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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
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    1 \2 E8 i; M5 \  ~* h解释的不错5 Z5 m, [% a8 w* F7 a4 |

    + ?# h2 R. Q& W$ f# _  q" o2 W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。& l. @/ r( q2 ^$ J9 j3 X
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    关键要素$ W0 g7 ?6 Q9 F; f, }$ Q# N
    1. **基线条件(Base Case)**
    2 Q, {4 p/ `% p* \# j! ?" T0 r   - 递归终止的条件,防止无限循环
    9 r- ]( v3 L( x: t   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    0 o' H6 A4 E; E& p! Z5 `
    8 y! L0 Z8 H9 d" h2. **递归条件(Recursive Case)**. L1 J* d% V9 v8 |7 V. H: J( p
       - 将原问题分解为更小的子问题
    $ F; b/ Q# ^  q- k2 J& w8 i   - 例如:n! = n × (n-1)!
    7 B  N' j0 X; \% s- s( N& A2 ?0 s8 p. c: s
    经典示例:计算阶乘) H; [/ n; I7 ~! c0 ^
    python
    5 [  e4 K* r+ R0 Ddef factorial(n):) r1 h( A- e6 o- n, D
        if n == 0:        # 基线条件
    0 E. L' i) d/ {8 |' N1 U2 m        return 1
    8 E6 I0 M% L% h  z6 r3 B+ w. g  j    else:             # 递归条件0 N* @' d: Z! X& T0 O
            return n * factorial(n-1)$ |; q/ o! o$ p' _2 x
    执行过程(以计算 3! 为例):
    ; y( B! n: O; u4 K6 c' ]) Nfactorial(3)* S$ I2 ~1 A4 ~/ T' L: o
    3 * factorial(2)0 c  z' r& Z+ v3 H9 W% k8 l
    3 * (2 * factorial(1))
    : P0 T5 G- \# @% ?8 J- \2 `3 * (2 * (1 * factorial(0)))
    + e4 @; N7 m3 M  K4 J# H0 E; V1 O' E3 * (2 * (1 * 1)) = 68 \. q. g; s: f* M

    $ A9 l& l/ E: P3 A 递归思维要点% T. R2 R! X8 q+ {8 I0 s( N9 R3 Q
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    . h5 |! K6 D0 k# m6 }9 a) ]3 E2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    3 d: u0 ?$ b! c3. **递推过程**:不断向下分解问题(递)
    ( R* c) o/ L; R1 I* |+ W$ Z4. **回溯过程**:组合子问题结果返回(归)
    " F. s/ Z( o& z0 x0 D% x% Q* V4 P' j+ V/ _/ q- V; ?
    注意事项
    2 h9 w! _  E2 T4 ^3 ^必须要有终止条件
    1 I1 p$ f8 ]2 ~" l递归深度过大可能导致栈溢出(Python默认递归深度约1000层)2 G2 k( {7 r3 `, W/ E1 g
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ( x4 W1 z9 G6 P+ g2 d( w尾递归优化可以提升效率(但Python不支持)
    / j* L' I% a$ O* d. ^0 r: R5 ]
    ( ]* _1 Q! W  A7 } 递归 vs 迭代: ^/ k% Q: Y& _+ B. N5 m$ [% O, {; \
    |          | 递归                          | 迭代               |
    2 \0 w3 X8 x7 c1 f% X& a3 o9 f|----------|-----------------------------|------------------|
    $ y) [% b: n; t6 r% [0 d, ?| 实现方式    | 函数自调用                        | 循环结构            |; j' o  K8 H' m/ z( K
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |6 c# x8 r3 p* J7 `) S6 C& p0 C
    | 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 `+ A( e' G9 s8 a# W$ U# d* w9 e| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    5 J! h3 B, y1 B8 h. p; }( E9 O% t4 P; C7 |0 d1 O+ I4 }
    经典递归应用场景
    2 l7 U8 F2 Q9 {% Z1. 文件系统遍历(目录树结构). k# V, W8 j+ M
    2. 快速排序/归并排序算法
    0 Z6 v5 [  v4 Q7 Z3. 汉诺塔问题! L# L7 w& t9 N; I4 U) B
    4. 二叉树遍历(前序/中序/后序)
      Z) G% W1 m8 w2 t% V1 M- S5. 生成所有可能的组合(回溯算法)
    7 o! T3 \  E9 O; v# E( h( |8 R0 K* P) N! d5 s8 P7 L; P6 ~
    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,, h) D, j, _: G
    我推理机的核心算法应该是二叉树遍历的变种。
    - R; R# p4 r( m1 _* D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:7 O6 d( ~2 Q9 c! `) m( t" d
    Key Idea of Recursion, u* w. D: m7 {; C" Q3 j
    6 @" u0 i7 `/ d4 D
    A recursive function solves a problem by:$ `* H$ g5 j2 r' e; ?

