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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
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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
    6 ~: B( u. F# C# s: K: f% N1 Y1 f3 q% u# f' a
    解释的不错
    7 U, i! r, q" Q& c0 }& ^- j: k4 q3 \9 b. j2 M( |
    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。6 L; H9 Q! x- D% O, x

    7 I7 {$ N5 R/ E& W* N 关键要素3 e. Y1 h' h+ [  d$ a  c5 i( k4 N
    1. **基线条件(Base Case)**4 ^8 J) Y  v0 T- j  R1 t
       - 递归终止的条件,防止无限循环  Z  v3 o: Z: W+ x
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1; I6 `* n! D8 D* P
    5 i# T6 T" T$ r; P: Z  b* R" [& E! G
    2. **递归条件(Recursive Case)**3 `) ^: I! D+ ]# @1 w% A9 K" O
       - 将原问题分解为更小的子问题
    ! e* x! ?1 E  t, O1 O  @   - 例如:n! = n × (n-1)!- B2 F( \3 `: [; j: P. C5 o0 J
    $ y  h# p  P" v  A7 k# F
    经典示例:计算阶乘$ u  N& j% S$ X1 H: O8 ?
    python, O3 A/ B6 g6 D( |4 f& m2 v
    def factorial(n):
    : x# A8 i6 L* d4 R" e" G! R    if n == 0:        # 基线条件
    1 g  l+ x; }& M6 [7 f4 l        return 1
    , t) Z( c# G7 x$ a0 J: Q" g    else:             # 递归条件3 O  X0 I9 [9 {+ X8 w4 u% W  K
            return n * factorial(n-1)) J4 h* l4 B% c! J) v2 I
    执行过程(以计算 3! 为例):
    8 o2 F9 @0 [! U: ]factorial(3)
    9 m1 y4 s+ u1 z" A8 I3 * factorial(2)2 P% i5 N8 r& ]/ S. X! c) r3 q
    3 * (2 * factorial(1))  `4 K: H  ?! X5 V) ~
    3 * (2 * (1 * factorial(0)))
    / @* ^  |/ b& j' b3 * (2 * (1 * 1)) = 6
    ( f/ ?" q6 v& O3 @. t+ x4 Q0 K$ @; f/ |) e" _9 d( E( L# t, j
    递归思维要点& C  V/ Q: g; D2 q8 ^
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    4 G, ]+ ^. N% N3 A5 t1 X2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    , ]/ @- ~4 C1 l3 U* Z$ E3. **递推过程**:不断向下分解问题(递)2 R: o  d7 p* L0 L, H
    4. **回溯过程**:组合子问题结果返回(归)
    / a* m5 _' @. T# ^# O2 i7 F9 ~/ e5 Y: \" q
    注意事项# B' s% Y6 {2 K4 J* V$ q: n6 x
    必须要有终止条件# O  N* G0 l3 N% @3 I
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    . ^2 i% H) k9 K! _0 H某些问题用递归更直观(如树遍历),但效率可能不如迭代( b$ \9 ?8 l" n! K& a
    尾递归优化可以提升效率(但Python不支持)9 z% e0 F, t0 [0 U' u
    8 x% s; @- a& G8 n
    递归 vs 迭代
    6 b9 g. [. l$ d9 h( t  b4 d|          | 递归                          | 迭代               |- V* o+ T- C) Y3 V# }
    |----------|-----------------------------|------------------|
    ) b8 c. _4 G- i. T1 G| 实现方式    | 函数自调用                        | 循环结构            |' p4 `9 M2 M0 V' H, z) S. f! T
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    + H3 w  T5 `% M7 l| 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         |
    2 \9 `% ^+ |5 v5 V) V| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& o: O5 }! v2 A' Y' {1 y6 n9 L

    6 }1 D% H% o# ] 经典递归应用场景
    " d: \1 C; ?8 n* A1. 文件系统遍历(目录树结构)
    " V# d" {( j! n& |2. 快速排序/归并排序算法3 ^0 v0 v& F, q
    3. 汉诺塔问题; K) g( x# v# ?) A: D
    4. 二叉树遍历(前序/中序/后序)
    - R8 V. Q5 C6 i- V& F( Z5. 生成所有可能的组合(回溯算法)
    , e% }% h- }/ g$ w3 \* g7 Y' @
    ( Q% S' N" p/ [' W7 q+ N7 L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 01:20
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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,; f, h2 j/ A0 }5 L! G5 |
    我推理机的核心算法应该是二叉树遍历的变种。
    1 L. j/ U/ k. t+ ~* r) H( |" f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:
    6 z6 a2 P- a9 m" n4 h# sKey Idea of Recursion0 f4 v. S  [& S  Z- }2 X: D! O

