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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
    开心
    2025-9-8 05:08
  • 签到天数: 3 天

    [LV.2]筑基

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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 1 f1 e* u6 E9 b. L4 J6 E
    ' E6 p/ R+ h' I! Q
    解释的不错, Q3 a+ H$ {, b5 h; ~- o

    0 i3 W9 o' u& V/ r4 `* x( 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。1 c8 a7 J0 X8 |9 m% r, {8 s
    + f8 o0 S& l  y, ]$ _
    关键要素
    6 a, G7 Q/ @/ ~* O$ L1. **基线条件(Base Case)**
    ) i( a( o- g1 x* |   - 递归终止的条件,防止无限循环9 t! H' ~% R# B7 `& |$ R
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    7 Z" J+ G# b( d4 \+ O. K+ J: v  _; G1 S4 x$ B
    2. **递归条件(Recursive Case)**
    " O0 u3 h% I, d! v   - 将原问题分解为更小的子问题
    $ P( T8 ^& }& M2 b   - 例如:n! = n × (n-1)!5 S. ^! ]0 Z( u; }8 x
    % p+ W2 m6 [4 T% H
    经典示例:计算阶乘
    - `$ A- ?$ M1 e. j1 e, m) v" |python
    ( F, W( k$ @7 W* S- |def factorial(n):# j/ S8 B* i0 }- b
        if n == 0:        # 基线条件
    1 o4 F% F# `1 H+ M& B4 a6 B        return 1) J) B+ T2 U- l& g  K" Y
        else:             # 递归条件: o" _7 g" x; `* [5 z
            return n * factorial(n-1); C9 o7 n' q6 ^; [8 |
    执行过程(以计算 3! 为例):
    : j9 ^+ h+ L( G! Qfactorial(3)" _3 t, W& T$ `; `( s. ~3 N) a
    3 * factorial(2)
    # e  N6 ^" @# _. \) O8 N# h3 * (2 * factorial(1))
    # |4 ]' o7 f, ^' w4 i& b! }3 * (2 * (1 * factorial(0))): q2 D+ H% R, R1 n4 j0 f' e; n* U
    3 * (2 * (1 * 1)) = 67 b9 H3 N* P8 E3 W" p4 G8 \
    : n4 l4 G& ^: o* l/ ~! h
    递归思维要点% {8 h: V$ W  ^" a/ V5 I5 j
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : T0 @/ S: b# }# W9 F5 g9 u/ [; T2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    3 y! {. P' g& m% r6 |3. **递推过程**:不断向下分解问题(递)' y3 a7 B, r. S
    4. **回溯过程**:组合子问题结果返回(归)
    6 N1 g% b2 J/ r) X, I' D7 [
    + i2 ]2 i  m6 F 注意事项0 h: q% U; E, [$ o" C+ m9 v  ^+ X  a
    必须要有终止条件$ P8 Q! }6 d, i4 Z; J+ q
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    3 w# ~6 R, F, h1 U/ i某些问题用递归更直观(如树遍历),但效率可能不如迭代
    0 ^0 j& ~$ G8 |) p" X  B尾递归优化可以提升效率(但Python不支持)1 e1 ]# T8 [% o$ [, G2 p0 w+ L
    & W2 I. e' J. U4 m% e
    递归 vs 迭代! ~: c1 Y6 W% p% Y( Y
    |          | 递归                          | 迭代               |
    1 \. J+ q, M  U  k1 W/ M. u- [|----------|-----------------------------|------------------|
    ' u9 }% B' f- j. O, L| 实现方式    | 函数自调用                        | 循环结构            |
    0 k" M' U7 G  S( U| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    1 o% [5 k6 |' J8 v2 X| 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         |% w( W6 w' Q, R+ E
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    1 G/ B  C1 o+ f8 A/ p" [7 N8 |
    + W, K5 {$ S6 t# _0 N/ Y3 N, a 经典递归应用场景
    ; u$ j% t: d- ?' j1. 文件系统遍历(目录树结构)3 u3 y9 w3 g3 t! [
    2. 快速排序/归并排序算法
    6 M) k6 B5 A; s6 g: f3. 汉诺塔问题
    % e2 h. N& R3 ~( ]& T4. 二叉树遍历(前序/中序/后序)
    ; v( s( D6 D8 q+ z/ h5. 生成所有可能的组合(回溯算法)
    5 f1 v1 c  Z8 ^* D$ P7 p3 ~& Q3 x4 _6 b! b6 X2 g, h
    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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  • TA的每日心情
    郁闷
    5 天前
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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,
    . u& @/ P* f( w4 s我推理机的核心算法应该是二叉树遍历的变种。6 s: o+ z8 e( D" }2 G' C+ P
    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:
    2 l; G8 t# H, R, ~Key Idea of Recursion, t% ~! l# q( A" a2 D) |& `# ?0 }

