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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
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    解释的不错
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    * k, y3 P" W2 r. n4 S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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    关键要素8 m, p  V2 {9 |- J
    1. **基线条件(Base Case)**
    * R+ ~! ^& v% h( w! V& F8 j3 [. [  n   - 递归终止的条件,防止无限循环3 r( U, q$ k; b; H: M6 U& W
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    - v- v; I# U4 U3 L8 m( X. i( M! C; Z8 q; M9 J" s! o
    2. **递归条件(Recursive Case)**/ o9 m8 [! R5 W2 M
       - 将原问题分解为更小的子问题
    6 @7 _) x" A) D0 f   - 例如:n! = n × (n-1)!+ S3 X/ m# O. M+ k- s7 |
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    经典示例:计算阶乘6 R/ q" d- t# _4 C4 X
    python4 L3 ]  s; X3 Q
    def factorial(n):8 A3 t0 k. |% _( F7 f
        if n == 0:        # 基线条件8 p9 \; K* W8 V* \
            return 1; P& Z7 ?4 f$ c+ @, l
        else:             # 递归条件
    $ T6 ]9 g! C" w' `        return n * factorial(n-1). O; N( P& H5 u$ x
    执行过程(以计算 3! 为例):- x  @$ W4 w! W- z4 a
    factorial(3)
    . A$ [8 K, a& o. k( v$ Q: z3 * factorial(2)
    . s1 S, S+ Z, H4 q/ _% a3 * (2 * factorial(1))+ ?1 P1 Q* ]; ^) W) u. h' Q* s
    3 * (2 * (1 * factorial(0)))
    0 }: O7 l7 [1 e8 f3 * (2 * (1 * 1)) = 6
    $ i; L: s% h: P9 Z
    & I0 B4 e& ?4 c4 F 递归思维要点
    / m  V  J" S+ B- N; h1 n' `! K* T1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % {7 b3 Z- d% |5 q' ]2. **栈结构**:每次调用都会创建新的栈帧(内存空间)5 r1 J( w' q3 d$ F6 ?. |
    3. **递推过程**:不断向下分解问题(递)
    & M) K2 G! Z% y0 F5 q4. **回溯过程**:组合子问题结果返回(归)
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    ; s2 G- P/ R9 v$ G2 ^$ k 注意事项
    4 r0 |, j; c- _1 T. P4 ?& V必须要有终止条件
    ' C& B3 B( W- }" P+ Z% L递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  U$ ?8 l: ^2 ?- @
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    , t: F2 A1 Q/ m3 k尾递归优化可以提升效率(但Python不支持)- J% ]  k3 w, m$ H* G* T4 w' q( `
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    递归 vs 迭代1 t/ U4 c% `6 U) Q3 z; }
    |          | 递归                          | 迭代               |; ^: {* g/ y2 U2 R
    |----------|-----------------------------|------------------|
    % S2 e- p" a* e. v  L| 实现方式    | 函数自调用                        | 循环结构            |  m$ l6 P+ |* ]! z, {" n  R* n
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |- c3 ]8 b8 a: N& J: p
    | 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         |
    9 ]' }; X$ x, L; h4 m& Q" o* W9 W| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |6 e& ?! [$ L# a& P2 N
    : X- h! y4 W6 A9 j/ P
    经典递归应用场景$ i( b9 T7 p4 b: C# W
    1. 文件系统遍历(目录树结构)
    0 T, }! N/ w' M# {& g' X. G+ s8 g3 ]2. 快速排序/归并排序算法
    # N3 k3 q: M" z3 Q" }  w( i) o3. 汉诺塔问题. S9 i( f, E7 J; e$ h1 M  W1 k! {7 S: ]
    4. 二叉树遍历(前序/中序/后序)/ u: [9 C, J3 b& N/ y
    5. 生成所有可能的组合(回溯算法)) j1 ]8 O& G, v5 B6 b3 C4 A3 o

