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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
     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
    # M  X3 b6 c: K1 F
    + H  ^6 }0 W  k# `) [8 z解释的不错
    9 Z8 O9 m% K: R) Y% G
    : E4 H9 w  G; _$ a; \5 {  A, L* S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。
    % d) F3 ~9 E. E0 [, O$ Z5 K: B# w9 C
    关键要素
    4 W$ q' S# I# U' v- s4 A* M, \& H1. **基线条件(Base Case)**, }1 F* p+ ^" d. h
       - 递归终止的条件,防止无限循环8 w, p/ _3 M! f' ~  `+ z8 L
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1. o; h8 g6 t; i) f5 {! W9 o) |
    ( P3 s" m( v4 u2 e+ L
    2. **递归条件(Recursive Case)**% j( `9 Z# v3 \3 _$ Y, g
       - 将原问题分解为更小的子问题! V4 {4 c, T" L& ?7 L# r
       - 例如:n! = n × (n-1)!( w# J$ R% S, S% r
    ; z+ _  B$ l% v" h
    经典示例:计算阶乘
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    def factorial(n):
    : A5 s  j8 o1 {. O" U- v( A    if n == 0:        # 基线条件; i0 Q* |, }6 w
            return 1& q1 c7 G2 a/ R( |
        else:             # 递归条件1 q- U7 Q; R( k$ r* Q/ h) K; O
            return n * factorial(n-1)7 y1 o  c- ]: z+ h! P
    执行过程(以计算 3! 为例):
    4 O/ C5 {# V. {* S  Wfactorial(3)
    3 ]- L* Y" F9 x6 P* Q* V3 * factorial(2)
    ! h: {3 \+ o5 r0 [3 * (2 * factorial(1))" D1 N1 @4 |+ L9 [
    3 * (2 * (1 * factorial(0)))
    9 `: `$ P8 x) z$ v7 |" R3 * (2 * (1 * 1)) = 6' {: ~& ?" `, f

    . N4 `9 T- ~1 G. B3 k: F, m 递归思维要点7 S  ^- y  J0 p3 o  d: T" o
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑  m' f4 Z3 y% A3 [5 _4 f' v
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    & c* r& {% }; Y: m) E- t* m7 @  Y! h% T3. **递推过程**:不断向下分解问题(递)
    2 {9 _7 u4 Y) U* L% n: }4. **回溯过程**:组合子问题结果返回(归)
    / `( M+ e) G, C; X7 J
      c" H  e1 j  Y2 z! R# N; m 注意事项
    1 ~- t( U$ C5 i) o5 A" L必须要有终止条件
    7 v0 p; ~& j* J+ N递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    8 L3 i2 I7 c& |, `: Z0 R" `( s某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ( A1 n6 W; P$ E. X3 k9 o; K) g尾递归优化可以提升效率(但Python不支持)
    ( p9 R+ Q; I' S5 m5 v6 g1 x0 o% ?: C( H+ r2 R. K& Q6 N. K5 f
    递归 vs 迭代
    ' }2 j6 D1 m8 x|          | 递归                          | 迭代               |
    ' `; U: P( K. T& }|----------|-----------------------------|------------------|
    : H( K- ~# |$ y% `: E" n# T* ?| 实现方式    | 函数自调用                        | 循环结构            |6 o& E, r9 Z% }; F8 J1 E& f$ m/ {
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# d- X7 s' J; T+ D" `6 ?  y
    | 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         |
    4 a0 u1 q0 v+ d0 w| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ( `/ }$ M0 }  N0 Z/ i# s2 K' n5 }% r) G; Z
    经典递归应用场景
    4 N! l/ \2 e( O% u7 @1. 文件系统遍历(目录树结构)
    , f  C' r: b) i; A7 c) N6 Z2. 快速排序/归并排序算法
    * |6 h. Y; j" o7 l7 n* Z3. 汉诺塔问题
    9 u) M& x% I# c6 K1 }4. 二叉树遍历(前序/中序/后序)
    # a4 E8 r* l: V% J0 }- e( t& P5. 生成所有可能的组合(回溯算法)! s+ ^5 k1 L6 |

    ' f) l9 m/ t* X! T/ 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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    郁闷
    18 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,, w, {4 \1 @% c( F! z
    我推理机的核心算法应该是二叉树遍历的变种。
    ; s# X  y+ @: V' u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:
    # A# B! V/ _! HKey Idea of Recursion
    & u6 e( H, [9 I. s( q4 ~# t9 D9 t0 K/ a+ r
    A recursive function solves a problem by:+ G( N: V/ N1 c' l# T8 q: P
      f! \% E. h2 ~* b* ]4 {
        Breaking the problem into smaller instances of the same problem.8 J+ \* {4 Z& s1 W

