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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| 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 ! d8 U1 y! t  m5 J5 @8 W6 c7 ~
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    解释的不错
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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。
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    4 J8 W. c+ H3 |  k 关键要素; N! P0 U3 a/ I
    1. **基线条件(Base Case)**+ [& G& v0 g5 R. ~
       - 递归终止的条件,防止无限循环- X0 N. A  z1 q2 J, I( _/ g$ \
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& a- w" T6 C% @4 Z' ^) [' D

    . g- E' W) i( R2. **递归条件(Recursive Case)**! }7 v# r! O! u. M* B
       - 将原问题分解为更小的子问题
    1 s/ E' n2 ^% _6 U0 q5 k   - 例如:n! = n × (n-1)!
    4 P$ D) @% [2 T3 }
      f7 K& |3 f& J# B 经典示例:计算阶乘( b9 p3 D8 D% Z9 Y) H8 f- a
    python+ G9 x! F7 m: R& D
    def factorial(n):; V7 a) A+ Y" S' x, W) S2 q
        if n == 0:        # 基线条件
      h, r9 H3 ?! _( a$ n; c        return 1
    4 \9 p4 w# u% _    else:             # 递归条件" h  t! ?, ~5 G  g: a
            return n * factorial(n-1)8 o) [! B9 i0 U8 e
    执行过程(以计算 3! 为例):
    9 f; y- \6 i3 v$ o2 Y3 nfactorial(3)
    ' a6 X2 M+ d2 Z8 ~# }3 * factorial(2)
    ' I8 A# Z  r2 S/ h2 u) @3 * (2 * factorial(1))2 e  u( z* e- I3 j! j
    3 * (2 * (1 * factorial(0)))
    $ ]& x( v9 D0 ]% Q/ H9 D0 C6 @6 U3 L' o3 * (2 * (1 * 1)) = 6% d- R7 Z. I. D
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    递归思维要点) w/ c/ ^$ z! j: S' ]+ M
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑! ~2 `" A* K, i& V, `9 z* d0 q
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间); D/ @8 }4 q) y. _& M' O
    3. **递推过程**:不断向下分解问题(递)
    % j, ~" D5 f+ O, C0 u4. **回溯过程**:组合子问题结果返回(归)
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    0 z5 d1 ?. F+ }; z* S; u 注意事项1 F# [) T$ a3 S& Z: X
    必须要有终止条件
    " r* x7 D+ x  f) }, ^) i2 D7 w递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& ?: t: l% v" s& `
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    2 `; P: X7 j* F( P4 @尾递归优化可以提升效率(但Python不支持): A4 D# _% x: F. X3 ]
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    递归 vs 迭代
      j6 w$ y4 F3 O|          | 递归                          | 迭代               |8 K1 Z3 g  s  Q3 P- u& G
    |----------|-----------------------------|------------------|; q& [2 f0 R2 p1 q5 Z+ ^5 n& V
    | 实现方式    | 函数自调用                        | 循环结构            |
    6 ^2 }) e& H6 V6 [- t0 R! d* _| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    8 c- r: o8 k! ?$ y1 l0 T: z5 n| 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         |5 f) A* T! [: R4 w+ w6 C$ ~
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
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    / `- V0 Y) q! R, N: d 经典递归应用场景
    . l1 p+ ~' H& j: S, s1 ^) M1. 文件系统遍历(目录树结构)# a0 W* }% z& I0 w
    2. 快速排序/归并排序算法1 Q# ?; ~) k& h( _( d0 W+ o
    3. 汉诺塔问题3 K1 l+ ?8 K! U5 D9 \* Z
    4. 二叉树遍历(前序/中序/后序). C! Q% |  n# M  \4 m6 w  P
    5. 生成所有可能的组合(回溯算法)
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    & J% A$ c5 W6 Y! A$ m3 K2 \6 S7 Q; P9 Y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 06:54
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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,
    2 {, v# a, f4 s2 R我推理机的核心算法应该是二叉树遍历的变种。- l3 D6 N! j" P( U# 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:: G5 p8 ^; K. A- h& N) x' r  ]
    Key Idea of Recursion
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    A recursive function solves a problem by:
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        Breaking the problem into smaller instances of the same problem.* \. Z  f! @& u- l, `
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        Solving the smallest instance directly (base case).
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        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function
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    9 A/ {# _  G& |: k% Z) |  H    Base Case:
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    ) W4 i0 j' B0 U( B2 F8 `        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            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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        Recursive Case:
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    ' b+ X3 F6 g+ c0 n        This is where the function calls itself with a smaller or simpler version of the problem.( u! m3 K6 A7 t$ [) g& f: ~5 J/ ^
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            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 g* O' P/ M, P2 Y& ^* Q5 i! M' G

    + G5 @/ [# X/ W& NExample: Factorial Calculation
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    ( C% A* f' A: Z* j; ^+ k8 CThe 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. a/ J6 T; v3 I( M# R7 L

