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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
     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
    * I: z! x9 L6 @4 @% E0 b# [5 P  w! e7 }# I6 `5 t& I. B
    解释的不错
    ( ?/ t+ `5 V" k: t+ L  s7 S9 \4 U* }( {
    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。( `; f* O; i; L6 b/ V: g( w7 m1 D7 M, _

    . T) z  F4 N0 b8 `7 ]3 w4 u 关键要素: k  e9 L) j# V$ B
    1. **基线条件(Base Case)**
    ' N$ M' a+ s# ~" l+ \6 f5 p6 a, l; O   - 递归终止的条件,防止无限循环
      z6 {7 L* J" i) H   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1" N/ T( I. A. Z7 L' ?" d% Q
    0 u8 D- b( N! q
    2. **递归条件(Recursive Case)**
    / D2 z# R# P& r+ Z$ `- O   - 将原问题分解为更小的子问题/ d; @% M& k2 g
       - 例如:n! = n × (n-1)!  e* V  k, N" X& O9 l

    % b. f/ N& p% U& ?$ K$ B& y 经典示例:计算阶乘
    3 r, A$ q" ^3 x% P9 t- W2 T8 Opython$ s9 {, P7 B6 p. m4 m
    def factorial(n):
    0 K; C! R/ p" i! J& Z7 f+ C, y1 C    if n == 0:        # 基线条件
    3 h5 J3 I: [; z6 I9 S        return 1
    ! d8 L2 [! r4 x    else:             # 递归条件
    ' R( Y) u0 S" S, D: H/ e! r0 ]        return n * factorial(n-1)
    0 y* I! c5 N9 ?" c执行过程(以计算 3! 为例):& g6 Y, ?, o2 s  }$ Y
    factorial(3)+ a$ H' h* b8 U3 E1 N' I$ H' o
    3 * factorial(2)9 p: ~  J8 H; U. ]' ^1 {
    3 * (2 * factorial(1))
    * j! O( |- V5 U) ?" _, L3 * (2 * (1 * factorial(0)))
    1 ]: A5 Q5 J  o6 a' Y* k  c! T3 * (2 * (1 * 1)) = 6* |. j$ o$ d+ U( o- Q
    0 H5 z, j7 ?$ l  X0 o" j0 K5 t
    递归思维要点4 i9 }5 N- I2 t( Q/ l
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    6 q3 |* V8 O( C3 u8 X5 u2. **栈结构**:每次调用都会创建新的栈帧(内存空间)2 y  q/ y( z. N4 K) n7 J9 Y
    3. **递推过程**:不断向下分解问题(递)
    3 S* t; z2 W: _4. **回溯过程**:组合子问题结果返回(归). Q( x, t! z8 A2 K( k
    6 w1 A) N! w# _! v) @  H
    注意事项
    1 y# o" y+ |- o9 D+ w; P必须要有终止条件7 k% S. J: P+ K5 p. ~. u( T
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)' }5 U* h" ]6 x) J1 Y
    某些问题用递归更直观(如树遍历),但效率可能不如迭代  D' N3 q1 }- p% h( o8 K
    尾递归优化可以提升效率(但Python不支持)
    ( e# W* T# G8 G% c' P  d& `$ F  ~4 h) N; ?
    递归 vs 迭代
    3 G' K2 n1 H( W|          | 递归                          | 迭代               |
    ; z# _5 Z) j; _- R|----------|-----------------------------|------------------|) s) b5 O) O: _
    | 实现方式    | 函数自调用                        | 循环结构            |
    0 v4 {& I' d- ~: _" r| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |6 _7 z1 B  E) d0 y0 [4 [
    | 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         |
    $ d" T0 R: S8 y| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ' |) \- P8 N+ W/ I2 K1 C5 Y; ?1 @7 @7 r  C% I6 q
    经典递归应用场景
    : d6 J4 y3 v) i$ I3 g( |% Q! Z1. 文件系统遍历(目录树结构)! W% {( W0 J* L, N. ^8 `3 F( o3 I
    2. 快速排序/归并排序算法
    + z3 G& w4 c8 L- K: j8 [3. 汉诺塔问题( l% E- G* E! d3 A7 ?$ `# g- f$ J5 n
    4. 二叉树遍历(前序/中序/后序)2 i/ B9 p) N% G
    5. 生成所有可能的组合(回溯算法)- N" k: J: @  B

