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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
    # [. w" T  P6 H2 x/ c0 i: I; l# Y1 C
    解释的不错, H! C! P9 N' k+ }
    ) P4 t" a  K* _# R6 w! H* T" ]
    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 C9 h4 L# F  Z) i1 Z
    / p2 W8 g2 D- n1 `3 v( C$ C
    关键要素
    $ _( w1 M/ f8 ]' r, C1 r$ U1. **基线条件(Base Case)**
    $ I% C% r8 D$ g" {   - 递归终止的条件,防止无限循环7 z9 t3 H$ `4 K* Q
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    + q" M: y! Q& \% f% g
    ; @! n7 z$ y6 r9 ~) `2. **递归条件(Recursive Case)**
    $ i+ H: H: \3 M- T3 f4 s3 y! A' Q   - 将原问题分解为更小的子问题9 E9 t& Y' f# v' h
       - 例如:n! = n × (n-1)!
    - F% \+ r& q! T( m# [  E* ~" ~( U# b1 p0 z* D5 L+ C7 f
    经典示例:计算阶乘
    / G! y; i9 p( Cpython
    7 Q& }% F/ T6 _$ {. `def factorial(n):
    . B) u/ R# H5 \! u    if n == 0:        # 基线条件" [# e( M0 @: E- t; ~/ ^3 t$ [
            return 1) u: L& X" z$ v$ o( i* @/ }' L# P
        else:             # 递归条件/ E5 E) K& M- X: d8 O: Q
            return n * factorial(n-1)
    6 ]$ ?# `/ e* r; ?执行过程(以计算 3! 为例):
    1 ?  r! q1 E# Efactorial(3)0 K- m  R, u, D1 B+ J% Y& ?
    3 * factorial(2)" c) ^4 V) C8 B. E
    3 * (2 * factorial(1))2 x1 m2 B/ z) r. U# c5 X; \! ]
    3 * (2 * (1 * factorial(0)))' a8 s$ ^$ r$ N+ t  J
    3 * (2 * (1 * 1)) = 64 z4 z8 P! h4 _0 w$ k0 d

    * g) h3 O* n+ G  g 递归思维要点
    : e' j; i: Z1 B7 w1. **信任递归**:假设子问题已经解决,专注当前层逻辑; c) Q, n. v; b- z- ^
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    7 r, q+ g: j5 K* H8 i3. **递推过程**:不断向下分解问题(递)
    8 ]3 f1 W0 n: v4. **回溯过程**:组合子问题结果返回(归)  Z' d; V3 Y# s$ ?) [

    / s% v5 {3 q" v5 y 注意事项
    8 O: Y) m. ^, m# H6 R/ b必须要有终止条件# I# G9 k3 i. ?1 b
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)$ g2 L" C; n& _0 d
    某些问题用递归更直观(如树遍历),但效率可能不如迭代7 h& k6 d/ }2 S0 h) [) P, W
    尾递归优化可以提升效率(但Python不支持)1 f; W8 @/ {7 P, g- G6 q' n7 H

    4 R4 P4 o' J# x- ^ 递归 vs 迭代
    1 _; n7 Z$ g  i|          | 递归                          | 迭代               |
    ' k. ~  _& w! i% h8 e|----------|-----------------------------|------------------|7 h4 M) S) v, g+ {) B
    | 实现方式    | 函数自调用                        | 循环结构            |! B+ v8 D% E0 n
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |' P% e. {" C/ U* E' ]$ Y3 W' {- S
    | 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         |$ w5 R/ }" \7 {6 D* v
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |9 W# I2 {' o1 x
    - G) R# z6 ]% ^: y1 l# O1 q
    经典递归应用场景
    7 H: G% _. {# L" ^1. 文件系统遍历(目录树结构)
    0 z5 M: k" A7 e: D" g: r2. 快速排序/归并排序算法! n6 ?% _) [1 B. O% F
    3. 汉诺塔问题
    $ N' L& |# `2 {9 ~& `4. 二叉树遍历(前序/中序/后序)
    * J; j. S% s7 a$ h0 L5. 生成所有可能的组合(回溯算法)
    + I! k4 O% s* L5 B
    * z( l) y  f* t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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    昨天 08:02
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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,9 r  p; i; T) v$ K# j. c
    我推理机的核心算法应该是二叉树遍历的变种。% A+ p( S, [/ m$ z$ ?1 g: e! y+ Z3 `
    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:
    & ?" f; S2 o( j- O, T- ]Key Idea of Recursion' ^; S6 n9 K; q& O1 d; C4 w% C

