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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
    0 x. c, E3 c1 |. Y8 F1 t) ?$ ~0 \, P! @9 R4 O  _
    解释的不错& h% q* j1 ?, b( t! p
    3 [( n+ |8 L+ D3 i& d( M. 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。- u. N$ k* }6 R# Y0 W! [
    8 ]4 {) s) G9 F* N9 w! h
    关键要素. l( D7 _& d2 S; {% R- D# L# D
    1. **基线条件(Base Case)**
    7 I8 p6 \; N0 o3 Y   - 递归终止的条件,防止无限循环
    6 p5 ?, j. B/ Q7 j   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    . F  L' w! M$ c- m* ^5 x% \) C$ n; a& X+ P4 f
    2. **递归条件(Recursive Case)**
    0 U  @4 a! d. E' C, g6 k2 w7 A( l   - 将原问题分解为更小的子问题: H: e4 |5 S+ ^) H
       - 例如:n! = n × (n-1)!
    2 m0 k- h7 P' H- d3 n4 s, C  ]
    % I- Y6 I5 L6 @# a5 k 经典示例:计算阶乘1 K, j1 l4 d1 }5 l/ j+ N% Z
    python
    / v9 y2 Q# i* N+ R4 Z- V4 \+ |def factorial(n):
    $ F8 e. p/ ~6 o! u7 ~. C, l" H1 d6 i: \    if n == 0:        # 基线条件
    ' b3 V  ^1 r, b- N- l# N: J% l9 Q        return 1/ ~- y6 E* J2 O& |4 a# p
        else:             # 递归条件9 `7 V0 N% j0 I  l6 d! g
            return n * factorial(n-1)
    / t$ }/ T; s4 U4 e执行过程(以计算 3! 为例):  a4 y( Q/ F9 ?
    factorial(3)
    1 K1 Q7 l1 }/ }$ e* c2 |3 * factorial(2); N4 O, U. x, L' z
    3 * (2 * factorial(1))! P9 Q( }5 N# m6 [. h& |( {
    3 * (2 * (1 * factorial(0)))1 \# Y; z2 ]$ G# m, i" t8 h
    3 * (2 * (1 * 1)) = 6
    4 }2 H1 r  ^# g% C; H5 M) M) \6 H1 g$ s' {' Q
    递归思维要点8 L2 Z0 w$ t) B" ?
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 @- n+ {' h- E' o, p
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间); @% S9 w6 m+ A7 E- g6 U0 @. [6 {
    3. **递推过程**:不断向下分解问题(递)6 G; y4 E$ D3 R7 f2 n' l. J2 n
    4. **回溯过程**:组合子问题结果返回(归)
    . S. _8 i8 V) ?) T8 x1 f4 x& V" D  u3 u$ n) H/ Q
    注意事项5 I. ^( X6 |, l6 q
    必须要有终止条件
    . \/ o. s( p. z- h8 R4 A4 W递归深度过大可能导致栈溢出(Python默认递归深度约1000层)% y& j) D, ^8 ]' j6 q8 N
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    / ^" l; k* Z2 \/ F尾递归优化可以提升效率(但Python不支持)
    - l$ l$ F; y, [' w& T8 w3 S2 C
    4 r4 A+ q$ @4 Z# L& z% C 递归 vs 迭代
    ! `: J5 e# r, p# {3 M' }|          | 递归                          | 迭代               |% j- X) l( H+ E4 T9 B) G$ \% x
    |----------|-----------------------------|------------------|
    : @5 n) R% b. {  Z6 p( V- S  `$ P+ V- p| 实现方式    | 函数自调用                        | 循环结构            |
    9 D( M' l( d! ^0 K0 E7 x1 y| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |. E% B* B  [% 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         |# T0 r8 x, Z- J9 \4 \+ M
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    # f' ?, m2 t( L$ g1 ~) z
    6 R0 d( t3 q9 [" O3 h' ?$ l 经典递归应用场景
    + L, W  V( l% A$ m8 \0 |4 Q1 z1. 文件系统遍历(目录树结构)1 y: F0 z7 }, q8 R
    2. 快速排序/归并排序算法
    ' h2 z" `2 N- K; H3. 汉诺塔问题
    2 `! y; D/ Y5 L# k) I4. 二叉树遍历(前序/中序/后序)
    7 F: x+ E+ i8 e0 a  U5 A2 |" m! O5. 生成所有可能的组合(回溯算法)% |) n$ G1 R# A& u
    & w5 I0 g% b, l1 ~) o4 s
    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,3 u: o8 V5 v2 J6 v: \7 S& Y. U) d
    我推理机的核心算法应该是二叉树遍历的变种。
    ) m7 h* T3 w) ]; ~9 f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:: y" G/ E7 S7 V1 D! v" D4 {
    Key Idea of Recursion
    5 {- M, Z; D! B% @, K7 N3 O9 n* C; _7 p; ^6 t7 S7 H
    A recursive function solves a problem by:
    9 q- J, q# i: h& O- ?& }5 C6 q& J! u% K' \0 N- e7 `. W. Q
        Breaking the problem into smaller instances of the same problem.% f6 j1 Z4 ]5 T+ n, W
    / L9 i$ r4 v9 M' G
        Solving the smallest instance directly (base case).
    ! \( B6 J: W  C! m, Q2 ^. V$ L# b5 }  M4 b2 r; F
        Combining the results of smaller instances to solve the larger problem.
    , l, l: V( O6 l/ c% V, B( o7 D3 q" b
    Components of a Recursive Function3 L7 [) H# F2 m  o; X  w1 B5 h
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        Base Case:( ~! R% W- D* N- U8 F% [# T. _5 Y

