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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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    2025-9-8 05:08
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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 2 A% B: E8 p2 N; ^
    # D  }# K5 s( i
    解释的不错' h5 B/ n% \& J

    8 X* D5 w1 D4 ]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。# N; O% M0 d1 \0 ^2 x, B$ n6 h

    / r# @4 `/ [$ M 关键要素( q# {5 ]3 D8 S5 l
    1. **基线条件(Base Case)**7 G9 i1 V, Y* y( {, d
       - 递归终止的条件,防止无限循环
    # x" \! y, E7 P5 \) L6 Q   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1* l7 r0 V7 ]) Y8 E( B" B
    , l5 t5 p3 K' R. M; M; _+ J
    2. **递归条件(Recursive Case)**8 G$ C. l6 m$ |' R2 X1 @
       - 将原问题分解为更小的子问题
    : n" Q) D$ S4 C7 D   - 例如:n! = n × (n-1)!
    $ ?' o# A$ F0 Z1 E. W+ L1 h9 ~. ~2 p
    6 p) [& ?" d# m$ |. `: u 经典示例:计算阶乘
    / T( k+ N7 r/ U' N* R  y6 Zpython
    ' L  m6 {! @+ _' z8 G! tdef factorial(n):6 I/ {" a  r. X! O# P% h! m) [
        if n == 0:        # 基线条件
    4 `0 \) L  E! M        return 1" W" L+ c5 L2 b% }- c) A
        else:             # 递归条件
    . S& k  Z9 r* d        return n * factorial(n-1). z* F- e8 V/ p
    执行过程(以计算 3! 为例):4 G" N8 S6 Q+ N1 f5 T0 h  b& K/ o
    factorial(3)/ f* s# o- ^* }3 I! e( w
    3 * factorial(2)
    # @5 G1 x  Q# v2 N* o7 @4 v: l3 * (2 * factorial(1))+ S) O2 O/ e: P
    3 * (2 * (1 * factorial(0)))
    ' G1 E( V2 }- Y2 \5 d6 Q1 e3 * (2 * (1 * 1)) = 6
    # y8 Y* Q* o' a* Y: l* g6 T1 w4 \# Y% \/ p! J2 J3 p5 q
    递归思维要点) c2 }& g; Z& g% l4 D& |) F1 z
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    # ]6 L: r- `) ?" h4 v2. **栈结构**:每次调用都会创建新的栈帧(内存空间)9 R! s1 M  T7 w4 |# O" D' K
    3. **递推过程**:不断向下分解问题(递)5 @% i' c3 @  u- @  j
    4. **回溯过程**:组合子问题结果返回(归): K5 G" i  V' r. A

    4 H/ ^6 U9 ~" I* c0 H" o4 ^7 W 注意事项! a/ h' L+ _) j# J
    必须要有终止条件+ N1 l, _9 b3 t6 i0 S4 f
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层); C& l3 ^2 _; z5 F  b4 _9 c
    某些问题用递归更直观(如树遍历),但效率可能不如迭代1 ~( B9 C7 }; G/ j6 f
    尾递归优化可以提升效率(但Python不支持)( u6 P1 |4 g7 t; X6 _3 }0 O, e
    0 K. h$ P7 A  ^0 [
    递归 vs 迭代
    * H  G  b& V, W' x( _9 M|          | 递归                          | 迭代               |5 K8 @& H1 g" ]" V* ]0 k4 ]9 j% A
    |----------|-----------------------------|------------------|
    9 X; y$ x$ l4 q: `| 实现方式    | 函数自调用                        | 循环结构            |
    ( I; S6 L. b( _* v9 q# ~  H$ e| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |& x8 L/ ~' {; v4 J
    | 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         |
    3 e, Z: K4 n, |8 q9 W! t| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |( x5 r( W/ s0 s% i' t

    ) E+ K* [1 o( m 经典递归应用场景  I6 J" Q+ g. ^1 r3 B6 \6 ]5 g
    1. 文件系统遍历(目录树结构)8 ~5 i$ X/ c$ ]" I9 I
    2. 快速排序/归并排序算法7 B+ g% Z0 N. \8 q' g/ S' D- S
    3. 汉诺塔问题: m1 |+ C+ U& E
    4. 二叉树遍历(前序/中序/后序)
    7 E& Q! Q4 s5 A5. 生成所有可能的组合(回溯算法)
    + w& [: t+ w6 a# \9 r4 }4 f! R! n. q. f% n$ I" V, g3 f
    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,
    2 y, U: T! s, m我推理机的核心算法应该是二叉树遍历的变种。/ i( f9 f! U0 \; y0 \# 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:8 F) B: _6 v6 J* K# J# T- b
    Key Idea of Recursion8 `# w- _3 [9 C9 m
    - m* g  e3 t3 K, W6 ?( ^/ ^
    A recursive function solves a problem by:5 f+ o1 s4 m0 Z& o! M0 {
    - L) _7 i  g# k) D3 |
        Breaking the problem into smaller instances of the same problem.* @; @2 \- D4 g5 F- v% }

