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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
     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   k+ [+ c- V( y- Z" w( s

    6 S7 V$ z9 A2 c8 p' T解释的不错. H- [6 T/ C' S5 _* w
    + ?: m* s& m6 N- B5 O
    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。- i& _# R$ w# [0 U
    2 B- C* G! S" @# d: l: A
    关键要素1 D( l- c# C% S1 ?1 J
    1. **基线条件(Base Case)**
    * F- c5 o& M6 _, ]) L3 p) a" H2 j! J   - 递归终止的条件,防止无限循环0 o6 |1 o0 P2 v0 _* H8 c; N( s: \- u
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    / m( h4 g  X- `- }* M1 K: q2 l
    2 L$ M. p- l! E# Z$ {$ J5 v2. **递归条件(Recursive Case)**
    1 ?: S- N3 J% k   - 将原问题分解为更小的子问题
    7 y) p$ D+ J" ~# y# X3 |7 w   - 例如:n! = n × (n-1)!
    . D0 e$ S: i$ l8 b2 J. ?) q6 _: W6 w7 O' e
    经典示例:计算阶乘( Y6 R7 U- G$ w/ F! g7 ]& \' `
    python
    - k4 |8 |/ V( b9 d; L: a3 odef factorial(n):
    : v: L9 n" B! O    if n == 0:        # 基线条件, X  o+ @  i9 t: L
            return 1/ _* M8 P4 H# [/ L
        else:             # 递归条件
    9 T- `6 M) x3 q        return n * factorial(n-1)
    4 n& K3 a! K  C5 Y9 j, s执行过程(以计算 3! 为例):
    - \: [6 L/ \, ]0 C8 y/ C1 W4 tfactorial(3)
    4 O; p6 Q  x+ h( d7 Z+ ?( K, E& b3 * factorial(2)
    $ o9 A4 v) K" ^3 * (2 * factorial(1))& q& ~" B$ H& h, a# R
    3 * (2 * (1 * factorial(0)))0 ~' w. @9 U* |% R9 k6 {
    3 * (2 * (1 * 1)) = 6
    ; O' N- l1 A# [$ |4 O# h# d" p1 m( K4 @) I* l1 |
    递归思维要点
    " a( `7 u; ~. J2 E1. **信任递归**:假设子问题已经解决,专注当前层逻辑, v. n& Z# P: L2 B$ u  j# v
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ; \8 X0 \5 ]% L1 D3. **递推过程**:不断向下分解问题(递)
    & p2 [0 l8 N+ H* g& B- W. N7 f. C4. **回溯过程**:组合子问题结果返回(归): @, u9 |5 q' _4 s) L; j1 G

    3 U, i2 R" j, t$ k 注意事项
    4 ^) Y( p9 B1 J* C必须要有终止条件0 Y, p2 R( X5 y4 z) Y% Y
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)$ k: N- ]  H+ G: [* @" W
    某些问题用递归更直观(如树遍历),但效率可能不如迭代: b+ c* {; Z+ h) k% E
    尾递归优化可以提升效率(但Python不支持)" w0 ^% W; U8 S% r2 l
    3 M0 Y& b( @. v
    递归 vs 迭代( [- v, c/ Z1 e) \4 \' k, a) b
    |          | 递归                          | 迭代               |
    - f8 A$ b: U  G% @|----------|-----------------------------|------------------|
    ) U) q3 ]$ D  T8 J- i0 _| 实现方式    | 函数自调用                        | 循环结构            |
    . t! v& R6 t* t! f4 G3 P| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |) C- H8 g6 c. K9 P
    | 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         |: E$ [0 s, G4 }1 \* m5 R
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    , T) i- w, o# M
    9 W+ }. {6 ]5 H- N  j+ A  D 经典递归应用场景. g/ D4 a4 N8 ^( V% L: g
    1. 文件系统遍历(目录树结构)
    1 G' r$ V/ `9 _2. 快速排序/归并排序算法
    ( n/ @! u& k% H5 ?% a4 W3. 汉诺塔问题( Z( l1 K; Q5 h1 B& v
    4. 二叉树遍历(前序/中序/后序)  }6 n5 w; `- a
    5. 生成所有可能的组合(回溯算法)2 g; {8 s" U( b+ ?+ M) ~0 Y" F2 A
    " S7 Y$ o7 k' ~" m' K5 ]- s2 ^, r
    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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    昨天 06:27
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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,
    : C5 p% P9 C% h我推理机的核心算法应该是二叉树遍历的变种。
    " m* z3 o* R1 B* H6 g  c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:# X- m& A2 h: j1 J' T  g
    Key Idea of Recursion6 z8 ]+ f6 O, o" c+ W4 X
    ) N2 `3 c. y9 J( g8 s. z/ t2 w+ c1 x' s
    A recursive function solves a problem by:
    ( t) f3 o. j) U8 [0 }7 p3 L' a/ F1 G- A
        Breaking the problem into smaller instances of the same problem.; U; k; o# X) N# W" d
    ( ^/ H5 G7 g8 E' q
        Solving the smallest instance directly (base case).5 Z$ `( D: D8 O. H( P& w  j
    - z2 N$ t3 ~" }% T8 h3 B
        Combining the results of smaller instances to solve the larger problem.
    ( q$ |4 \2 w( S# D# m* n( q) v
    ) R; @* N! s0 C% m: Y1 V" L% C3 TComponents of a Recursive Function
    . ^" K/ M5 S$ E$ Q* p7 X  @
    , u5 W# Z1 C' }, U# F( x    Base Case:% J1 I  s' s- D: S6 p+ D