    ; b1 ]/ }& [$ s1 G% X    Breaking the problem into smaller instances of the same problem.
    5 H4 o. S$ J" R9 o; d3 P8 i- P7 b0 K* X: Z  y
        Solving the smallest instance directly (base case).9 z* q$ R- }' U# [: g% C

    ( C' p) M$ L# @1 |, _0 I1 W    Combining the results of smaller instances to solve the larger problem.6 B- T% @8 i+ {
    2 F) v2 _1 h; h. `8 [1 {1 t
    Components of a Recursive Function; t) g3 k6 Q' Q& D& M2 D

    " S: O; Q0 j& I* \    Base Case:" Z3 t9 f1 q. z( e" \5 L0 \

    1 @: U$ t! x( V# r& F2 G        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.6 k5 B) \* K/ l3 i- ?- |' R4 g
    ; z# ?, |7 t, w3 F2 b! z( j
            It acts as the stopping condition to prevent infinite recursion.& k+ z. s& `6 b/ d

    ; G+ v8 A' Y% c! H3 t        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.& T7 n3 w0 Q  i7 f
    " A3 Q& p" Z! s0 g& {1 _2 z% `8 H% {  K
        Recursive Case:
    . ]2 {9 |, F- x% D% _
    3 M9 l0 K) h8 u" y, J1 j        This is where the function calls itself with a smaller or simpler version of the problem." u9 |: X! e9 J; ?
    " \, c* K0 i% w" q& l+ H( j& l" x
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    , _1 c" u3 t9 h+ z0 K% a; [1 c0 ~2 w  ^2 V
    Example: Factorial Calculation' ~1 R5 a9 V9 t/ g

    - B$ b3 [& ~8 r0 BThe 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:
    . C* f4 y) t. J( W: g8 a2 X, a* A
        Base case: 0! = 1: b" E0 `6 O2 O$ O* W+ @3 b
    / H, e6 [0 E& X, X  L! |
        Recursive case: n! = n * (n-1)!1 k3 r0 ?7 z( p: X/ E) r

    9 S# b* A% P6 r: i$ g% R6 YHere’s how it looks in code (Python):9 Z) _" O. x- ?" Z- P8 Z
    python
    * N; f6 e. P* e$ I
    $ T  q5 W6 z# J
    % q7 |" r: i/ b4 H2 X  ?) ydef factorial(n):# o- ~7 e9 o* l+ P1 B
        # Base case/ r( |( K) X4 ]( B
        if n == 0:
    6 _) ]6 ^* W* a2 r: C0 R& v# b        return 1
    $ A" W3 q7 C5 |0 h0 g    # Recursive case
    0 A* }; q, N7 X$ m: h: P    else:) ?- n2 r' S# a5 k* V7 w
            return n * factorial(n - 1)
    1 E3 G" C6 J1 I% H' _0 y; j* }0 t! o$ k9 ~9 d  R8 _% `
    # Example usage
    - ?) U/ s/ [. c& Rprint(factorial(5))  # Output: 120
    $ B  Z2 e5 C7 @7 T( q9 G& q1 y" e( R6 Q( n) d* z; I
    How Recursion Works
      _7 e/ {1 g" F2 B. T3 z
    0 w: B  U8 d9 T4 X    The function keeps calling itself with smaller inputs until it reaches the base case.
    * ~5 k, D* C4 s0 c8 y: w
      h2 w/ W! ^7 A4 t4 b: `    Once the base case is reached, the function starts returning values back up the call stack.0 A! g9 ?1 V, B! z+ w# R
    % Y# G. l, D- u9 `% G1 Y8 ^
        These returned values are combined to produce the final result.
    $ Q, L0 p1 b. o
    / t0 e! C/ W9 C* ]; |For factorial(5):5 Q: w/ a6 {2 D1 W8 |( R
      Z" L: _5 v9 k0 |" c5 n

