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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
  • 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
    3 ]7 {; V& k0 Q* `0 ?$ M$ v/ a" l2 h" u2 w8 l2 P
    解释的不错
    9 T9 K$ F( P- J" i
    7 a3 G7 t  r7 M7 ?1 Y$ e! C3 N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。
    8 w- [6 A# x* G  u  V4 x% [& }3 e4 w( i0 c6 u
    关键要素
    ' J+ Z- M7 c/ C) j' k0 |1. **基线条件(Base Case)**
    0 y# c# P  T" A( H$ p* w   - 递归终止的条件,防止无限循环
    & r; a) \: G- @) G$ R' h- b/ Z% Q6 p! Z   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 11 r4 C6 U' y7 L
      {- {7 u" Y, v4 Q: i- S* V( O
    2. **递归条件(Recursive Case)**8 M5 K/ T$ Z% Z4 G
       - 将原问题分解为更小的子问题
    2 t% A- @* D% K! I4 k. o   - 例如:n! = n × (n-1)!
    5 q9 ]- s! @0 d1 m& N$ G, I+ Y
    - [4 h, [, \& x 经典示例:计算阶乘
    4 y$ x; j' |# Cpython' }  x0 b; B, f8 M. D4 {
    def factorial(n):, q  |* T5 S9 N# g$ w
        if n == 0:        # 基线条件% U, U8 A9 e- W
            return 1! X  ]" m# l2 k* U, z  M
        else:             # 递归条件
    6 C0 y# V" X1 s2 P2 E1 E9 R        return n * factorial(n-1)! a7 a5 J. P2 r# ~8 g
    执行过程(以计算 3! 为例):' ~# D# b( a& a; _* G
    factorial(3)% f) }7 N; }+ g6 m1 K; T* q7 Y
    3 * factorial(2); j" Q3 r; U8 U' m+ u
    3 * (2 * factorial(1))
    2 @1 q# S3 [: K2 F# ^6 k: U3 * (2 * (1 * factorial(0)))2 [+ y; @: l( {9 d7 _
    3 * (2 * (1 * 1)) = 6, d7 b- H- M4 n
    " F. e) O5 O: @
    递归思维要点6 B7 W/ ?0 @! L6 i6 o# I8 K/ M
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % W9 z( J2 ?8 G2. **栈结构**:每次调用都会创建新的栈帧(内存空间): `9 g' _9 K  ]" v# o
    3. **递推过程**:不断向下分解问题(递)2 G4 G; e$ h8 \; V* J
    4. **回溯过程**:组合子问题结果返回(归)% e) U! Q( }! ^
    . e" n" _& q. M8 k, Z
    注意事项0 z5 u8 F/ f/ M( ?
    必须要有终止条件
    ' N1 N6 X4 U6 Q8 l递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    * w! k2 K# u$ W3 H. g  ~某些问题用递归更直观(如树遍历),但效率可能不如迭代0 g8 B  R6 t7 E0 }
    尾递归优化可以提升效率(但Python不支持)1 \# r, x4 o' y9 C

    5 ~9 F  X( R0 e 递归 vs 迭代
    1 Y! w; E' V; Y: \( k" A1 C|          | 递归                          | 迭代               |
    2 m5 ]: w2 ?5 L, P|----------|-----------------------------|------------------|
    8 {/ X" O/ O! ~  K1 k0 c# {| 实现方式    | 函数自调用                        | 循环结构            |: g9 B& E$ B" m3 M/ |: Y1 _& M
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    7 Y/ p/ s8 y& x) m$ a- C; `( G| 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         |
    7 |* \% k0 M- o3 L) L+ C% B| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    / E* \* o5 w) ], f+ v  C
    ' |- a* y& p+ n& Y, p# z: a 经典递归应用场景3 ?  y4 B# Z% `( o
    1. 文件系统遍历(目录树结构)+ V* K( Y2 J! H! I* |' [
    2. 快速排序/归并排序算法7 Z  K; p8 v$ T2 o
    3. 汉诺塔问题
    6 `) P0 a* ?; f# R/ x4. 二叉树遍历(前序/中序/后序)1 T8 C/ @" \( u. ]/ G
    5. 生成所有可能的组合(回溯算法)0 n5 C1 D( [. J# n0 b