      {$ u, P$ X9 t2 z4 \        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    & d5 z; i: q  A4 N9 B% [9 A/ ~. F( M7 ?' b
            It acts as the stopping condition to prevent infinite recursion.6 H, o! P9 s3 V% i. S2 l

    * q* _& h4 f5 I# T: s        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ' N( o/ h+ t" I7 f+ Z4 D( Z4 A: C3 w9 P) Y9 t) X
        Recursive Case:" f8 p+ |8 Q: _( f1 Z+ o, [1 n% ]9 I
    ) x# r. {: A6 X% ?
            This is where the function calls itself with a smaller or simpler version of the problem.+ U3 H5 J9 R5 f1 H' j6 _* ^# p

    4 G2 b! U; R# d) i( G$ Y: X        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ! m$ o5 X% `4 y( P) u: h7 X! R! ]. p% G) Q2 S( T5 l* N) v
    Example: Factorial Calculation; u2 n0 K, u! _
    1 E/ O" G4 `5 @" {" i
    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:" z, R) Q8 Q: x8 N& v6 t. l  C  o
    5 {# I& O, A" l+ |
        Base case: 0! = 1
    , Q; j: d$ q5 e6 B; S+ Y! |
    ( `( {: ~& f" ~: N9 p. ?0 [6 I    Recursive case: n! = n * (n-1)!% X/ w* d* o3 f0 u, J
    ) {8 N" B) f; e2 t
    Here’s how it looks in code (Python):9 Q1 Y* Z) u  k& O( A' u1 @
    python" r% B# Z- B" V
    1 _) r. [' U- v2 ?0 b# }- S

    5 _( o4 [$ R! d! Q3 R, ndef factorial(n):" u  H4 @: C1 H# u! a
        # Base case5 M3 G3 ]6 |& G4 U
        if n == 0:
    / p) N  h" y! h; d: }1 q) T        return 1) Y5 m6 `$ h: E
        # Recursive case  q/ H3 [2 H# J# X( m
        else:; X( U8 s  t! F4 E0 m( `2 y" p
            return n * factorial(n - 1)0 c( N) {2 b* a- m. V1 V! O
    3 }5 ?8 Q. Y# E, j$ W# ?& k8 Y3 G/ r
    # Example usage
    6 Q1 n- A8 d6 C3 r# L5 g  L+ Tprint(factorial(5))  # Output: 120
    6 E5 X( M3 I7 j3 T) X8 K
    / E6 b9 I1 {6 [; c+ Q; q. E' h8 YHow Recursion Works9 D; e- X9 U7 l) D3 k+ o+ c% U
    ! b2 n; L" P! k" C- j
        The function keeps calling itself with smaller inputs until it reaches the base case.7 o' k  o( ]2 W5 _! T
    9 d. v/ |+ s3 |6 _) A
        Once the base case is reached, the function starts returning values back up the call stack.
    5 W& q: Q( I6 g# u; r) K& N) k9 u- L, G* x% ?& q' y: k- O/ v" g
        These returned values are combined to produce the final result.( R0 C( g  x: ?

