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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 ; v5 L4 C2 v+ K9 I' F& A

      \5 P$ d7 a0 \1 p解释的不错- F6 c2 o- ]% U2 \& B. [/ t

      }0 t$ R) }: h0 Q0 z# P( Z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。
    4 p% F7 ~6 Y3 h% e' J
    3 h& G0 Z" e9 g 关键要素
    $ J& ~  w2 S9 q5 ?+ i4 d& Q1. **基线条件(Base Case)**; d: V5 Q3 S6 l: j( W! m) p- n
       - 递归终止的条件,防止无限循环/ ?0 t8 H" a7 y9 @+ e
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      ~9 R1 @6 }1 w% d7 {6 v# o7 Y' \& t$ o3 K
    2. **递归条件(Recursive Case)**! f* c/ ^' H  }: c
       - 将原问题分解为更小的子问题1 L6 W' }2 q0 U- ]6 x
       - 例如:n! = n × (n-1)!+ r# V2 h  y! O+ U9 C

    ( f! U% D# l' {9 o5 @' m4 K 经典示例:计算阶乘: t) X4 f" f. I: p7 ]& N  S
    python- |0 {% d% h4 U- L& g5 @4 ]
    def factorial(n):7 h* v4 z6 R+ `
        if n == 0:        # 基线条件/ D$ A) L6 ^7 _; S; b" e) v9 X
            return 14 g7 J' m+ C) V9 p
        else:             # 递归条件9 m$ [  ~* \9 G: b' E
            return n * factorial(n-1)
    . m* [) p! J2 `1 F8 C0 l. z( _执行过程(以计算 3! 为例):, ?  k- _8 ?; r& W! I
    factorial(3)9 O( i, P# k2 x* q
    3 * factorial(2)
    : g: d; l2 }3 E$ f: P+ O3 * (2 * factorial(1))9 ^) H! y+ i5 }8 G3 ?5 X6 ^
    3 * (2 * (1 * factorial(0)))% d4 ^! d% r# C4 Z9 N: R
    3 * (2 * (1 * 1)) = 6
    ) x$ @) H* V3 s, `1 Z6 Q2 ?
    ) u, E% D8 @7 s. |6 n 递归思维要点, [& z8 R) q8 j
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑1 L; c/ m1 Z2 W( J9 h
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    * O0 u' {6 a) L3. **递推过程**:不断向下分解问题(递)
    # @7 X: L3 H* N  \5 ?+ T8 [4. **回溯过程**:组合子问题结果返回(归)( `: `5 A# X0 j3 v' P& g. X
    , l1 X( p4 L: Y+ ~; H4 Z8 `& S
    注意事项
    5 G: N' d- C* e+ f9 M8 y2 c必须要有终止条件
    " |& n2 @2 N+ I. v6 c* u$ _1 Q  M递归深度过大可能导致栈溢出(Python默认递归深度约1000层), o/ ]# c# _0 U* d8 e
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ( d% I. y9 L7 h* x. e2 |尾递归优化可以提升效率(但Python不支持)* x% u( }; I- z% f
    8 d+ n1 R9 |9 ^
    递归 vs 迭代
    2 A& |) a' [# w* j2 c- g|          | 递归                          | 迭代               |
    3 g3 l: b' X- V. x+ K+ J|----------|-----------------------------|------------------|+ u0 }6 R! Y$ q" h3 O3 i& G
    | 实现方式    | 函数自调用                        | 循环结构            |
    2 Z1 k, m6 f+ a0 P5 E) }| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & d' {5 ?! y& M0 c' e. O6 L| 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         |. K- c0 E  {/ k
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |: e+ i  U( F8 v5 Z. N9 {2 n+ l9 s" S

