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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
     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
    8 J* C! K) ~' s6 G( g$ X2 H# W' F) r5 [/ }: |( n7 y
    解释的不错/ D8 v5 V- k5 |( p
    - Z. x+ V- p4 N6 T6 K
    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。
    2 h0 J( t* f: V0 z( Q
    6 ]5 m5 j& c; h0 Q7 a 关键要素; d# V4 D2 L+ b6 m* ]
    1. **基线条件(Base Case)**
    : u& Q) ^1 e  Y; ]$ A$ P; I   - 递归终止的条件,防止无限循环/ x; _  R9 v+ s6 D* c
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    + {- \0 i  H$ R8 {% j+ r0 G$ p0 c1 M, b3 o
    2. **递归条件(Recursive Case)**$ N. N: E  ~" d8 _8 \
       - 将原问题分解为更小的子问题+ \( d/ p* |. d, r" M$ i5 T
       - 例如:n! = n × (n-1)!0 p; l8 Q$ \2 N2 e# v7 V

    3 T! [6 p4 L, v 经典示例:计算阶乘
    # S3 L$ c; p  |5 spython
    , F- \, W8 U9 E6 Z4 p/ rdef factorial(n):  V6 u% N; m. X, Q2 C
        if n == 0:        # 基线条件) E. X' p4 Z- k+ p# U# k( q! g
            return 1
    " W1 y) t8 j  {    else:             # 递归条件; z& y/ l8 D% c  h  d5 f+ c3 H+ |
            return n * factorial(n-1)) b& Y) d6 m7 B& X; N! m
    执行过程(以计算 3! 为例):
    ( T; e; F) y$ @4 [; r* \: x' G2 d- Xfactorial(3)6 V4 E7 E" _3 I
    3 * factorial(2)6 g- D" c9 p& u9 `: y
    3 * (2 * factorial(1))$ P/ }; D' i* M- o* C! a
    3 * (2 * (1 * factorial(0)))* Y9 [& D6 I* M7 g2 ]3 I
    3 * (2 * (1 * 1)) = 65 w) o+ |3 `2 ]/ S8 P% |

    % H4 t0 x4 A1 h& \8 U$ f 递归思维要点3 m& a- x2 f, R* _  s
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    9 B+ f# u# f5 j$ F" P& g5 z2. **栈结构**:每次调用都会创建新的栈帧(内存空间)7 H' T, k% T+ l4 a9 E# A
    3. **递推过程**:不断向下分解问题(递)
    , ^# i& X. n2 |9 i4. **回溯过程**:组合子问题结果返回(归)5 W* G2 J0 w9 K* [

    ; N" m% u; T5 ]6 T8 V 注意事项
    . [. l' x) b6 s必须要有终止条件
    8 H- L( Z& o8 o, Y* M- _3 ~递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    + ~0 `$ k/ p2 J0 }* @某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ; S+ a( J7 W! ]5 a  C5 I尾递归优化可以提升效率(但Python不支持)3 F6 Q  x* Y! k$ C
    ' E) X- z  a7 E! p3 U3 r
    递归 vs 迭代- \* x# `1 q7 @3 e% X4 c
    |          | 递归                          | 迭代               |( L- c8 m- N1 {, i
    |----------|-----------------------------|------------------|* Z# S8 w% U, U; ?$ Z0 `
    | 实现方式    | 函数自调用                        | 循环结构            |
    # L& n1 B2 a. u$ o6 K+ c' s$ [: W+ H0 N| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ( e4 t6 f& h- W4 G7 w; Y| 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         |
    ( E8 d6 [# X* k/ h| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ) f8 W8 T) l/ ]+ G* J2 I; Q* l+ R" u1 l% B$ t
    经典递归应用场景8 e  B4 }0 F- W/ ]5 y
    1. 文件系统遍历(目录树结构)2 u: Z; f) \- d/ B* Y7 t" E
    2. 快速排序/归并排序算法
    9 `( c- M- `  F  T) z  S3. 汉诺塔问题
    7 e7 g3 @# q7 S/ d( i8 `8 L: O* n4. 二叉树遍历(前序/中序/后序)' J) v7 ~3 O8 H/ C# C% M' s% Q
    5. 生成所有可能的组合(回溯算法)- M* n: C* K9 `# E# B: g

