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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 0 P+ w9 x* i4 }. U

    4 E) L* k6 R1 u. _0 d- H$ x解释的不错4 v$ B* {; h9 n& {/ `" x
    / H. Q& F- m3 h+ C: }+ P, p+ l7 y8 s4 e
    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。) p: x# l6 ?3 {% o  [
    $ j; }8 Y! ~. e
    关键要素
    1 `8 i* `) N7 A! r( N- H1. **基线条件(Base Case)**
    ( C& j) J* h6 g# c& O   - 递归终止的条件,防止无限循环
    & k2 N8 C$ d1 t& o( W- K7 m+ O   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    6 A/ k* A4 L2 A$ g
    * P! f$ n& v  E" T2. **递归条件(Recursive Case)**
    & q) m% H, ?+ g, E. K6 i, p$ p; T   - 将原问题分解为更小的子问题
    8 z6 o/ A/ J8 a/ c( g   - 例如:n! = n × (n-1)!' a+ ]5 w1 N- \' q; Z& ]

      L- {+ x* L0 y, j 经典示例:计算阶乘- i# D8 {6 ^* r" w5 ^. o
    python
    + |4 }: l9 ~; ddef factorial(n):
    . m- i* D( Q0 M( W% S- m    if n == 0:        # 基线条件
    # K# |. ?4 A; B8 o# H6 s' J9 a        return 1* O; _/ _/ _% w8 N8 H# U0 e
        else:             # 递归条件* }  d. I' J2 U
            return n * factorial(n-1): A8 a0 P& Z8 A
    执行过程(以计算 3! 为例):4 T1 x3 p% Y$ s- m. {% i
    factorial(3)" B8 r6 H# N4 y8 F6 K# l
    3 * factorial(2)
    + X1 R6 j0 n+ ?4 M3 * (2 * factorial(1))
    . C* L& V1 r7 M2 J7 ?& v3 * (2 * (1 * factorial(0)))
    ) H2 Z$ U0 C, e7 g/ ~3 * (2 * (1 * 1)) = 6, @' r( |9 d0 s0 D7 ^

    6 Z$ R* K3 z  l! U 递归思维要点
    . k2 t* Y" A; [% D4 L1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % \6 Z7 W. K7 g% Q# @: }6 ^; h2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    5 D: [5 G$ f% ?! f, u' e5 U$ Q3. **递推过程**:不断向下分解问题(递)
    5 t8 F- w4 {! B& g) b" @9 J* }4. **回溯过程**:组合子问题结果返回(归)
    ; H5 E/ C& S$ v& D& O% e
    ' _( |( W( z* B4 P6 \ 注意事项
    5 e$ U: ?# `& M5 h5 ]& `# k1 T必须要有终止条件
    ; [* A8 V% O' u3 ]6 B9 p3 F: F递归深度过大可能导致栈溢出(Python默认递归深度约1000层)- G8 y- v! Y- r3 |8 m
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    8 O: x& a5 g2 v8 l) U8 y" N! @尾递归优化可以提升效率(但Python不支持)
    6 m3 z6 |5 ]& C7 s: J, P" c# A) F5 f0 `
    递归 vs 迭代1 d! C! |- n2 j1 E# L6 B/ K  s
    |          | 递归                          | 迭代               |
    5 a6 T: C1 ?. n% b$ T|----------|-----------------------------|------------------|8 ?% o5 U1 z' G, n# x. W
    | 实现方式    | 函数自调用                        | 循环结构            |$ E- D: f4 T2 t2 a' ?
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    4 O1 G1 A# J- o6 s1 k& I+ h| 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         |) q, E0 y* J& b, |
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ; d+ j. x2 Z6 Y* z& j5 w. }4 `- T8 N' U8 k$ q5 j9 n
    经典递归应用场景- W' T3 G$ R& J& K6 D+ W. F
    1. 文件系统遍历(目录树结构)6 U0 x3 `* r; N8 D" Z
    2. 快速排序/归并排序算法" Q, H4 a9 m) ]- j, D1 C
    3. 汉诺塔问题
    ) m4 [7 k/ x$ ]( G- i; C! V0 i4. 二叉树遍历(前序/中序/后序)
    8 H+ j& [$ f1 ?7 e2 x- G% P5. 生成所有可能的组合(回溯算法)
    & m9 L* O  x+ E' {: a$ P
      X2 }7 |1 b, Y6 S  @1 A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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    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,+ p3 o- F. Q4 y, L  y
    我推理机的核心算法应该是二叉树遍历的变种。  Q0 F( Z. ~0 l7 m
    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:
    : U4 q0 s" a# _& d3 b; B; \+ UKey Idea of Recursion
    2 L- n5 w7 g) J- L" E; O
    2 n1 }1 |0 L2 q2 N7 p1 T6 QA recursive function solves a problem by:* l. P+ L# M7 Y0 j. v8 W
    * F/ q" E7 e. J5 p' }" u; x
        Breaking the problem into smaller instances of the same problem.
      y+ }8 Q% I2 p- y$ M6 x" }' Z% U0 w) B% |* ^: Y0 x5 Q
        Solving the smallest instance directly (base case).
    ! A, ~! m3 v! j
    / n+ Q2 `- ~0 ]6 j" W5 d& L- }& M3 l    Combining the results of smaller instances to solve the larger problem.% \0 {% R6 [- L3 {' D2 B
    8 n6 [- I# G7 h
    Components of a Recursive Function
    # H# s/ G( B* c; H3 D. L! }: ^
    , d5 Y: l/ q  c- L7 F    Base Case:
    $ w9 j& @) D% F# I# }% J2 u1 `  U, v  z& }; \
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.3 O8 F5 o+ t1 U& p( W+ O

