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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
    4 \, J. i3 v0 ^" s2 k3 W6 o  I2 t) Z9 S* K" \2 L
    解释的不错
    8 E+ E  ~* ~/ K
    9 o% u, ]8 ~$ t' X9 s& t% F0 U; R2 W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。$ S5 h$ w  ^. t
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    关键要素; t8 @; s8 ^5 w8 C
    1. **基线条件(Base Case)**
    1 g3 O5 m6 d/ a9 I8 N/ s   - 递归终止的条件,防止无限循环9 O  X; B! B" i
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 17 M3 H) R- S, j+ i
    8 X) n; F1 ]: Q( \5 G
    2. **递归条件(Recursive Case)**
    3 [8 M' A7 X9 E   - 将原问题分解为更小的子问题( C) J9 I+ ~( ?, C8 {2 W3 a
       - 例如:n! = n × (n-1)!$ W- R  Z) [8 V4 q6 v9 _. j5 B0 k
    % U# r2 `& B/ R8 _  n
    经典示例:计算阶乘
    1 G9 W% L% g- W* p1 o3 h3 }python
    9 n* C1 ?3 t" I  _def factorial(n):
    0 N3 n. P6 K/ m    if n == 0:        # 基线条件
    ) m7 g# |* ^# }$ `2 I        return 1
    1 G; P4 a7 P" |4 a7 k$ y3 e    else:             # 递归条件
    6 F2 j1 x" L* {( m$ q1 k  Y        return n * factorial(n-1)1 N" R1 q$ @- A1 n; ^/ C
    执行过程(以计算 3! 为例):
    * u( O# M, H3 sfactorial(3)& K. p0 k2 `4 D
    3 * factorial(2)
    ' [5 p% w! o. s2 R( D* b3 * (2 * factorial(1))- l9 Q4 i: R5 _
    3 * (2 * (1 * factorial(0)))
    2 _" D8 T& x; N% e* l3 * (2 * (1 * 1)) = 6
    - B4 n# x6 w2 }, t
    8 o+ B8 W! Z' x# |" Y$ ~5 v 递归思维要点# M+ H- B! e0 D. v/ ^( X4 b
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
      L8 [: j4 d" C' ~5 c2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    % T1 F+ c/ F  Q- V3. **递推过程**:不断向下分解问题(递)
    / D" v8 g& `4 Y) U4. **回溯过程**:组合子问题结果返回(归)
    - O7 d, I' K/ `; `% |8 t  \4 K7 m% |1 d8 w2 r: b6 f( ~
    注意事项
    ( z7 h: }* I$ O7 o3 A4 v* r必须要有终止条件7 g: z# u% z! \5 E- l: \! g
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    $ u+ |7 T( h* Y% Q1 N某些问题用递归更直观(如树遍历),但效率可能不如迭代
    # j1 p1 y( p. a7 w1 N* B! T  N尾递归优化可以提升效率(但Python不支持)
    : K% B+ k, I  b4 `6 r: [+ f6 @. M: Q
    ' s$ z) |$ B  Q! |7 s" c 递归 vs 迭代
    4 B9 R, [" e* _# [! \|          | 递归                          | 迭代               |
    + r, y5 s7 g5 y: G7 [$ V7 j2 {|----------|-----------------------------|------------------|
    . N% }; K+ |+ c4 f, P4 z' t5 n| 实现方式    | 函数自调用                        | 循环结构            |
    2 F5 x5 r( ]' C' C! q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |: c( I; g) ~* ^) a
    | 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         |
    - k9 |/ t' T% _# w% {| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    % s0 `& T1 a+ O4 Z: U) k, m" U, C& ^! w* A* b% R
    经典递归应用场景
    ! [8 F& Y, x$ s6 y' m1. 文件系统遍历(目录树结构)4 Y2 Q9 C! t/ i' ^
    2. 快速排序/归并排序算法+ M5 I4 B5 m/ M. V+ M; u1 w
    3. 汉诺塔问题8 T% G9 C- d+ n9 W' \- K0 r
    4. 二叉树遍历(前序/中序/后序)
    6 ~9 v% H; }- O. f( H9 \2 z$ I% ^5. 生成所有可能的组合(回溯算法)7 Z2 {' i* j0 d
    2 f9 A. M$ @1 z! Q  r+ d
    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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    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,; j1 W/ I3 O4 B
    我推理机的核心算法应该是二叉树遍历的变种。
    6 u3 v# Q+ {8 ^# G9 W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:
    + A/ V3 [8 i& M4 y8 qKey Idea of Recursion4 m, J. A. U" Z5 }: h4 g& I
    - r8 C) L; E3 K( @6 A5 n' t
    A recursive function solves a problem by:7 r0 ^0 y" A- C
    6 F% i+ W7 w4 ^5 d8 j7 j  A
        Breaking the problem into smaller instances of the same problem.3 h( b' _6 w9 k: x6 l