    2 e7 `! `& f* F. G! 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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    8 小时前
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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,: z' F* G" |7 q; \& E1 z
    我推理机的核心算法应该是二叉树遍历的变种。
    / }% P5 P& m$ G7 d6 L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:
    0 v  z1 \3 S  P: e  pKey Idea of Recursion5 F" {! E( x# F4 H/ P
    ) t$ Y" g) G% Q. n( |+ [, ]$ [
    A recursive function solves a problem by:
      A6 H& i0 @: _$ T
    7 A1 h/ Q) E; Y4 b$ U& k    Breaking the problem into smaller instances of the same problem.0 C( {4 Q1 T* o1 ^1 }# d

    1 H' m: `2 b2 T    Solving the smallest instance directly (base case).
    0 |6 a6 ?4 p: Q& ?" K. P* |' W+ u1 L, a! K2 u
        Combining the results of smaller instances to solve the larger problem./ I) s  {: }1 n/ U" `+ B
      I& {) B7 l  X* X9 R! T4 W! u0 ~
    Components of a Recursive Function) P% z3 u3 w4 A9 T
    0 C$ U& Q' ~  x! ~3 q  o
        Base Case:
    1 i" q8 ?* A3 n
    7 s9 ]  J! `3 m$ @4 s3 f        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    , `- R" J: h) f% j- U; n& M0 u6 Q) s# m7 P7 V* x- r8 O! Z) V; E! O
            It acts as the stopping condition to prevent infinite recursion.
      ?1 u; A2 V! B" z
    + r. d9 s" H1 r* b1 N  P; b5 u. T8 b        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( h( X2 o  v' h

    4 W& {! A  q2 l, x% k/ A- R$ c    Recursive Case:
    $ f$ ?8 o+ Q/ d5 I
    6 \9 I, f; f8 S- J, y4 c        This is where the function calls itself with a smaller or simpler version of the problem.
    $ S3 U6 x: D- Y* @: X" O6 d4 U2 W
    : E( ]3 A8 A8 b/ H3 G6 q( J9 r9 B        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    0 D: q. b6 Y$ _" T8 N' t' p
    ! E: Z0 }5 k  F3 l1 K6 ~0 ?Example: Factorial Calculation* f6 v0 P# t  n& s, q
    5 H6 x9 E4 F& |( K' G! V) J
    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:7 F# h) S" O+ E- G: T" w
    ! k6 n$ ^( v# E: M$ L
        Base case: 0! = 1
    4 ?5 W* X+ O" o1 a! h  \/ [6 y1 c9 N8 j/ ], |4 ]. T
        Recursive case: n! = n * (n-1)!
    ; @2 h$ @: L7 m4 D9 d  @6 C+ o( `
    Here’s how it looks in code (Python):
    8 G0 A, m8 }# b; M& ^python* P. z( Z* r& L) p
    ( V! q, |9 U% u8 G; v; T6 j$ Y

    $ v% A) M; h' [; x+ adef factorial(n):& f3 @8 t3 @% w
        # Base case
    & C" g4 P; u- N    if n == 0:
    3 x0 Y0 @. i+ Z6 a; V1 Q        return 1
    8 q  |5 w: v; a) [- ?/ r4 X    # Recursive case
    $ I# a- k7 _& _2 y    else:
    1 R! S, l7 m. k2 r/ a2 V9 f7 g        return n * factorial(n - 1)
    1 B/ M. [; @# s* M2 y0 D/ ?; o  ]! r! O- i4 b
    # Example usage# i. C7 M9 S/ L
    print(factorial(5))  # Output: 1204 A1 x$ T, V6 ]

    $ E' k' }. E1 [9 h3 ?) gHow Recursion Works- n# z) ?) s( [0 C6 d& H& Q

    6 X2 l( Q& u* G% X    The function keeps calling itself with smaller inputs until it reaches the base case.
    * U- }' z: ~0 }3 Q0 E8 M3 ^& j6 C+ t( ]. ^5 q! @
        Once the base case is reached, the function starts returning values back up the call stack.
    5 L9 `/ E6 q( l3 U6 s
    % G1 |# ~7 y5 {& ?1 Z8 d* Q# P4 O    These returned values are combined to produce the final result.0 E/ G5 a* ~  s( C" K' ~/ C
    0 R* l$ M2 o; A; R, l
    For factorial(5):8 r% {, j, X* [3 R7 X8 J

