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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 Y+ n( l; P7 T# U2 z9 e7 T3 _
    " n# T* Z: ]1 [+ c& D解释的不错" \/ ~8 N% Q6 l, B: D
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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。
    1 D8 V2 l  d# F% W* s+ P( l$ Y+ Q8 D
    关键要素
    3 u; s, ?$ W3 m: o, \( g1. **基线条件(Base Case)**% o  z4 |8 I( e) |: W6 x. j
       - 递归终止的条件,防止无限循环- t+ e. S. w0 c
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1  D, A$ O* E$ @
    5 A. c. W- p; P$ S- f( U# u
    2. **递归条件(Recursive Case)**
    9 W4 z! S( d' f' D+ _   - 将原问题分解为更小的子问题6 p+ m( D+ p  B
       - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘+ H) C% W0 _8 l7 L
    python
    & c/ J& f8 y( O8 h3 I5 M7 m& t1 [8 ~def factorial(n):
    / a: ]1 u8 t7 B9 }. I8 v    if n == 0:        # 基线条件( x, M" t; H. {! E  @6 d
            return 19 v, j/ q" v) t/ P! O
        else:             # 递归条件
    3 |# O5 I" \( Y% Q- Q% t        return n * factorial(n-1)& U/ g6 Q; f6 @) ~' R
    执行过程(以计算 3! 为例):( u. ?( Y' n7 ?8 q3 i
    factorial(3)$ m" [9 B6 g$ q# X5 U1 _& m) N$ R
    3 * factorial(2)# P' A. I& h7 o5 o: H: k
    3 * (2 * factorial(1))
    & @, Q8 b1 j$ e; |: G3 * (2 * (1 * factorial(0)))8 A' e4 J; L1 o2 a1 }% H
    3 * (2 * (1 * 1)) = 62 x% {* G; t3 I. [4 k, }

      m8 F1 T# u# `7 j. {0 |% O) ~ 递归思维要点
    2 W, ~6 m: r; @1. **信任递归**:假设子问题已经解决,专注当前层逻辑* K# Y6 J; w& ~$ t+ s! _
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    * p3 J0 q) F+ K" t4 k3. **递推过程**:不断向下分解问题(递)
    " Z& r5 g1 H. S5 g4. **回溯过程**:组合子问题结果返回(归)
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    ' D/ B4 k$ ]) c+ @9 j1 j* j/ k  ^ 注意事项$ l0 K$ |8 c* w" g
    必须要有终止条件
    0 ~/ l: x- ~$ b; _0 h3 M# z递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    " [9 J0 a% ?4 U: _5 {# {某些问题用递归更直观(如树遍历),但效率可能不如迭代" A- t7 ?7 _& i
    尾递归优化可以提升效率(但Python不支持)3 A' W" K/ F) ^- q& J4 `/ v
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    递归 vs 迭代" w! S/ v& d# R# ^: @% L. J
    |          | 递归                          | 迭代               |" P. p  l9 _1 I0 q; _
    |----------|-----------------------------|------------------|1 A" E3 @& s" z: o
    | 实现方式    | 函数自调用                        | 循环结构            |
    " y. m9 D( @4 `+ m) Y; K| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |" ^$ Q3 Z: ^/ L# M) u" {) I  }
    | 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         |6 B, y3 [1 D5 i2 }
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    : j% y* x; l" L+ k- \  q
    ) G: K9 q( N: T 经典递归应用场景
    ' ~' G/ c. ^6 Z" J' W  r1. 文件系统遍历(目录树结构)
    1 }) g; u" F8 P  Y# X2. 快速排序/归并排序算法
    / M' C! b  e, W$ Z7 l  l2 d3. 汉诺塔问题
    3 D$ u. K3 u" A8 u1 I0 H% y. ~4. 二叉树遍历(前序/中序/后序)
    + j2 n& x5 d7 }; s# ~2 E5. 生成所有可能的组合(回溯算法)6 _' ]! C1 o, ?1 g5 ^3 b
    4 G$ J: K" z9 C& |6 ?7 a4 l4 B! P$ S/ _
    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
    慵懒
    昨天 09:58
  • 签到天数: 3148 天