    5 f5 E' F. Z9 `; r" M+ l    Solving the smallest instance directly (base case).
    ' t& }8 ]! s  F) H$ V5 R) |2 O! D& G
    9 e( _" u0 S+ C5 G' O    Combining the results of smaller instances to solve the larger problem.
    ! a3 I+ F& G+ ^0 U4 E' g4 |0 K' Z3 x4 x& O. h& B
    Components of a Recursive Function
    1 d4 J6 d. m. ^  Y
    5 }( y/ o- b% J1 Z0 u    Base Case:
    / ^- F9 r" Z) x) \, X1 h6 @$ x/ f, c2 S5 ], j0 O6 Q
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    % }3 a1 S" G; K3 }) z  H" M! t
    7 \- G2 \$ Z7 W' Q$ {8 Y7 U8 C        It acts as the stopping condition to prevent infinite recursion.
    0 P7 m" x; J+ j" C9 e
    " l7 G0 W1 f. M2 c        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ! ?6 [1 u/ Z% r( l
    6 |7 ~! W! _/ r, T    Recursive Case:
    % O+ X7 J! ^4 m6 q/ i) f# ?% i& H2 f' N- m  f
            This is where the function calls itself with a smaller or simpler version of the problem.
    # E9 ^* C/ j# n" ~% |3 I  r9 O: J2 m! u, \& J! K) d2 j5 z$ s: q
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    + {3 o9 f% d: K8 O. _
    1 p' m* Z  \3 lExample: Factorial Calculation
    - A  f; x6 y; h  ^2 e2 E# V0 `  [
    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:& I* c- x" _/ y# a+ L0 q8 {4 |

    ' U- i6 X9 q& {* y# w5 {- y    Base case: 0! = 1, {& c+ f. e) w6 I* S. S

    + v9 Y! a, P5 w    Recursive case: n! = n * (n-1)!
    0 z0 i: c8 ?, l# z8 w
    9 o* J7 S% P( m2 tHere’s how it looks in code (Python):
    : E" A7 [3 E6 Epython  G% X- v3 M. Q/ Y' @
    2 W- P, O+ _: W: a0 D8 A5 x3 n3 E

    5 G0 K+ Q2 q- t3 k0 q! t, g/ jdef factorial(n):
    - m2 y& ]7 t0 e) i: A    # Base case, l5 n2 h# Y. w% v% Y3 ]/ K
        if n == 0:* N7 K, V5 ~7 u/ K1 Q; ?' e5 h
            return 1& V! N0 C7 J8 T. ?
        # Recursive case
    $ q8 P: M8 c* e  A$ _  [& U    else:& z( u$ [$ O' y$ d
            return n * factorial(n - 1)! q, z; a' w7 k1 K! t/ J! @
    5 M; G1 m% w) J/ K$ S9 v
    # Example usage
    % Y3 ]* l# }, wprint(factorial(5))  # Output: 1201 z7 X; Q. P* p1 ?

    6 a3 Y( R. j) `1 i0 p2 WHow Recursion Works* T+ x1 P1 ?: a2 p) ~

    6 s+ T9 z, j8 v# A    The function keeps calling itself with smaller inputs until it reaches the base case.
    8 C" L, P! I% j" I7 f
    ! h) Q4 `- j4 a! E8 A4 n7 H    Once the base case is reached, the function starts returning values back up the call stack.8 ]  |0 B, n5 Y. v7 l. v

    7 G! q+ g$ S8 X* q    These returned values are combined to produce the final result.: [+ O; g' r, K' i( l2 U0 f9 i

    ; R) [2 q5 |# I! ^. h% ~For factorial(5):9 X. R6 O8 Q6 R* X$ h+ G
    & g+ @( M3 W$ i, G