    , [/ q. r! F) E: u7 Y1 v 经典递归应用场景
    2 O8 n: F. y9 R8 r* L/ G1. 文件系统遍历(目录树结构)" x+ d" t; M" Y4 M
    2. 快速排序/归并排序算法
    2 i1 k8 l7 S; g! j* j" c+ r3. 汉诺塔问题  s- C% q! v; o, w8 q1 S2 d, ?% Z
    4. 二叉树遍历(前序/中序/后序)8 v/ z% y+ s" Y5 q5 n* i
    5. 生成所有可能的组合(回溯算法)$ f, e# D% {% A+ Q; Q
    / \# y# a2 W: j0 R' i2 G
    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,! S: a7 E7 H, _& O- w2 H! T- h8 v4 F, Z) m
    我推理机的核心算法应该是二叉树遍历的变种。# u8 @* Z( k: H/ |7 H2 B" 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:
    3 e" p  Q( K1 g* X; i1 }( rKey Idea of Recursion
    ; e8 [: U& n, n# u4 s
    6 m* A! U7 ~: U' _A recursive function solves a problem by:' v2 ~9 d% _$ W. u8 K0 ~( j
    6 w* u4 I3 X  O$ V7 U1 m
        Breaking the problem into smaller instances of the same problem.
    # k" r  C+ h( w$ N+ Z4 o) f) i4 A* M9 G- I0 z( X+ {6 J
        Solving the smallest instance directly (base case).2 i. e& ~1 q' d; r; V
    & K' L" X4 a, q' \0 _/ K
        Combining the results of smaller instances to solve the larger problem.
    3 x8 ~+ \, F% A& c& x1 a" D' }- }. ?% ~' Q: h; G' D
    Components of a Recursive Function& B: W) [( C9 L; \" T1 c

    : C: N: k; a+ |; a3 K! R5 G7 |- k    Base Case:& B0 X9 e3 ~8 d* O+ _

    ( i7 v, r; ], v) g        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
      X/ d: ^0 z+ X5 x) Z$ P7 U/ V6 i& l' C( |* Y2 ^
            It acts as the stopping condition to prevent infinite recursion.
    # Q# [$ e. l3 M2 N9 m5 ~2 }$ V$ I- ^) ]  Q
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    7 f5 w- v$ {: U4 k+ W- T; K3 Y4 l4 y, @9 ^: G
        Recursive Case:0 q" x% _- {+ w6 `" g
    ) i5 X" O" p) b; C
            This is where the function calls itself with a smaller or simpler version of the problem.2 W  B* a( h  e0 Q, v1 y
    9 b) Z% K6 V5 C& D" @! c9 Z9 p! G
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).0 D4 P' B9 q! H$ i3 `$ G6 J

    ( }+ q% W4 `$ g8 OExample: Factorial Calculation
    4 F8 m7 S6 j# E0 I$ Q, j0 y9 Q4 z' H! i! N
    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:8 l3 L2 }6 G+ k

    - h9 s4 t$ V: G% D- }) j  t/ U    Base case: 0! = 1
    ' b; b0 }, r3 R! U7 n
    6 K: ?# \9 Y2 y0 x; y+ g# N    Recursive case: n! = n * (n-1)!
    / {/ _- B5 R" {5 G4 y! [$ y  Y  }( H$ P4 V- V! V
    Here’s how it looks in code (Python):
    # D; u' ~3 b9 F  L1 Fpython
    - b5 S3 G4 @4 X  x7 `: q# \$ F
    + `* z. j- h3 b0 \' X6 N9 U8 m/ D) A5 T' G- ?. D
    def factorial(n):2 q9 l0 K5 w4 \* S! a
        # Base case+ n3 _+ z9 w5 G2 G. p. t
        if n == 0:! q& u9 E5 K5 Y# {
            return 1, T$ \2 ^2 H, j$ j4 [
        # Recursive case& p, [! P( d0 n+ A3 j
        else:
    & @! ~; g* B( o$ g2 K# X) M        return n * factorial(n - 1)
    9 l7 B* O0 b& h5 Q* Z* H# p
    % l' B- j! n0 u' h- Q, d, ~1 A. B4 {# Example usage' {6 z: F1 T6 y9 R
    print(factorial(5))  # Output: 120
    ) t; u# }% a( W$ z. x* Y# g! Z" `# Y3 b$ C. {, \
    How Recursion Works
    * g3 h% _: z$ S3 {( |9 C/ \" K1 e; s8 K3 o: U
        The function keeps calling itself with smaller inputs until it reaches the base case.4 U7 Y: d  l/ I! d2 r

    8 R6 l3 v" m4 \7 i( S% j    Once the base case is reached, the function starts returning values back up the call stack.
    : {# L( D( }5 U0 v6 f
    ( {9 e: q' ]4 X' p    These returned values are combined to produce the final result.
    2 _* v; _0 J1 b9 v/ H5 [! z9 `9 o1 }: u2 W8 K
    For factorial(5):: M) M) I, N1 F+ Z