    # d0 t" m) b- E# CFor factorial(5):
    6 Q! I: w/ ]: E# n8 s) V
    1 o" K" f9 O: M. V# g0 a
    ( [2 G! r/ k1 ifactorial(5) = 5 * factorial(4)
    & ~  @. g1 q' cfactorial(4) = 4 * factorial(3)
    5 c9 a( k0 ?3 ^. Y$ Bfactorial(3) = 3 * factorial(2)) }) _' G$ g3 I% P5 r) X& _& P
    factorial(2) = 2 * factorial(1)& b: ^, `0 i% [( ^6 Y+ s
    factorial(1) = 1 * factorial(0)7 U" V9 R6 l7 w; ^8 x( ~
    factorial(0) = 1  # Base case
    1 G5 T1 l  B+ N5 A9 |) ~" T. n. X% K0 d& Z
    Then, the results are combined:
    " j- c: t7 u+ s& v+ k* M' F/ k. G' |+ i' ~& N: F% h0 n
      m& d, e  X1 K( @
    factorial(1) = 1 * 1 = 14 r; I9 ]7 u3 a- j2 N
    factorial(2) = 2 * 1 = 20 w. `7 k+ \5 ~9 Z
    factorial(3) = 3 * 2 = 6
    % N# V" X% c6 j1 M0 Hfactorial(4) = 4 * 6 = 24
    2 K- F: m5 `- {* X/ Xfactorial(5) = 5 * 24 = 120
    & o# [3 O5 V3 r- N8 a3 ?( n
    9 E6 G$ C- ~3 q5 X9 KAdvantages of Recursion
    9 m, y  h6 d1 h
    * r/ l! P' X! u5 }    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).2 O+ h$ A* A% W6 X7 `
    ) M5 `5 j9 s+ D5 l3 s
        Readability: Recursive code can be more readable and concise compared to iterative solutions.  S" U2 N4 s( T( H" d+ S
    ; C" |) a3 E( W5 q+ _8 T8 k
    Disadvantages of Recursion
    6 M2 X. `# a9 i" I* [3 F9 |+ w2 i$ ]- ?7 D
        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.
    ! X' Z8 Y: `3 p+ e0 B+ Z' R% J
    , s6 l2 a9 f& Y; O3 R5 c    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 [7 T2 I* e3 I$ A$ u. Z2 D: |, k# q- v( x* M
    When to Use Recursion0 v6 a1 R" O0 a. J

    1 S. {6 l. _% i* }$ B    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)./ H2 u7 b- Z, B* K  b! C4 Z

    8 I" s1 s8 i  q, }4 z    Problems with a clear base case and recursive case.
    5 X( b% ?! I  b
    , o! S" ]+ E3 |( c: {  jExample: Fibonacci Sequence( q7 C; A2 R6 e) x4 s) o0 P8 s

    % a" u! I6 e5 ~2 ?  t, m  B8 |The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    0 A8 Q! Y0 N% A' X6 y% g6 x$ z) ^8 Y+ V/ C5 A% S4 h
        Base case: fib(0) = 0, fib(1) = 1
    ' n0 {6 Z# S+ g( \. D- W' i0 E/ P, m. F4 }" `  X6 y( \$ O
        Recursive case: fib(n) = fib(n-1) + fib(n-2)" B$ c, p4 T+ D/ F

    ; ^( d. u( }+ u1 h1 vpython9 n: w. h/ [( N

    " k8 S4 N! Q$ s# ^- J+ Q$ u+ P0 ?
    / X4 }6 t! Y/ ]  hdef fibonacci(n):' c3 _( _/ i2 b! S) g* B# r( |
        # Base cases* {+ V. a; D; G2 Q. p( `
        if n == 0:
    & V; r: g* q( u( W9 W/ [% w        return 0' p+ ^" H2 L0 G- v
        elif n == 1:
    ) U& Y+ _& D' p3 q: x1 a! A3 R1 K+ d        return 13 A% i! j( o% C5 }3 ]1 L4 @6 h  g
        # Recursive case
    * ]) |9 |. q$ \" B) V    else:- g- u/ Y3 I( T+ d/ x, T, Y4 _
            return fibonacci(n - 1) + fibonacci(n - 2)5 A6 x: \. A6 P) l% O/ i

    7 Y2 l9 }1 `' w  S. c# Example usage
    : E5 M1 c, U: R7 }$ j3 Gprint(fibonacci(6))  # Output: 82 w/ G" R8 U! f. m( y

    ! d. q9 J+ b7 b& @5 _; b$ RTail Recursion
    . s! Y  E& {  `! {  Q% Y# v0 @& p, x* D7 X; `  j* H
    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).+ H6 D9 j, F9 N6 Z; U0 W
    - E) [# O  H) ^2 d& 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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