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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
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    7 x- K* V* X: \# M! a0 [' x" S解释的不错
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    ) J5 e' j6 u5 F3 k& I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。8 c- f: \, x2 t0 M: w3 m
    # \" O5 I$ Y3 |% V+ x+ T
    关键要素
    ) {( C! r! R1 v$ q: N1. **基线条件(Base Case)**
    $ e9 b) I9 o1 k/ A5 J, Q/ r2 G, F   - 递归终止的条件,防止无限循环
    9 a0 O8 W* t; N! d  C+ a   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    $ k! N3 {7 F8 i
    5 s. M7 `, x3 d" m2. **递归条件(Recursive Case)**) x( W- E% O/ C( w. x# N# Y
       - 将原问题分解为更小的子问题
    - _: Y* z8 c% b" _+ k- T  Z2 V   - 例如:n! = n × (n-1)!8 |, q/ V( D& e1 m
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    经典示例:计算阶乘1 R/ u& [% f: Z' i* t
    python3 X. K- S  V2 x4 [( Q
    def factorial(n):% _9 b4 N9 [% w, e+ t! X2 s
        if n == 0:        # 基线条件. L" t" w/ d) Z; ]6 |2 X1 t! L
            return 1) P$ k6 X0 O" g
        else:             # 递归条件
    " ]  E- l( O1 d) j        return n * factorial(n-1)
    , |$ ~3 W: g5 G2 S* p执行过程(以计算 3! 为例):* A$ v0 N" b8 ]- K- F5 B  [( l6 k
    factorial(3)
    ! E3 T, f; u! M) T0 W3 * factorial(2)
    ( h2 R7 ]( F  z% ^% K% A3 * (2 * factorial(1))4 u1 R) j$ D" {, m" Z# x, b! i) S
    3 * (2 * (1 * factorial(0)))( ?) U$ }0 P7 ^1 j8 |" e
    3 * (2 * (1 * 1)) = 6# z# u5 N' l5 A; E5 t$ b( B0 R8 y

    & r8 U5 x# e) S' L8 i4 f. x 递归思维要点1 J  w7 R/ }1 d8 I2 Y
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑2 E  v' q! h3 c, a* r  E
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    4 F: ~/ s# ^7 x+ l9 h0 v3. **递推过程**:不断向下分解问题(递): e- F; `: a5 W
    4. **回溯过程**:组合子问题结果返回(归)
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    + @% t0 j; h! J  a$ z/ p 注意事项( M9 ~& u& O3 V0 G, T9 t; a7 Z! [# d
    必须要有终止条件
    4 R, L" N' A  U9 W- q- C. U递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    , `7 Q$ |* I! d6 Q' l+ G4 I  v某些问题用递归更直观(如树遍历),但效率可能不如迭代
    / `" q: ~1 H" m+ w7 ^. {尾递归优化可以提升效率(但Python不支持)$ q) r, n; ]% z9 J  t" ^( j

    : N+ ?4 ~9 Z' y& M5 x' S 递归 vs 迭代! w! ], i" |' v2 F( r! t
    |          | 递归                          | 迭代               |
    5 \( c9 P( [' `" _: a- l|----------|-----------------------------|------------------|
    % M# y* ?  g# J| 实现方式    | 函数自调用                        | 循环结构            |
    & W/ t* E5 l% L  }' s| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |  {" s- P  U8 d
    | 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         |, N6 O4 N/ W$ O  o' |7 P1 z& k
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& S4 _* f- l, e+ e! [7 }9 P+ E