    * B7 @* @7 o7 A6 ^- q7 ~' c$ G1 x) R' |% K7 P( ?
    factorial(5) = 5 * factorial(4)
    & n! o* w! w. W- ^* afactorial(4) = 4 * factorial(3)
    : ]. O! u  K8 r6 nfactorial(3) = 3 * factorial(2)
    / f; t# u" j4 y- e" M5 `factorial(2) = 2 * factorial(1)5 E' x# ~5 U, U5 v5 V: r/ E" Q1 o
    factorial(1) = 1 * factorial(0)
    ) T1 Q9 d, `9 }7 H- \9 ]factorial(0) = 1  # Base case! g) O0 A7 o8 d" q& E  I

    " T+ S1 ?- P) T7 fThen, the results are combined:* L) e. d/ s3 Y9 v6 a, [

    6 e3 _$ [3 ?( ~5 D: o% Y  `! v4 I1 U9 e4 v7 N. W( ?, `2 G
    factorial(1) = 1 * 1 = 1
    % L! u2 L; F. q5 |" I5 m" F( Hfactorial(2) = 2 * 1 = 2
    # v3 ]- R* n; t/ l4 Z* y8 {factorial(3) = 3 * 2 = 6
    . x/ ~9 S+ G- {7 v4 E% v6 c! C7 ]/ }4 ffactorial(4) = 4 * 6 = 24
    . `: ?; `2 v6 k- D8 a5 Hfactorial(5) = 5 * 24 = 120
    # [- ^, R4 s, W. M5 f: o6 q" J
    # ~' s( S+ u# s. J5 G' O' E# D; `Advantages of Recursion$ r/ e' Z: b- l' S7 x$ z3 G
      G9 D2 E* y8 d! m% \& e# l
        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).6 c4 X, D( }8 @+ \  o

    + H9 z6 z+ f1 u8 F    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ; B5 M0 j* M8 e4 ^6 A
    ( x0 W" B) z! h! PDisadvantages of Recursion- @9 P% s4 k( z

    + y0 Y" O/ a  i2 f% ]    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.! I4 j! e3 }/ i/ E
    2 Q6 {. q4 a" Z
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    / Y6 S. X9 h1 F8 z& ~5 ^" }4 K; y( d3 H1 K( w
    When to Use Recursion
    $ Y4 ]3 M5 b$ J, N: A& \0 H" _: T  R- j' u9 q& T6 j
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    & U+ _" x- f9 o# g  i# s5 s- e8 q( m0 H
        Problems with a clear base case and recursive case.. Q' c, X5 l3 H$ V  u: ]. C

    ) F, s# J, A, h7 F" BExample: Fibonacci Sequence3 ~. C! _/ Z' H$ T5 x" E

      }6 \# I" t: }% u+ j" ]: O* ^The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:- N7 l9 x( j0 d+ `# K0 L
    , |$ ^" Y5 E* f" e5 ]/ }- p( O6 `6 e
        Base case: fib(0) = 0, fib(1) = 1
    : U; L' y0 x: s. ^" q( I4 J
    " a/ c7 g- @$ m% F2 w, N    Recursive case: fib(n) = fib(n-1) + fib(n-2)! w9 J& [& N( w' e

    / b" e3 K9 ~. y5 k% Kpython
    ! H# X# F9 w7 q! ^  ^5 z) ?) `' [7 y' {" g

    9 \' Y8 o' U' Ydef fibonacci(n):
    ; A7 V& e: A8 `* c' C+ c. J    # Base cases) c# f& j, i) _1 l3 P
        if n == 0:
    / \0 w' A* G+ x3 I! E+ B        return 0" U* j7 t2 z% C) p: @, s8 K
        elif n == 1:
    / h7 U% Z$ c5 r$ {        return 1$ Q) }6 V5 U1 z- {' |7 U1 t
        # Recursive case
    8 M, {/ y, z* z# h7 V    else:0 `. w& L$ H/ K7 _% C9 y$ y, F
            return fibonacci(n - 1) + fibonacci(n - 2)2 |# a" \* Y" L* {% I  F

    % E# n4 Z0 {* r8 T6 c/ w! u# Example usage: o8 H* v; K1 B. _5 z: i
    print(fibonacci(6))  # Output: 8
    " k0 s# P- b& T# A4 u) T0 C% {  O
    8 _; W1 u8 i* Q8 ^$ vTail Recursion
    7 {( Q" q6 u/ E) {3 A9 ]
    6 ^; u, H9 }) B$ `2 a: n0 a" \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).
    + I4 f% a  g$ _
    8 s, j$ q/ i, h' T2 f* j6 yIn 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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