    0 g$ r# o0 x2 W    Solving the smallest instance directly (base case).0 P4 @3 c+ c. Q1 @" r7 e$ C

    & f  ?$ M2 U# o    Combining the results of smaller instances to solve the larger problem.1 Z0 w) \, f) I. }

    1 I+ z! j; P. R& x' ]8 h% l' r3 dComponents of a Recursive Function
    % _! |, P. F: g3 e" q/ E2 H) w  M- J
    % t! e3 H/ K2 ]. g5 E4 M2 a    Base Case:
    ' p, b# y7 ?) l) [* W& j3 w4 l) x) q2 ^
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion./ ]) c4 Z& m6 r" i2 B( Z0 `/ j
    ( T( ?% ~1 i$ ~7 y
            It acts as the stopping condition to prevent infinite recursion.
    0 _9 f. c" `) n3 q5 \
    6 @4 j( w9 l5 x        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.9 a/ w9 }8 G6 s* n0 p( U6 H/ f

    " ?8 I5 v0 \, x4 O; N7 M    Recursive Case:- Y9 c" v( I1 V+ e

    0 J5 o" D+ P6 q( {0 R        This is where the function calls itself with a smaller or simpler version of the problem.
    / T% ^% `0 x& j$ R& e# @7 ]2 b8 ~' e7 S4 Q3 f: Q2 ?+ c* r7 H7 h( v
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ; ?) R2 W6 \/ x0 I% v1 P$ l+ M. I% R( E: ?! I5 r5 o% v7 l; u
    Example: Factorial Calculation* m# H8 p0 t$ S7 O, |' n: Y) Y7 f

    " Q! C1 b' y( p  |% 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:: {1 p3 E' ?; D- K
    ( S4 D+ M/ S( p* z* }1 `' }. w
        Base case: 0! = 1* n4 ?* \2 Z! Z' F- w4 Y

    ; M* q7 Y# z( y" R    Recursive case: n! = n * (n-1)!7 ^& m6 N8 R, J. B, g0 I1 C) G
      o7 ~& S8 n) E7 ]$ c
    Here’s how it looks in code (Python):( \: l' {" b- B3 B' D  b4 T( N$ c
    python
    3 I. V4 U( u' G7 U9 j. R" ]& o
    # D+ j5 X+ S* ]7 {; N) C, x9 a' L, v3 A5 r6 O- k" s
    def factorial(n):4 s( f. z% p4 D& H9 e) u- c: T# C
        # Base case
    - y9 `% P- Y1 P  s6 f( C    if n == 0:0 U: K! E8 `. l! N' |2 k
            return 1
    2 T# z5 I1 ^) G! _2 n% x    # Recursive case
    8 F, k' q* R* H: G' `    else:
    , @; ?7 A# Y1 ~        return n * factorial(n - 1)9 q* B/ f8 Z3 ^) }  e. I: }. B$ A

    ; T" m4 V( x( l3 J  u! S% a# Example usage
    % O6 w7 L8 I+ P- cprint(factorial(5))  # Output: 120
    , o, q1 p7 k9 o( P1 l
    ( o. M, w2 O% R1 ~- K. ^- qHow Recursion Works  W- `; E- F" m3 g/ I  k4 N

    7 f6 @" T' k0 Q) Y8 }' n    The function keeps calling itself with smaller inputs until it reaches the base case.1 D9 {% r) W; ]+ y% E