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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,6 l/ q8 ^. S* {, m- ?
    我推理机的核心算法应该是二叉树遍历的变种。
    + S! A. E4 m0 i0 V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:
    : l' n% T; Z$ E. S6 |8 l6 |Key Idea of Recursion# U, L/ W4 H$ H8 n" ]

    ! H9 u9 S7 \/ m' w9 N6 ^A recursive function solves a problem by:
    0 e/ s6 m9 J% a6 j) o9 C& R  ~( k; t3 ^5 m7 Q; D, q
        Breaking the problem into smaller instances of the same problem.% q5 t* m& ~6 h
    ! U9 Y$ T  z5 T; m
        Solving the smallest instance directly (base case)., l! w1 \/ k& q! t5 ]8 v5 Z
    ) R; T6 i$ w0 w
        Combining the results of smaller instances to solve the larger problem." S! A" l0 z7 T$ R

    : H' w& D) w1 U* S8 x5 _7 N+ dComponents of a Recursive Function* E- n% S3 U; K* f, {/ V6 Z
    $ j% F& b0 j( M; l
        Base Case:
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    4 `& j3 X/ G6 i4 W+ P+ i3 U7 y( [        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    3 C7 G$ p8 A9 B' O  }' F* R% {; P# ^6 g# f# R" N  v
            It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    5 i6 y: m) d8 C0 V) U3 ^& M0 r: B    Recursive Case:
    $ G! |. y7 V: G7 s9 n( s+ `# x0 _7 ?+ Z6 Q& k/ x9 {* C
            This is where the function calls itself with a smaller or simpler version of the problem.
    ' n; c0 F/ ]+ E+ ^: P8 _# m- u  R. p) m1 F& J# K! \
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation
    % h+ W! p" K" Q9 k- H) C- R# _: ?* d% o/ }
    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 w6 x7 }) R1 B" |
    1 p' M# x  k8 x
        Base case: 0! = 1
    + u! l2 O% G- ^
      M3 D# u7 C' W% U& _% y0 M3 \' {    Recursive case: n! = n * (n-1)!
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    Here’s how it looks in code (Python):- x) Z# `4 g8 D  G' k- H4 @  {
    python
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    * F5 [& \1 H# R7 V/ c0 x  y1 H8 i/ V( W8 g! ~7 i% @
    def factorial(n):3 [- a( X# F2 I& z7 t- B
        # Base case
    $ J: M& e1 _& R2 o0 x    if n == 0:
    : _' `% V; j9 l+ q, N$ i        return 1$ O: H4 b6 S- G. ^- Y, S
        # Recursive case. U. y- A% s3 V. V  Q; [
        else:
    # m; P  x  L2 n2 F9 L        return n * factorial(n - 1)* E9 S4 B$ M9 u6 ~. n6 M% Y, M

    3 R4 T# O; `+ K) Q4 W; e6 N6 F3 w# Example usage  b( I5 b+ |  }/ t) J" B1 ]
    print(factorial(5))  # Output: 120
    4 F0 f; O  m$ s7 \- i
    " _, j9 l8 l$ S9 e5 N9 V% yHow Recursion Works7 P* u! p; C- r$ _( d% h

    ) z& y" f& U% C$ [    The function keeps calling itself with smaller inputs until it reaches the base case.
    & d! u) e' Y' z" _+ s; b
    : N( G) r4 M+ \( R( x    Once the base case is reached, the function starts returning values back up the call stack.2 o1 j4 H* y4 _/ r$ u& O

    ! F  s) F8 T. O) }& v( ^    These returned values are combined to produce the final result.
    : ~1 P* K; D( ?: s+ ?0 _- t9 H8 T) W& C3 {" m7 n9 }# _3 d
    For factorial(5):
    2 k+ G& i0 v& [' N: e8 ~2 K7 c" _% X' {+ g

    7 Y; |7 J: g/ _4 @( @! dfactorial(5) = 5 * factorial(4)5 L2 a- o/ x5 n2 A* @3 R0 A
    factorial(4) = 4 * factorial(3)6 i7 U8 x( f' X* Q4 i$ k7 l' W& b4 q
    factorial(3) = 3 * factorial(2)
    ( x# B6 m0 E. M, [factorial(2) = 2 * factorial(1)
    4 B3 [# _. j! _' ufactorial(1) = 1 * factorial(0)) f3 g2 E. S* g4 y
    factorial(0) = 1  # Base case8 ]8 J7 U4 T  k. t8 b" V/ I