    ' |6 y) f+ C7 x3 f        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    7 ~$ k- g+ N/ f' V) V, q
    - L; e: ^) p0 p5 Y; a+ R0 t        It acts as the stopping condition to prevent infinite recursion.
    6 E& O' s# `) l5 Z& V2 x. s. w1 p2 y3 O9 a& _
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 g% p- X) l( Y3 c, A7 c
    . K* q: R2 k2 m
        Recursive Case:2 o* c8 j8 Q* [

    + ]. S) b+ r; u$ V; n+ N( P% |* F/ j) J        This is where the function calls itself with a smaller or simpler version of the problem.
    4 C: G+ O' f3 A1 c7 A
    4 F& m6 T6 P( F/ w# t. ^        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).$ r" h& v; e8 P$ ?3 y; c

    5 D( q3 B" O1 E' E0 |Example: Factorial Calculation
    2 x* I8 ^1 z% y) I/ k; l# d* [) L) Y3 D( e& B
    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:
    + C7 r- f8 b1 v+ Z! h& c& Q% V  Q
    # r: H* J; e9 _( h- t6 E+ v& ^+ J    Base case: 0! = 1
    9 R% O6 S$ g( m+ H6 h
    - l+ v! |; W1 j' a1 B  F    Recursive case: n! = n * (n-1)!( o/ x' i; A% G7 Z0 f1 x& Y
    ; Z2 Y! s; c; ~1 w) w. Y: g$ _, `
    Here’s how it looks in code (Python):
    " N' J8 N! ?3 F0 npython
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    6 C) Z  l6 j) _* B+ z3 G# Q  t& X. Z( X- I+ N) d; ^  E2 F
    def factorial(n):
    / s4 p4 o! B+ \  e0 ~  x    # Base case7 {7 P0 Z. u7 W* L2 a
        if n == 0:
    ' L8 v! k6 L# a$ q- m        return 1( t! P+ k- I2 J. C
        # Recursive case
    ; a, E6 K( A$ l4 A0 A    else:0 G3 v4 i7 U3 d
            return n * factorial(n - 1)& @  i! r" s/ \" w- A1 A

    % d* B" L# Z+ w) _4 y4 ]3 E# Example usage* H( C0 l' p: L3 x
    print(factorial(5))  # Output: 1206 I/ X1 W9 e6 ]) V6 b; l8 c2 F

    ( O6 U( v& G0 S4 G) b- sHow Recursion Works6 s1 D# a1 z, C
    ! G1 D# J  x, }! q) q3 E2 j
        The function keeps calling itself with smaller inputs until it reaches the base case.
    3 ]4 q* k" w% C
    & b( [; H" {" R9 O    Once the base case is reached, the function starts returning values back up the call stack.
    ) H; [. [3 `+ t" \8 H- {1 f) r( ]: R
        These returned values are combined to produce the final result./ r4 U0 N9 G" j% w/ A" O# a

    : e. p% V! ^* O" e5 V) }7 w" I+ r  TFor factorial(5):* {% \9 ]5 u7 P- d' r+ E. o% Z

