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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 + d. r* j6 X, p/ D

    * s6 o) Q0 s: C- E. r解释的不错" _' Z5 T9 X6 l; y9 X) Q" }

    ; ]% U7 _" u- @4 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。
    : B& d$ O# T, F' F( O
    $ `5 Q. V; T. f 关键要素/ v6 @( c* l& W2 ^0 T
    1. **基线条件(Base Case)**
    9 H; h" n) L0 h2 a. t/ ?   - 递归终止的条件,防止无限循环9 n3 z: }. I/ D" d5 _2 q* w$ `) V
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1  I: x$ H+ x: x  `

    3 f& p  [& w9 P) X2. **递归条件(Recursive Case)**
    + v, K; k: C5 j   - 将原问题分解为更小的子问题5 D. O6 U2 F* ~& A
       - 例如:n! = n × (n-1)!
    & c3 m. v- j% H7 X9 ~4 J2 I9 _5 D6 B0 B/ b1 g7 p
    经典示例:计算阶乘
    # V& o4 L8 u) q) Opython
      K2 r7 d7 N+ r/ ^) ]def factorial(n):
    ' _  K$ j; G8 M) q  d" l) S" B    if n == 0:        # 基线条件5 M$ G) D& ~* N5 B$ l
            return 11 t" e; B7 g) Y7 a4 [  n( a5 _
        else:             # 递归条件
    " a8 P; S8 u9 a7 V& N. s/ U3 `        return n * factorial(n-1)6 F, i) U9 X* E$ ?
    执行过程(以计算 3! 为例):; q7 s9 n7 Z+ F1 ^, m/ \$ o0 l* S% ^7 Z1 e
    factorial(3)
    % s. ]9 s  ~0 Y6 U3 * factorial(2)
    - p6 T& ]4 X" A3 * (2 * factorial(1))
    % h9 E. ?! Y7 x6 Q3 * (2 * (1 * factorial(0)))
    ) _: K* l' J  o6 Y" O8 n; x3 * (2 * (1 * 1)) = 6
    ; R# S/ z/ @9 `0 U) g6 L) z. \, b4 @+ k4 U5 m7 E/ e
    递归思维要点; C/ C+ r7 T* I6 k& o. W: l
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    5 B( ^* l( i1 A' n3 B8 v2. **栈结构**:每次调用都会创建新的栈帧(内存空间); E2 K5 u0 S2 k- u% Q
    3. **递推过程**:不断向下分解问题(递); \# y# b" o  U
    4. **回溯过程**:组合子问题结果返回(归)* o- _# a8 m- D0 M- X8 {

    ' p. g5 T% |- A  v( } 注意事项2 [8 t4 j: y6 k! t& ]
    必须要有终止条件
    . j$ j# }3 l2 }/ R递归深度过大可能导致栈溢出(Python默认递归深度约1000层): A% P- J9 B3 L' ]! W# z5 H* o3 |
    某些问题用递归更直观(如树遍历),但效率可能不如迭代1 p# y! j/ d, N
    尾递归优化可以提升效率(但Python不支持)
    & f1 V) J( X# t
    ' N( N9 J0 f7 W* q 递归 vs 迭代. Z9 x2 n( U1 Q  ?; E1 r3 y
    |          | 递归                          | 迭代               |4 i* p1 r' v. [) W! c* @
    |----------|-----------------------------|------------------|0 |) r# F- k5 C* d' B* C
    | 实现方式    | 函数自调用                        | 循环结构            |% o2 o# Y8 d- m7 `  I
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    , V3 y: Q/ z- r) u* A+ I| 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         |  W! R5 o" [  X2 o4 H
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ! f7 f' n- H8 k9 Y/ U8 X, \
    % a5 Q6 M  D, J1 e- X7 J 经典递归应用场景" `' I: Q! y( Y$ q
    1. 文件系统遍历(目录树结构)
    / l5 s2 ]. ?( }5 S/ b5 K/ A- c- j2 E2. 快速排序/归并排序算法1 [! w! p. e, `) D3 L1 ?- w% Y5 D  c! w
    3. 汉诺塔问题: Q* o+ O$ R, l# U
    4. 二叉树遍历(前序/中序/后序)
    5 d: F* {& r9 s- A. U3 [3 t5. 生成所有可能的组合(回溯算法)
    * U. c! W1 ?6 a1 z: t% s3 w1 @- G! e2 \7 p
    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,9 u$ b; E& d7 T- h6 t6 N5 B1 w
    我推理机的核心算法应该是二叉树遍历的变种。
    5 g& y+ m7 p( M9 j2 G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:5 y) B1 n, w% R0 N4 I
    Key Idea of Recursion' H, |0 ^! O8 R% L0 U

