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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 / G: `! ]! [% y, O3 Q+ a
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    解释的不错
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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。
    " j5 ~7 y7 q: |& y. p1 M3 `' f; N* v" ?) ^0 r3 S. f0 Y
    关键要素
    + ^; K8 Y" X, N3 ]* P7 e  `! Q: M1. **基线条件(Base Case)**% g, V, [' p4 \3 P, @( Y
       - 递归终止的条件,防止无限循环
    3 f  V8 L1 K& t  a0 G6 t  {   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 14 d' l* U, [6 L. E
    2 ?; o4 w1 |- t1 q) E
    2. **递归条件(Recursive Case)**' i( U& f* z6 N6 }6 A1 T
       - 将原问题分解为更小的子问题$ D0 @  {' ?5 x' N4 {7 V# E
       - 例如:n! = n × (n-1)!
    ' ]. |$ }" s1 p! R$ T# a) w* Z% p" x4 E  \5 E& J  g: m
    经典示例:计算阶乘
    & B; Z! d3 I; \7 X- q1 Jpython/ T1 V. Y: f/ g$ w
    def factorial(n):
    ; i/ ]. G. O. t4 G    if n == 0:        # 基线条件+ x4 C' I, u: ?" V- `8 P" [
            return 1
    # b% r, \1 ~" G% m  C    else:             # 递归条件
    0 X, Q+ x0 H* R" c  L4 Q  B        return n * factorial(n-1)0 _/ A8 L, D0 q# T9 F+ O- U, ~
    执行过程(以计算 3! 为例):: a9 @- @9 D" Z7 X/ `
    factorial(3)
    : B' L7 R) m3 K3 U$ O( Q2 ?3 * factorial(2)
    . T8 m; \- e) ^3 * (2 * factorial(1))
    0 Y, b4 @$ R& \8 ], g# Z2 s3 * (2 * (1 * factorial(0)))
    * I8 _& Y+ n8 U( _# N6 `5 P# ~9 d3 * (2 * (1 * 1)) = 6
    ' c3 N' E* U. e$ ~/ z8 k. G% j# _# n  X5 v
    递归思维要点& [. L. P; X2 A
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : K3 S: `8 o" h. O: W; c2. **栈结构**:每次调用都会创建新的栈帧(内存空间)/ ^, p8 Y7 Y" y  b# s! B. v
    3. **递推过程**:不断向下分解问题(递)
    1 i  V2 K9 ^$ [* X4. **回溯过程**:组合子问题结果返回(归); R) L1 A  K- q1 |
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    注意事项
      o) ]! z8 ?& B必须要有终止条件
    2 N; h7 i, e7 Y+ G; ~  X递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ ~# J# s$ R! ?% H  Q
    某些问题用递归更直观(如树遍历),但效率可能不如迭代: M  ^3 l$ F( s4 ]; T" t: d6 v6 n
    尾递归优化可以提升效率(但Python不支持)
      S* d4 W# K3 h1 \  B7 T+ |" p3 ]
    ( C8 @( N  Q# ~1 a; j- T* b 递归 vs 迭代
    9 Z! n0 C, ?7 Z|          | 递归                          | 迭代               |
    - W' i8 |# q9 k|----------|-----------------------------|------------------|
    3 M+ {2 E3 O. S6 X2 P& k| 实现方式    | 函数自调用                        | 循环结构            |, x6 G( _. Y+ }2 G
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    " }% R  u$ |4 g8 ?. D' 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         |1 i4 S- I9 N+ F4 ~" T  A
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |6 }9 g5 t; Q5 c7 W

    % V7 l7 M& C6 W/ }; [2 S 经典递归应用场景7 B+ j/ j* E3 w  W
    1. 文件系统遍历(目录树结构)
    ) V1 X9 F2 r, C% A2 e( t* V7 i2. 快速排序/归并排序算法
    6 G. {/ X3 d" m1 \; t3. 汉诺塔问题
      a3 L& `! l* p6 d- W4. 二叉树遍历(前序/中序/后序)
    ' q6 E9 G7 N$ k$ B* ~5. 生成所有可能的组合(回溯算法)
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    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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  • TA的每日心情
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    前天 06:54
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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,
    ( I3 ^; W# S( l5 M我推理机的核心算法应该是二叉树遍历的变种。( |  }' f& }! I
    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 c! |2 s$ E. v6 W! |Key Idea of Recursion7 ~$ z% K, X5 W$ K8 ^
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    A recursive function solves a problem by:
    ) a0 [9 |) H' @: R3 S6 E$ O1 r2 T8 H3 D9 N; P8 M' h# }* i- M
        Breaking the problem into smaller instances of the same problem.( t2 S) x6 V5 a/ O% c8 ]
    + U, z9 o% K! Z- v4 W
        Solving the smallest instance directly (base case)./ p; j3 y9 d- D" X4 C/ z