    7 O5 `- E1 F* U, `        It acts as the stopping condition to prevent infinite recursion.
    , Y# k% f! c: c! i# N+ s9 m% ]2 v7 h& `: h/ b0 S
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.: U9 s/ u  V3 U) I) }1 F
      K5 u/ q+ `  T1 b
        Recursive Case:& a8 E( [* G( A5 K$ |% i3 A/ w0 {; H
    & G3 E. B8 S! {
            This is where the function calls itself with a smaller or simpler version of the problem.8 c1 J. b0 S$ I7 O, S$ C

    9 O4 J) d2 B9 i! z        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    7 }4 x% p. E/ v! n. ~6 k
    7 @: P0 ]0 [& h/ Z' T; q7 Y1 r; g% |# nExample: Factorial Calculation. s+ N3 G; U3 s2 }/ O

    9 Z; Q$ `+ ^2 _# a4 X& tThe 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:
    9 H9 C% a+ U2 h- {' J
    3 j0 P0 `3 N$ y  }) z) ]4 ]5 s+ m    Base case: 0! = 16 y2 n8 R; y9 K4 b% h9 K

    6 O6 U# m4 F$ d0 V    Recursive case: n! = n * (n-1)!, P; h9 A3 s4 J6 G% E0 x& M) F, ^

    2 m) \- Y. s0 T: _$ s; VHere’s how it looks in code (Python):- p4 X" P& i" R. i! R% ~" Q" Q
    python8 d2 B7 `' U3 i) @& W" E& ?( S

    , o% k# x& K4 ?" Q6 N) x, F& I! _1 e
    def factorial(n):' D3 x1 h2 e; {" v: B4 c
        # Base case( X7 l% Q; J+ N1 m5 Q0 D# `
        if n == 0:
    " _% N- {4 O5 X' F5 x7 J        return 1
    4 j% ]  ]. x! G! s9 T    # Recursive case
    ' L0 Y7 [( b2 `/ n9 V, F- Z! d5 E    else:
    # J: O, N) S) i1 y$ Q4 G! J# K$ S        return n * factorial(n - 1)
    9 A. x, I4 t0 ^6 {+ D+ ~% H* V5 c& M  T; Q( L0 N
    # Example usage
    , ^! s! \0 A$ G. H" fprint(factorial(5))  # Output: 1209 O% ~+ P1 U1 @) R* ^% }

    ( m1 P" u1 O/ j/ m3 L! v/ Q3 lHow Recursion Works
    & W3 f) j/ H! W! f% ]4 M+ g' N2 ]2 Z6 L" k+ L( T2 }: B
        The function keeps calling itself with smaller inputs until it reaches the base case.4 M4 H0 J0 n( s- K9 k6 X
    # l7 `# @+ O7 e! R
        Once the base case is reached, the function starts returning values back up the call stack.
    2 {0 {# ?  C' r4 I& Y  w) |; o; [8 L' v: f6 z7 Q# e
        These returned values are combined to produce the final result.
    1 L1 f- P7 ?: X5 H, ]3 [) L' H  X, P/ s* R6 v! L% X
    For factorial(5):6 y8 h6 ?0 q0 R+ P, }/ H9 b3 v
    0 X9 F4 ]& i$ D0 L0 R( n& W) Q
    7 x2 v" H% C+ V! A8 s! h9 b; s) C
    factorial(5) = 5 * factorial(4)& ^+ l" |1 o* R4 r& ?
    factorial(4) = 4 * factorial(3)
    2 F( l* ?8 Q2 F& d# N# U& i  q$ Ofactorial(3) = 3 * factorial(2)0 H# u  u3 G2 `. P/ H' d8 f
    factorial(2) = 2 * factorial(1)
    . z9 ^$ F/ I& b" ]factorial(1) = 1 * factorial(0)% G5 L) D3 c' m0 |$ s& C
    factorial(0) = 1  # Base case  ^# x5 t+ M" @: ^) |& b4 c4 a4 J" F
    2 D, J. L+ g2 s% U8 O$ V
    Then, the results are combined:: P1 N$ [( {" n4 b. E2 \  i
    ) G2 g+ W0 p' h. a