    % L! Q0 B% u9 p; \5 _' |A recursive function solves a problem by:/ s( _* f( z8 h

    ) z) n$ y, y$ ~9 x& f    Breaking the problem into smaller instances of the same problem.' t6 D5 u7 a+ a/ Q# m& T

    1 j! m( v. _) r: Q' i6 ?    Solving the smallest instance directly (base case).
    ' X2 W( u7 g# c! H
    , H1 v. M* \2 R. V8 ], \2 X0 C    Combining the results of smaller instances to solve the larger problem.
    / w4 u" U7 V3 ~/ d
    / B+ h& {" U% O9 j6 h0 |Components of a Recursive Function
    " ~( }# O# ~8 j" q  I, @5 L- Q
    ' a) C3 [5 _7 `! S$ D    Base Case:
    2 d# u2 g# G  L  O. K# j8 A% I- ~0 D8 ?$ w
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.- `  v8 k3 T/ @7 ]+ g/ Y

    , l2 `3 {" W5 ?  j" _: D, |; a        It acts as the stopping condition to prevent infinite recursion.
    / o/ Q0 b- H' e7 R6 u2 B  f7 z/ \4 }  G
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ [" X- o% E% z, r5 H5 {

    1 t: p  d. W4 w4 T& G1 p    Recursive Case:
    & ]5 A0 `# U6 {# ~2 R, [1 N
    # W) R, C) y& c1 u0 j        This is where the function calls itself with a smaller or simpler version of the problem.0 h+ W/ Z$ u( Z5 y' Y

    9 @# k  ~/ \9 {! J! Y1 I        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ; p* c# o1 B7 r  j* @$ z8 }% i6 i4 Y6 C4 x, I# {/ B
    Example: Factorial Calculation
    * P5 X% `' M: [+ C9 ^' Z7 v, c" u$ q' V" c4 S% Q( O) f% ]
    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:0 ^( h. ^9 C- H# A% {9 |

    " U2 ?; o3 G9 g  F# a1 d, v" }    Base case: 0! = 1
    6 Z' K, g( g9 S. t3 r1 z3 s& R, p; Y: ^5 w" K8 V) S! H
        Recursive case: n! = n * (n-1)!
    3 K& D' I) Q9 E* f, L, Z" @; f: l& Q; ~0 c  L$ L
    Here’s how it looks in code (Python):: n* @% s  ^  V! V
    python5 i/ V, R% y1 Z

    5 g3 T3 ?4 C4 L$ V& F4 I8 U/ [8 |( c& P
    def factorial(n):5 B  Y9 }* b, p! P; t
        # Base case
      p4 n3 o) r. Q7 |    if n == 0:. G; v; \, r* ?. X2 `% Y2 L
            return 1% `8 h* F  j6 g  x+ z
        # Recursive case+ W4 J: i/ c+ M" L7 A$ O5 G8 K3 |
        else:
    6 y1 |' a& s/ w        return n * factorial(n - 1), z2 W2 D* y: h% T; u% H

    7 P) F) {; _: }9 e# R# Example usage+ X( E+ O  [2 S
    print(factorial(5))  # Output: 120
    0 k* E* N" C. t/ I) \# V7 L1 J7 t
    4 a! K& p! V/ y9 I! c3 THow Recursion Works+ e; ^9 X- I" ?; m. U9 ~) b' I# s
    + C( T  q' q# C, @5 O* D* E
        The function keeps calling itself with smaller inputs until it reaches the base case.
    & Z# m8 m; i& x! M4 A
      U: e  j7 i. o3 V1 {    Once the base case is reached, the function starts returning values back up the call stack., Y0 ]% y- a' b0 `$ H+ J4 R

    ) ]  P) g$ l9 u) n6 H9 [  W  t  }    These returned values are combined to produce the final result.) U1 t7 D& H; m% p
    . u0 K8 o: K! Z  j5 n1 i% W  K
    For factorial(5):5 s6 |5 U% G; K/ W4 O, \) y

