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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
    / k6 {' P$ a! \+ q' h$ \& }; U( F2 B- r6 I) f
    解释的不错
    # L5 e# |6 v- n5 s" A" s' n
      r7 n; A  M  n/ w/ `- Q# A4 x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。
    ( d* w: L8 d( O# X7 D- M* l4 u- S: `) }! t' v9 V
    关键要素
    7 x6 O5 A9 _' @4 z4 X+ k1. **基线条件(Base Case)**( B, r) t" ^6 y1 g5 [7 J
       - 递归终止的条件,防止无限循环, v5 R1 u8 l  j, ^
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ' X, H* v, Y$ `% V. r" m4 C9 c7 z) J
    2. **递归条件(Recursive Case)**
    3 t& Z4 m" e1 V% Y5 l4 m   - 将原问题分解为更小的子问题
    " ?" ]( P; q7 Y" }   - 例如:n! = n × (n-1)!' `' F1 ?9 o$ L# \
    . o9 n- W6 ]+ U' G3 o
    经典示例:计算阶乘
    & b8 @% D2 _- u, Y9 wpython
    8 ?8 N( o: U5 e$ `; i4 i# E, b9 L; mdef factorial(n):) n0 J1 U9 }# X, i' e
        if n == 0:        # 基线条件; L  t3 @5 u  E
            return 11 }/ D# C0 D9 \, C* {2 `( E
        else:             # 递归条件
    4 f- y' x# u  L/ c0 M$ r8 P% j4 C        return n * factorial(n-1)
    0 B( E9 v, T2 Z- o执行过程(以计算 3! 为例):
    / K* B! Y* T+ Sfactorial(3)2 I& l4 R9 V6 E
    3 * factorial(2)
    ( f) \& ?0 x3 b: g% [: z% ]3 * (2 * factorial(1))" n9 a: P2 `9 U3 N! p0 {
    3 * (2 * (1 * factorial(0)))' O5 v/ Z/ y$ I* r2 a( o
    3 * (2 * (1 * 1)) = 60 P$ g/ v; T' [  b  l" O
    8 Z" v5 o' S5 H! u# N0 L
    递归思维要点
    ' c. j" F8 i# E1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    * d8 I3 s" C% ]: |5 K- U8 h2. **栈结构**:每次调用都会创建新的栈帧(内存空间)) w0 J3 F) y; q
    3. **递推过程**:不断向下分解问题(递)
    ; W0 c0 I% M7 @9 D4. **回溯过程**:组合子问题结果返回(归)
    , g+ w1 l% u+ [. b2 [( }( E: f
    注意事项+ e; l. |) d; n2 T- m' `- L5 D
    必须要有终止条件/ m- ]. p) u6 D6 {) C7 N; c
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)' s$ j5 O* T% Y$ F; F
    某些问题用递归更直观(如树遍历),但效率可能不如迭代+ N2 w; ?2 V7 Z; U3 m5 ?7 [# O. i
    尾递归优化可以提升效率(但Python不支持)
    + H+ H( F$ U1 P) Q+ S0 |6 C. j# A2 P
    1 ?( p9 R6 H+ A' \ 递归 vs 迭代0 x7 l4 Z  @$ c* N
    |          | 递归                          | 迭代               |
    ' ~- w1 d+ f* M5 [|----------|-----------------------------|------------------|7 x) u  r3 V  D- Z$ q) m( g- b& ]
    | 实现方式    | 函数自调用                        | 循环结构            |! E' ?: r) X0 {; W0 @& P
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |5 p4 R- ]; n6 J, ]
    | 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         |
    " f9 A) S0 W; Q9 s, x| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |2 P: C8 a& f+ M" b+ u' B
    ; L0 f, j. D+ L5 R3 N/ C
    经典递归应用场景
    ; z2 G% n2 V  ~+ m5 b' ~* r1. 文件系统遍历(目录树结构)* M  q6 D! i( ]4 q7 o" |
    2. 快速排序/归并排序算法
    ' M' g. g( |3 V: W$ f* g: V3. 汉诺塔问题
    ' P8 _: X; o& [" L) q4. 二叉树遍历(前序/中序/后序)
    # G# c, a. D& t8 O5. 生成所有可能的组合(回溯算法)6 L) ^1 O- n' y# c) O3 [" j