    : h" x4 N0 B8 ?9 A( C1 y/ G# s0 gfactorial(1) = 1 * 1 = 1
    & s4 J" O+ _8 l8 c/ b- S& [factorial(2) = 2 * 1 = 2  q3 N9 o, s$ n3 v4 ^  y, f
    factorial(3) = 3 * 2 = 6" {, s1 {, ~( n3 v
    factorial(4) = 4 * 6 = 24( `) i- G! h+ L
    factorial(5) = 5 * 24 = 120
    " G. J, D: P! w9 c0 x& |5 k5 j
    1 e+ x5 y  ]5 wAdvantages of Recursion
    # @! `9 }4 U+ y9 S) k- o& T, ?* N+ G  h, `+ E  N* t
        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).
    ! j) C3 b4 s$ ?& x- S; T/ u& h5 U  V# s0 X$ c
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    + w) V" ~" |0 w9 N- H9 U& n4 {+ Q8 y: c
    Disadvantages of Recursion0 D) e& ^3 P* K4 j2 u

    3 i: w2 _: ^" |8 @6 W: H    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.! A% s5 u! Q9 I
    . v) w0 {6 l) J1 N( j* n$ Z: z
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
      i' P; ?7 M; G
    9 V( F2 ]7 Q9 U: eWhen to Use Recursion
    - D/ J( h, L" y2 s- E' y2 S  @/ i5 N) t0 X2 N2 B0 |, \
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ' L0 C7 {1 G0 M! ]/ y
    ) I- ~0 J' V; D9 @    Problems with a clear base case and recursive case.
    9 {) I  n: p! y( r6 q8 L# V; x1 K4 R
    7 _5 J, |7 q7 k# a/ aExample: Fibonacci Sequence
    " b2 v4 I) K& ?% ^5 k% b! ^+ B
    * S* G( K  f3 g. c/ @2 W6 s( LThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:" {- L/ w: h% m1 q; l. Y8 H4 h9 j- _) h

    / m8 T! a, f5 S2 X; o! u2 W    Base case: fib(0) = 0, fib(1) = 1. D" M- Y' g# X8 @4 c6 |# h

    ; X! r' o; r  a% E% c+ T. T& r    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    0 d, Q# R, S# f4 `( W/ @
    5 n. e" U3 {8 _7 z9 g6 Xpython* O8 t1 ?/ x5 a( z/ ~
    5 r" h- u/ ^; P2 v1 `& `8 {5 k  o$ n

    ' {, F7 D( y' `9 Ldef fibonacci(n):
    5 A9 [% d6 ?# _, A' s    # Base cases& T7 [$ @4 {, A) v: }" \! \+ `
        if n == 0:
    0 {1 |! j* [- X5 F. |% r& j        return 00 y8 q4 X, J* Q, p+ O
        elif n == 1:/ [' |5 H; j' k( l
            return 1( d! L9 O7 k  [5 R1 k: F) V
        # Recursive case
    . Z% O1 |5 A2 |    else:0 M% `; e  Q0 M9 g9 B
            return fibonacci(n - 1) + fibonacci(n - 2)0 S1 ]3 D- _# W" P! h- d

    2 z( c( l7 x' x) ^7 ]  F# Example usage
    7 @2 X* Z  I' M7 V1 z7 q% J* Kprint(fibonacci(6))  # Output: 82 b+ _/ V% Y- ?3 r6 p- u
    ) ^2 O5 p" ~; N
    Tail Recursion
    $ K# ~: H, x3 s4 W+ y: k$ F9 m7 a
    # [4 \$ k& [# e. r$ qTail 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).- b6 A  I, i& ~1 }
    " ]! g' t, S4 x" O9 V0 ?6 j
    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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