    : E3 @7 s0 u  o! M; P- T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

    昨天 09:55
  • 签到天数: 3196 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,! d5 i' E* e0 D) U
    我推理机的核心算法应该是二叉树遍历的变种。
    ( ], _/ O. C# K. H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:
    # r, s5 Q: x8 J& p5 eKey Idea of Recursion* ^1 k- G. ~! {# p) C
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    A recursive function solves a problem by:
    / K; m0 D( Z# X9 j- |! t6 v: ]* Q' `4 U# A: S' u
        Breaking the problem into smaller instances of the same problem.
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    " b* Q! J1 X8 k$ G    Solving the smallest instance directly (base case).
    6 r& C* \# N- i, ]' Z
    1 S4 c( T* a; b/ F    Combining the results of smaller instances to solve the larger problem.$ N- q4 U' q! c5 K; p) @) V
    9 i3 r: y* \" ~
    Components of a Recursive Function
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    4 u5 a; k4 |- L0 W) u/ F2 ]( {; N    Base Case:! f8 `  g* I2 m

    ( a/ F4 }: W+ \' s        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    * j! {3 W5 F1 ?$ D4 ?; u& a  A& O+ _8 X
            It acts as the stopping condition to prevent infinite recursion.3 h4 v+ R( {6 K: \/ O- h8 P" Q

    9 \8 ^* H; t! F( |0 |9 d; S5 `9 Q2 A        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ C4 R6 e$ P( R5 g; `) E
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        Recursive Case:
    4 [' L  z3 S/ I! K) Z4 U4 p% i6 l4 V# g6 K
            This is where the function calls itself with a smaller or simpler version of the problem.
    5 d" \( u) F. L4 f. |# N* c& a- }" a7 I& M5 h9 F
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* z/ B* G2 j. [1 ]
    ! S7 O6 c! M+ v# x6 Q
    Example: Factorial Calculation6 @, @' Z) D1 S5 b0 X
    7 C& @# t- e' A: O5 L; P
    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:* {5 _- K# V& |$ w
    6 W' ?" V/ r0 {, N3 z  |, ]
        Base case: 0! = 1
    1 ?, {7 B  b  J; d, |8 h$ G
    , e# s( K1 ^. X- _7 l8 J) f    Recursive case: n! = n * (n-1)!
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    Here’s how it looks in code (Python):8 G" M* r& M- g4 g5 K  `8 B1 T
    python
    $ h5 B% g  D# q2 G
    6 P% W* I4 R) T$ r; B9 K' J( o6 J; r
    ) f- a  j  ]8 E& Vdef factorial(n):
    8 R3 a0 B/ m6 S7 D* _    # Base case
    # S' V  I& \2 S$ X1 Q! |( {    if n == 0:9 l$ p( a" ]( I+ I+ X7 T4 y: b
            return 1: {* L1 }3 Z# i  `$ G
        # Recursive case
    " q" C: v: }8 j: j$ {    else:& s6 z) R2 r+ c1 z. f9 a* g
            return n * factorial(n - 1)7 s  w; d+ O- `: }3 T8 v
    & i6 q0 X/ F4 Q+ B: `" S9 |
    # Example usage' J- @* D' f0 o* l6 B2 e; H
    print(factorial(5))  # Output: 120
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    How Recursion Works2 r0 \# W6 k  m$ ?, Y4 U5 M
    4 z, a% y9 P4 h' U7 _
        The function keeps calling itself with smaller inputs until it reaches the base case.8 R4 h1 H& r. Z2 u2 l/ g5 n+ e, f0 n

    ) I& o' U# [' u1 O& a; \- P) w    Once the base case is reached, the function starts returning values back up the call stack.- v. r8 n' U/ ^5 P8 j3 v

    5 g/ n1 u3 k  O( O6 D    These returned values are combined to produce the final result.
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    5 f  y  _  _# |For factorial(5):
    - h$ a: v* f( I& ]- U3 c0 B5 h" v. l! S# S4 o
    ; j4 L5 Y) Q7 r0 c
    factorial(5) = 5 * factorial(4)) A0 M$ Q8 D, N+ i3 y
    factorial(4) = 4 * factorial(3)
    , G. L( o, x5 l, v" Dfactorial(3) = 3 * factorial(2), Y- c  ^; _0 n4 P# O. d
    factorial(2) = 2 * factorial(1)) `0 w. }! X6 S
    factorial(1) = 1 * factorial(0)" ~% M- ]0 v6 S( w& J$ h
    factorial(0) = 1  # Base case% ^3 F6 ~/ U5 t- r$ K