    9 ]4 |: m8 a. l3 k- r) V& g; f: z$ u    Combining the results of smaller instances to solve the larger problem.
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    / T5 `. ^( x% X+ P! t+ lComponents of a Recursive Function/ r  a2 F) V, T: u0 P% m
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        Base Case:
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    1 [( ~+ c$ H0 ~) c; M3 y7 t) h        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.  T9 ~+ R. u3 _- @' ]6 w; @: d$ n
    3 s# H2 e: g, Q8 D$ p+ U4 U! }
            It acts as the stopping condition to prevent infinite recursion.
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    6 e2 r6 O3 G8 x8 {8 k+ k# t! I        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( w# |2 A3 u* d1 G" [

    ! ~& m, P0 c' o+ c9 e+ `, n    Recursive Case:1 y- {8 {8 k" w( m5 j& w# c

    ! ^1 H1 {& y; k: W' J: p        This is where the function calls itself with a smaller or simpler version of the problem.3 [+ c- M3 q9 K; |

    ! v% ^4 o% e% g7 Q        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    3 {$ X$ V& @; M0 c0 g6 J
    4 q7 b0 Q# ]7 B4 sExample: Factorial Calculation$ F/ b! G9 z, {

    0 D: I  Z& t- u/ c, Q2 v# iThe 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:
    + b( ?9 L& F% O% k1 U7 D* `  B2 C! q) ?& a9 h, z, B) k  y
        Base case: 0! = 19 H1 o2 C2 c* ?* G/ h% U/ b
    ; a9 N1 D& @( M8 F5 l7 L
        Recursive case: n! = n * (n-1)!/ U+ E/ Z# D. a; p% Q+ Z+ w* X  w
    7 N& C) v' R" Y3 V. W2 y
    Here’s how it looks in code (Python):. X, c4 A6 }+ K! a7 b% h1 b( |
    python; ~0 s. S$ j8 B0 V( m

    - P: k; x+ ~- r# Y4 u% O1 A
    + ^" P" W: H) w' Udef factorial(n):) D, q0 R3 {4 a) m2 ?# r
        # Base case& q4 i' @% l/ g2 G
        if n == 0:
    ) ~) T0 X8 A% @' K        return 1
    7 B4 N% z- A2 H2 x, H    # Recursive case
    * w! G" E/ T! c$ D% `* I' y9 e* ?    else:1 Z; l; m* i) P) s) K1 p0 C) k
            return n * factorial(n - 1)
      ]  P$ G1 @+ d+ H3 }  ~5 i
    7 @" ?, g" b4 P) l8 r# Example usage7 o- Q# S% i/ t
    print(factorial(5))  # Output: 120
    7 u! L1 N/ D+ |; r% J
    # }6 f% g' l) ]& d, v6 IHow Recursion Works$ o' {6 s! p- c+ X; a. l1 n

    - @/ x1 r8 b3 N, |, x    The function keeps calling itself with smaller inputs until it reaches the base case.! ?. J0 S* Z* `$ a# w- s
    0 R) T1 }3 G: c2 C5 [
        Once the base case is reached, the function starts returning values back up the call stack., Q/ p' v; v: w9 b
    8 |% ]( s0 q- j
        These returned values are combined to produce the final result.* t" G! G& n5 ^) ]- {
    * q  F9 z; c1 d4 l: `9 h+ `
    For factorial(5):
    7 ^8 I; P" v9 u; c3 \5 `5 ~" i2 Y  }% Q/ W& s