    # I8 z5 ~7 R: ?) w8 W6 R' Y    Once the base case is reached, the function starts returning values back up the call stack.
    ) l' [3 \, a1 w- f) w( x
    2 q% @3 ^" r2 W. K7 H    These returned values are combined to produce the final result.
    . F& O+ b% `& M: u" q$ S' {& T2 @
    0 K0 T; S3 G( R7 @0 A9 KFor factorial(5):
    * [+ ^% [! V5 }% _
    ' l2 _6 R% r2 x5 e( r
    5 l. n1 e" L  Z; ^factorial(5) = 5 * factorial(4)6 W3 m5 l' f! }2 F8 G0 B
    factorial(4) = 4 * factorial(3); b/ M9 S# |: z' C) E/ v8 M
    factorial(3) = 3 * factorial(2)3 }2 i2 V' M# \4 ~2 A
    factorial(2) = 2 * factorial(1)
    ' d8 r  X# r0 b) B9 w. Q. Sfactorial(1) = 1 * factorial(0)0 e9 l6 d% Z% I% A3 R
    factorial(0) = 1  # Base case
    9 G9 w2 W2 i9 @8 I' F$ h
    # n$ G8 j" O* ]  m( fThen, the results are combined:
    8 q, _7 q' i! C% s( T- G5 \+ [5 v0 |6 r: Z6 ~/ s' E

    1 v2 e6 p- J; a7 {) E+ W) i$ Z0 _factorial(1) = 1 * 1 = 1
    . n) ~' m3 K' j8 a: d  Xfactorial(2) = 2 * 1 = 2. T* v" K) Y6 i3 X1 u0 p: s
    factorial(3) = 3 * 2 = 6
    : P* C! ?. F' H, Z/ }3 L0 ^factorial(4) = 4 * 6 = 24
    " Y  x4 U) d5 O( d# l5 `8 P; j# Efactorial(5) = 5 * 24 = 1200 J6 ]/ R- h/ @4 ^) H$ n! J
    2 J/ W  j* `" y" X
    Advantages of Recursion1 ^9 V" y. \) A7 {1 }" _4 D
    6 z3 D# S, c( o5 u
        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).
    # ~. C; q7 n* ~' O3 m7 U9 M1 r6 o4 L. @/ K+ Y5 g
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
      y, A8 {3 W7 C' H& f) G9 g1 }5 g+ U5 L. J! ^2 D1 c3 P; O( t
    Disadvantages of Recursion8 V9 X0 b. j/ R* E2 M

      A8 N/ `' F0 O# p7 Z    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.$ V! ?( ]6 h* g1 g7 J
    + w6 k: b/ f4 C0 G$ m- {
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# R) K9 m3 R+ q) v# X# D

    4 y! i; j% U) N( `% h$ gWhen to Use Recursion# N# O9 |* l$ S: T1 p0 x% ?

    7 q  N* m4 j  h0 F6 w' p    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ' W' f& Q: m- V4 O" A& P" R+ _0 z) w2 U* W2 T8 w
        Problems with a clear base case and recursive case.) g3 y* c( \/ o7 e) a0 Y# @
    ) R% a7 W. x$ ]9 G2 i
    Example: Fibonacci Sequence
    4 u3 \7 y" d6 Y! h; l: L8 R
    , y9 }* [7 J2 K$ IThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:" D' [$ g5 j! |  F

    4 \, P3 y+ e' ~- [4 q/ p, K6 X6 g    Base case: fib(0) = 0, fib(1) = 1
    7 }& D7 n% t: a4 W5 W
    ! M7 e" ^) _: h! V3 ]& i    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ( v  G( d2 w7 n5 G2 b- p
    7 L: L$ L2 l! ]python
    6 c8 f% B$ o0 q$ N1 L$ _7 x6 y' ^' v7 F
    , z! k" \/ J) |  s) i8 A
    def fibonacci(n):% Q# b3 c2 O( x$ @& @
        # Base cases
    1 |& f: n' @2 @: w3 Y/ [/ |    if n == 0:
    ; Z3 v5 u- y' X! t1 u        return 0
      {* [3 A* X& T5 Z0 t  M    elif n == 1:
    $ P/ T( p4 q1 |/ [: `; b' e        return 16 d9 h8 C0 X/ A# t% ~- P2 b4 b
        # Recursive case
    4 `5 w% c1 p' u, d4 ?2 n    else:5 C, l; }* _9 N0 k' \
            return fibonacci(n - 1) + fibonacci(n - 2)
    + N8 m( ]2 G1 y* q
    9 W+ w# v! {) p$ k7 s  R7 W$ G# Example usage/ S. Y& e$ D! u8 W
    print(fibonacci(6))  # Output: 8! a2 i" J) R* J' `: a
    1 T! ?( c" ]9 r' q
    Tail Recursion, I% O( N2 k2 s
    1 b0 F0 |* n0 T8 [* x# Y
    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).
    : H: ]2 J: o, d" M# r. x- T! T6 n9 I( I5 A+ ]
    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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