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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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    2015-11-30 11:11
  • 签到天数: 2 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
    ! a- \' j+ r" W) o* \  z, L; a- x( x5 G* d  y& B$ d6 {" T
    解释的不错1 P1 Q4 p9 ?0 z

    + o8 G! |+ J6 K, g5 Q2 b6 x: o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。
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    6 j* j4 d; @# s3 q5 e8 X) I 关键要素( Y) e8 I( \* o( G, \+ y% t
    1. **基线条件(Base Case)**7 m5 G+ s3 z6 Q# b- `
       - 递归终止的条件,防止无限循环
    9 S- w# N. v" V* ^$ c2 `! z8 G   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    3 _0 @  }0 P7 A2. **递归条件(Recursive Case)**
    , {2 `3 J$ i$ P2 I. G$ M% y1 r   - 将原问题分解为更小的子问题7 {6 @- U: z1 J* [/ Y
       - 例如:n! = n × (n-1)!
    8 n7 \" L. O) ^& ]2 o9 D2 |* F2 F8 o' R7 k$ l9 l
    经典示例:计算阶乘
    % J7 y$ f$ K" ~1 }python
    ( [: I+ y/ E8 i9 q6 [! Gdef factorial(n):
    4 U7 @  }" x+ h7 _! i& h    if n == 0:        # 基线条件9 V1 H4 W( `4 P. a
            return 11 g* ?. A% L' s& E2 ?& R/ o
        else:             # 递归条件  j) i' r4 S- s* `; `5 D* T1 b
            return n * factorial(n-1)
    ; S8 w; b/ w4 s* r# w8 R1 ?执行过程(以计算 3! 为例):
    # W1 Q' k' \( w4 c8 [9 cfactorial(3)" o; K7 e$ L3 q2 _& o
    3 * factorial(2)
    , @3 l( `' J  H3 * (2 * factorial(1))/ m  D, G% P) P# C1 q
    3 * (2 * (1 * factorial(0)))
    - H* I- F( i& l4 C3 * (2 * (1 * 1)) = 6
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    * l  [" j3 q8 S# |3 c' x2 q# K 递归思维要点, @. u1 i3 L, T" E& w- c& b) X' Q
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑& Z3 X8 U6 l# f
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
      z- w  K, s. n5 B3. **递推过程**:不断向下分解问题(递)
    7 }7 V9 M" y. C4 F4. **回溯过程**:组合子问题结果返回(归)! E6 N9 y4 Z) Y9 i( }" T

    ' `2 |" d2 F/ W) P9 J9 c# a 注意事项
    8 `1 c- {- @: _% c3 `必须要有终止条件
    . g1 [9 k) {5 ?' B1 {  Q递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    + Y* n+ \* n: S& f# z# h某些问题用递归更直观(如树遍历),但效率可能不如迭代( v6 N  R! d% l7 _# u
    尾递归优化可以提升效率(但Python不支持). z  e0 N  H* h5 y* x1 F6 P! B
    + h, O' O0 d7 G& v6 A* `: z' o
    递归 vs 迭代
    ! S% l8 {: u/ Y5 S6 ~, K|          | 递归                          | 迭代               |; U5 }) A; c6 h  E, i  M
    |----------|-----------------------------|------------------|
    : K% p7 b8 {6 V! K' G' ~+ X; N2 @* j| 实现方式    | 函数自调用                        | 循环结构            |7 K. c' U- N. H& h. V: n0 u
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |8 ^' \7 `1 z0 H* O5 t5 N4 \
    | 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         |0 o' r; }3 F- y: v$ t4 j( Q  T
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    # I- s  h9 }# Q( ^
    ' L) C0 o6 @. @& k 经典递归应用场景4 V4 ?' `  A! V
    1. 文件系统遍历(目录树结构)0 M. Q7 m. k! y1 O1 o. |4 V
    2. 快速排序/归并排序算法2 Y8 w  V+ S- s) K
    3. 汉诺塔问题' r( a# ~5 N( ?4 L* L
    4. 二叉树遍历(前序/中序/后序)5 y' `9 X. |! U# x( e0 m
    5. 生成所有可能的组合(回溯算法), o6 Y9 \$ x7 c& O+ E" \, n( [

