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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 : D8 ^2 l, u& v4 d9 b
    % k! ^, p  J8 [) u. h9 a
    解释的不错
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    # j; ?, X- Q% L5 L# _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。. E5 ^: D( x& ]2 {  v5 }9 H

    $ K, l1 n+ b; [7 r 关键要素6 O4 a8 C- e( S& s, O. U7 U; a
    1. **基线条件(Base Case)**
    / W0 e3 X2 G/ y   - 递归终止的条件,防止无限循环/ h9 R/ D# A4 @, |3 T! k
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    8 c. a0 Z2 h( S+ j  ^. {/ u7 C8 I8 Z# {  x# x0 w6 r) n+ X* ?/ q2 a  g) |
    2. **递归条件(Recursive Case)**3 C3 P6 y" _8 \7 a2 q$ m# c3 E1 a
       - 将原问题分解为更小的子问题
    8 h( {3 T7 N' X4 U- _/ ?; S" C   - 例如:n! = n × (n-1)!4 e' v7 ~& {9 _9 v$ n1 t

    # K6 X# q  o( L7 l! E8 P: j$ E 经典示例:计算阶乘3 D) C7 T/ r! x6 \- V( K
    python
    7 Q/ v, \5 o& a" Kdef factorial(n):" X3 C4 k, O4 t4 z: l/ E! q
        if n == 0:        # 基线条件; J( g9 C5 x! T( L
            return 1# ^( W! ]/ y6 n$ A
        else:             # 递归条件. ^6 }3 ^) ?, d/ E9 w
            return n * factorial(n-1)
    + w$ [- z/ @9 {7 q执行过程(以计算 3! 为例):
    : _0 g! w' [% J' s$ Hfactorial(3)
    2 S3 {, w- T( r4 \, ^; R. u- I3 * factorial(2)
    & w$ ]6 o$ ]7 X  e  U, {+ t1 X3 * (2 * factorial(1))
    % f( u2 c1 I/ Z3 * (2 * (1 * factorial(0)))* e  ~: X8 m+ y3 x: h8 v2 S
    3 * (2 * (1 * 1)) = 6
    7 A% B, g1 b" g
    8 }0 J, ~/ h* \$ m 递归思维要点2 T) A! K# H! Y3 F% r& E
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    # P# J8 p4 X6 X, @+ f  [2. **栈结构**:每次调用都会创建新的栈帧(内存空间)  Q6 T  [% S# _3 \+ g1 E
    3. **递推过程**:不断向下分解问题(递); i) }& \4 a/ x4 b' |7 ]2 X  ]
    4. **回溯过程**:组合子问题结果返回(归)
    * k& l; b% |3 f9 x
    - V7 F" g9 h2 y 注意事项
    ; Y6 H3 \  V3 t8 `" c& o必须要有终止条件
    5 m. Q+ r4 U) H% }' g递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
      i4 u! n  D6 R8 z1 L; x% @/ m2 a某些问题用递归更直观(如树遍历),但效率可能不如迭代& ^! y) g# h% W
    尾递归优化可以提升效率(但Python不支持)4 y8 |8 s& T$ v

    9 g! w1 K4 F. k 递归 vs 迭代% X3 C0 H" J+ J2 h/ V6 _
    |          | 递归                          | 迭代               |" P3 O2 v1 z  z2 p- Q  {
    |----------|-----------------------------|------------------|9 @/ A' y: ^+ j; ~6 {+ R1 E9 e3 w
    | 实现方式    | 函数自调用                        | 循环结构            |$ c" Z5 G5 G( f! V
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |+ r  Q9 m4 K) ~- f
    | 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         |
    * V" J) p+ b2 q9 T; ^1 M+ d4 ~| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |+ V: N8 t7 `; r7 L
    4 X! n# N$ J2 ?! U- B$ b  q, s
    经典递归应用场景" g9 q, V5 l/ q5 }( E2 M
    1. 文件系统遍历(目录树结构)
    - ~, F3 V9 v" n  p. r  w2. 快速排序/归并排序算法  [% V* y8 p/ f
    3. 汉诺塔问题* ]# i1 k. t. p: W$ W
    4. 二叉树遍历(前序/中序/后序)) z, [2 y8 Y- {! W' Y* ^
    5. 生成所有可能的组合(回溯算法)0 H; t, }0 c% c5 C