    - z/ c% [1 |  S( eA recursive function solves a problem by:
    . C/ a& O2 a& R! Z! l) r) x: Y! j5 L0 N, m9 q6 v
        Breaking the problem into smaller instances of the same problem.1 [2 x0 _! o6 M* m1 I3 o

    / e8 f; I, h; Z4 @6 \* }& V. g    Solving the smallest instance directly (base case).7 \5 d3 ~% W; ]0 s
    0 U( h9 u3 f5 J# t9 R6 O  E
        Combining the results of smaller instances to solve the larger problem.
    5 G2 [0 n2 H$ O, A4 {9 W5 Q! c0 U9 k, L* u! q
    Components of a Recursive Function6 e6 I: g0 k  r4 p4 d' {8 K, M
    ! H0 r3 Y. R) k& I) n! U( ^  ^& d) v
        Base Case:
    ; ?' L' {5 k$ e, q$ M2 R$ L: J" ]/ I. o( w# G* g9 ]& }5 J( _! L
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.! r9 |, \2 X3 i; S

    6 U$ y+ _; ]. D" X, n6 x        It acts as the stopping condition to prevent infinite recursion.; N. m) b, c: }+ Z- N3 g- o1 i7 C

    : m3 Z0 o* Z( D3 @8 i, [        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    9 F+ p+ h4 v1 t. W3 k8 i& [( b) N& K* O9 j, q0 X9 _
        Recursive Case:7 ^* k0 V" I! ]

    ( I$ |8 c' d( Z, f        This is where the function calls itself with a smaller or simpler version of the problem.+ }5 M$ M+ b' m/ d; h# L; v
    . z3 y, h- p# p! {
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 \* K+ r# K- j2 G* N

    ) T* t! v: p; ^Example: Factorial Calculation3 }9 _; G+ s$ i" ?7 r
    ! d8 j' }( S+ x
    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:
    2 B* R4 T# e, X1 q  ^0 A  Z" x( Q& o6 p0 }3 N% Q2 w% U
        Base case: 0! = 1
    * n: Z/ S4 T4 p7 S
    * E8 C6 A( R! H' G; }; S    Recursive case: n! = n * (n-1)!
    * Z& i4 M/ ^& `7 A/ }6 }  W! g) W- `" @
    Here’s how it looks in code (Python):. e) d/ ]6 G0 G3 R6 Z. L, y
    python
    . g; k/ b5 w. i  [3 n: B' S8 N0 `, M  {9 Q3 z; J3 K8 Q
    . S$ ]0 d. c2 ^" P0 [8 C0 M
    def factorial(n):
    & V$ h* ]3 f: p4 Z3 A( p    # Base case1 R! i5 Y: u  r' p0 d
        if n == 0:
    ' K. W% i& Y( `8 \# C+ o! k* A& r        return 1
    0 G9 h/ q# _+ M) x" z% ?+ C% k    # Recursive case1 Q6 b' f5 Q2 c0 X
        else:
    7 a' U( |0 U1 @# g% s$ _9 E5 b        return n * factorial(n - 1)  T5 k  ?9 a- t' u
    ; {) q, C" U+ g* D0 B3 |
    # Example usage9 n' b0 _4 Z+ o: K1 h; R
    print(factorial(5))  # Output: 1208 J) R: o# [8 V7 f' Z/ y/ P- ^: |' p
    8 }  y1 |6 h9 p
    How Recursion Works5 Q, `5 D+ H3 x1 A
    7 r  b+ T; f& }, j  R1 f- i
        The function keeps calling itself with smaller inputs until it reaches the base case.
    8 t5 w( r! h8 @+ P
    7 |0 t* d4 G) i; R9 q5 R5 A1 V/ _    Once the base case is reached, the function starts returning values back up the call stack.
    ( P/ e& S4 W+ I& S( T
    1 Z: p! _9 A' P* p7 E    These returned values are combined to produce the final result.- C6 h% A2 U  n6 Q+ i5 a9 p, C1 L
    % y1 O- M# N- O+ s" [
    For factorial(5):* I8 j3 D3 K; C8 g8 c4 _

