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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 6 m" c6 Q0 g6 m
    4 G4 w* D+ ~! A7 E$ M* y+ I  `
    解释的不错
    . [4 D+ K6 V/ j7 t6 L0 T+ h8 i( d4 i7 w7 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。
    $ t+ E$ L  v) t3 l- i" `4 r. q& L$ E9 E$ V, ^3 d
    关键要素
    3 w/ `  {' h% r! E. Q$ W1. **基线条件(Base Case)**
    ' W  m- M/ _8 A5 t; E, R   - 递归终止的条件,防止无限循环9 U+ _" R) ?0 s; _, r4 `
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    1 p+ v; U. [0 Z, F" P. C* w0 y: G
    5 i5 g3 J9 u1 p6 R; [2. **递归条件(Recursive Case)**
    ) B& M# h! k" W' d; K2 }: K   - 将原问题分解为更小的子问题  ~+ i5 C; Y$ w
       - 例如:n! = n × (n-1)!; _0 ]9 ?- P( ~9 I" F; b9 g: s

    & y( z5 A4 b: s- [/ `; X+ Z/ |- v 经典示例:计算阶乘
    * U& P7 E9 f5 \python0 x8 U: K" @; ]& s& ~
    def factorial(n):
    # ?7 `% H7 T' R& s2 @4 u9 H    if n == 0:        # 基线条件7 i, k6 M" b2 W$ z0 s' t$ e
            return 10 X+ o1 T9 c* O
        else:             # 递归条件+ H8 v# q0 w9 P8 \3 f) i9 Z
            return n * factorial(n-1)
    : x5 Y+ a& k5 B/ L/ I* J7 b执行过程(以计算 3! 为例):4 Q0 @9 b6 V4 Q2 W" _
    factorial(3)- Z- L0 f0 B9 x- `2 c
    3 * factorial(2)- A+ a' N* H. f5 ~' ?) D  M5 h
    3 * (2 * factorial(1))
    . f0 N4 s+ X5 I6 i4 z% {, L3 * (2 * (1 * factorial(0)))4 R2 f9 I- @% A  l) ^( @6 U2 A
    3 * (2 * (1 * 1)) = 6
    ; B. C. K1 i5 u7 \" `$ P% Q* ^
    " \2 h2 O4 R4 e# X/ R- O 递归思维要点
    - D- {7 K  j. j1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    2 ~# v' P& g- R0 R2 c2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' J6 u  _" u$ P/ A* Z8 ?
    3. **递推过程**:不断向下分解问题(递)
    $ _1 Z) W( v* V9 q$ X4. **回溯过程**:组合子问题结果返回(归)" r+ [/ e* z+ P: r. W) P; Q: w

    5 Q: e, o. k' Z5 H6 S: U1 i 注意事项
    . s, L- w% Z  u6 A: F( n必须要有终止条件
    ( r/ Q) A' V+ T5 n! t+ x递归深度过大可能导致栈溢出(Python默认递归深度约1000层)* c3 s5 L( b' d
    某些问题用递归更直观(如树遍历),但效率可能不如迭代) A" d5 I) ~, y0 D# P! `
    尾递归优化可以提升效率(但Python不支持)
    4 s- Q2 W* C0 b. ~) [. [
    ) g9 N$ ]+ f2 u$ }- G) _' S0 j 递归 vs 迭代
    ' R& ~: N. _$ z* T) C7 ~6 o|          | 递归                          | 迭代               |
    & G! J7 u* K" x3 _|----------|-----------------------------|------------------|
    6 _2 C% o5 Y+ f- Y: v5 @' V5 z7 i| 实现方式    | 函数自调用                        | 循环结构            |0 P" t/ G; ~+ ~7 t% \, A
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |) Q/ K/ |( c5 e/ E  X
    | 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         |
    6 j4 ?& e5 Z  b. P) u3 R# P| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |) v0 F1 k; A8 X, F. R3 W
    5 P" _$ v( x- A8 u
    经典递归应用场景
    " s) \# f1 @3 e$ I- P8 U1. 文件系统遍历(目录树结构)
    7 x7 U, @; k$ B' ]& f/ O( {2. 快速排序/归并排序算法) s* l- L! v( z: P8 p
    3. 汉诺塔问题
    * z$ b' N: W5 E" X4. 二叉树遍历(前序/中序/后序)0 z" h9 I' ^& q: w& c$ c3 k  |
    5. 生成所有可能的组合(回溯算法)
    ) @& E( L: \4 q* }1 d+ k' ?1 Z+ \% Q& k4 O8 c0 D
    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,
    $ Y+ t; m. B( ~8 I1 q/ E  S2 W+ u我推理机的核心算法应该是二叉树遍历的变种。
    - }% N# [( `* J8 `' c9 q6 i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:
    . ~" i2 @& [; T0 j# G  ZKey Idea of Recursion
    9 h1 N8 H  s% l1 ^2 g) |
    : @7 m& B5 [! |7 \% X# H2 GA recursive function solves a problem by:) J: F1 p4 W( M; p9 l3 I; I