    $ t( ^8 E1 ^9 x' t1 E4 `6 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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    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,
    6 w8 V! U; W3 B5 h" l2 Y% l+ F: ~我推理机的核心算法应该是二叉树遍历的变种。( U7 g8 A" y$ a/ `9 o
    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:
    : a3 {5 Z; @  t. V2 d& QKey Idea of Recursion. P. _0 d# @9 s2 M: L
    3 c1 ?0 d/ e# b  e+ Q- _4 ^
    A recursive function solves a problem by:4 H/ s! L' ?- j! G: F
    3 ?" f8 M+ W/ g$ A- d( F
        Breaking the problem into smaller instances of the same problem.4 V! X8 g( K+ G' H9 L4 n* \

    9 j( B6 W1 V  T; U8 D    Solving the smallest instance directly (base case).; t0 k, G' ^# [" e/ i
    + z; k0 A8 d5 {) H
        Combining the results of smaller instances to solve the larger problem.. m+ y4 _/ E7 R  Z) R  W1 ~
    ( y7 \2 g0 b: t% n8 G) R
    Components of a Recursive Function
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    7 ?: y4 G3 o7 Z, K! g" _    Base Case:( C% v' ^0 |& d# U

    ! q' w3 {' m4 j! C) G" B        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.- K& A) v3 ?# F; Q, d' i) p% f, T
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            It acts as the stopping condition to prevent infinite recursion.1 {9 g$ o/ P6 e0 M' M9 v

      q4 V2 r. H# C' `, d) q6 a( n        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.$ x2 n" S( G7 J

    - a0 @% ?* E1 v+ t5 k    Recursive Case:
    : O* E: ]7 G5 _0 a+ L* M) e8 O, `" d/ P0 H! t
            This is where the function calls itself with a smaller or simpler version of the problem.8 f2 s; s' h3 w

    4 Y2 e. u6 }/ {$ X. T$ ]; a        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).0 p  X: |9 |( t, f; x
    ; v" j7 S) [# J8 Y; n: q: c: K
    Example: Factorial Calculation! w7 e0 D& P. w  x0 u

    % ~. Z' X. q( M! PThe 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:
    , q- u7 O3 a8 D" y+ z3 D8 |
    & M& C) r. f9 K; I4 R* z4 A    Base case: 0! = 1! ?( D2 F& U1 D/ d$ R+ O, W
    8 y& H$ P& J* M* d. F
        Recursive case: n! = n * (n-1)!
    , c+ j; h& L- d# ^. D
    & M. d0 L4 ^. AHere’s how it looks in code (Python):# e) y7 b, n8 W: I& |
    python9 ~  q4 Q+ c3 U$ p5 Z

    - _6 G+ o1 L2 i) ]0 w5 p" V: z1 B1 x$ M0 u
    . n1 ?2 F0 }8 Z1 _, s9 j* a# }; Odef factorial(n):
    . v& y( P. _: \$ g- g: Z, O1 ?    # Base case
    3 @7 ?. T, [8 ?0 d6 I8 U0 ^    if n == 0:; d+ m7 o8 M; c1 ]. M. C
            return 19 l6 Q3 E7 B) O: n; p
        # Recursive case
    * l4 ?( {7 }2 q  k4 p( ]    else:
    4 |0 k& R" N( z9 {1 W: n        return n * factorial(n - 1)
    8 J/ R( d- O5 Z& l8 v' ]% b8 a$ W1 r" Z) r- V0 p% p* s8 F
    # Example usage
    . a& n. R+ e- W! m7 h) Cprint(factorial(5))  # Output: 120
    8 V/ J4 O. o+ z% y
      i4 c5 i8 \. L; D+ F. R4 h% R3 Q1 gHow Recursion Works
    + t4 m* v. r$ ^8 _9 y( I) i* N% V( L# L" U; a0 T
        The function keeps calling itself with smaller inputs until it reaches the base case.
    $ c0 @% }6 E0 w) h
    : j% p5 V/ r5 _' k' _; z    Once the base case is reached, the function starts returning values back up the call stack.
    ) z2 E/ B! H0 B, i$ ]7 d0 T: s
    3 j9 x7 F6 @- D. `5 I) V    These returned values are combined to produce the final result.
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    For factorial(5):' d* c. d4 e1 a$ J" p3 t1 n- h( j
    4 E8 n6 K5 S5 I; r7 ], _4 I( O