    & s% S$ P* l% @& x/ }A recursive function solves a problem by:
    ' u) f1 \! p9 f; O+ Q; X! C! x, s0 ^( G; Q
        Breaking the problem into smaller instances of the same problem.. f2 b$ |3 P! j$ n" X
    " t" ^6 g6 L( D# y+ a- p: B
        Solving the smallest instance directly (base case).
    " b2 g0 m, R) H  C: F) G& l$ |' x3 A7 l  z% p
        Combining the results of smaller instances to solve the larger problem.1 D+ S* t2 O# x

    8 Q( x7 N  u8 m( CComponents of a Recursive Function
    % Y6 z( T0 N2 q* t7 L  r; ?& `4 ?8 c: U6 p, }" O7 S1 v$ F5 R
        Base Case:* \" b! s3 a1 e8 \7 o7 D- p3 j* e
    ( _1 G& o# Q  @) t: a6 c
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.# q) K; s# A2 C1 L

    * t9 a$ i0 P% e1 Z        It acts as the stopping condition to prevent infinite recursion.5 t$ F. |# Y8 L$ b

    3 D. M* M0 W) ~& `( M7 q! h$ o; {        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ( Z2 H  b2 G* H$ o! e0 o( s
    ! j6 o, b2 J# u' v$ E) E    Recursive Case:' L( E: e0 @7 j

    2 V+ b- ~! R  S( x/ E  i. B        This is where the function calls itself with a smaller or simpler version of the problem.! S* M5 i* O- o: v+ V$ A

    0 c* s4 ]6 u# j4 _* U        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." R+ L: y  [$ j0 Z7 [
    ) Q* Z' v3 F0 d+ ^+ P% ^, z* y" a
    Example: Factorial Calculation: H9 Z" u6 r1 x6 e' [0 |$ n

    9 ]+ F9 ~* y0 B$ R6 NThe 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:
    9 I+ o. U: A; m2 P  Y* u- A. Y, y# m: l; y$ `! b
        Base case: 0! = 18 n0 m* h0 j* J0 Y% u

    / g$ I# z6 b" i, o. h9 P8 p" N    Recursive case: n! = n * (n-1)!
    0 }3 y0 a! R9 z- g; `
    7 P) |9 F8 U5 w6 q6 e# n3 Q- b5 LHere’s how it looks in code (Python):% ~1 O  }! D8 c
    python* v5 I8 F6 f4 {* C/ L, i( R
    % ?  {4 ^1 |) D3 J
    + B3 j) A/ ?  M9 w' I, R
    def factorial(n):+ k, a' i8 X) x/ C
        # Base case" S* [% V0 Q4 o% {- o1 r. t
        if n == 0:
    2 V$ R, }2 t9 {% s' M% E6 }' m        return 1
    ! |7 V% U( E+ t4 r+ _7 }    # Recursive case
    - ?  u* c. V: T/ D5 Q  `    else:5 V7 D* r0 `) Z) e
            return n * factorial(n - 1)
    # d, T0 Z- S6 q8 ?- z5 N7 F
    2 }0 |: d. Y6 ~. A# j5 {/ D' ~0 Z, F6 `# Example usage
    5 c) j% c: Q+ I1 h7 j0 Kprint(factorial(5))  # Output: 120/ e+ V) _7 S, ]6 e0 ?: R

