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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| 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 " t2 X/ H2 r! l9 b6 ]& V
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    解释的不错
    7 t& h1 `* [+ M) ]. M
    6 g0 r; w8 n" M0 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。* Z/ B( f9 I6 h2 \, _& H

    3 r& W8 P: a/ C1 p" }! x) e- f 关键要素8 _2 r6 o$ O: H; @0 V
    1. **基线条件(Base Case)**% p$ {$ ?- H, v) n
       - 递归终止的条件,防止无限循环  s- c8 k4 C, A" v2 g/ `5 S* x
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    7 }3 \3 C. a: _9 F& r2 z' S
    3 ?; ^% t7 H/ U, i* D! P5 ?2. **递归条件(Recursive Case)**
    - j( J( T. T0 y9 J+ l2 ?   - 将原问题分解为更小的子问题
    1 @/ o6 e5 R. b( W   - 例如:n! = n × (n-1)!. d! e/ _7 E- d9 w

    6 X, I1 b8 |$ }( n% d3 Y' i 经典示例:计算阶乘( R' f& @" K4 b" a6 s1 Z
    python
    ) d9 N" E7 E, Bdef factorial(n):. U6 Y. N/ Z/ x7 p! _* O( V2 l
        if n == 0:        # 基线条件
    . N. ?5 w# L& Q+ q1 I        return 1
    / a" x2 E# Y( ]& ~- w( T    else:             # 递归条件/ a( t1 [' m! `1 V8 N) s4 ?
            return n * factorial(n-1)7 O9 U; y9 \. {/ w% m+ [+ r
    执行过程(以计算 3! 为例):
    " c* n- n' ~8 u/ Q6 Dfactorial(3)
    ) l+ y! `( K2 @8 `; m$ g, _3 * factorial(2)
    3 J' ^5 ~: f( Z$ W+ B7 L: E3 * (2 * factorial(1))
    / K$ Z! F. F( ?: f9 T& Q( z5 p3 * (2 * (1 * factorial(0)))
    9 j8 b+ z" D% ?3 * (2 * (1 * 1)) = 6
    5 @4 Z1 _/ c4 `6 e
    1 _6 g9 ]3 p- v* i 递归思维要点
    + h& y# t2 {; o2 F& L8 _  E. d) N1. **信任递归**:假设子问题已经解决,专注当前层逻辑! J& l2 z* i3 I
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)  C4 d/ D+ f0 Z0 _% P
    3. **递推过程**:不断向下分解问题(递)7 Y' z$ F% C! V- h
    4. **回溯过程**:组合子问题结果返回(归)0 Q0 h( s- Z( k7 C
    . P# D7 B0 x9 |' z; i% y+ h1 K4 I
    注意事项
    ! W. H! i3 u2 ?( w) G/ _必须要有终止条件
    8 V; C( r0 c3 h递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    2 a* w7 P$ K" C: ]5 r- _* Q某些问题用递归更直观(如树遍历),但效率可能不如迭代& G# D2 F  u7 J( Y" E: ~& U2 O
    尾递归优化可以提升效率(但Python不支持)
    " I) ]6 `3 ?3 }- R8 H2 O7 }
    # ^1 Y3 q. P) a' f1 U. r: t 递归 vs 迭代2 I: W6 L) v/ u, \# D2 ]! V
    |          | 递归                          | 迭代               |3 s! C# a1 s" n/ }
    |----------|-----------------------------|------------------|" O5 J( G8 Q# b7 s
    | 实现方式    | 函数自调用                        | 循环结构            |
    ! L5 {, y$ ^0 [* w. || 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    1 N1 ?! l* ]' p3 v5 A; |6 f# y0 r5 A| 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         |
    & p7 B- h% b/ V| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    8 W( i2 I0 A; ?1 r8 [, c! c" _6 ~$ y  e& ]* n7 s, y3 I' H+ O- `7 V5 d
    经典递归应用场景
    + w4 P# f% W0 N, [1. 文件系统遍历(目录树结构)% f6 |2 c* l& {) o3 m# D
    2. 快速排序/归并排序算法/ F3 ?  T7 H- f% j9 p
    3. 汉诺塔问题
    8 W8 \+ X( b- _+ D, c* x8 _* I: M) O4. 二叉树遍历(前序/中序/后序)
    ) m9 t# u& H  ?5. 生成所有可能的组合(回溯算法): q: G/ J  s, K$ i
      D1 @7 i. n* L" ^
    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,, U7 ]# V/ K* E
    我推理机的核心算法应该是二叉树遍历的变种。4 y; M5 N; t+ \
    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 d5 T- C; v* ~1 P6 q
    Key Idea of Recursion6 K/ s" d9 o1 @9 u$ y
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    A recursive function solves a problem by:
    3 ~" h2 K* z- W8 |8 h% T/ n0 s- U, k, O' `9 C; b5 B
        Breaking the problem into smaller instances of the same problem.
    " a' L6 H7 L3 j+ I( e% s: d
    / ^4 |1 d2 z, v    Solving the smallest instance directly (base case).- L# p3 k8 _0 E5 @3 F! s

