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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 % Q9 k4 n' ^2 l3 d% F& h* o
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    解释的不错
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    5 a. S; _$ x0 P2 _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。
    ) ?4 ?4 I$ K; m) f7 a
    . I: {1 j. R% f3 M  c 关键要素4 ^4 F6 a, |, l: g" x
    1. **基线条件(Base Case)**
    1 z0 k2 A* e8 @3 P! U* ~   - 递归终止的条件,防止无限循环' Q% C3 Y5 s, K9 |
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ' Q; {$ `* p, I2 D) M9 M
    4 x7 A5 O* d4 o: b$ T/ n$ Z" q5 m2. **递归条件(Recursive Case)**
    & v! n. i0 K% ~1 E8 L9 a* `   - 将原问题分解为更小的子问题# H3 ]6 W' y) u6 H' r& T
       - 例如:n! = n × (n-1)!" t, H6 Y5 `! d' B& }' V7 B

    , x3 W+ J0 j6 ?. f. t3 c5 \' | 经典示例:计算阶乘3 q# j& H4 s% ^, m' f
    python
    # }: ]/ U5 E- z" p$ Pdef factorial(n):8 J: V2 X0 y/ e4 o
        if n == 0:        # 基线条件
    * d! `$ G' x9 K, K' s/ N        return 10 X9 Z6 R' E; G1 a. ^
        else:             # 递归条件+ h" P. w9 u3 c) b* o
            return n * factorial(n-1)
    * O& `2 Y+ t1 X) k& k  t9 S执行过程(以计算 3! 为例):: p; h  ^! I/ o2 c) N
    factorial(3)
    8 ~9 L& w4 |2 L! _( @3 * factorial(2)) k  X1 n1 f. F* U9 p, v" f
    3 * (2 * factorial(1))! h4 P5 U% g4 t; C2 H
    3 * (2 * (1 * factorial(0)))' Q3 o+ b  ?3 G' n4 H5 |
    3 * (2 * (1 * 1)) = 6
    1 Y2 m* E# l/ u8 u8 P; A
    . S. @5 o0 l- E$ e; a2 O; U 递归思维要点
    & C: ?5 \% I9 O+ c1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    9 I0 u0 T+ A7 g4 W3 g! I4 X5 D2. **栈结构**:每次调用都会创建新的栈帧(内存空间)* h0 f5 D( \: j! i1 y, y% k! P, ]
    3. **递推过程**:不断向下分解问题(递)
    + m- `- E' z9 z2 E9 g  n4. **回溯过程**:组合子问题结果返回(归)
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    7 I8 O1 y; S- |# R 注意事项. W8 v6 O8 Q" c, Y  d5 u- T+ M: e! J
    必须要有终止条件5 [6 l  @4 f- u6 P+ o3 ]; m! H
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层). W* h7 ~8 L8 i$ \0 m
    某些问题用递归更直观(如树遍历),但效率可能不如迭代, P! U( l! U& d
    尾递归优化可以提升效率(但Python不支持)
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    # m, `# v3 b& R& h* ^: \$ @ 递归 vs 迭代. z9 Y5 M8 H# @
    |          | 递归                          | 迭代               |
    # f# e; V8 x6 T. k* i! {8 H7 ^$ i|----------|-----------------------------|------------------|
    / y7 O4 z' l6 j5 Q5 s4 w# Z| 实现方式    | 函数自调用                        | 循环结构            |- g  }/ @- P9 q' ?5 [, Q5 f. z1 k% n
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |/ j+ b$ w7 ]( g- T% R
    | 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         |
    0 Y- K+ n3 R% b1 k| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    5 Q- A+ v0 _8 d5 A5 ~* {. M) Z+ h# p4 R( T% d
    经典递归应用场景. f9 ]* I& J, ^0 q) [5 n
    1. 文件系统遍历(目录树结构)
    , n7 m2 L8 J) b' b2 l9 h  A, t( ~" [2. 快速排序/归并排序算法
    9 u; Z' J; \8 f+ O: t8 Z$ F3. 汉诺塔问题
    : @) \" N9 T! k+ u  S4. 二叉树遍历(前序/中序/后序)- Z" E$ ^* L) s& A, G( E
    5. 生成所有可能的组合(回溯算法)7 |6 H* ^& `' z" z6 U; w