    % C- i" T6 J/ g5 c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,
    6 C; g; k6 T- l" Y我推理机的核心算法应该是二叉树遍历的变种。9 K$ {# s& x4 G9 `, n
    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 f. Z- G0 ~5 @* i+ R3 o/ P- T2 L
    Key Idea of Recursion, D: @# h3 p/ Y
    1 ]- ^0 q* m2 M% R* a" v7 j
    A recursive function solves a problem by:
      Z8 A% z1 d# O/ s
    0 d2 w5 z' Q; b2 E  k# e    Breaking the problem into smaller instances of the same problem.
    & R/ k% p9 t& `- |& q/ {( y
    5 S! J5 N# p# g( S8 N; a    Solving the smallest instance directly (base case).
    ! f6 z. b7 ~4 E5 y: q) X: g( h6 r" p
        Combining the results of smaller instances to solve the larger problem." X' Q# o) w1 P' i
    ; L- |$ N" _$ r- c" g( B8 @7 C
    Components of a Recursive Function* j: F: u) C; T* O$ y
    0 w1 P6 k& C0 O; h& k% s* |8 q
        Base Case:
    6 c: P1 }# B8 F9 o$ M8 p8 x4 ~5 J$ L( l
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    # Q5 B. i8 U9 L; x! i$ V2 }9 L! K8 h5 Q* z
            It acts as the stopping condition to prevent infinite recursion.7 L& T, S; d6 n2 w" c7 C

    ! E! ~0 x+ @' w        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    & o3 i/ N, T* P7 \* Q8 A, c3 c
    3 A/ ]" h& H  ^# J, y8 l    Recursive Case:8 c1 K. N3 M/ f0 e

    3 s- j0 J- F3 z2 u9 G- r        This is where the function calls itself with a smaller or simpler version of the problem.2 B: T1 w5 f! I1 _  h/ D+ \
    $ n6 O+ C, d, L2 `" ^, w" J
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    3 N/ J0 x, [+ _# H( T& T( u
    ; O( Q# v* y) C9 VExample: Factorial Calculation( c. X2 y0 D- X: P4 d
      j! r/ x% u. `; w
    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:
    3 S! \3 {$ a- D% `+ N8 _4 x% B4 W% u" r7 q. T) `# a
        Base case: 0! = 1
    ( E6 w8 Q3 `' q4 ~, Z. z# L" q) x! Z6 J$ ?
        Recursive case: n! = n * (n-1)!8 f1 _1 R  H, B- x) G! R6 k- i
    $ P  A! I% U, T( j
    Here’s how it looks in code (Python):) b+ m3 T4 O/ B. R$ f
    python7 a6 g8 }. f# t7 H

    - H2 M5 Z4 Z! U+ ?6 }& |# o( M. H; ?/ y/ a" }, C- X
    def factorial(n):
    ! b9 w! @5 l% a; ]    # Base case
    " N6 Q# ?7 @/ x2 i6 Q& E* F    if n == 0:
    : Q4 O- O# G6 n        return 12 |  R) h; Z( Z; Z/ i/ ]
        # Recursive case
    # ]  g  q$ o+ O' K+ A    else:
    3 U! z% c+ z+ _: i$ w, y        return n * factorial(n - 1)- G$ p+ C. E6 n/ }& i: q

    + z4 P, w4 A" @( @6 e) k# Example usage
      v* R& g! e! P5 J+ I% Cprint(factorial(5))  # Output: 120
    5 }6 G+ p/ P) C. d1 u, W' T9 ^: R4 x+ j  i/ w8 Z
    How Recursion Works
    4 I# y& x! s/ X: h% X! E7 Z; y
    : |, R8 g. L9 p    The function keeps calling itself with smaller inputs until it reaches the base case.
    3 U* \) j) M. \, ^4 U, j4 i
    " V1 J7 h! b4 h4 a# P2 M    Once the base case is reached, the function starts returning values back up the call stack.
    # _& b( Q  p' U1 n% H' n+ {
    ; |" U6 J* v! K0 x, N/ e    These returned values are combined to produce the final result.$ p' Y3 c& h1 |4 \3 _; W  H$ o9 V) W
    * z' ^/ C! q7 ~* w4 Y, \  e! l4 x
    For factorial(5):- l. d+ r* w  R1 E