    + ~# Z# s' K/ R, B  Q& U2 a    Breaking the problem into smaller instances of the same problem.% ^+ b7 l! a% b( H

    ! [2 L6 C2 @/ `% \: B/ A1 P2 u    Solving the smallest instance directly (base case).
    + u  i- K' I* H7 p3 I4 \# _. e: n& l- D' A9 S; N  J1 a
        Combining the results of smaller instances to solve the larger problem.0 J- k9 T( X3 \! q
    0 X4 h4 E9 _& y& z
    Components of a Recursive Function
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        Base Case:
    1 Y) u1 B/ X: a; g' s- v9 ~: g( v/ ^% t& v" u4 |
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ' _3 N, _: y9 f7 z: W0 j4 c4 l
    & V0 M# q0 K* v  j) G" U        It acts as the stopping condition to prevent infinite recursion.- n/ x6 C+ v; P; e

    / h6 D; L% @. _/ h        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.: Z% `7 g6 e# `' f+ h
      _2 d! R7 Y( {
        Recursive Case:' c- n% ?5 A# k' L/ B/ K" a, c  M

    3 H& p1 b* E; q        This is where the function calls itself with a smaller or simpler version of the problem.4 y+ Q) q! j0 J! a; ^4 I
    5 j  z( R# t1 V/ g) V
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ! d  S- w3 H1 a
    5 k% m' }1 k9 IExample: Factorial Calculation( C# p4 n- e& _% |5 f2 t" k
    & v$ `7 B# }: |7 {# q9 E/ H' t
    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:9 T9 |$ q# v8 E. a; c
    # N# j" i! C9 i7 N: {3 T
        Base case: 0! = 1( b6 @  u* G# ?" b9 ]. {  j

    $ O  T, ^* i* M- U3 X& i    Recursive case: n! = n * (n-1)!" a' b! y; v9 M2 Z

    . D. ?" P. I2 Z9 }- ^3 xHere’s how it looks in code (Python):
    1 v8 d8 ?, D, T2 M1 K) D% C/ Epython) O, G3 o' ?! @* K& O
    ' P1 r: l$ b; X: }" Y* T
    ) g7 F5 _0 K& }
    def factorial(n):. |% ^7 j  {% _$ E
        # Base case% h4 M1 @7 x$ C& u
        if n == 0:
    " H$ ^; b+ T4 c6 M1 m% W        return 1( f4 ~$ j6 u  O( k% A# s8 M
        # Recursive case$ B( h# O, k% Z5 o
        else:# m# s) y- @1 q4 l
            return n * factorial(n - 1)
    6 J8 A  s$ L. o2 D+ d7 Y
    - I, x1 H: r& H/ o. y# Example usage
    " N# |* o0 J8 S! d9 h) ]. L; O8 m) J. wprint(factorial(5))  # Output: 120! `6 |' s  w4 L  b& @& d" }$ r
    ( o# R& B6 I9 W2 g" r" g
    How Recursion Works
    # G: y3 f) b9 ?. K" D% i- l
    ' y/ U6 h- ~2 D6 J    The function keeps calling itself with smaller inputs until it reaches the base case.
    % \) _  K/ N7 \- p  a% U$ j
    $ p0 ]& \5 L8 i( T# m9 e    Once the base case is reached, the function starts returning values back up the call stack.6 o# y( i5 k9 j" l. L& z- o% i

    0 K7 ]3 H) v* T+ {! P    These returned values are combined to produce the final result.
    0 x; V+ H  Z/ k6 l: M0 o9 Y0 w, t3 E  E& F
    For factorial(5):2 d& a. `/ G7 u& L7 S: ]% n" e
    # d) r% M/ F' D% [8 ?