    6 }5 `7 P1 o6 x1 y6 V0 ?5 D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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    昨天 05:49
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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,
    ' B2 a8 u0 d/ }我推理机的核心算法应该是二叉树遍历的变种。
    - q! n) p4 s. D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:
    * A( _4 Z$ O, BKey Idea of Recursion
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    9 W9 {# B9 [0 J1 N: W7 UA recursive function solves a problem by:
    & o. g- O, R1 O$ Y+ Y/ G0 s# \1 B) W& h" G9 @/ R6 t4 Y9 h, y" q3 U
        Breaking the problem into smaller instances of the same problem.
    4 J* s: I$ g  D* @: n6 _$ F
      s2 Q: W! ^. b1 ]' V/ d' h# J! m    Solving the smallest instance directly (base case).! s) L" h8 ~5 u, c
    - {! S5 o( a6 `
        Combining the results of smaller instances to solve the larger problem.
    ' d( \/ I- S  S0 ^' k2 e1 o4 Y5 C0 w7 _
    Components of a Recursive Function0 |+ N* E, K( H9 d3 {: q
    & }7 G! K4 W: Q4 C  W
        Base Case:
    # T3 F( @! H: y  `
    9 g* Y4 }' ?2 ]4 V, q        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.% h# d( |# x# Z6 N1 ^
    & d: U4 G1 S& `* W) |# G4 D- E3 _
            It acts as the stopping condition to prevent infinite recursion.
    6 w0 I: ^# j  c; P7 j( m" }' t/ Y; q6 N" E" |# w7 |9 \3 }' f
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.7 V6 _5 ^2 b- T/ a4 L( P5 e4 O

    & o) i+ K0 R8 w3 j/ d' l1 D    Recursive Case:# {- {0 I3 P! K& x8 V* R& ?
    # l' Q* l/ c: z; O. Y
            This is where the function calls itself with a smaller or simpler version of the problem.
    * E; w  `8 @8 O9 |
    ! R1 b; ~) q& b7 R        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    " j5 \8 q; m6 ~+ \. l& W8 e
    , M" T" v9 x6 k# ]  P2 MExample: Factorial Calculation
    ' T% |& Q" o+ T) D6 B: |1 w2 @. g
    4 G3 s8 Z: ^% `8 D* Q3 E% }: YThe 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:
      |& C! b3 t, i/ @3 f3 N0 \& R8 i5 x; A
        Base case: 0! = 1
    % G6 U8 U, Z0 `! q" Y2 Q) Z; _+ U) u1 r# Q3 d6 W5 K' p
        Recursive case: n! = n * (n-1)!+ {3 b2 O" W6 j4 U/ L; n

    3 s1 u8 M; o4 d$ R, BHere’s how it looks in code (Python):+ Q! L; D9 |/ y% i; m6 O9 u0 J  W7 \5 G
    python+ p! c6 z; h% C# a$ ^+ G

    1 s' T' i1 [( |: x6 H- l! ^
    % B$ c, z0 `+ Tdef factorial(n):
    ! V  k  z4 G8 m7 N; ?- d  A( g    # Base case
    & {9 S1 N" X' P    if n == 0:
    , o* X4 W( A1 o. W        return 1
    % o1 R: `4 W  N9 Y4 F* U    # Recursive case
    . E! l2 @7 l' Q8 J    else:
    / T+ Q4 `  b+ x" c8 `6 K        return n * factorial(n - 1)6 n! ~3 y6 v  O6 ?" z; ?9 O
    : q- R, l: h0 x4 X  ~- l2 c
    # Example usage
    7 `! K$ d2 t% y; d) g( L2 \print(factorial(5))  # Output: 1209 F: I# p; o/ r$ Z3 v

    ( l8 ~& k% R8 A$ [9 `9 iHow Recursion Works
    ' `! u. }. |- O% p& V
    1 `" p) k/ ?5 t& `% t4 C    The function keeps calling itself with smaller inputs until it reaches the base case.
    0 l' h/ i8 ^1 Y* `( H# b
    ' f7 _# g% J/ k5 B( V8 |5 K& j    Once the base case is reached, the function starts returning values back up the call stack.
    ' ?# o9 ~. G1 Z. S/ O$ D* N+ G: ~: R
        These returned values are combined to produce the final result.
    / I- x' o5 u# }+ E8 p- I$ q; Y  J  }8 j$ L% ?& P
    For factorial(5):% ?- U8 F1 z& j5 i+ j. N" o
    / T- Y. u$ l$ r  X) H* W% M1 L( D