    # z' u9 ?3 u: `9 u4 m+ SA recursive function solves a problem by:1 @8 z# E9 |9 e8 [3 w

    , b9 P) i# s+ L% m' V    Breaking the problem into smaller instances of the same problem.' R8 H/ ]# Y- E: \, C9 o
    4 n& l5 ^" H2 x; [6 ?& }
        Solving the smallest instance directly (base case).8 d( ^! S& z* `: \/ J; H3 ]
    2 _& Q. {/ ^( y. @% g- ]
        Combining the results of smaller instances to solve the larger problem.! @" V$ [1 J4 A$ H9 V) w
    - ~2 P  U2 O* e. W+ o: `8 v
    Components of a Recursive Function
    " R* Y( e, J& L7 u7 |
    & a- H- i" J" s  X) X. l    Base Case:. ?) O& I8 [% Q4 l
    ! b  ]9 ?3 x  e9 @& r% m
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
      X' K! ]( T4 H" Y4 r8 c/ B9 j/ S: s1 _+ z- w3 H
            It acts as the stopping condition to prevent infinite recursion.7 w( l- b0 `" S  b
    ' f" g4 u, z  V; r/ s0 R5 ]' |6 M6 a
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    , @; b  R3 L8 s* |. T5 X
    # `5 \# y6 `/ |) h    Recursive Case:; M, y/ r3 x! f, d; G4 P- P1 `3 G
    $ C: ]/ K. _$ E. Z0 |
            This is where the function calls itself with a smaller or simpler version of the problem.+ S$ ^1 Z( _1 c8 M; s$ `/ M

    - C, \' Y( P9 ?! w; k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).1 `$ ~: Z+ W/ a. [0 W5 X
    ; }$ m' R" k! L) m. ?0 a; p* K% D
    Example: Factorial Calculation4 W' R% h4 O5 g5 n6 t$ k3 T1 Y1 F7 v
    0 [7 R' O$ ^+ W9 n
    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:
    % H+ Q& a- H2 V$ ]3 e* j
    1 L# d. R& ]3 |9 |    Base case: 0! = 12 m: L! u! o0 U$ I9 ~2 _$ P

    , \" N6 ~. L. C+ t+ f4 w1 u    Recursive case: n! = n * (n-1)!
    % t4 m3 E) m  h
    6 o( h8 [4 o- H# i) v, [! t+ PHere’s how it looks in code (Python):" F7 a* t2 p- `+ u9 t
    python
    4 h- R- s& T' Y; Y8 q8 H; F+ `* e8 u' D# ]/ H  k3 a

    8 H+ {" v- o$ u  i, A4 @def factorial(n):
    4 Z& @7 L3 X$ O4 y( F; n6 M    # Base case
    6 E  t" c7 J5 ?    if n == 0:7 q( b7 I6 g; X/ x
            return 1
    8 T3 F4 x7 q' s) |: Y+ x    # Recursive case; ?$ Z. T, R" G. ~
        else:' _$ @& ~( G4 @0 Y( R; x9 I
            return n * factorial(n - 1)& `) Z! L+ }8 D, ~% Y: C

    4 H( @  m+ P" _3 p: l7 O# Example usage" u" W+ P) s# n5 d# h
    print(factorial(5))  # Output: 120
    9 @! w( f0 Q  p( Z7 g9 o* d7 n) h* r3 ]* A
    How Recursion Works
    7 F* a" P" y1 O, j: d) z1 M! d. \8 K% p. p; V, x$ G* \
        The function keeps calling itself with smaller inputs until it reaches the base case.
    5 s* k# J+ K' Y; ~! O) S/ K! n9 q
    ; ~- z0 h! ?8 T8 R3 Z    Once the base case is reached, the function starts returning values back up the call stack., \9 ~2 }, c0 F5 @4 z' P

    * v& N$ T3 C  ^7 q- P! A    These returned values are combined to produce the final result.
    9 s$ F# F4 G) W6 Q; V- o
    ; [4 P1 I& A- L& ?For factorial(5):- e; [: [) T4 V0 E# I