    ' r* ?  o2 ~' ~- L/ `factorial(5) = 5 * factorial(4)
    * j. W5 m9 c9 @) {, Efactorial(4) = 4 * factorial(3)3 |# R5 [) n/ p0 y, M- K  x
    factorial(3) = 3 * factorial(2)
    + l6 `( r8 r5 Y+ e5 @. P: pfactorial(2) = 2 * factorial(1)9 q$ d. ^" L6 {! n9 P
    factorial(1) = 1 * factorial(0)0 N: U* m/ Q6 k+ S
    factorial(0) = 1  # Base case! y4 |# j0 s  J* L; v

    : X+ D+ E3 E9 [) K9 ~Then, the results are combined:! d! I' F4 {1 a5 p  H+ l; C! W! q# J
      c6 T( _6 ?( ?/ s

    + k: Z+ a) H+ `: ^  sfactorial(1) = 1 * 1 = 1* q2 @8 _+ T) i/ u6 x1 t
    factorial(2) = 2 * 1 = 2
    ; A0 p+ r: r5 {" q6 {4 g% i* Efactorial(3) = 3 * 2 = 69 d/ X( S6 i9 L$ l
    factorial(4) = 4 * 6 = 24& v* F" J5 l  q
    factorial(5) = 5 * 24 = 120
    6 a6 J. ?) n& W- k( Z
    ; B0 K4 S' W1 W8 Z# G& h$ N! mAdvantages of Recursion
    : W3 l5 s2 s* a7 q( P8 _- m5 z$ A* l- J0 y
        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 @# v- m( @2 D- v' Y# k" V: p

    , h5 }7 r( `% F6 l. O# P    Readability: Recursive code can be more readable and concise compared to iterative solutions.. {# F5 i* G* u- Y( ^/ p

    ( P/ c' B* p9 y7 ^. wDisadvantages of Recursion, r5 F: W" C) g1 @: Q  F% k
    : K  S$ x5 w6 I; e6 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.
    6 J5 ~. _# n4 D" y/ R1 S8 Z$ E1 h" r
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    / |5 i  U$ T. j# E2 C9 _7 F" h$ f( c
    , @/ Y* J* J3 u8 [When to Use Recursion
    % e9 z- Y3 u& V! o. U$ C
      v( D0 t$ |  x! N; d    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# o- p; `% `3 [" t/ ^- p/ z2 C- f

    2 d5 N8 a5 v3 N' f- U0 w7 O7 Q    Problems with a clear base case and recursive case.; M. ^" z6 ?" r* U1 @7 u

      k0 Z  u- p/ T- c( ]! m1 H- cExample: Fibonacci Sequence
    ; z5 n6 u. x7 a/ c/ ~4 W# j# e/ P& x
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    1 |. \7 g4 y# ?* Y' q: \  w6 ^6 P
      x3 G: ~( J: m* s+ k$ B+ G1 m( X4 ~    Base case: fib(0) = 0, fib(1) = 1
    % p# k3 X; _3 }# t" Z/ e% m, S' G: H& u: J
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! g* l  h: X( F) K! G) z9 [/ s# W4 |. l# q/ }8 F/ I
    python9 n8 I  z/ B# {8 O- T4 Q
    ; g) b0 N. O7 Z$ b; W: \

    ( b0 M+ k  u. O. h. {" Q" V4 \def fibonacci(n):# L4 \$ V  V# O/ C
        # Base cases7 n$ v. a3 e' o( Y$ {- l
        if n == 0:  g: S' W4 t. |
            return 0
    7 g% K8 I! N. k% Z& ]" X( |, S" R6 u    elif n == 1:
    . J+ b5 |, N9 v        return 1
    # p" P( {' j3 W# A& y+ ?    # Recursive case
    6 s' t! f  G( A% h. Z    else:( J9 D) n% {" O
            return fibonacci(n - 1) + fibonacci(n - 2)
    % B8 U+ u9 C8 F1 }1 h
    ) x& a# \4 R) @' q5 E6 F! w4 o# Example usage( k$ P) W& J6 T8 X
    print(fibonacci(6))  # Output: 8+ s  {/ A( w+ v0 g1 T# K, j+ Q
    0 D/ q. z; I) v0 |2 ~3 X" C/ z; K
    Tail Recursion9 {. g. u8 m+ X
    & R8 q' b& g7 @4 y( ^
    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).
    ( p7 R3 ^7 u) k9 o3 I. R& b& m" m  J4 R* k
    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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