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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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  • TA的每日心情
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    2025-9-8 05:08
  • 签到天数: 3 天

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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
    3 b  h* N  ~* s* t- r, G" Q% Y5 N$ P  z/ G
    解释的不错* v  k% P0 Q7 W$ V3 b" S- m% ?
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    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。7 L3 |$ o+ `# D' R  T( q0 C
    8 D8 h4 O( A& E% S" E! Q
    关键要素
    3 T/ Y6 p: Q) H+ g& C1. **基线条件(Base Case)**$ F# K  }/ M$ Z: L
       - 递归终止的条件,防止无限循环
      P1 w: k/ K+ S   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      Y* a; V8 L9 ?! a+ a
    - m! g$ V0 _8 c2. **递归条件(Recursive Case)**. X& Q* ~3 J- s$ j9 i3 P
       - 将原问题分解为更小的子问题9 v/ q* N/ C$ |# T7 y
       - 例如:n! = n × (n-1)!
    8 ?/ G+ }2 D; C$ N
    / L5 a; g* Z* _8 v) w. E' Q0 D4 S 经典示例:计算阶乘
    - i& k( }! T  j/ hpython
    0 B( S5 i5 s3 [2 Tdef factorial(n):
    . u* p3 g4 I6 `8 I    if n == 0:        # 基线条件
    3 M- V# u* u  I1 ?        return 1
    - m2 L; g0 G5 B' S' ]% t9 S2 k( v6 }& p    else:             # 递归条件
    . S, X; Y3 `; \7 C& e        return n * factorial(n-1)
    # V7 B$ f% h# N" t执行过程(以计算 3! 为例):# c9 w8 \0 d; O: B, F8 B
    factorial(3)5 X, \! E7 b; K* ^+ u( E0 d9 j
    3 * factorial(2)
    / {1 x2 ]# `, w1 n8 s% u3 * (2 * factorial(1))" G) \  ~! ]! Z0 d4 ]
    3 * (2 * (1 * factorial(0)))
    ! I( J* c) x+ r3 * (2 * (1 * 1)) = 6
    6 G( @8 m5 \. w4 {, O6 `6 W) \; W1 z7 O0 l0 m  Z1 A9 A
    递归思维要点
    8 F: e2 e" `. l1. **信任递归**:假设子问题已经解决,专注当前层逻辑, D! p& @8 @# @" U7 ^9 ^! d
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    9 Z* K8 p8 R9 S, w- l5 i; l3. **递推过程**:不断向下分解问题(递)3 o2 E5 @5 V1 p7 e8 w  x+ s
    4. **回溯过程**:组合子问题结果返回(归)4 I1 }/ b0 Z- i1 a

    - e: o- N* B( U# K' H; i* k 注意事项; u" h1 ]+ M9 R3 F7 v( R! E0 `
    必须要有终止条件. @5 s  F! F& A& C! |* Y) O
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)7 R: M- T3 X* I  q. I, T
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    0 R( f: Y2 Y3 y# \尾递归优化可以提升效率(但Python不支持)% t3 W2 D" }$ W; L0 [( _0 I+ t$ G

    ; }0 S8 H& ^/ D$ E" x 递归 vs 迭代
    , h7 B) c9 P( D$ c0 X0 H|          | 递归                          | 迭代               |
    ) S9 k5 z$ S/ I+ f' d; e& |. ]8 y|----------|-----------------------------|------------------|
    0 q7 t; T' w2 ^4 q| 实现方式    | 函数自调用                        | 循环结构            |
    % |9 Z6 H1 g. U" s& i! B6 n| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |5 l2 `4 `, q) F" g
    | 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         |
    4 P9 c& A. j2 M& A| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ' F; w/ D3 _# i1 }) p
    3 o& L# E3 j/ R/ x8 v: y 经典递归应用场景1 [& L" \" V1 \0 I) U( |$ @3 o6 T% S
    1. 文件系统遍历(目录树结构)' f: t1 R+ E) w! [' o# E  H
    2. 快速排序/归并排序算法
    2 ~* h  k2 f. M7 ]& b3. 汉诺塔问题
    3 R) J$ U7 R; v' [2 u4. 二叉树遍历(前序/中序/后序)
    " R7 j3 s( w4 b7 W9 v5. 生成所有可能的组合(回溯算法)
    # D5 D+ M9 T  `' |& ^. z8 U  Y8 K" t# |3 I/ L9 W
    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,
    , m. d: [, T7 J- ]我推理机的核心算法应该是二叉树遍历的变种。) P4 Q0 }* t$ a5 K* }  g
    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:
    3 C  ~0 u1 y& k+ s; p3 AKey Idea of Recursion/ D0 T7 O6 Q! \3 a! p, s  ^1 O