    6 T# x+ z, Z5 H* U- Gfactorial(5) = 5 * factorial(4)
    2 g6 k: S0 _3 p9 x( [6 q% l5 U, t; Bfactorial(4) = 4 * factorial(3)
    6 i$ _. ~, q* e7 D# Ffactorial(3) = 3 * factorial(2)  a; ~$ v! b* w' ~. @1 y8 [
    factorial(2) = 2 * factorial(1)
    ) q6 w6 d7 f( r- ~( Z* P, J$ Ufactorial(1) = 1 * factorial(0)
    4 J' k  I( s4 S7 d7 _factorial(0) = 1  # Base case
    . V+ p6 E$ J% Y6 W1 s: o# K: I1 A! ?3 }8 S+ c% a5 a
    Then, the results are combined:
    ' ~8 p6 N$ I. _1 ]4 q
    $ Q/ r' w$ j' l) f: ]9 h
    0 @. x4 |8 D5 s  b8 A) ~9 dfactorial(1) = 1 * 1 = 1) d* S9 Y: w' a7 A+ a- V5 [- |
    factorial(2) = 2 * 1 = 2! {: q9 f! H+ }/ _# k
    factorial(3) = 3 * 2 = 6
    & m4 |4 U1 L$ o: s5 `" t8 }factorial(4) = 4 * 6 = 24# g3 E0 V" A/ y( s2 L
    factorial(5) = 5 * 24 = 120; l- ]+ z$ K" g: b' G. _# t4 o9 }

    0 [5 t, C' l7 ?5 z3 XAdvantages of Recursion
    , ]- S7 G' o& b% s; Q
    : P. y  ^' n* B/ m    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).
    3 b, Y3 N% ]5 z8 ~7 K* R! I& s
    * S1 k9 I" u* X3 F    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    , ?- A" l2 r2 s! U0 F8 Z' u; Y$ v; @  d& A
    Disadvantages of Recursion" Y: S0 D2 F0 O0 @! n. r

      Z% W8 E6 \; w' `4 D    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.2 ?; J, s; v* c$ @1 x9 [) y
    * ?, j  w- L& X/ u, n# W" Z1 `
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 u  N$ P, f$ f6 _( L

    3 m( _" u" ?" q& s4 NWhen to Use Recursion2 k7 g7 p; h4 R/ G7 {" x
    + z6 \% n. e! M) g7 n/ o0 Y) X
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ; p' z( |) Y9 i4 a) w% H6 n) L, |2 a! D* Y3 T9 q9 Z1 V
        Problems with a clear base case and recursive case.7 j- K5 ]9 Y6 u" i
    % X+ k7 c% s9 J- j  Z
    Example: Fibonacci Sequence( _8 P& r% v9 b4 q8 N* o. I
    % U( _# f. d8 v; V7 Q( B
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ! o4 j9 ]" F9 ~; W5 s: }  T1 L. O% X6 e6 ]5 z) Y! m" A
        Base case: fib(0) = 0, fib(1) = 12 \: H' d% \  [/ f! ]. u- A
    6 l9 _* [4 K7 M9 S6 ~/ ?
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    " s! }0 G. k" J7 q$ x6 I0 A  e
    3 i0 l3 [! k8 U9 b* X1 s# tpython; s  [0 a0 j4 Q( i

    ; ?3 L! k8 m' _, z4 z7 f, K  p$ O, p
    def fibonacci(n):
    4 n' J+ W; ?+ Y    # Base cases9 j( Q+ ]% K8 E% }
        if n == 0:
    - n6 i8 e" e1 P6 [$ Z        return 00 X  B1 q0 O3 F2 A
        elif n == 1:& l+ L, q: r/ v1 Q. V0 I
            return 1
    * e% X8 B$ o  I. }  z    # Recursive case0 v, k# X/ E& o' t4 J5 f
        else:
      m& d- ?9 y* F# j        return fibonacci(n - 1) + fibonacci(n - 2)
    7 i5 Z: r5 R6 S
    0 Q# Z1 x  t7 K% T' y+ M; K, K# Example usage/ H! C6 Z$ h: W8 c
    print(fibonacci(6))  # Output: 8/ U) j0 B* G& v$ h$ \

      D9 T6 p  c/ S, L0 qTail Recursion
    7 N9 ]7 J% N5 c  n4 h* F
    + d* G( m0 B8 s0 L# @- R: WTail 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).+ n" n6 x) r: ^) y

      v2 p9 O8 i& m; c" IIn 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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