    1 N1 g+ F- Q% g2 T- E8 d
    " s* x7 n, v) Z* \, r3 Y& _0 Xfactorial(5) = 5 * factorial(4)* U! j( [9 J3 o
    factorial(4) = 4 * factorial(3)
    3 s5 e2 @( x7 J7 ?' j0 xfactorial(3) = 3 * factorial(2)
      |8 z9 t: p2 s8 ^. Kfactorial(2) = 2 * factorial(1)4 b3 x5 s: Q& E( A9 U7 j7 A
    factorial(1) = 1 * factorial(0)
    ' \7 d$ G$ `0 j) ufactorial(0) = 1  # Base case: l  z( F4 G5 D  Y; l* R
    - z" V$ f  g3 s) `
    Then, the results are combined:
    . X$ u* L6 _( f0 [
    # t7 x; J" r1 Z" o( I6 P& \5 e" N3 o9 O' V$ x' z
    factorial(1) = 1 * 1 = 1, F, ?* H9 Z! l& d4 ~
    factorial(2) = 2 * 1 = 25 d3 g* T+ Y+ t1 P
    factorial(3) = 3 * 2 = 62 V0 r* m7 V  a( _& q/ {: \
    factorial(4) = 4 * 6 = 243 v9 y3 Y7 [3 `0 \+ R  K/ c( E
    factorial(5) = 5 * 24 = 120, C4 V& K, z9 Z4 |( V1 j7 ]
    ) }. M& |4 L$ [% w
    Advantages of Recursion
    3 }6 I$ T0 u' a- O
    2 r, J; q/ L) g7 R    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).
    * \' e, D* s3 g$ n& j) K3 G6 V% \: ~0 C- H
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
      r' a. \3 |6 j! i. _' G& X+ q) D' @6 D- F
    Disadvantages of Recursion
    6 [5 Y& ?( }8 k; `4 f: Z: c' P
    1 H; t; Z/ g/ @* g5 Q  `0 d* u    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.
    / _/ j$ L1 \" d& [- a  ~$ L' w' [) H3 ]4 M" V" Q
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    & T! I+ K; c- i, J  b% i) \
    / T, D# F$ g0 E; q8 KWhen to Use Recursion5 O5 r; `0 I/ [3 O
    $ ^. K" H! [- ?: f7 p
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).% G, c0 w, u( w8 S& ]/ k2 K7 O. Q
    , c, P) F- B! g. G; x8 Q$ f6 v
        Problems with a clear base case and recursive case.9 L2 a" \( c/ e7 U! I% B1 }

    + `$ K' J3 ?/ b& L; _  mExample: Fibonacci Sequence
    ; i# r1 t% ^% \8 ?4 `1 H$ X, y& H1 b' H
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:; L+ `$ N* w7 V" W' L
    ' }3 `2 X  X! A( c& I0 ^" l& w# r
        Base case: fib(0) = 0, fib(1) = 1
    ' {" O/ A% R& P4 R' ~& I
    & b0 A+ q" A' D    Recursive case: fib(n) = fib(n-1) + fib(n-2)& m8 Z  N# o8 Q. u& O9 H

    + d* f5 E& L" bpython8 B% i/ l+ r5 I0 y8 e$ W" x( f

    . f. u4 I1 }* g6 |
    9 R/ N, B4 b  b6 z- Zdef fibonacci(n):  o+ n" ?' O5 `% d+ j
        # Base cases
    ! y& e, [; G5 ]0 i. [/ \; L    if n == 0:
    " r4 ]- `$ g2 Q0 J        return 0
      @, g+ r$ w" {: f; O) N7 k    elif n == 1:
    9 i0 M" ^- z0 J' G' y# |        return 1
    * S$ G) D/ q& m0 s    # Recursive case4 ^5 d2 n- T3 Y  G
        else:5 H2 B2 k/ ~/ I) _, ~
            return fibonacci(n - 1) + fibonacci(n - 2)
    6 f/ D+ `7 g% u' r% f
    0 @% i* t6 G. Y' C" u# Example usage8 @* _$ V8 S- Q( r) _, n; a! ~- x
    print(fibonacci(6))  # Output: 8' N% J2 Q% A! G0 h, L% o
    ! U! A& q! o# F( A9 \) @& O: C
    Tail Recursion
    ) e( k% L: ~8 G, B) c- A7 E9 ?; c% A" C  m' B
    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).  |% C5 i% M2 p1 Q% V

      x$ e3 i! T* T) GIn 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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