    # d$ n/ ?, \& {9 H8 ?# r! h+ r: A# i6 i9 ?" k7 A7 Z3 G8 M
    factorial(5) = 5 * factorial(4)
    2 V2 I2 \% d. I1 U, m2 Kfactorial(4) = 4 * factorial(3)% T( T- c5 Q- d' d$ }. Y0 X
    factorial(3) = 3 * factorial(2)$ f' A" K2 S: a: ?! J* s9 ?
    factorial(2) = 2 * factorial(1), z( F3 i, ~' e: t* E# H
    factorial(1) = 1 * factorial(0)1 l/ M+ e5 Y" m3 N8 p! }6 ^0 c' T
    factorial(0) = 1  # Base case$ c6 H: `" \9 _% i. t# B

    ' D2 `) q: V& a7 x) M. k' ZThen, the results are combined:) U! s' f) o8 ~8 b* J& q; a) m
    7 a6 U& u' F: B- L8 v

    # a0 L4 m( \9 b4 n' C, S5 \# S7 g5 X& b3 ufactorial(1) = 1 * 1 = 1! @8 O4 h' m/ |1 J3 z7 `2 q6 I
    factorial(2) = 2 * 1 = 2
    # P; l* r* G7 K5 E1 ifactorial(3) = 3 * 2 = 6" {0 y: x* j$ A6 \9 X
    factorial(4) = 4 * 6 = 24  A$ Q+ |' A  k+ s. Z
    factorial(5) = 5 * 24 = 120
    7 ^5 J& \* c' `7 Y+ [$ e8 B) j0 U* N8 L& q. I! `
    Advantages of Recursion
    : f6 c( V. J3 `
    4 V, n6 @5 ^7 m$ {8 _# Y. }& ~+ o' h7 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).
    # ?5 y1 [+ }' F$ r& m+ b; n7 g8 U% J  R4 j% v
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    % J4 P) T* H4 j$ a7 h; S5 U$ o  _! d; p4 s$ E2 O& Z# {
    Disadvantages of Recursion
    7 d# ?0 B* Y6 _2 J6 ~- C7 ]3 g. @2 ?6 I
        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.
    " c  Y, Z7 f/ @$ h' J
    6 L8 R8 ]$ t% d4 s! J    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    * o+ A3 l. E9 c6 r3 `- H1 c1 z4 E5 M" A
    When to Use Recursion
    9 y4 E7 M% t  m: c, d* H6 a
    4 {+ f1 G* q4 S' R2 {3 r    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    " i: e+ m" |' ~! B" h, u0 b& }' T- [1 Q8 K/ [6 V
        Problems with a clear base case and recursive case.
    . e! {  s3 Q2 n# @& A6 \' o+ U
    . q! m! R* Z- KExample: Fibonacci Sequence
    - ]% x( L3 F* r. z0 [2 w5 H' c8 q+ T' s- s4 v( v. H; f
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    : p2 E9 s" }: X1 z& [( x/ Z8 T2 s1 W5 c/ m( W: Q  @
        Base case: fib(0) = 0, fib(1) = 1
    " E3 g! D/ s# C& T( ?& `; j0 e3 Y3 D" a! {0 K
        Recursive case: fib(n) = fib(n-1) + fib(n-2)' x/ a+ ~) K$ z5 _: t
    1 {% z, b9 J. Z$ `! w: E
    python7 Y! z; ~& n7 n' v9 M

    . G. [3 y( k- Z0 I+ D! _6 V6 G$ {- o. I
    def fibonacci(n):
    # {1 L4 m7 r4 f1 w    # Base cases5 d1 J. v6 s' x. J" u7 o# w
        if n == 0:" i: T' c) ~/ V" W4 S- b
            return 0% Q# G5 b+ E! k8 u+ A
        elif n == 1:
    4 n- P. ?* j& X. l" o9 S2 f* T% y        return 1
    , O% C7 u% T$ e( F( l5 b$ m    # Recursive case
    1 v# \0 N, A7 D, _% W1 D; m2 t    else:
    - Z: I, a+ g$ V# Z, A' L$ v        return fibonacci(n - 1) + fibonacci(n - 2)) k5 ^9 N3 z9 \- [

    % I' T6 @# ~$ G# Example usage
    ' k! H8 L7 u* _2 eprint(fibonacci(6))  # Output: 8% t, g  }' I: q6 W) M" g4 N
    0 z3 [) U8 c6 Z+ e' e' n
    Tail Recursion
    ! x3 R: `" r- p( q. _! _/ h
    4 L2 s* n3 Y! ^( q' @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).
    & [: Z6 X. Z$ y- D
      Q7 w9 l. _0 fIn 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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