    ' @/ q+ C9 m3 O" ~2 K& \* d' f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,
    " j7 v* W; Y, B! p我推理机的核心算法应该是二叉树遍历的变种。8 o* w, t6 X5 h) B
    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:2 Q& a& a, C& S" S2 i! B* o' ~
    Key Idea of Recursion" W( z6 V7 _; K# N
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    A recursive function solves a problem by:
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    4 f% e" ]4 I4 \) R- ?: v3 @+ ^    Breaking the problem into smaller instances of the same problem.
    5 h* b4 m( m  O+ r5 T- q3 b% n0 O& z  B: x* b% P
        Solving the smallest instance directly (base case).1 Z9 s! m+ K! N; w/ s; T1 {
    6 }+ n$ x1 A& y" o3 \9 w, H' c
        Combining the results of smaller instances to solve the larger problem.6 @9 w8 m+ Q; q+ q# D3 o

    % f# F* ?. k3 \Components of a Recursive Function
    - ^9 L. v2 i' k3 y1 h" j
    ' \2 i/ H; i, i1 S- l5 I+ B    Base Case:
    0 e0 B. E1 B; O, H! o- D5 o" Z4 P( }' m5 Y5 [, N" o2 P0 W
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            It acts as the stopping condition to prevent infinite recursion.# Z3 z4 N# k; k9 F: ?" a/ ?! c4 {
    % N) d4 m+ g! g# S+ z
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:
    / ?2 h5 j  p0 x) \+ g7 M2 A& l' ~9 Z! O4 L/ n
            This is where the function calls itself with a smaller or simpler version of the problem.$ l# Y2 D# Q% i2 \! b( f" z0 H

    2 \3 R# G* W) A7 X2 y2 T        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., T1 w; b$ r  k2 r/ C* n( F

    + v& E0 k) t3 V( lExample: Factorial Calculation/ `9 d6 u* V& f/ B5 e
    ; ~8 j) x9 S9 K7 g5 Y5 ]
    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:! f$ y! i, x3 u* h" u) w
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        Base case: 0! = 1
    4 _6 C5 o. v0 k; p9 k. ~' u4 T& G: S
    " B4 h7 ?6 I* W+ K    Recursive case: n! = n * (n-1)!
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      ^& V/ ?' {8 p' j+ C; a( gHere’s how it looks in code (Python):
    . O5 e- x7 m% ?9 O0 @python
    5 }! u4 m$ v, w. K" }6 ^' F; i9 M5 T* J' S$ B3 i- W- S

    ; L: x# D3 r3 udef factorial(n):
    8 q8 u) r4 s/ g$ b' K2 W! F    # Base case$ M" g% L+ {1 p3 e( l' Q( ?
        if n == 0:" |1 x) v1 f% B' O* }
            return 1+ t9 [% M* K3 O- g
        # Recursive case
    : j1 i% B* C3 l" |    else:
    * H/ L' j  ]; j2 J1 I        return n * factorial(n - 1)# O; c3 F- k1 Y

    3 l0 F3 n: r8 V" T9 e# Example usage& ^( ?6 V7 Z+ _: ?$ b1 Z8 }
    print(factorial(5))  # Output: 120
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    2 M3 D3 L! r! J2 `/ t  e3 mHow Recursion Works4 Q) I8 Z6 c  p, @: U4 C% j8 o3 X

    2 e5 k( r7 f2 `/ L    The function keeps calling itself with smaller inputs until it reaches the base case.
    5 C4 e6 |4 o1 h% L3 C9 M
    3 C& ~0 @- b" k$ C" ?3 m" z$ B/ |* `* \    Once the base case is reached, the function starts returning values back up the call stack.+ v% J+ o4 R. S8 y2 ^! \4 A
    " B% e& a1 J! |9 F3 @
        These returned values are combined to produce the final result.2 h* d: u5 c! A& Y6 K( u  [& l* g! C1 _