      T5 q$ v! I: o7 e* Mfactorial(5) = 5 * factorial(4)" B& k( H8 ^# g0 Z8 P8 m
    factorial(4) = 4 * factorial(3)
    ) Y! H3 y$ @( b. t+ {5 ]factorial(3) = 3 * factorial(2)
    " c% y; W7 m2 b0 ]5 r1 ]( @factorial(2) = 2 * factorial(1)( e& L" w5 f7 ]/ J' M4 f
    factorial(1) = 1 * factorial(0)5 H( o  B) {& K# C3 \* H
    factorial(0) = 1  # Base case; |5 \8 C- L5 N$ ~
    2 F6 L4 m7 |8 s& s
    Then, the results are combined:
    $ v5 a" n( s2 D* S; |! N! K! N! e3 B9 a9 G8 E
    ! z. {3 J: o4 X. m  O; c: Z
    factorial(1) = 1 * 1 = 16 L2 N* H. R$ V# a3 ]
    factorial(2) = 2 * 1 = 2
    * ]6 ]4 s5 Y3 f/ I! V% hfactorial(3) = 3 * 2 = 6# v0 a5 c( G7 r- w/ c6 q0 |  _* }
    factorial(4) = 4 * 6 = 248 {0 E0 \. e: ?- W
    factorial(5) = 5 * 24 = 120
    3 _7 O8 E; D! M" n/ x9 K0 J5 d8 _( ?  G9 Q& ]/ F7 g
    Advantages of Recursion6 |  q# ~5 V# f  \
    . L3 k  }$ T4 f4 u; P7 {
        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).# \* d7 t) ?" E3 H

    ! @( r6 ^$ k% C/ [7 z. r    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    + e( h! |! u# o4 ]2 m9 J5 C% b' `  n+ |/ V: T, |, T
    Disadvantages of Recursion
    $ }) a( k4 o6 _/ i0 z8 O2 f$ Z
    * l& U: U7 |7 q% j6 h" ]    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.4 w9 R' `" M% d: _. I
    ! V' O5 u2 v7 L1 K0 a5 ~
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)., r1 Q" i. Y+ z( k" w+ w/ u
    % F; `( |% ~+ o& Z7 ~' O
    When to Use Recursion' r9 \1 y& w3 C0 G

    . d  {$ D1 ^8 p    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    1 l9 ?8 D6 {; q' r& C7 A. Q5 F9 a/ a' R, C- `+ X2 }
        Problems with a clear base case and recursive case.
    + G, \$ i- S" w+ ?$ b, `( ?  a, T' J" ], D7 e0 j
    Example: Fibonacci Sequence* R2 E2 d: C! n6 g

    8 k( q0 P$ e5 J4 g6 K' z* sThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    1 ?4 g8 t, N/ o# ]: p2 G$ e; {# j/ ^/ P, y9 c5 ^: |. ]" ~
        Base case: fib(0) = 0, fib(1) = 1
    8 R; v' M0 T& o1 Q: E2 Q7 w1 w9 ?/ L* M& V) ^" k( y# ~# X
        Recursive case: fib(n) = fib(n-1) + fib(n-2)9 E3 M/ x4 V: R+ c/ D9 d

    ) O& @& y4 B1 z1 \% [python
    5 H% p8 x6 }& M8 b. Q
    2 b* H0 a; O& q9 s+ x- ^. h# P2 \. [0 N8 m
    def fibonacci(n):
    ) m! Z" g2 p; M: r: M( M. |    # Base cases
    9 u  X/ v5 @% s3 N. d    if n == 0:5 i$ d2 ~) [6 d
            return 0" i$ L- q/ a; ]2 ^( K- U" s
        elif n == 1:& @9 v  C: ]. \, N, x) n2 V$ o
            return 1  J) T2 ?! {# [5 Y
        # Recursive case
    / {0 k0 x, J; A2 H, ~    else:6 b4 N; P: j- i+ H
            return fibonacci(n - 1) + fibonacci(n - 2)# G$ a2 u% L* x0 i, g1 m
    ) F" h. E# K7 t5 t) j
    # Example usage
    4 v; t. X' g! K# U& B' j- iprint(fibonacci(6))  # Output: 85 x2 _! X3 R1 B& x1 [' {* g9 j

    7 O4 u. D+ X2 {2 sTail Recursion
    - n2 H9 f& H0 H
    4 u- o, A; @9 @& E9 cTail 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).
    ; u8 A* B+ w0 S1 s! T) U
    $ r9 d' F# w. x) [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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