    $ o; n) Q0 E8 k9 z    Base case: 0! = 1, o8 y$ S5 D; c5 B$ M
    " y" a" |: j0 _, w- F' G2 }$ z- L
        Recursive case: n! = n * (n-1)!
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    5 @! `& i. _+ }& b+ l2 LHere’s how it looks in code (Python):! x$ {- J- Y* f1 _  T: ]
    python; y; W) v$ s2 U& N

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    8 \3 A2 m0 ?! s2 Pdef factorial(n):
    . D8 w) p: Z5 E3 p. d4 M    # Base case
    2 ]. _4 u* M0 [& i    if n == 0:3 c4 \% X7 r1 X0 @0 o
            return 1
    : f/ Y- z# P0 n# [- v& U$ r8 a    # Recursive case
    9 M1 E1 C: _, I& r0 t# M4 ^    else:
    : p" ~7 f/ ^2 X* M* t! H- L        return n * factorial(n - 1)$ L0 t  J' w8 o9 s7 K1 \
    0 h! }7 I$ P$ B) S
    # Example usage! T  i8 H( M; d% v; ?
    print(factorial(5))  # Output: 120
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    How Recursion Works
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      q& O, p; G; D$ Q    The function keeps calling itself with smaller inputs until it reaches the base case." ~/ w* c7 a, e6 ?4 A3 w: T

    . f4 u3 I+ P# L* L. p# K5 I    Once the base case is reached, the function starts returning values back up the call stack.
    , l8 A5 x* ^9 x& b. J
    8 D4 O/ M/ g) v# S/ g4 T8 w' V. r    These returned values are combined to produce the final result.5 {0 g# X4 i2 \5 S# q5 J
    % |3 e. V$ L  b4 D# [2 \/ M! I2 ^
    For factorial(5):
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    7 A2 |0 R) f. y- X. ^; H
    factorial(5) = 5 * factorial(4)7 z2 y& @' J# o, N  e; f
    factorial(4) = 4 * factorial(3)+ U* C, T. K9 W" V; U* @
    factorial(3) = 3 * factorial(2)
    + Y6 e7 C/ a2 E' @1 p6 }2 Hfactorial(2) = 2 * factorial(1)
    % L+ A* }7 R) Bfactorial(1) = 1 * factorial(0)
    ( F1 O- z1 x& Zfactorial(0) = 1  # Base case
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    + `8 S$ R. O1 U7 i$ h4 |2 DThen, the results are combined:+ Y; w/ \. m9 B, S
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    factorial(1) = 1 * 1 = 1
    5 |+ J# w: R% z1 m. s3 T  kfactorial(2) = 2 * 1 = 2
    ) q, y" a6 {: g0 e1 V2 Kfactorial(3) = 3 * 2 = 6, R8 }# m2 ^8 ~) M3 {6 ?0 U
    factorial(4) = 4 * 6 = 24
    9 V9 l+ ~% R5 yfactorial(5) = 5 * 24 = 120
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    Advantages of Recursion
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    * `# `& p  L2 S& z; D) z    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).) G8 b* B% m2 s- U# ~

    + g( |9 p* |! _% I2 B* H    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion
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        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./ E- {. X; h* U( C2 q
    3 q& R9 l- J+ ]* f) `
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 u4 v6 s: v; ?2 e' }$ i9 M

    9 r, W. i* X9 Z0 E  K* [7 yWhen to Use Recursion
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    " u7 w6 [) _+ n" j( z8 D6 [    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' ^7 l& f. q  r$ ~) @+ u, f/ R
    # ~. F0 I: n1 x% x0 f& O  @
        Problems with a clear base case and recursive case.2 X9 c* R# F2 C% {( ]
    # C1 t4 w( s/ L6 g3 T: n  g8 I7 K/ Z; Z
    Example: Fibonacci Sequence
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    0 O6 n2 b3 H; v& t9 |' e1 ]The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ; e7 @6 W% y' U9 A4 d/ @% E2 l$ E8 Q, X* \. u( k) e
        Base case: fib(0) = 0, fib(1) = 1/ t9 d! S8 z& s, V  U1 V% M
    . A8 E3 d# U( g& N7 c) E. _; C0 x
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python$ K% z1 J9 a( z5 B
    - i: v9 v3 D: I$ t) y/ S3 _5 k  K- ?% O

    2 W& L% R+ F9 d! I0 q9 Ddef fibonacci(n):$ @# Y( A. g% I5 D3 T
        # Base cases, a% T. F0 @5 ~
        if n == 0:. ~( f( g  @8 i1 ]
            return 02 A  D( j8 }0 Y3 J! q
        elif n == 1:
    6 z- I6 ~6 i) x  u' h        return 1" C1 ]4 p/ j& `4 n# m& ]
        # Recursive case8 B  v- [1 n8 g3 F2 V& ]5 c
        else:# }- Q" g/ U; T( b6 @
            return fibonacci(n - 1) + fibonacci(n - 2)
    : z' w7 v: e' f9 l0 x7 c: i$ `- a- t
    # Example usage
    $ D# J, T5 V! A( G! Nprint(fibonacci(6))  # Output: 8
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    Tail Recursion
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    % {3 C& d: F, v+ xTail 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).
    # ^' g- `( u7 X
    # B' I7 B3 g9 h" {7 A- OIn 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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