    8 f4 @2 [; I, b  d+ U" ]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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    13 小时前
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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,
    4 U. T# D' m) t5 e0 o我推理机的核心算法应该是二叉树遍历的变种。: E6 n" A& G/ A
    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:
    $ j7 L' w1 Q4 z0 F( zKey Idea of Recursion
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    - Q! K( O0 y5 N. @4 G( mA recursive function solves a problem by:
    % @& I9 w% t, J7 l7 |
    " i8 n: p/ p9 r8 u  ?    Breaking the problem into smaller instances of the same problem.
    3 k: V7 v, X3 x( ~3 a+ B. O8 P" d0 B  d& O6 j  x3 l
        Solving the smallest instance directly (base case).
    ' |; x0 ?3 [$ O5 L
    # J. y* G* }8 }    Combining the results of smaller instances to solve the larger problem.1 F+ u$ q# P/ t4 `

    " {- x( u9 {# ^  V% tComponents of a Recursive Function. R  h  y# G. b# O  g0 x

    3 w) o7 f, Z9 _- ?    Base Case:
    0 a+ z/ |4 s4 K% H  `  j5 h
    : w; V7 @/ z0 R. l$ K' M        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.  ]. K# b3 ~9 q) A5 V9 `' u
    8 ^, F& U5 J  H2 p; H: V2 T
            It acts as the stopping condition to prevent infinite recursion.
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    + |; }& T% X2 t        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% l: `  ]  y; \& H9 Q, f
      |+ a) |6 j; Z8 S' Q
        Recursive Case:4 S% u% x" D6 I) F- C7 f

    . ]) `  V( S6 p% d        This is where the function calls itself with a smaller or simpler version of the problem.
    3 c) L$ {: C* ~( `
    3 s# F$ O9 t9 w* o9 y( _& i% a2 O5 Y        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    3 m3 u+ j$ k' k' |5 a7 D; P% Y$ hExample: Factorial Calculation
    / ^  d+ p' U$ V
    2 B5 Z4 q$ R& x& l* I/ }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:7 `7 k% Z. S! \2 X* P6 J
    ' d; g; Z3 x3 N% [5 E
        Base case: 0! = 1' `  ~: c1 a/ w, y, }

    0 M3 i4 \6 \2 O1 G9 F$ E    Recursive case: n! = n * (n-1)!
    7 `% O1 C; m2 Y4 M9 f5 G7 V, w1 _# X0 Z" m
    Here’s how it looks in code (Python):
    8 Q& \0 @/ S2 m; ]/ n' E! Lpython
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    4 Q- S' G% v1 v' K: t! tdef factorial(n):
      p1 L5 s! F# j: ]+ g# {; b    # Base case3 x) p% ~, v2 D- J4 G
        if n == 0:
    : o0 C4 J# K7 t) W        return 18 o# o/ x' j" e, V% D% Y  I2 v
        # Recursive case
    4 A; e  _% X4 G    else:
      o4 |' ]& c# D4 O& [        return n * factorial(n - 1)
    6 }' @, F$ P. M9 ~& E$ q9 }' o
    ' V; X8 z1 }/ x# Example usage
    . G# z1 T0 o( r$ _7 J; ^print(factorial(5))  # Output: 1209 K! m/ L( M9 p/ x& v: K
    ) Y+ t0 d# }# P6 U: M
    How Recursion Works) g0 T6 L9 o0 N) n

    - N$ C5 h: e8 X- c% ?    The function keeps calling itself with smaller inputs until it reaches the base case.
    8 {4 L  o9 M6 @/ F( \) d( H2 @. e( V; W5 v0 D6 T5 y4 S
        Once the base case is reached, the function starts returning values back up the call stack.+ \; R! I3 Q5 T  z4 Y; i
    / u1 ?/ s7 W; Y6 r
        These returned values are combined to produce the final result.. ~# P) k! p) E' @. B) N: Z