    : X( {4 D7 [1 x 经典递归应用场景& R* b9 y4 ]4 T
    1. 文件系统遍历(目录树结构)+ b; I- a) n3 G! }) b
    2. 快速排序/归并排序算法( s$ p- r: l- v3 B7 Y, g
    3. 汉诺塔问题, H1 K) ~1 l" l. X
    4. 二叉树遍历(前序/中序/后序)$ C5 j. h  q; y' u
    5. 生成所有可能的组合(回溯算法)" {5 U* A8 T4 \; U% Z; v
    6 K- r+ M% b" I3 ?. Q5 i
    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
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    昨天 10:43
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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,
    ; j7 M6 J1 [$ M3 M- R我推理机的核心算法应该是二叉树遍历的变种。: @5 J. X% o  h
    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 f- o6 s, x0 y4 `Key Idea of Recursion
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    1 K+ Z1 t. j  z4 n. W& A* h  WA recursive function solves a problem by:
    6 s4 ]- s' {- B, d  l2 }! q. J3 {6 n2 G
        Breaking the problem into smaller instances of the same problem.1 p4 n& R1 f) ?  o4 c& j6 _
    3 W  K, ]" ?3 R. b8 Y
        Solving the smallest instance directly (base case).
    ' |- w# ~' A; n6 ?1 }1 O9 o3 Q
    ! j, j* m% B' Z% w" k9 F! H    Combining the results of smaller instances to solve the larger problem., B. z2 a1 t+ F4 t9 \/ E
    8 H! [3 g" I) \1 j  c4 b( B
    Components of a Recursive Function7 Y; W, I% O4 ?) R1 V

    + c9 d9 Y7 V, G$ N; `    Base Case:
    + s; P. P2 W% q$ a8 ^+ z/ C) g( D
    / ^, [# C) K1 M6 V        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
      C! t, z8 r% R  a% g
    ) A+ B' I' Q- O        It acts as the stopping condition to prevent infinite recursion.
    : ~( S: G8 R2 h# U* d, D! V5 V) N- K
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ; _9 B: ?/ u! P+ }" Q
    ( ~& [6 |# ]6 f  `( s- F' U1 L    Recursive Case:' A+ N$ Z6 N8 v2 D+ v
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            This is where the function calls itself with a smaller or simpler version of the problem.
    $ y$ i/ h+ ~1 }- x9 Z' c) {; l8 \# V$ s; d
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ' i+ {: Q3 ?: P4 T. y- I, K9 A. P; @1 H- w+ l$ p! W# z
    Example: Factorial Calculation1 _% b6 t0 x+ B2 O
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    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:, d3 G: m: c* d
    " n5 Z1 t+ B$ ?) E$ s0 U' l
        Base case: 0! = 1
    - R4 ]# u$ k) D# L' C$ I& [( v1 `6 q/ w* T1 n: N* o6 c
        Recursive case: n! = n * (n-1)!' i2 U9 j$ T3 L+ o
    ! u- m- g. V' J' a
    Here’s how it looks in code (Python):. _. l( m  u% c* `1 S5 V
    python: m# Z7 H9 X+ L5 e- Z

    & u$ \4 h$ w; w; x+ S  _; W. f% r
    def factorial(n):
    1 \* [) E; s5 m9 I) Q4 u    # Base case9 q3 V  L8 [3 S
        if n == 0:4 p6 G* `( [: C* b, I  }8 O
            return 1
    - S" K! ]4 d3 f  o2 t0 ^    # Recursive case
    1 ]5 a. m7 U+ w1 O8 T( z    else:
    0 P) B% x8 }! ^4 q6 Z        return n * factorial(n - 1)
    $ ]$ [# U4 {5 k! V1 A1 R. \
    9 x3 Z: S# \) s9 `, \5 f, ^, x# Example usage
    % j. g3 D& m- e7 w; S8 t3 K/ ~print(factorial(5))  # Output: 1207 I/ u6 v. X/ @
      T0 F4 k) c2 H4 }$ C& ]
    How Recursion Works" a' X! O& G4 ]* H

    4 [  }/ v8 C) I    The function keeps calling itself with smaller inputs until it reaches the base case.$ M0 r- P9 K' `2 j; s