    1 ~: X/ ?1 l, C4 ]7 i2 U& p/ ]+ O0 rfactorial(5) = 5 * factorial(4)
      Y6 |: A% I" R# R4 _' sfactorial(4) = 4 * factorial(3)
    # b+ z3 T2 f; h' q& K. Mfactorial(3) = 3 * factorial(2)2 M1 Q* W. m5 X- W% }  q3 a
    factorial(2) = 2 * factorial(1)% u& l7 u9 t) a
    factorial(1) = 1 * factorial(0)
    " v5 W) k6 z, i  N) ?7 Efactorial(0) = 1  # Base case, h: e8 k. W3 {. I, N( h6 J2 O

    . x& F7 L( k1 @% x) ]$ {6 M6 DThen, the results are combined:
    8 C" i. ~. E$ m: o) q" @
    8 d/ A; m  N" N# L. b( l- ?6 E0 W8 Q! }2 R0 R
    factorial(1) = 1 * 1 = 1
    # |4 \& L! D) D2 r) o2 Wfactorial(2) = 2 * 1 = 2, o/ X4 T% ~' k* a. c+ g5 Y
    factorial(3) = 3 * 2 = 6
      B/ Q4 J2 |7 p2 i6 Pfactorial(4) = 4 * 6 = 24
    " _, j- l# u6 \( [4 [factorial(5) = 5 * 24 = 120
    5 u( Y' @1 F2 M, T2 q7 y7 ~& ~7 v; b- n! Q8 b- z6 O- N
    Advantages of Recursion) W0 T9 c' g! J+ c2 f" N4 c& T, y
    2 g5 R, N: F8 U, A$ H! m
        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).  j( q2 X6 E' g. P

      k( j( B5 h  t: o! X$ D  y    Readability: Recursive code can be more readable and concise compared to iterative solutions.( }; x- E, e# \" M" b+ g
    3 ~  T* O5 k0 z2 ~, q
    Disadvantages of Recursion
    , p8 {! |; e) U" z2 c
    9 q) D8 f% ~# d4 e, P7 B    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.
    % s0 W4 w) Q7 c- |, V. k! F. H5 Y2 R# [
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).' c+ R2 E: |+ A) L. r3 q8 U, Y

    - e5 o2 g. g$ RWhen to Use Recursion  c; r9 |+ p% D. I6 M6 y- b( G. x
    $ U+ F# ~' ^, v3 W  ]1 i1 y" `
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    " t9 t, h" ~( L( n  J! L) f$ q/ P3 l+ T/ k. J8 N
        Problems with a clear base case and recursive case.
    9 d" P% b6 m9 }3 Z
    ; ^7 R, z* ]1 a$ JExample: Fibonacci Sequence
    $ p1 W" b9 S* Y- ?! q+ P6 j: o# E7 j: r; Z0 h$ z
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    , r7 X% m! [* K0 ?
    ( K- \$ R0 c; J3 K! |0 ^, i    Base case: fib(0) = 0, fib(1) = 1
    ; `6 m  P: r# P8 L5 P. i6 ]; n+ q$ X
    : w8 d0 h) W; \2 I' q( T/ w' A    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # b0 ?5 C1 Y0 m& h) G9 t  r+ O% A) A# ~( D5 g2 ]; v# b" Y4 |* O
    python/ f, i" K7 D; U9 E

    - S2 V* b! C* V+ r- J4 F) @5 {% y$ _0 V& v
    def fibonacci(n):, {4 [! [; S. i* J4 z- k# N( h* }1 [) x
        # Base cases
    6 l, K; Q1 H2 H/ A/ Z# [    if n == 0:
    $ u8 R4 ^+ z* O/ }$ i        return 0
    ) l2 K" {5 O4 G- r1 s* D    elif n == 1:
    + h4 G0 O1 P5 a        return 1
    , q% |- |5 u! B& C    # Recursive case
    6 U# w. `; y" k; V9 c' k2 C    else:
    9 b! Q2 h0 w/ ?4 h9 x, w% @        return fibonacci(n - 1) + fibonacci(n - 2)
      ~, u9 r/ q( r- O) V* x% `% K. c0 Q. G7 W) @# J9 j5 I7 v) A2 u
    # Example usage6 ^" d6 p& f) x( E6 I3 d1 S* H/ G
    print(fibonacci(6))  # Output: 8, W1 J8 O( Q; }: d6 c* g1 s4 s- {

    ' E2 [. X  h" O6 ITail Recursion
    ) I8 F/ A4 b7 ?* T# t
    $ ^" |/ u0 t* K' ITail 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).5 V3 L( m# M) R
    ; x1 {& \" o& s  J. o
    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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