    " T2 j" W( h/ ?) P4 L" u, Zfactorial(5) = 5 * factorial(4)
    . I: _8 A$ ?& ^factorial(4) = 4 * factorial(3)* p' F4 b' L6 {9 u  |2 d
    factorial(3) = 3 * factorial(2)0 C4 G4 o9 F& }+ u
    factorial(2) = 2 * factorial(1)
    - |* R- ^4 R* Yfactorial(1) = 1 * factorial(0)
      t1 w9 r9 k9 E% m/ e3 `factorial(0) = 1  # Base case
    ; d: U  r1 l3 O# g" z) q3 ^  ^8 Q; U+ u3 z9 E
    Then, the results are combined:
    + U1 M6 V- Z. w* w% H$ H6 d  A6 m3 E
    / K7 V/ b0 [/ |5 E4 U
    factorial(1) = 1 * 1 = 1
    6 \2 n, J7 O' _! R9 W3 u+ S! Y" \2 Efactorial(2) = 2 * 1 = 2
    0 A# k8 J- c, k# u4 c0 Q& L' q; Mfactorial(3) = 3 * 2 = 6
    / g' e1 x! {6 h5 R, c" U4 b$ xfactorial(4) = 4 * 6 = 24
    7 v: p0 M9 c% C; \- Q7 ?factorial(5) = 5 * 24 = 1203 l4 }4 k0 q5 d! ]4 i1 k
    5 G1 C. }( T# M% q' e+ a6 y
    Advantages of Recursion
    * _4 L5 ~7 g3 q, |/ O/ S" ~! u( k4 z2 v3 u
        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).+ z0 q' \- [/ H1 g) g5 O

    3 ?2 g2 m: K+ v2 i$ E    Readability: Recursive code can be more readable and concise compared to iterative solutions.: n; B: u$ M6 _
    ! _9 k0 i0 R" s; w/ I! v4 e
    Disadvantages of Recursion, V! n; \" _: x  P2 f6 |

    / _/ q- D9 |3 b9 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.' p5 k! [. Y/ O2 A2 {1 U. W7 \

    $ J5 K* a! Q6 j- D6 V) h2 f    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).3 m1 b5 n) w5 o7 x7 A

    2 p) A+ x+ D) eWhen to Use Recursion/ {9 V+ `- Y2 I. P- M

    / P0 ?+ k6 w1 M    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).* r7 v& v4 w* s

    - g0 o/ @1 [1 \7 v/ t( [3 i6 |    Problems with a clear base case and recursive case.
    3 q9 j5 F8 h! B  Q- l$ Z& ~4 v- I3 W- E1 f9 g) O5 X
    Example: Fibonacci Sequence8 E- f  ^3 y: ~) s1 ]' ^

    / b( @5 P+ A6 S/ d/ n: N0 XThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 S, J9 o  y8 [( L% h: g

    9 a* n# T; F  I9 k( Y" N    Base case: fib(0) = 0, fib(1) = 1
    4 T: g3 E. H' i# `' O- k/ V# I1 z  F" R" J- q$ B  k
        Recursive case: fib(n) = fib(n-1) + fib(n-2)3 T- O: z* r. c0 p

    ( E4 u0 c5 a& t8 lpython
    - _5 {6 c7 U8 u
      p2 U% O7 s: F" c" X! \; E5 n) \5 U& z" F7 g9 J9 q' O
    def fibonacci(n):' B: [+ r5 Y9 E$ ?
        # Base cases
    / y8 y8 a$ ?) Z8 P    if n == 0:
    1 G$ u* [9 \0 K        return 02 n: M' x! _$ }$ W2 d- c' Q
        elif n == 1:/ h! }" q" e2 U! k) L( w
            return 1
    $ J" K% _; `. ?+ S, b    # Recursive case
    , H' r  c/ X7 o! e! D! r/ l, `, k3 \    else:
    4 Z4 ~8 h- b' `        return fibonacci(n - 1) + fibonacci(n - 2)" H! _0 i8 x) z* x$ t9 t* ]

    8 X) J. H5 h, e  i& S# Example usage( V- ?8 _/ m3 v1 U3 J
    print(fibonacci(6))  # Output: 8: u( K0 _( `$ b5 T; [
    4 H1 {. H8 `! t2 |' r* q4 V, T
    Tail Recursion5 A/ k6 R  t0 H# o  b+ T* `
    / r+ r+ q& z" J: U+ v' e
    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 G7 W/ j4 M- H% `3 Y7 ^* u; O2 l& u" L' f" 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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