    ; w6 M% u* g7 v4 _0 O+ YFor factorial(5):. v. s) t% r, a
    & |  u/ B& I) o- v9 j. U
    % v# T% @8 P% E  h3 B
    factorial(5) = 5 * factorial(4)
    . v9 m6 b$ G0 _3 p5 c$ Efactorial(4) = 4 * factorial(3)
    ( @; m" i* n% X: D% Rfactorial(3) = 3 * factorial(2)  W- L8 W$ o) y, X' O. Q8 Y  _' L
    factorial(2) = 2 * factorial(1)
    ; L& ?, f5 p+ k3 R1 tfactorial(1) = 1 * factorial(0)
    ; B5 W( d9 @1 g' C3 W, u, |factorial(0) = 1  # Base case
    ! z* I  |! a: C  j! Q* B2 L
    . J% |2 E7 y& T( e/ i1 E, sThen, the results are combined:
    / y2 s+ q# @7 J$ W8 d$ l" B+ W5 X5 ~" s7 m/ u

    ! ]2 b1 l  [  }: nfactorial(1) = 1 * 1 = 1
    , M0 k& ~$ W# L, _3 Y+ U6 V* Bfactorial(2) = 2 * 1 = 2
    1 C) [9 a" G0 h8 E5 \factorial(3) = 3 * 2 = 6
    1 L* G! `5 t5 i& zfactorial(4) = 4 * 6 = 24" h8 [" W8 W# L$ {( |: x- \4 D  |6 B
    factorial(5) = 5 * 24 = 120, F9 p* g: U5 p1 X; Y
    , u! d; k' M0 X
    Advantages of Recursion  w# W. R9 }% R3 H9 ?! `- r
    : U3 I9 b5 ~! T8 L5 m( w
        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).7 ?; X& _- [+ d: D1 a' P
    5 w+ n+ i( v4 y2 h
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    2 X7 }% S  Z, z; U1 @. A9 f1 w1 ~# f0 ]
    Disadvantages of Recursion
    5 {; b" H1 m6 Y: d. D; l8 N
    9 x3 s" h) @; Z. O- f5 N* i4 f& ^    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.
    * q9 ^) G; N% g% a% T5 ~
    + T, L5 j& ~* I- w& Q2 E0 l$ p% S    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ( C) T% r0 n5 W- T/ c
    6 ]& ^: E1 [! ]  l1 h. DWhen to Use Recursion
    9 N) K7 d# h0 p& s( s! m- t  ?; x2 ~0 ~: Y1 Z
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).) S- `: x  g0 _

    " B0 y3 c" F, Q. ^) j% W1 ^" I    Problems with a clear base case and recursive case.
    3 O% z: b1 C3 t/ P, [/ l
    6 c6 Q8 y& Y# o" V. [2 aExample: Fibonacci Sequence/ v1 k& R  V3 }

    : Y" g1 J0 e% Y- O1 ^9 OThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    0 V& P! ~/ T7 i2 h7 T% ?
    6 y% Y% i% \3 u! T4 C- u    Base case: fib(0) = 0, fib(1) = 1" `# R" g# [7 o1 ^( f0 `
    & E" P; A& i+ w9 i# b
        Recursive case: fib(n) = fib(n-1) + fib(n-2): h# f( k, N+ W. h, x% {) t

    , P/ c7 r1 _7 S3 w+ j+ I! Bpython
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    5 `  m5 o' p8 v* c
    6 [* G/ Q% R9 Q0 hdef fibonacci(n):
    ) D0 G5 s0 R$ Q( _5 u8 `  l  V. `    # Base cases
    - F$ ~7 j% M5 j* ^: U    if n == 0:! h4 [7 m- F- N$ l
            return 0
    5 }& Z3 W1 O; p2 J2 ?, r  K/ R1 j    elif n == 1:! N; Z5 I& H4 B$ z0 J3 E) K% |
            return 17 h# x0 q/ u: M7 x& I: h$ Z' p( l+ a
        # Recursive case
    : G# ?& d  N  r1 m: E    else:( \4 Q3 H2 x) o4 n
            return fibonacci(n - 1) + fibonacci(n - 2)
    & P& |( n) g" P6 X- S# L* \9 |6 f) z9 \& K- e$ O
    # Example usage% Q& G# @; z7 O$ z
    print(fibonacci(6))  # Output: 8
    8 X* `! j. E. J5 b, A$ X/ M# @1 k1 s1 j/ v! V, W
    Tail Recursion
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    # c2 Q+ Z7 V+ N+ G! S4 d- U1 TTail 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).0 ]) Z3 v9 U( f! z9 U
    9 H# ^2 w$ ?" e# l# e  q) q
    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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