    " E! n( w7 E. S! `7 m' S) U  W: L; y7 t: r# Z  R
    factorial(5) = 5 * factorial(4)
    0 k; C+ t9 Z7 p' d. tfactorial(4) = 4 * factorial(3)
    ) t8 K& X' p0 _# ?  Dfactorial(3) = 3 * factorial(2)1 r9 h; U/ I5 D6 b1 d
    factorial(2) = 2 * factorial(1)
    0 l- t3 V( h7 t9 }: q( t; Ufactorial(1) = 1 * factorial(0)0 }( U1 X. Y; S8 ~$ z5 h
    factorial(0) = 1  # Base case
    8 r/ y. D9 p0 z* @  a: t( {. T, V* I9 ?" A& j4 z
    Then, the results are combined:
    # u1 C6 X3 W! v2 j$ c3 N8 N1 N+ e" [5 ~, T: w

    $ V! B7 X$ G, {) dfactorial(1) = 1 * 1 = 17 z" o0 s+ t8 d9 W% {0 K
    factorial(2) = 2 * 1 = 27 b) z& q# n( n
    factorial(3) = 3 * 2 = 6# _) k: V( _3 T) Z
    factorial(4) = 4 * 6 = 24; a- z% N7 v' ^. E: k* B4 R
    factorial(5) = 5 * 24 = 120
    # V) Q8 ]% X; B' H8 ?
    3 p4 l6 B# F* H$ _  wAdvantages of Recursion
      h1 C. ]. J( T9 c7 U+ M
    - M4 f: o5 E; [5 T0 \. W9 I& j    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).
    " `7 E! }' K% f0 c! _7 r% `0 ]# n. S/ M5 B2 R
        Readability: Recursive code can be more readable and concise compared to iterative solutions., X3 K, V+ n  @9 l& J5 g+ h2 u5 s

    4 {2 k- P( p0 g/ {( FDisadvantages of Recursion. g2 J" Z- X( F$ ]) {: i

    7 v: X" @( \: ~, ~! x  ~    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.
      h2 ?: F  s, ~4 F
    0 T. v6 O# B: Y  W% `    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).. i" d& t0 M) W

    $ t# R) M1 z1 d9 S" y% mWhen to Use Recursion2 l# K) v1 O) W8 T: a  W' @
    3 H- u+ i3 x2 Q4 p2 a
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    6 R4 G9 t, Y9 N+ E4 q5 [7 C; d# F$ n6 p
        Problems with a clear base case and recursive case.+ A  e7 t8 v" Q6 E9 f$ X8 {. ^7 ~

    $ s/ i5 j& D) s5 V0 TExample: Fibonacci Sequence
    - w% L9 {5 ?! j' q  I- Q; c' }1 v: C: X: h" t( d
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    : Y- p: S7 `9 K
    : _. B! Z- @& _. J: ~' ?3 t7 s3 a    Base case: fib(0) = 0, fib(1) = 17 {. Q: F, s( @% C7 v
    9 x5 `1 [6 H& Q# X/ W+ W
        Recursive case: fib(n) = fib(n-1) + fib(n-2)1 f" S+ y# A( ^2 e0 N
    $ x: A6 t6 _+ b
    python/ W! ~7 h  C+ l+ \9 r6 E

    9 T4 u2 P4 D! R, A, x/ {9 D# }, G
    1 u" t! g5 e- Kdef fibonacci(n):, t# @, \; M: a% C* j2 T6 u
        # Base cases4 b  v) `+ d: D/ x( S/ }
        if n == 0:
    9 T$ @4 w1 c6 \: @, K; U        return 0. P* _) g: t$ j5 x; s
        elif n == 1:& H7 h+ ?8 e% W: i4 p
            return 19 ?* M; ]9 s' K3 s% _- g$ L# ^3 W
        # Recursive case
    9 \; q  g/ @' F3 H  h& e& r    else:
    4 M% u1 H; ]; [! X& `/ n        return fibonacci(n - 1) + fibonacci(n - 2)4 R5 M% u1 K3 l* ~/ e

    + V& e/ o+ U5 G+ f' Z7 Q6 `# Example usage! @4 e8 i' r2 g
    print(fibonacci(6))  # Output: 8( C/ e, k# n0 c9 c% C- u9 M1 m  J- P

    / s3 R: z' n$ V2 K) ~Tail Recursion: H. c# ^5 U4 Y8 i3 T

    8 ~5 k' q* I2 F/ UTail 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).+ |% f# [- G  |6 Y

    7 }  S9 o+ @5 f9 M3 K2 y0 CIn 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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