    : H! M! k6 g8 }, Z8 c! [9 ~7 l, b: N9 ]    Combining the results of smaller instances to solve the larger problem.
    % x5 N; y7 x1 O) @$ X$ E  `# S' F% G5 y: b5 o$ T
    Components of a Recursive Function0 |) C( K/ V. M
    ' ~# ?' s; t- m- X
        Base Case:
    & `4 o$ C; N+ J9 _% C0 |; A% Z. Y9 L) n$ C# J. G6 F
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion., @8 H7 {+ B$ K" ^) G* S
    . x3 W' c8 ?' ^) M$ a; A9 s
            It acts as the stopping condition to prevent infinite recursion.
    ) k. U& i, ^. M% h8 Y3 Q- p, m7 S& m8 x) y+ G$ O: e
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    , l2 [. U. z7 @  T' S
    2 \: R% c; ~& X1 Z: R0 {    Recursive Case:+ {7 Z$ S; m4 s$ R. M/ G( B+ J

    5 z' p$ `: Z  v5 C3 E  w        This is where the function calls itself with a smaller or simpler version of the problem.' c1 i% w/ M3 e! J, N
    ) |$ _6 l. W) ~% c6 i1 W) R
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    0 g4 \: k1 i* G- Q; ?+ R8 zExample: Factorial Calculation) D( Q1 A) a2 n
    , F" u' E8 ?9 b) D
    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:
    1 |, s3 P! q3 q" u9 T( Y; `2 }
    2 w* o- p0 Y: a+ p1 F    Base case: 0! = 13 _( |- ^8 `; f. V! C
    , n, S1 r) F8 |3 C+ x
        Recursive case: n! = n * (n-1)!
    4 k: Q2 c& f5 W" q' e) h9 \7 o. y' f# l$ R  l+ f. [! ^
    Here’s how it looks in code (Python):3 x- |% W( |) e$ L/ x4 u- u  [
    python6 a) _6 ?* d4 W" o
    & f; a& M- ^8 w- j% c! e) X: ?

    / @- j! Z1 o4 p; k. v9 f* Mdef factorial(n):
    , r. u% I5 I5 K$ ~9 `. u) `    # Base case
    - [4 w6 Y1 {6 E1 j" w    if n == 0:( ?4 q# [7 w1 z1 R: k
            return 1( V4 |: \6 E% D4 \2 T
        # Recursive case
    ; u4 u) @3 F# l0 \% b6 F/ \+ Z    else:( ?: C1 f5 e8 O3 p# S
            return n * factorial(n - 1)
    6 x- s+ K7 Q7 @9 t' J
    4 N% d: {7 v2 t7 x5 L  f# Example usage
    * N1 N9 |% `' \  u9 k* b$ Aprint(factorial(5))  # Output: 1206 E+ Z; w/ m0 l* }* B% d4 W
    , B" p' j1 }! q( B# n1 w0 b* l3 ~
    How Recursion Works7 j; `1 H. u. S: ^. z

    - s! O* Y5 K" c    The function keeps calling itself with smaller inputs until it reaches the base case.
    $ A2 ]. R2 D. f( t. p& {) O: ?- O4 }; {/ d, _  C$ z! r- K# a- g' o& r
        Once the base case is reached, the function starts returning values back up the call stack.
    2 {5 E2 @( D% F
    3 }$ ?4 H  q- ?$ h    These returned values are combined to produce the final result.
    - [  m' x, K* y; t' f7 v/ p3 i9 z  ]
    6 y; B4 Z# K1 u/ c1 lFor factorial(5):" {" p3 g& O+ T5 s
    6 J% v2 o$ l. ?