    9 r5 Q, |7 N" }& E) O4 nA recursive function solves a problem by:
    . A  C9 H% D, W. p, Z8 l
    : v; }1 T) [0 T  j- g, d6 Y, V  \    Breaking the problem into smaller instances of the same problem.2 o  Q, r( f" f" z3 h( M; p. x

    5 L4 B! i8 K$ D; D$ U" _    Solving the smallest instance directly (base case).& U8 V! Y3 X$ O- y0 T7 F; e
    9 m! E3 I4 K) C
        Combining the results of smaller instances to solve the larger problem.3 z, u# N0 z& O/ _5 h7 C7 f

    9 S. T$ n+ G$ M8 |- r% g1 ?2 HComponents of a Recursive Function
    ' b# j- l0 M- B  I9 K  l2 M) B( N0 B" C$ d( m2 b/ T# |% P
        Base Case:
    7 i+ @+ R: y2 ^' H) o7 d1 g! G
    2 O8 X, E: E! B* I+ w9 |        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.2 o1 ^8 t% S# V# X
    6 `% C+ f4 _+ F1 ?! c$ ?5 T; W
            It acts as the stopping condition to prevent infinite recursion.
    5 g' s% P6 J0 M) s1 Z( f( S, h) Z3 f6 ?: k' M4 Z* u7 h
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    1 |  k) {. p* }! ^1 v1 V& _! p7 r' r% U9 d: A2 c) q0 s
        Recursive Case:  r# y! g3 v' [4 K( j3 U

    7 [! C4 a8 B" ?* U' J$ }' i        This is where the function calls itself with a smaller or simpler version of the problem.. W' z! _( C* C/ W0 Q& N5 p" C5 {( v
    & v- S1 m- U6 l% q
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* m9 C' C" x. O. N- S! ^! ~- S. ^

    ! [% M* B+ d1 iExample: Factorial Calculation
    2 P1 ?5 R; |) o; b! k3 E' q4 ?% C+ i# D; @4 }
    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:
    ; d; `! H1 p" T. }0 ~7 P# V$ N3 z- ^' s; I$ U5 }4 {5 g5 K
        Base case: 0! = 1
    2 I/ x( I$ h3 s/ y5 s: o0 w9 D
    ) b2 S# p! x$ ^/ G( a    Recursive case: n! = n * (n-1)!  g$ s$ o2 @4 [
    8 |+ P' p# B! S" K  E, T
    Here’s how it looks in code (Python):
    $ }  a! J. ~( S7 y: y( ~python
    $ @+ l1 a/ @* L1 H, J6 `/ I. s; W' V. n

    0 c& I3 e2 E' o% Wdef factorial(n):
    ; g7 K% G" n4 Y, I8 R' {    # Base case: q6 i. p$ C; L# P
        if n == 0:
    - }2 p  Q" f. n; @0 g  n6 d/ W/ h8 m        return 1
    / E. ^+ ^) Z6 {" D    # Recursive case
    ' R7 o6 A" @7 K% M7 j$ P    else:
    % O8 ~% `. ~  F        return n * factorial(n - 1)/ w. M2 J* T2 O$ }8 Q! [' [9 Y
    * {$ c/ y+ `; m- P  a) A
    # Example usage
    ( [' L! V( T2 k% u0 s0 {+ p( N& Hprint(factorial(5))  # Output: 120
    3 r& L* e" O4 r" {' X1 {" b& P  Y0 M3 P$ ^% o5 g% q6 F
    How Recursion Works
    # U4 S8 x" T* t. |( S# x" P" a( o: F0 q! n
        The function keeps calling itself with smaller inputs until it reaches the base case.
    9 q- r1 D! ?. j+ F8 ^1 T: h
    ; b; h: R, w5 c3 x- U8 n4 [    Once the base case is reached, the function starts returning values back up the call stack.; g% Y* u6 j& |2 o. i- S. u" w
    : x5 k% y6 \& ?) ~- ]9 G! H) v
        These returned values are combined to produce the final result./ P/ V: U) j8 D" y9 ~. l% L( ~1 t
    ; ^: c7 a+ \3 d. I
    For factorial(5):
    ( c: ]9 \: k& ~
    9 w7 r& G9 J# Q5 |$ C( G; Q: r$ T9 ^+ t7 P
    factorial(5) = 5 * factorial(4)" D! G/ S% t/ V! _, `$ W6 K  d
    factorial(4) = 4 * factorial(3)# h" _( V& j" l4 A/ T
    factorial(3) = 3 * factorial(2)
    $ F6 l' ]) Z& o! pfactorial(2) = 2 * factorial(1)
    7 j) P% a9 N7 Mfactorial(1) = 1 * factorial(0)+ k+ B7 R+ C! l: z3 E
    factorial(0) = 1  # Base case. A7 _1 z- t( r* n; ]: Y% |& g