    ; u( J0 U/ i, b7 O! v
    % Y, Z: U- x. n- V' _factorial(5) = 5 * factorial(4)* v; X1 u. S8 V% t7 O
    factorial(4) = 4 * factorial(3)
    & K( i5 t4 W) s' {* Q0 f9 ?factorial(3) = 3 * factorial(2)1 \0 r% B; [4 z& V
    factorial(2) = 2 * factorial(1)& n: Q% H; n3 ~/ n( d' n0 r
    factorial(1) = 1 * factorial(0)5 o0 P; r# F8 b# i. D4 A
    factorial(0) = 1  # Base case# @0 a! y7 N; w: ^( m. O5 a

    ! U) L! v& T4 [3 MThen, the results are combined:/ W, g: V! K& V4 ^- u
    5 M) M, d$ T3 n+ L3 \

    ) y' r* p- y& E8 @' V6 ufactorial(1) = 1 * 1 = 1
    ( d) T, B2 ]7 N) A( b' dfactorial(2) = 2 * 1 = 2+ n5 V9 g# c4 R
    factorial(3) = 3 * 2 = 6
    ' q+ Z, o8 U& L1 q* O% X: H' zfactorial(4) = 4 * 6 = 24
    5 t' U( \  C, f, ?9 I4 m7 x6 m+ Jfactorial(5) = 5 * 24 = 120
    ) X* ?3 B: H+ a6 b( k, o: {/ X: @. `8 }2 c/ k
    Advantages of Recursion
    4 @; T" r& A! a) O
    0 E* j5 M# F6 a9 R8 O1 a    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).5 a2 g7 n! M1 ~8 H2 B

    . T7 X& s! L  @! h8 |7 B    Readability: Recursive code can be more readable and concise compared to iterative solutions.1 ~- x& v# [8 G" }% U" V% ]
    6 ^5 W, Z! Q) o! ~& {# }0 U' H& n
    Disadvantages of Recursion$ C( Q4 a" Y' @) H$ f
    ' W1 W( S( r; T/ b9 z: C
        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.
    9 d# N: S+ L# Z4 Z2 o) y+ Q8 W# ^) i$ }- T
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).' I$ @4 t5 |+ K4 x, `; K/ h$ O9 l
    & o9 l* J7 Q$ ]
    When to Use Recursion
    8 k- }1 v' R& l, @* l& V: e9 t! g( d+ v
    6 _  m4 Z) m0 l, {" m7 ^    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 [; B: K! t; @: M% Z8 j, ^
    - Y: ?, r1 e/ t3 E- ]
        Problems with a clear base case and recursive case.
    ( z+ w2 u- t! _- c) w2 E8 U0 }1 s& z7 ~2 q
    Example: Fibonacci Sequence/ V3 l% Q2 \7 F$ l
    # E: ?. {: S# ]  ]: f8 H. h8 e
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ( E' g) q- Q' j9 e
    3 G1 H$ G- I4 G8 D- i* s    Base case: fib(0) = 0, fib(1) = 13 g6 e- ]  N( ^
    7 M  [$ R3 n' p) E% O
        Recursive case: fib(n) = fib(n-1) + fib(n-2)$ Q4 a0 {& y1 J5 m
    # c  h8 h. E6 s! r. E! ?0 U$ j3 v
    python
    . s( M- n6 Y) ~7 G2 ?! R+ m/ _  ^
    4 X1 F- @. m# V5 }& _3 X* ]
    0 x- [1 Z( Q/ j- X3 Bdef fibonacci(n):; D, {0 L& O+ @, g
        # Base cases
    / y, `; B( Y5 ?5 |    if n == 0:
    * e6 w8 d; v* i( o7 u, O( a" n+ N* O        return 0, J0 i( {4 a5 E7 U
        elif n == 1:
    - i  Q1 |5 ?1 l        return 19 U3 G4 H7 Q5 t, I
        # Recursive case" P9 |6 X7 m& R9 S% \5 n6 k
        else:
    ! H$ b" ^5 d$ S/ j$ G* ]! v        return fibonacci(n - 1) + fibonacci(n - 2)
    1 F' N0 v4 K& U5 ~1 {. l, z) k) e3 _$ L1 {  z
    # Example usage
    , U, ~1 H- h7 r* G& p6 Gprint(fibonacci(6))  # Output: 8) g3 {6 t% B# y( A2 T8 n1 A

    ' L' m+ ~+ e; h; O- f) c% LTail Recursion" v" C7 v2 B( g$ K+ H1 p
    ' P1 n( M- s) G' r
    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).
    " i6 Y" o3 b6 Z7 P
    . A0 Y$ r( }& o  Q" N- lIn 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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