    6 ~/ U  S) I) AFor factorial(5):
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    4 v  b* ]6 ?( I  ^7 i, y, W4 V$ v% Q8 j+ z
    factorial(5) = 5 * factorial(4)- X# v  Z/ p' |9 O4 ~! n  N& U9 ~
    factorial(4) = 4 * factorial(3)# U+ @) {& @: O9 h. a5 t8 {
    factorial(3) = 3 * factorial(2)
    : z. Z, _" }0 W5 H4 o, t6 ]7 @3 Xfactorial(2) = 2 * factorial(1)7 i9 F! W  s$ y0 l( a) c' i
    factorial(1) = 1 * factorial(0)
    % a( |- a  i, s0 u8 F" s  A3 H+ j6 Ufactorial(0) = 1  # Base case; P0 z2 y+ Q/ [

    $ A7 v0 o  N  `6 H# `Then, the results are combined:
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    5 D9 }- Q7 \2 `: i( M; j& T
    , d; J& c& e& a: n  g" i2 `7 xfactorial(1) = 1 * 1 = 1
    - r. Q9 ?8 {8 Tfactorial(2) = 2 * 1 = 2
    7 u+ s5 h3 d- |+ ~- A9 M0 |6 ?factorial(3) = 3 * 2 = 6! I" v. U# r$ h, P
    factorial(4) = 4 * 6 = 24
    1 A: U. z' V" afactorial(5) = 5 * 24 = 120
    + l3 z4 h2 k1 r0 m+ \
    1 _: g0 w2 m  }) K" I" ZAdvantages of Recursion
    7 E. ?9 z( \1 Z3 u" e1 v' \
    7 h( H" q! T+ p; J; `" k& D    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).+ i  i  d, X  z# l
    " V; a! t8 n4 ?* U+ Z7 C: O
        Readability: Recursive code can be more readable and concise compared to iterative solutions.# ?2 T" a8 \* n. G+ n
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    Disadvantages of Recursion$ n4 d: C/ L, p6 g% h# S
    1 I$ ~# S+ @3 C* w* ~
        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.
    6 n/ D+ Y4 `, ], ~* M" L% X7 V! q; B1 T/ y& C/ q- e1 S2 n
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 c5 S% X) e% [7 U+ p: u( h3 d

    3 \* `. m/ I, B4 ]- J! q# JWhen to Use Recursion! x4 P! Q' c, @, P' P* v4 Z

    : q- {! f6 i6 e" c    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).( w2 Q5 B$ [/ P/ N
    ' g5 t" ~; `8 |7 W$ {& p
        Problems with a clear base case and recursive case.
    8 x+ t" [2 ^* U' f, N4 r8 H# T5 @5 n2 N# |. O3 h) E, m$ W
    Example: Fibonacci Sequence
    # q' U' l9 H, h6 h: z* ]4 b- G1 y3 k+ J+ ?3 u8 J8 Y
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    0 r4 n+ H2 }2 I" P' B! ~; O; E+ K. V7 ?7 S* h6 W/ d
        Base case: fib(0) = 0, fib(1) = 1( m2 f+ w# K* U1 F1 i

    / w% }. r9 u* G: A! e5 ?) H6 h    Recursive case: fib(n) = fib(n-1) + fib(n-2)/ R6 O9 {0 W1 O( I
    0 C& l, H) {5 y
    python& }* i5 `5 o* {8 ~. Q! p8 m" J
    $ i/ i  }7 E( y- P1 |0 ^7 y

    $ \# G1 v8 n/ O. T8 F6 K1 mdef fibonacci(n):8 p! @% v& j0 k+ e+ _
        # Base cases
    & |6 T$ G) B( g( r" q    if n == 0:
    4 l5 Y0 S( G9 t9 S0 ?( O0 `        return 0
    ' l% V% ^/ K, w/ a! K; Y    elif n == 1:+ {! D( p* ~" W; c
            return 17 p2 ^' s$ a1 n. c
        # Recursive case' B7 }4 q0 C" u  S
        else:5 v2 w4 o% u! s- [7 H% L
            return fibonacci(n - 1) + fibonacci(n - 2)# b, u" b& `9 P" }% u
    1 L- Y( l9 m  u& I3 A
    # Example usage
    9 c) t5 g6 U) k- {7 h+ w: Bprint(fibonacci(6))  # Output: 8- z6 u, ]9 u) a- g# C2 O
    % {3 T* D  k' t6 _
    Tail Recursion" m5 V) R9 x  z/ U* ?& h

    $ O$ u* h! @) i8 QTail 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).
    * X0 D. ?: J7 N6 a/ I, g
    1 f& a' D2 i6 H9 r1 N+ b; iIn 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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