    ' s( E: a% }8 A6 a) [8 d8 _Then, the results are combined:
    2 K8 H" ]/ {& G, z- i) ~! [8 Y/ w9 r! i- Z( J

    , F; K% p8 }; @6 Bfactorial(1) = 1 * 1 = 1( g% O7 u, W( B! _! f  I! c
    factorial(2) = 2 * 1 = 27 y+ r8 a# o9 i
    factorial(3) = 3 * 2 = 6
    5 ]- n, }3 \1 m0 g# z  ?factorial(4) = 4 * 6 = 24
    1 ^( e! o3 D! l  ]6 Bfactorial(5) = 5 * 24 = 120
    6 T# v9 F$ _7 i& q! q$ M$ _8 g
    , _& y/ q3 S  BAdvantages of Recursion, Y$ i- Y2 `1 p& n
      p% {% X; q! x+ \
        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).4 g$ y( {+ ]9 |" h  B; I
    $ |: o3 w# t$ c; J. x: V
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    6 t, z6 k0 x7 P) L3 \, ~# [6 w
    ; [  [$ X$ L! E* T& G- `/ sDisadvantages of Recursion, ]+ t) @- s5 K& s

    3 {) {( Q$ A8 j8 `  U+ H    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.1 _' ]1 A  \* p" m9 B  |) r
    3 D6 _6 P0 V/ p9 g
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).& i" M& q8 o! P' I
    6 B6 }1 w3 K4 q! Z( L$ d
    When to Use Recursion
    : m+ A% @: l/ P% I( J2 z% W6 P3 c) f; w- X
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).* }1 w1 n7 E9 f6 y  E+ {

    8 i# M& k' {5 N$ c: W# u! ~    Problems with a clear base case and recursive case.
    ( I+ l" c% k# L2 A/ a9 y
    5 T6 N% A: h( H8 O% a5 G6 Y1 VExample: Fibonacci Sequence+ `# l$ m4 u( Q1 Y4 @( [; }- g

    ; X7 t! u9 x8 {6 M9 Z) o( |9 X; o% pThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 H2 @3 B9 ~+ Z( {

    # |3 J8 h  s9 u7 o2 j9 a    Base case: fib(0) = 0, fib(1) = 1# d0 \% ]8 G- s9 e

    * v. {9 E$ a1 A' v5 o0 S( F. p    Recursive case: fib(n) = fib(n-1) + fib(n-2)2 [! [! |1 s& Q0 K

    7 u3 k" E% I: M; P8 ppython
    : w6 W% ?+ S! B: O5 z8 L
    4 T9 N$ U3 ?% O, o% \% \7 P
    . G+ [0 k9 P& u* t- edef fibonacci(n):/ Y9 x' Z+ Y! s) M& [# u
        # Base cases$ R# u* x: Z- Y, i
        if n == 0:( \; H, J) ?* O6 ~' M+ C1 {
            return 0  j5 f% u; C7 E% {  W
        elif n == 1:
    " X2 U6 w8 O0 w" _( I$ G        return 1
    9 l" n2 K6 e: T/ v7 e. E* g    # Recursive case
    / S' X7 f! g; s0 @; m7 _' [- I    else:
    3 r& }) Z6 H9 G1 s0 q8 O! D# n# I) z        return fibonacci(n - 1) + fibonacci(n - 2)
      x, ^4 R3 K* P4 h( ^9 X7 f3 S, w4 y$ F4 o# p. j
    # Example usage
    # B2 B" f0 T  y: U& g4 U+ Zprint(fibonacci(6))  # Output: 8
      {; l' s; x& D& v+ `& |; i6 A& y2 N
    , l9 ~* }5 N( W0 OTail Recursion
    , O6 E  e; f, @  Y9 b
    6 \5 A/ z0 A9 L% u' OTail 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).
    7 U" P  C5 X( T4 q# J5 o& z. |0 [8 [% Y
    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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