    $ T/ M4 |4 W/ K; ?+ n, Z( Q; fHow Recursion Works  j$ \& S6 n7 e! ?! }% d0 {

    $ k" O: g( j9 q! x- v    The function keeps calling itself with smaller inputs until it reaches the base case.! k% L( `( H% q& f
    - Q- }* j' l+ R6 v$ }
        Once the base case is reached, the function starts returning values back up the call stack.9 ^. c1 `  {6 m  D4 C3 Z
    : y0 _, P: e; @  ^9 r
        These returned values are combined to produce the final result.
    7 W4 n( B5 B! d2 o/ v$ F: ?0 B0 W) A  }' o! c
    For factorial(5):
    4 h2 z) u, Y9 @* B* Y3 B  s, Y' j+ J5 G+ t0 T! v: g: d6 o2 Q) h1 z; F
    % \) e8 j2 p1 c8 w* \& N$ i2 A
    factorial(5) = 5 * factorial(4)
    6 T8 j6 j( m: _8 b0 ^3 Wfactorial(4) = 4 * factorial(3)
    2 W2 s5 k; H! u% i. Q- J2 Vfactorial(3) = 3 * factorial(2)8 |9 _  `: R4 ]/ M! A
    factorial(2) = 2 * factorial(1)% z' l9 S* v3 ]7 H, m& `+ Q
    factorial(1) = 1 * factorial(0)
    8 y$ u  W' K, E+ g) @/ wfactorial(0) = 1  # Base case
    ; K1 L9 n4 V7 t& s: W" O9 y. E5 T# \8 Q/ u
    Then, the results are combined:
    3 W9 _+ @5 E: R5 b; ~- o" v8 E: [4 G# x; |

    ; L4 e3 M: @  U5 n' C9 ffactorial(1) = 1 * 1 = 1
      V9 J9 S* Q, O+ Kfactorial(2) = 2 * 1 = 2/ i0 j5 r) l$ Y1 {: t: c0 L: H2 L
    factorial(3) = 3 * 2 = 6
    4 y1 S& H/ `; Gfactorial(4) = 4 * 6 = 24$ W4 k  M4 ]3 t4 y. o$ h
    factorial(5) = 5 * 24 = 120  x6 Q1 @$ ~, ^( G
    5 b. C8 F7 [% h, n
    Advantages of Recursion
    % V5 N. T& m! G% |4 h7 e0 _+ H: a% M5 |+ k+ f, Z+ y! D/ |6 q
        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).
    . |3 B/ M$ I& Q8 n% I# c4 H
    " X2 k. \1 U) [, q5 Z# [$ ]    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ! T2 C0 U  y# X, e+ z" @9 [
    6 S# ^2 \0 I, }& BDisadvantages of Recursion3 O- {1 C# h' q8 K7 u. E

    * L/ m& c: J, o. h    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.  @% }+ K! d& m& H
    - U% \/ S  ]8 S5 w! e3 }: g) h' R
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    5 c% y( i! O) U
    / L( B5 D& R; u1 P% HWhen to Use Recursion
    : Z" [+ @6 D' r/ b# ?1 f  b/ N: d! Q- l
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    # N8 K4 s# j) T# B
    6 q# q! N/ i% B) e    Problems with a clear base case and recursive case.+ g& `3 K! V% _* g5 r
    ; L& u( r+ B) b; E9 Q
    Example: Fibonacci Sequence
    . ~1 f% P+ P. r. K/ N
    ( x: X7 \& }. ^The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:; F3 R$ L0 F2 g# ?
    1 e$ W- U. z% l+ L: `% F2 J. Q6 @
        Base case: fib(0) = 0, fib(1) = 1& `* T# R+ u! F/ f( N( g7 y8 q, Q

    ( r+ ?; j2 x/ u0 W- P9 t/ b    Recursive case: fib(n) = fib(n-1) + fib(n-2), x# {+ A1 \: j
    5 X- w$ B8 o' g$ C' q+ C
    python5 F% H) c3 O0 {, J; k" ]

    3 o* l) \4 a' b) o
      ?* ?  H+ D% Qdef fibonacci(n):
    % b2 Q8 Q& h1 I3 m    # Base cases
    ; Y# U- A5 X& e/ k9 f6 O    if n == 0:
    ( W1 A4 h' G- P, x8 T/ v4 m        return 0
    2 o  V' c( |2 M) x. H4 g    elif n == 1:
    ) }" i; v# C% l# c/ Z+ @" ]        return 1. ~( ^' ?4 z0 x/ B4 m! r3 t: L
        # Recursive case
    4 {" t: q3 _' j: _- x    else:& G* b4 F( b! [, F3 C1 J
            return fibonacci(n - 1) + fibonacci(n - 2)" w; H3 F5 y+ @+ s- f
    6 v! B. v( a2 r$ \2 \
    # Example usage( K; R! }6 m6 U* ]
    print(fibonacci(6))  # Output: 8& t2 ~# J8 ]0 D
    " K2 i+ r. `, {
    Tail Recursion) i: u1 P: U/ p. a% C# d

    ! R6 J7 g- Y7 o" B- X5 WTail 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).
    5 }4 B( `3 B& y  q( C+ g: f2 T$ ?  Y) x3 F$ m! [
    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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