    ; W6 Y# r, e4 ^! U5 i/ e2 w7 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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    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,
    % h8 v  A. ?  C0 |( h我推理机的核心算法应该是二叉树遍历的变种。
    : d1 h0 K. F5 R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:$ z& ?& X2 F' ~- Y  t- ^. [
    Key Idea of Recursion
    , {; j% [: L/ Z* j* p' I
    # A+ N- J' U9 X, G- ^$ tA recursive function solves a problem by:
    ! D0 ]+ A- K, k4 }1 j
    & S( m* g( X! A5 [8 m    Breaking the problem into smaller instances of the same problem.( W' `! e) o8 p7 @$ i! [

    . A3 K7 e5 y/ o* m8 {    Solving the smallest instance directly (base case).' v' c4 Z5 j, n  O% W7 K

    ' F! H. f- {' [4 g    Combining the results of smaller instances to solve the larger problem.
    % f2 E: u) o- m! v
    ; b, W$ Y* Z+ ?1 W$ e) eComponents of a Recursive Function" h; i: J" c( U

    3 ~3 E& X  }/ y3 `    Base Case:* B- k7 d( L4 X3 A
    $ Q5 _; e' d! T3 {; H
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ! G/ _: _" o+ Y! w! ^
    8 m$ L, @5 G2 P+ h& r) u: \        It acts as the stopping condition to prevent infinite recursion.
    + |" ^1 n( L4 K' {
    1 X0 h! n$ y; i# }6 h        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.4 ^# E! E% }* ^
    . |# L' l# E% c( @4 N
        Recursive Case:! M# p  [& s! E7 ?- Y$ Z# n- J- G

    ( Q4 @/ A( N, M# L0 o        This is where the function calls itself with a smaller or simpler version of the problem.% s2 T8 F3 U. U2 U3 ]" t

    2 s$ b; V/ ?/ y: x0 a5 H) y        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).  [9 n5 w) s, F8 N

    2 S' P; h* m4 {3 R/ |  H2 p7 [Example: Factorial Calculation! _; a+ Z. _6 b: \5 G3 b

    $ e) T9 T4 G$ T, B% ], 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:
      }- C  J  m" y$ x" T, ?* _6 l7 b# D2 ^
        Base case: 0! = 1
    9 C6 z/ u$ O1 ^) ~" N/ h3 u4 Q$ b  \; ^+ u  r3 ^& ^
        Recursive case: n! = n * (n-1)!
    * F# R' Z' t% Y. w3 o: Z2 l7 s" \) A  C; ]+ `. n
    Here’s how it looks in code (Python):
    " \5 ^3 n: A: _python" y1 y$ n- t! b% `9 \

    + Z. r' {! F$ Q9 x9 m$ K2 H' `
    $ @( G: r% \" c% Y% l5 i8 @' Ydef factorial(n):
    . R+ r# ?0 R  k! z2 r3 P    # Base case
    0 L% u, m$ h  E0 R9 `9 O5 x    if n == 0:3 v. {, Y  x: c9 u: g$ g
            return 1
    % G4 Q) g% c8 H6 Q5 l9 T    # Recursive case
    / ~; [% b. P4 ]3 o) Y    else:
    2 w! f* X% M5 |; k3 U        return n * factorial(n - 1)% V1 T4 G* P5 i6 ^. Q: P) t$ b

    4 F. D7 h! J3 W1 V# Example usage
    $ t0 y2 _+ G% O, Vprint(factorial(5))  # Output: 1204 C. _2 t! X& b; ?4 P8 i$ r( ^

    , q5 G1 C5 Y* C5 @4 [, ]/ CHow Recursion Works
    5 r4 j9 U6 k( D) h$ s5 ^- n% V- t: d$ [+ l+ U) l. E
        The function keeps calling itself with smaller inputs until it reaches the base case.
    4 Z' n4 o" N5 ^) B; C- F  w
    & i1 s  m* E( |9 ?# f  A    Once the base case is reached, the function starts returning values back up the call stack.( a' G5 l  E7 H0 Y; f' d. X4 v

    ; ]3 V) S- x/ ]! O    These returned values are combined to produce the final result.
    0 K) L" j5 Y  B/ G/ ]6 M
    ) ~7 {  h) W' ~For factorial(5):
    ; Q; o) c" W( P& T3 w8 _2 ~. |
    ( s# D# e7 D& Q6 F" P3 r; B1 F
    8 m) i6 j% h4 D1 ?factorial(5) = 5 * factorial(4)
    5 o' c9 w0 T% N; v+ Y8 h  ?3 Vfactorial(4) = 4 * factorial(3)
    & x. E) Q9 e) F" g: {: B5 wfactorial(3) = 3 * factorial(2)/ \6 s, f2 g( S+ n, Q3 {
    factorial(2) = 2 * factorial(1)
    9 S" d0 v$ N5 l  Y' ~factorial(1) = 1 * factorial(0)( M; l% S: R6 s! \; Z
    factorial(0) = 1  # Base case6 G5 h3 H, U) C7 W) Z5 \' X# {
    * [9 j7 n, ~8 C; y! D
    Then, the results are combined:
    * x8 C, r  U) t3 ~  z9 c1 _1 ~+ @/ r! R0 n