    % a5 j0 w2 n' t- q" P! `1 c    Solving the smallest instance directly (base case).+ _6 J( x- w3 Y& S" M2 C

    / K5 [( e1 @& o. C+ N3 ~9 b0 F    Combining the results of smaller instances to solve the larger problem.! S" i' A% ]2 l% I" f4 S
    $ y  r9 ~- P! o& n, N( k) s; D+ l
    Components of a Recursive Function
    3 ?5 |! y+ ~/ A- p  c2 ~8 T  l' j5 t5 @3 J. {
        Base Case:8 e- x+ H- [' Q. j7 u* Z/ J3 X
    * m  d1 i3 U; v9 H& v5 W( I" Q
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.7 ?3 `4 Y5 Q4 A" B+ I& h! t

    $ \4 x( }; s4 f  `8 ^7 p# f& L" i        It acts as the stopping condition to prevent infinite recursion.7 l) {, _) h7 F0 Z3 y% {( l! a! r
    1 C" E. ]+ r5 t; D& H5 h
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.- G2 C4 h8 {- C0 p# A& ]$ }

    6 h5 o' P$ J; u; q    Recursive Case:
    3 R& I. Y2 I; g, Y( |# ^# R9 [* r1 a! W1 m: K2 Q8 Y
            This is where the function calls itself with a smaller or simpler version of the problem.
    $ Y. \7 M) O& t1 w+ @+ G5 l+ W, g, A3 w: b2 }- m
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).1 h% ~5 G* h8 _4 b/ b/ k, {( `  _
    # p: x6 U/ Y8 L5 t
    Example: Factorial Calculation
    " {; ?0 ^0 A2 k+ }5 y
    " o. g! Z0 L% `8 l1 gThe 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:, {' x% E  h" G

    1 A1 [/ H: V3 r1 P' F# m    Base case: 0! = 1, p4 l5 @. R) Q
    : z4 k. _' z+ C/ F. c
        Recursive case: n! = n * (n-1)!
    # N" i& m/ s2 ^( f9 ?  I) K! u/ ]; A# q1 ^' n
    Here’s how it looks in code (Python):
    ( j) O- @1 e8 ]5 Q9 gpython. P2 K- ]+ t0 f* u! V" ^

    7 L7 h% [+ X. o$ b* D& ~& g& {& ~. D1 m
    def factorial(n):
    1 q/ G: F" _% \3 u0 n4 j    # Base case6 {2 W) h2 K7 |) h% G/ B+ `6 N8 k
        if n == 0:  Z  e; @3 x6 v: v4 ?
            return 1
    ( e0 |4 R' s$ X# |' v1 d    # Recursive case# N; j8 S2 U- Y7 G- r
        else:
    2 I% G" o1 H2 R, s- [        return n * factorial(n - 1)- h* [* s- \' _! E$ h  S7 L

    $ C: L% }2 `) t8 [4 R# Example usage( z( v) j* m( k0 k, G% }
    print(factorial(5))  # Output: 120$ F4 q9 I$ |8 F* t

    & O# o4 N/ d. G) q# sHow Recursion Works* d% h: t% K! p1 G3 W
    0 k8 n. M/ L+ Q6 E3 c, ~! T
        The function keeps calling itself with smaller inputs until it reaches the base case.' W% x) h; s# F$ t8 s" U* \
    0 ?: ]! J# X, t3 H! l% ]2 s& o  S/ L: ^
        Once the base case is reached, the function starts returning values back up the call stack.
    , ^8 B1 d- ~4 R4 y
    ' s) A( V- k5 m8 }2 t    These returned values are combined to produce the final result.
    . ^) t3 e& r! T! k
    , |1 }! @& v# [# H- [For factorial(5):1 x, i# {% I0 ~5 z$ l  P+ l4 r