    ) A. O& V4 s" _& m0 P2 ~! a. N9 M$ V0 y1 U
    factorial(5) = 5 * factorial(4)3 K- t+ B( D' g, g
    factorial(4) = 4 * factorial(3)
    : N/ u" }: D' D+ tfactorial(3) = 3 * factorial(2)+ V0 C$ `$ D$ e1 m
    factorial(2) = 2 * factorial(1)
    " k5 f2 k$ E. ffactorial(1) = 1 * factorial(0)
    ; L) S' u7 v) `( u( sfactorial(0) = 1  # Base case6 C0 U' v7 E4 K* k7 W! E; E( H
    $ e& _( q$ o7 n. b6 ?& e" H' u, Z
    Then, the results are combined:7 q5 H5 k7 x! D3 z$ Q

    ; A9 _" I1 k9 Q0 G' D& a: f8 ~
    5 v5 O6 u0 I0 c/ T, `8 Ufactorial(1) = 1 * 1 = 1
    7 C+ w7 S5 o& X( ufactorial(2) = 2 * 1 = 2
    ( o5 b5 _; {: sfactorial(3) = 3 * 2 = 6. x5 K! V! D1 D5 i
    factorial(4) = 4 * 6 = 24# x" g- p6 e6 h0 O* J. {) @5 \$ }5 \
    factorial(5) = 5 * 24 = 1209 ]: J# J  y. [; G+ ~/ u

    8 \, z1 E" l% o) ]2 i. h; O0 TAdvantages of Recursion7 B4 q0 g, }2 o& V( e1 J  ?

    6 P. f/ [2 O; b- [2 B6 l: G    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)./ P' \0 T1 h3 D( \7 u! [
    % E/ A+ Y3 i8 t% C4 a; `" ^
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    , u& m, u4 l  l, q, B! m- ]: D& Q  H) C% ]  b) n# C; \
    Disadvantages of Recursion
    $ [( {& {9 W  m  D1 F
    7 f, L0 G: i7 C    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.
    * o) t& B% v7 w# R8 b
    ' e0 [8 S# N# r# ^1 p. X    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    + W0 @+ i- ?4 h
    & v) K7 Q4 m/ l2 WWhen to Use Recursion
    - M4 F8 L; q/ L! y3 |2 }8 [- F) z* v/ {" \) S
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)./ A  I8 k4 d1 J0 G$ _

    ! D9 H$ ]- {! A# y% m# L    Problems with a clear base case and recursive case.. s1 P7 P3 m' p: k

    7 Q, f( U  K1 P6 tExample: Fibonacci Sequence4 x( [( ~8 W3 H' k& R& O, ~

    # M+ J; C6 t8 XThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 f# T" b9 x8 Q7 d$ K1 p: ?/ v' G

    % D' ?7 Q! J. {    Base case: fib(0) = 0, fib(1) = 1
    , E+ a4 l2 m8 `
    , n! n) s, j& W5 g8 q' H4 u$ P    Recursive case: fib(n) = fib(n-1) + fib(n-2); ~2 b8 Y6 E3 v. @4 G/ _( ~4 H

    & q, V  V: X8 f5 l7 H$ Upython+ k* y& A% k3 P0 A5 g: T# T
    ' t" e4 J2 a* B1 t: z

    0 Z7 w* D: P4 ddef fibonacci(n):
    / r* E9 X+ T3 c8 t1 G+ K    # Base cases8 h: P" `% t6 q" R, f( y* ?% L' j
        if n == 0:4 C/ s" V7 g# Q% x
            return 04 S' d7 I7 L5 Z& t" B
        elif n == 1:! L6 Q6 [. R+ B9 Z
            return 1
    ' w, i! I5 d3 F' T    # Recursive case
    : C8 G  }. d3 @  P5 G* I6 O/ Y% \0 K3 t    else:
    % W# O5 L: z) P- _        return fibonacci(n - 1) + fibonacci(n - 2)0 [# r' S4 y0 G+ l6 A* ?
    . ]7 `6 Z; \8 @
    # Example usage
    7 D1 Z( b! J8 C1 V3 x  M* wprint(fibonacci(6))  # Output: 89 B/ W, U7 k+ E' v) I" Y

    8 c' [7 C$ z4 J6 i+ }Tail Recursion% X4 i5 u* y+ N7 x4 X: M8 K/ L
    ! w( }" j! j, m5 L
    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).6 e3 E. ^! Z  W4 S
    " z- u6 A. D. I2 F
    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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