    % @9 D6 i8 v* A/ Afactorial(5) = 5 * factorial(4)# C! M% z" q+ P8 t& \) N8 I
    factorial(4) = 4 * factorial(3)( T- I4 q7 w- U7 i5 T5 n; P, i
    factorial(3) = 3 * factorial(2)" [7 L' r- k! U# a7 n
    factorial(2) = 2 * factorial(1)
    ) o  E2 h( r" x) m) G( z- |; Gfactorial(1) = 1 * factorial(0)
    / O! B1 a. u% p  dfactorial(0) = 1  # Base case. [/ i3 H" b! ~- P+ A5 q) Y3 D
    ' o. O+ l) @+ I2 \. C
    Then, the results are combined:
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    9 C8 b; W7 i/ m% q4 mfactorial(1) = 1 * 1 = 13 ~! V( a) v! k4 ?
    factorial(2) = 2 * 1 = 2
    8 [+ }9 z1 o3 h0 F2 Nfactorial(3) = 3 * 2 = 6
    * a/ S% q$ q( ~: ufactorial(4) = 4 * 6 = 244 e4 R* w/ ^, N; @
    factorial(5) = 5 * 24 = 120
    4 P  G* u1 Z$ j
    8 g; c* i6 a/ l% F& R% `Advantages of Recursion7 L' s2 Z4 j1 ?2 Q

    , X# Q. W# I" G/ T6 n0 r/ o, ?    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).( S% e, L8 A6 A$ i3 h% l
    + l$ k& t  ]- c5 w& i
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ' S. R/ ^. q; q0 n: A
    ! s, x# e) y: p' eDisadvantages of Recursion/ D1 k0 x, j. x: m+ D. |; M

    ! Z" k; v4 `. N/ N8 i% Y    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.
    ' G1 T4 Z! p6 |% E. j
    / N! Z( w' y( w" b, S3 |    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).6 d: F- A# b& h8 _5 u2 x$ X0 p8 C. e

    ) k$ ?& x* _9 a# \. TWhen to Use Recursion
    4 W+ E1 z, B' K$ f! J1 w( @+ Z! E5 Y/ T1 ~
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).9 [/ n3 \2 T$ j! Z. l6 ^
    9 z, E+ z4 }& `5 n$ \& C& H
        Problems with a clear base case and recursive case.
    9 ^4 k; B8 y  ~6 g3 ~8 G; Q) U  t8 W+ N! p6 }, u( l
    Example: Fibonacci Sequence
    6 \; s& X6 U% d
    1 X0 }- T; B- yThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:' k+ m+ K6 @8 Z& m" G& t

    9 N; S0 r/ t/ ^1 {2 @" T+ l' u( ~( M    Base case: fib(0) = 0, fib(1) = 1/ I# q+ V" S+ z  H/ [/ Y# c+ Q( u4 N0 V

    + \( C) s+ o; F2 ]! U    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    4 \8 W( Y* {' }9 K! d/ Q/ o4 A& h4 Q5 G  z5 ^8 O
    python
    . b9 m, D3 {2 d- ~9 _5 L6 t+ I# J  l; m" f
    + j$ R$ W9 o" T( h$ C, f5 ?
    def fibonacci(n):
    " G7 Y6 n- ?7 I    # Base cases+ A& H4 J0 w3 b# I! w
        if n == 0:- ?' |+ l1 D" ]* h* x- i4 S/ h/ P, H
            return 0
      c' `. E' `/ c  Z    elif n == 1:) {- }% v5 H! s# G2 Z
            return 19 }4 @- s6 v$ J% D! a9 Z
        # Recursive case
    7 G; w0 A5 A' _- r7 ^# `    else:9 O8 f! \9 h* {( N+ J! i: }  [
            return fibonacci(n - 1) + fibonacci(n - 2)
    " ~. q# t: M# }
    , m) ?8 h3 A  F+ ]# Example usage. s: G" ?- C0 i
    print(fibonacci(6))  # Output: 8; F4 b3 D. I6 u; O# P9 U, U$ b
    3 X9 K9 s6 @% h! K
    Tail Recursion
    2 v* S+ q4 c( t0 \/ y
    3 U& a. K# v2 y- I! p( n7 gTail 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).
    $ j9 F" C7 o0 N6 \( q8 X- b# @6 ?& C0 |6 Z3 D# U
    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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