ S( _7 I8 Q; p# H* \

    6 C! Z& M$ A  qfactorial(5) = 5 * factorial(4)) R$ E. D& g% s) C( T; X
    factorial(4) = 4 * factorial(3)/ j- h3 _$ Y* U; O# W
    factorial(3) = 3 * factorial(2)
    ! {) U- p3 x3 d( gfactorial(2) = 2 * factorial(1)
    ( o: x) Z% q! u" `, l5 W" Vfactorial(1) = 1 * factorial(0)$ T  `8 b! i# H* U* m
    factorial(0) = 1  # Base case7 \4 Q' N0 W8 u
    8 x( v' l. P( s0 J; ~, u# W; q
    Then, the results are combined:# r: N5 [" |5 z. Z4 i7 {

    ( p7 A. F, s% Y
    6 U  S' ^1 W9 p. j- zfactorial(1) = 1 * 1 = 1
      [2 \! M* b( F- sfactorial(2) = 2 * 1 = 2
    ! O* I' i! _3 e+ z) n  b( w; C( B, R: Dfactorial(3) = 3 * 2 = 6
    & ^( L5 Y* |8 \' R) u1 dfactorial(4) = 4 * 6 = 24
    % e4 ?0 m& {* Z5 n8 x1 F. pfactorial(5) = 5 * 24 = 1206 L! n3 f; C  f
    . u+ K+ X4 C# \$ O( G. b
    Advantages of Recursion
    + V. @: w, Z$ h+ C4 J' I* k
    / ?! a3 p6 u* _5 c7 o, U5 C3 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).
    # d# e  q5 B4 `3 e% w* q' \. Q  K' l7 Z! s2 f5 n3 t8 b
        Readability: Recursive code can be more readable and concise compared to iterative solutions.5 z2 {  ~/ Q- r/ q/ J
    , X) x' o2 M  k* _/ m" s) x6 w
    Disadvantages of Recursion& f: G- A+ n: \  h8 c1 }" F

    + Q8 R/ ^7 o( l6 G/ I    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.
    , F( p# ^1 t- K9 U  H. E. |5 X6 p- R2 x* }
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).( ]5 g8 h8 O5 `. y  A
    1 r- U7 r7 q- ^$ b3 F0 G4 [
    When to Use Recursion" m. I3 |$ P) J- s: c

    - L# t, @6 r4 v8 C8 e    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    5 K, L/ _: a  f% E8 ?6 i) a' U8 z/ _5 h" w
        Problems with a clear base case and recursive case., J) E: N) |2 e* a
    : P, A1 F3 p  c
    Example: Fibonacci Sequence
    + y7 \, n8 G) U: w
    ; Y6 Y% L7 v5 n4 \The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    , r0 s+ E$ c0 F& P+ I9 X) }
    / r/ T/ C1 i7 f- l3 T    Base case: fib(0) = 0, fib(1) = 1
    1 \- T/ T8 z& ]2 c5 _  C( g# K' Y& \+ Q  z# K
        Recursive case: fib(n) = fib(n-1) + fib(n-2). \  S; s( O) P6 [6 k9 j" o) @/ D
    # `6 f; H* B4 u7 p" J. }
    python
    7 y* `2 ]+ P# a( t; a& T2 d( M3 I; S/ S  D/ M

    # ~& @6 U/ p  d5 F6 qdef fibonacci(n):3 d0 g1 b/ I0 k9 r4 u/ d
        # Base cases
    ; {7 R! n5 C& ^- g" j  b5 Y4 L    if n == 0:
    1 Z5 t+ P( e+ e0 B( g  u& x- J2 Q        return 0
    0 Y( s7 `, s" R: B    elif n == 1:# p7 @2 `' \0 T+ p6 W
            return 1
      s* V$ b3 j0 g" L8 n    # Recursive case
    " r; U" F4 ~3 f1 D+ O' @    else:% Y, f5 b% \8 p+ t2 d7 X( b) _: e2 f
            return fibonacci(n - 1) + fibonacci(n - 2)
    2 v. ^; M7 I# t# L$ e- e: ?) F! e2 C3 L8 A
    # Example usage
    $ _# t6 j# k- ?3 i. {  i3 jprint(fibonacci(6))  # Output: 8. q/ }6 J6 v+ D$ h/ I; g8 r
    5 T3 O; ^' J' o% B
    Tail Recursion
    " r* h8 U  L# g1 |8 V; c! ?( U
    : Z' L1 m  y$ vTail 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).% R" m6 f  m. g& O( U9 E$ u' K

    3 }, P  W- w/ c8 W9 ?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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