    7 [% F* K% c# M" p8 `9 _; y& I  ~, l- e, Y+ Y4 A+ \
    factorial(5) = 5 * factorial(4)6 ?- J+ j  T8 p
    factorial(4) = 4 * factorial(3)7 A$ W7 U6 H/ ~5 W3 d4 Y
    factorial(3) = 3 * factorial(2)
    . G/ z9 e3 S4 kfactorial(2) = 2 * factorial(1)
    / a9 Z/ L( H5 c' V; Sfactorial(1) = 1 * factorial(0)
    6 P: t5 c; l- G; X( G% Cfactorial(0) = 1  # Base case9 y2 W, d; D, ^4 r

      }- f- @  Q) v: z0 ~Then, the results are combined:
    ! d+ X, ^0 {8 d1 g9 P/ K
    * u+ i1 K2 l' u2 }3 y, W6 n! q9 C* x& n. a' k! V
    factorial(1) = 1 * 1 = 1
    : T) `1 ]* F+ H  R  }* yfactorial(2) = 2 * 1 = 2' L1 R& w' G+ a7 f+ M% \" c+ Z
    factorial(3) = 3 * 2 = 6
    ) D7 y! N% G! s$ zfactorial(4) = 4 * 6 = 24
    + l: ?- g* C/ ]factorial(5) = 5 * 24 = 120
    + E4 I, G+ |! v5 R7 o* ~) _/ l7 p, f$ l$ |5 B8 n
    Advantages of Recursion
    $ M# y! Y6 S9 t% \9 w% O: g
    / W) Y1 ]5 o. `3 r5 n4 u& i    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).0 G4 Z. k4 ~& H2 M

    3 [$ b" n3 u' r# c, ]    Readability: Recursive code can be more readable and concise compared to iterative solutions.% b* ^% G  K* l( L
    4 _# }3 r: S7 m+ U
    Disadvantages of Recursion2 t. o7 E. v4 C4 a

    " R6 `: c  N6 A# H  k    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.) p+ _; y; i/ _" X% k# n: p  y
    / i8 S+ t1 Z( C
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).( m3 N8 q( |; c4 H
    1 U5 @$ C- c( r# R  X
    When to Use Recursion
    ; g7 n% V+ Y, T. `: G% M
    9 T3 |) O! d8 o. p8 y" t    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ' e. M- o( Y; f8 [9 M5 r( e9 y( R" d" `. Q9 b" [, U
        Problems with a clear base case and recursive case.
    0 p& Q9 b: e8 }& s
    / J; o3 _7 r% Z( AExample: Fibonacci Sequence
    3 |5 r# U, O0 p0 z+ s6 A5 S, m& k
    & i, m+ V3 p: E5 U& W# [3 QThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    2 r. u" X8 n- |1 d5 m2 V. c& O
    8 [; Z6 o, a1 v$ b0 q    Base case: fib(0) = 0, fib(1) = 1
    5 B/ N; N9 e; c' G
    & a1 W. q4 I' l# E; @8 Z    Recursive case: fib(n) = fib(n-1) + fib(n-2)% j0 Y0 M. |* ]4 i, L
    1 c! c) k4 Q! ~
    python
    % C' h* P" [5 c" J6 f$ J2 P3 g$ U$ k+ ~3 R" a6 ~8 k/ ?
      x2 w0 S5 u' J2 R/ p5 J
    def fibonacci(n):
    ( P( Q+ h* H, ~    # Base cases7 H1 z  ^. X: {' |- X; T
        if n == 0:
      u7 C7 |( K% |! ], ^' h/ e$ x        return 02 Y: y$ E( `5 q, O* ~) V* T
        elif n == 1:
    9 D7 E! t$ m8 D9 @  h        return 1
    9 V7 |+ Z1 G- ?% b6 b    # Recursive case" h4 M' d% _, N& O7 b4 f" j
        else:
    ) ^9 M/ B( y! o7 R1 ]        return fibonacci(n - 1) + fibonacci(n - 2)% J' w! n  e) [' N# U( ~" Z$ M

      |- T4 K( U; y: W' H1 B3 ?/ b+ M# Example usage
    2 K( P9 U7 i' p% f7 w7 u; Vprint(fibonacci(6))  # Output: 8
      w+ h0 r8 r1 u- i8 _/ B
    / A! G9 T" {4 l: |/ j3 fTail Recursion$ M9 G2 R6 y- m% t8 S' n6 Z
    * v) t; O% W9 Z$ J
    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).
    % I$ L& `$ T0 i4 N+ f3 ~7 `5 E4 L
    5 t& a6 n! x3 z0 m8 a: 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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