    1 j5 T5 }  n1 x# nThen, the results are combined:
    5 r( p5 ?- f  u; D  E9 x; E, E! I. I3 V0 Y( F8 E

    6 ~, }! K7 c  I$ Wfactorial(1) = 1 * 1 = 1
    2 |  d' [9 Q5 `factorial(2) = 2 * 1 = 20 k* o8 E2 p7 e9 O7 F  t1 r
    factorial(3) = 3 * 2 = 6
    ; x$ Z$ j. p$ t2 Nfactorial(4) = 4 * 6 = 24! b, f8 ^. Y- M! W2 {
    factorial(5) = 5 * 24 = 120
      ?! G/ \" H* G0 u. [. U/ C5 U9 m8 }2 r; }3 B5 g
    Advantages of Recursion. b0 ]7 N. D' I7 e' F
    : L: F, E) i1 Z7 D
        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 U6 p/ z0 O5 x* h& Q; D+ z) v3 @4 k! X
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    5 k0 ~# j3 x. y, ?% E2 M1 ~2 S& p9 G0 l  G5 e: N' c: X* m
    Disadvantages of Recursion
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    1 c% Y1 }9 `( _8 l( l( e0 _    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.
    . J4 ]. `2 i$ q) B
    & S! g# }2 f7 E    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    / a, J3 H0 _; A4 q+ o. c! A2 `2 o$ S7 ?
    When to Use Recursion: J% j( X) }, V, i

    7 c1 t8 @: Y- |5 B0 r9 D# y    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).+ h: W0 h: ?; [; K5 t/ M5 z

    8 ?% M/ ~0 X+ X0 r. w( i    Problems with a clear base case and recursive case.  B/ L' c  t. K# y8 }
    3 S9 f: w2 @% y2 d4 \% e; n0 w
    Example: Fibonacci Sequence
      Q/ G8 u5 n1 e1 H# a* _. M9 H+ Z
    ( A/ X6 [* {$ K/ Q2 IThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:8 v  ^2 x1 [+ G
    0 s6 y- [! v/ u7 K: _  }9 D( u
        Base case: fib(0) = 0, fib(1) = 1) V: s2 m: l# l. E

    ( h. J: @2 L& M: {$ G' C! O* T    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    2 I4 U8 y$ d' G6 m2 ?5 s# d" f) P7 M* h6 C
    python0 X1 ?# \" N: F% B
    8 S; F- y3 y/ \- I1 c
    ) }2 C0 Y( s, \8 W9 f
    def fibonacci(n):
    5 p. H1 z6 s2 \* Y& Z    # Base cases
    4 P  b7 c& l, c    if n == 0:
    ; D0 f8 S- k, k; j        return 0
    ' [6 D: e7 x% C- n    elif n == 1:
    5 M3 S0 M: ]8 }. o/ o+ U        return 1
    # V" M: p+ B' i    # Recursive case
    ) \3 P$ e7 |9 v  n( y2 w    else:
      X) n9 i3 L; `7 a* V1 V/ W4 J# M        return fibonacci(n - 1) + fibonacci(n - 2), h4 d, C* m0 H0 f& c- Q- g
    8 [+ w& `/ G0 _+ U. |
    # Example usage; W  p& N; A# {4 j+ t" Y% u8 Y7 }
    print(fibonacci(6))  # Output: 8
    * B0 \) f  u% v3 r' {& C# x! l! Q5 y4 [
    6 i1 O8 s6 h: U9 D) VTail Recursion
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    5 e) f2 |: Z1 A4 O- o, ]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).
    ' u# B7 v, h% B* J4 |+ R2 P5 e$ e) h: u1 N5 I5 E" p7 R# X3 r
    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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