    5 ^3 ?% O& z: z1 [( j
    5 k# X( _: m7 \9 T4 q/ R" w, R, d  Lfactorial(5) = 5 * factorial(4)& J9 @( c) A* M( k. M6 U! H
    factorial(4) = 4 * factorial(3)
    4 g( ^3 l$ _2 ?% \3 z  J; Sfactorial(3) = 3 * factorial(2)3 e& d0 K1 c+ z  V- w( \. [9 N7 ~
    factorial(2) = 2 * factorial(1)
    ! F9 L# Z. C2 H6 ]% L8 N8 M' F& ]* |) Nfactorial(1) = 1 * factorial(0)! R: z2 \) K$ R$ Q% y: R- k
    factorial(0) = 1  # Base case( J9 L$ E( V, I* q
    4 V0 \$ M8 s6 V+ F; N
    Then, the results are combined:# r) K! G# T# \- M
    6 P' F: c0 D. O7 G0 V1 E
    ( Q/ T8 W( t* `1 c
    factorial(1) = 1 * 1 = 1
    2 x% u2 m4 R* Q( G6 k$ Pfactorial(2) = 2 * 1 = 2
    ; m! X5 p9 a2 z0 n+ I* T& Rfactorial(3) = 3 * 2 = 6
    / b% [3 n. `7 C- U8 V: a1 \+ Afactorial(4) = 4 * 6 = 24+ m& t4 x2 L! r0 a: V- E3 p
    factorial(5) = 5 * 24 = 1206 @. b8 Z+ z! O& o0 c
    . Z& ^9 e+ D' r3 S' c0 W
    Advantages of Recursion: m" a$ S6 Y3 ^) m: H& s
    . J4 G& S( ]$ M/ D, Y; k" S5 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).5 o% q& [9 q7 m9 g* D( o2 P. b, r
    : B; d6 B( |% L7 Z
        Readability: Recursive code can be more readable and concise compared to iterative solutions.+ T# o  E! Z/ Q3 K$ O1 r. i2 b9 \

    ' D6 a+ W3 s( e, I% h* q" Q' @6 IDisadvantages of Recursion
    ' U) I, p; i0 M# Q& I
    * M, @9 o& u; E  V6 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.! L1 B/ k2 g  R5 U1 g/ Q: w- U; [

    / D/ r5 E2 O9 {' [: p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 E: D% t* |/ N% `7 q! t% u
    0 x3 H5 U/ q: M0 g. k0 U
    When to Use Recursion8 c* F2 A6 P. W/ }$ Q& V
    " R" e$ r9 O9 v0 n2 V  V* h9 F/ M  o
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    6 [$ F* I& w9 @4 i! q! }/ i
    : ]: G- l% L0 v; Y! w' q" A    Problems with a clear base case and recursive case./ |* z+ s2 X) }/ H) t: H6 F1 n

      S8 ?. g! e/ u7 B7 h) a  N* iExample: Fibonacci Sequence0 X5 f8 G8 {- i0 Y- c! a

    9 o0 ~7 l8 R) MThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 l4 b9 ~2 Y. r6 e/ j

    3 A$ G$ }+ G1 g* Y9 L    Base case: fib(0) = 0, fib(1) = 19 j+ U. d4 P0 J7 m% U8 N

    7 v: o6 g) O0 D- I5 I    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! |4 ^' S+ B  G* q! m4 [( G+ @# @
    python4 B8 d3 A; D1 \6 e+ l& y* i' v
    * J9 _2 e/ ?: d; _6 H. a1 l6 l
    / X3 I5 D, u  _- V
    def fibonacci(n):
    9 x9 E# R. o+ f    # Base cases# O7 X4 Y' j0 q8 l4 `4 m
        if n == 0:. S3 C: `4 ^( z& r# m* d& c! u8 ~
            return 08 |; h+ a7 o8 m; v, W; J& b* B
        elif n == 1:7 a/ t! t. T1 L! m1 y
            return 1; D- w- P6 \' c3 {
        # Recursive case7 E4 s; X- r* {7 C* P3 @4 Q
        else:
    - g) B+ }  E. Y. ^* P# t- o        return fibonacci(n - 1) + fibonacci(n - 2)
      r( _2 R3 S1 q' u8 a  k' F" c2 Y9 G. {9 R! V( c/ v! Y
    # Example usage
    2 E/ a) ~; @  [print(fibonacci(6))  # Output: 82 A: r3 S) K- c: @+ [6 {2 w
    4 K2 c+ M8 H' i; Z* N% J5 f
    Tail Recursion3 [. f! E' m. b5 Y' G: s

      f1 C0 {3 o, Y2 n9 e: JTail 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).0 b. I- P. z, _, e( Y

    / {) S8 c8 R) @. dIn 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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