    , y/ a% {5 L# Yfactorial(1) = 1 * 1 = 1
    - _7 m/ c0 }7 T& V8 {factorial(2) = 2 * 1 = 2: z2 n/ s1 z7 e! T6 p
    factorial(3) = 3 * 2 = 6
    ( A* w* e+ W' E# d$ Hfactorial(4) = 4 * 6 = 24
    0 |# L. ~6 B( L  nfactorial(5) = 5 * 24 = 120
    ) z! g2 ^! e# D7 I+ b, t, ~3 f! T4 n! K. }" t+ o5 \
    Advantages of Recursion+ |! L- k' I- N+ P! ~  W4 n. P" @
    ) V4 T5 X) j- j
        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).1 V5 M. B( ]& N2 i5 {
    . [2 W% {+ Z. ~4 F! \( k; h6 i
        Readability: Recursive code can be more readable and concise compared to iterative solutions./ {& \" E1 x8 U0 f' @+ }2 J# c1 ~+ o

      N* E$ H3 u& X. T# _: {Disadvantages of Recursion# Q* x4 p  a5 N' V3 m

    7 T4 W1 H0 G/ b9 I( C; ^+ v    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" h: `& @8 S. w+ ~
    9 ]. [+ }9 E: A/ t4 g    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).* _  f3 \; ?) x

      _6 |6 u6 M. x1 z2 r' W$ oWhen to Use Recursion
    ! J! t4 ^& A: U# v( n6 p  s7 Z$ ~9 m$ `" S7 j, b0 H" N. A- G0 u* {
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).4 U0 ]% S  `5 B$ |/ r
    2 M! b! U4 R: y8 j1 U3 N
        Problems with a clear base case and recursive case.
    5 u' A2 |) g# @# g  [( j2 k( b2 F) I& [* V/ K, H9 M
    Example: Fibonacci Sequence' D4 J) V$ x: Z$ l6 ^
    6 k2 q4 S$ b& o% V) [! u" x
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    - a3 \9 M9 ^. d0 I) Q! a" V# M6 x9 q; }' g2 j
        Base case: fib(0) = 0, fib(1) = 1
    9 u# X1 V: J6 A0 z2 Q! a! ]) I; F1 R& u  n5 Q+ i+ ^7 F
        Recursive case: fib(n) = fib(n-1) + fib(n-2)" h  \* A& j4 G- e, c! q# [( C$ H
    / C  o; [3 T0 g8 R) Q
    python) b  p5 S5 a; X/ [" _
    & j. s5 H4 M$ q2 k( q

    0 q" m6 i8 d1 Zdef fibonacci(n):
    " m9 R$ o2 o. [) g( u    # Base cases# A" F9 v( o" H4 t4 `
        if n == 0:7 ]/ O2 ~, ]8 Z  g1 Z4 }
            return 0
    * v0 U! e$ b% H/ k- x9 X- z0 Y) h    elif n == 1:
      b# Q4 h+ \& Q( Z3 R* E* S# v5 A, U# V        return 1& H" R* k8 |* k3 i" D/ @; n5 C
        # Recursive case
    7 C. L2 B7 r( b6 B$ b    else:  O' ]7 m& f) Y6 ]6 @
            return fibonacci(n - 1) + fibonacci(n - 2)2 J* q5 F% c, v9 I7 x+ P

    " G0 N  u8 [( L+ m# Example usage
    . g" Z0 f* C: z% X- g+ v2 F( b0 d& [print(fibonacci(6))  # Output: 8; \0 i$ s% Q7 I9 m. w& q2 D% d. W
    8 B4 w% X# b5 _/ |. t6 ^+ P' p3 F
    Tail Recursion: p2 K: _/ J* e" t

    + h2 B1 Z) l0 U& ]. ?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).8 H" w4 s" X5 A8 Y8 m

    + ?# c* Y5 n8 e0 k# g- OIn 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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