    & x! P' j" S& M+ l- i  {factorial(5) = 5 * factorial(4)
    # s* Z, a" |1 i+ i5 u1 ?factorial(4) = 4 * factorial(3)
    ' C/ q% O& A1 V  W! m  K% vfactorial(3) = 3 * factorial(2)1 m6 b5 o! l% @; l- m5 Z
    factorial(2) = 2 * factorial(1)  `/ U& r  E4 i  K% {
    factorial(1) = 1 * factorial(0)
    ; T  D5 g4 m- J$ _* M0 Kfactorial(0) = 1  # Base case
    : k% y* g( s9 ?) k3 Z( d, Z6 ~0 x% y; W7 q" T& b) Y
    Then, the results are combined:/ W5 u8 L$ u4 h6 H9 f

    % W" ]* k( M$ P9 t* x5 q
    ; v8 g5 G1 J# H; y9 U9 ]factorial(1) = 1 * 1 = 1* d3 c9 b$ B1 ^
    factorial(2) = 2 * 1 = 2# E( j) q" M$ n6 d
    factorial(3) = 3 * 2 = 6, M' N5 s5 t+ S- ?
    factorial(4) = 4 * 6 = 249 _0 O+ M9 Q' }) _' V
    factorial(5) = 5 * 24 = 120
    ) Q! O5 m+ V, I1 ~' Q& H' c8 Y3 [
    Advantages of Recursion
    8 w/ Z( R; R; r& Z4 B0 ?5 t$ p! r. p4 [, x3 V/ b9 a& C3 N
        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).+ `4 q& r9 i$ n; y; h
    & |6 b7 i: z1 j* A4 L
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    : e0 A, s& t8 }* i4 A2 n! [
    ( h$ o- R3 A) a! C# B# Y2 cDisadvantages of Recursion, V- i7 g. M: E4 M  V

    # z+ ]6 p8 O/ y! Q* O! x    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.
    " {( J5 Y) N! ]2 `# l, I
      @7 s* c2 D# V# i' u" @    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).; Z+ O1 g$ m: @( l) J* u
      K5 O. [5 X9 N) m* r
    When to Use Recursion" q( A/ ?. s' y, R9 x
    / ^! y0 C4 ~5 |  ?2 [7 a
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).1 z0 V7 }2 c. s2 A8 c
    & G( n1 P" P  ?  T3 }; o' I. r
        Problems with a clear base case and recursive case.4 r& P) G# k, E
    ( Z' u8 B# l6 y3 J* ?
    Example: Fibonacci Sequence/ q- c. r2 \0 u$ t- B# F7 [6 k; B
    / s% u, u- m: ?0 q, E1 M
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:- W5 v+ I# l. }

    2 b  j/ E/ e; z5 i. k$ O! e    Base case: fib(0) = 0, fib(1) = 13 D5 D0 ?4 P  a- b% P

    1 U  V- [+ M8 f$ Z3 B" M5 o: R    Recursive case: fib(n) = fib(n-1) + fib(n-2)8 Y6 ?& h$ o) T
    ! I! w4 \. j0 j: W' C1 }3 T
    python
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    & J- O' @! O3 c- V* }+ ?7 @. j' I. ~  d3 p, K4 H  w1 `5 O' x
    def fibonacci(n):
    " j4 E8 j: D8 s7 o- e! O4 d* x    # Base cases. q1 c- b! X2 q/ d: C  E
        if n == 0:, V: F# G4 S4 m$ e( x% e
            return 07 n4 E0 s3 M$ L5 B& T7 {
        elif n == 1:8 s7 @  u& k! ~; w6 [
            return 15 }" X: n3 e2 i( A- i$ u
        # Recursive case, E& N/ }) M' z0 R0 W% Q) O
        else:% i7 `+ L* I  R
            return fibonacci(n - 1) + fibonacci(n - 2)9 X1 N* M' L/ ~

    2 v; I0 J/ w$ v  A  R: I% M# Example usage' I5 c0 i% q% z! L/ R* U, @
    print(fibonacci(6))  # Output: 8) q( }  i# T8 e8 g4 A& @* H6 P1 G1 u

    2 \9 j- c, E( d1 K* [; Q1 ?$ nTail Recursion3 G, E0 v2 A/ R2 t

    ( `" P9 f) S0 F/ L5 m, k3 GTail 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 f* G! K" R9 e" T( q8 d, l+ O8 G% x5 a, j# G' V5 N
    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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