    6 X& C- X% ?* f: [; I- dThen, the results are combined:
    , O5 G7 r, N! ]  \" f  p) Z
    " C; r' f) y2 K7 F' r: I* N  l4 o  m! e' T7 E* M& h
    factorial(1) = 1 * 1 = 1
    8 D  j/ S3 [5 sfactorial(2) = 2 * 1 = 2" b6 [+ g# F( W% a; G
    factorial(3) = 3 * 2 = 6; y6 E" s0 k# [$ O  y9 u' K
    factorial(4) = 4 * 6 = 24
    6 O4 x/ P( u. |8 p5 Ufactorial(5) = 5 * 24 = 120+ \/ f2 Y+ x- x$ \4 M

    3 k( ?( X: H4 `9 I$ N) b) |Advantages of Recursion
    " o: E* P$ o5 Y8 V1 [* P* J4 `& v6 f3 X* s1 M* f
        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)." @% O6 G2 s- f' }7 b4 d7 ]
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        Readability: Recursive code can be more readable and concise compared to iterative solutions.  B7 [+ s; t6 X8 _- Z* X3 l7 s
    6 y+ `% ]% c0 F: G& B! c* _
    Disadvantages of Recursion
    & F, v% M: Q0 w
    4 h1 F0 @0 J. R$ ?0 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.
    : u% A' c/ \, V/ y( W& s" E- e- j: F) q- \' w0 Z
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).8 R4 W. y$ w0 U/ s7 @
    ! l% F  K" {6 Q8 [) G, c7 T( x) e/ [
    When to Use Recursion
    $ X5 b" G1 A% Q  F6 T
    1 z# n: v6 i8 m, H. x) E: T    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ) @- w1 m  {& O3 `1 @: V) ^4 v) u1 p) A+ }7 ?* ~) t
        Problems with a clear base case and recursive case.) h2 _6 P8 i) z
      a2 n4 ]; Q# N* H- w- l0 C* ~( x
    Example: Fibonacci Sequence
    0 H, k& j+ E* @* C4 q+ ~+ |3 p# ^, B9 R5 }1 Z* T
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    3 W& h2 W- C: V; A- U3 i( L
    ) p8 H: R1 _+ d% U    Base case: fib(0) = 0, fib(1) = 1
    : l0 }  l: [1 a; u- A, Z* p# V* [
    ( n: I: X  ~  S; d" Z9 y    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    ) A  G$ W0 A# _5 C2 b: U3 `! jpython
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    4 g2 t$ e$ C; U, c4 ~/ L" a* z8 A
    / u. @. f; g3 `7 o( l% _def fibonacci(n):% a3 S8 n& Z0 |2 }$ w
        # Base cases! `, K2 \" N4 g+ R
        if n == 0:; n3 D# a& R9 K5 m* j3 s. v! m0 k
            return 0% F* p/ w4 ?! ^4 M2 A6 E# _- k( I
        elif n == 1:
    / ~# k5 Y- a2 {( h6 J  ~! b* i$ ~        return 1
    " O5 J7 J! d4 f    # Recursive case$ z8 C' }) a6 h- y( ?
        else:/ B# k) k. c, P! T) E% i6 t, W! L
            return fibonacci(n - 1) + fibonacci(n - 2)
    / I; _7 j" {$ k  _9 P  `% x+ J5 e7 n% I
    $ o: @, `5 |$ U0 r4 o# Example usage4 y( q1 R8 l! s- T( ~8 z% c
    print(fibonacci(6))  # Output: 87 r% w" v3 i8 A. P, @( P$ ?4 y

    6 n9 |6 g( s* H$ [# C) \& JTail Recursion
    " R9 X+ w1 Y5 x
    " V) I) q8 I, }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).: d, X& D( G1 H3 ]7 u

    * X! s7 `5 F& |. F8 w& Y. O8 |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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