    , f4 r) s- z) V1 L( G5 O! j    Once the base case is reached, the function starts returning values back up the call stack.
    " t9 X( x. d) r, |" W" Q# B8 e$ ]2 D% M6 \5 s! b6 a7 }
        These returned values are combined to produce the final result.
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    : `8 s4 J6 B* R7 K" Q7 EFor factorial(5):# C/ O* g- H1 l6 R/ q0 o4 R; u! z

    0 g+ K' f3 \- x8 n' \5 S9 @
    ( J: A2 h  ^1 J" N, C+ y. T/ [$ Yfactorial(5) = 5 * factorial(4)
    " X! X8 f! b- `& X& A0 I( hfactorial(4) = 4 * factorial(3). U3 C7 M, V( g' p1 l3 ^0 J0 w
    factorial(3) = 3 * factorial(2)  \% l. y( K2 Z( v. A9 i
    factorial(2) = 2 * factorial(1)
    " N' T. u# V" H- p! q! L" xfactorial(1) = 1 * factorial(0)
    1 m: {9 {% N. }" H& Q) w- Z# V& @( Ffactorial(0) = 1  # Base case0 D. F: I7 W' ~7 T

    6 J- I$ c2 p' B( AThen, the results are combined:; ~" g, G5 n$ @2 t0 `
    ) ^- b' j' u, i' }! Z
    ! d/ q" f5 ?& f' f( u
    factorial(1) = 1 * 1 = 1
    ( H. f9 |) o9 [7 I8 C8 Afactorial(2) = 2 * 1 = 2
    & q, h4 L: Z6 @9 i6 O# D) R0 a3 Y) T$ Gfactorial(3) = 3 * 2 = 6
    ' \, e% h3 O% Y; m" k9 e/ a1 ffactorial(4) = 4 * 6 = 24, b! E# C, i* k6 z7 `3 X
    factorial(5) = 5 * 24 = 120
      H% N7 @9 P3 S8 [8 P- J( P
    ( `/ }6 B0 n* E5 L# g- q) ~4 h+ pAdvantages of Recursion
    6 E/ L: f9 N2 I' v+ |; g& S" @" f, O3 D8 N3 X' Y5 N8 x
        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).
    , R6 p7 z/ h) D# z$ c. M7 Q$ f+ `* b; P. J# E7 L
        Readability: Recursive code can be more readable and concise compared to iterative solutions.8 O: n) H" D1 \$ k" B, k
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    Disadvantages of Recursion0 M: A* T. n0 L& f2 ^, j1 l

    ( c: q  j- c1 F; i% ~! Y: ?    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.4 Y1 B# w. V% P: B
    $ J; k' i! ~- y) A
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).; r' |* u8 e- r1 w1 j. L" b

    1 i8 `( X, X4 k4 MWhen to Use Recursion
    9 h9 Z; t: J0 _* s
    8 ?3 ?, [. H1 m4 @6 a+ f: O    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ! f. Y# ^! g- c; T1 T# U7 j1 w/ c' k) M+ {& k
        Problems with a clear base case and recursive case., N1 _9 n) s, [" M/ Y6 t

    * J4 h5 l8 o3 c% ]Example: Fibonacci Sequence3 ?  l0 H- m6 Y2 e% _: L7 v" Q! f

    4 t  R9 ^$ F+ L9 l; {/ a) G7 E9 aThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * a4 b; N( L7 q" p% _" E
    8 @6 c; \5 z5 X* T9 a    Base case: fib(0) = 0, fib(1) = 17 z, H& I, {# k9 o; A

    ) w3 v5 E, i9 ^; u' ^    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    : k1 c( r5 x- N4 Z: T* @python9 u* \$ m2 W$ B7 ^! O  S: [# u) q

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    , i7 s1 l3 W( w* x5 P: Kdef fibonacci(n):" j- Y& h5 w( A  Z& X0 B+ v- o
        # Base cases
    4 L( \7 E1 R! w3 @* y    if n == 0:8 s7 W; \8 j; h
            return 0
    ; a. v# t. x* h    elif n == 1:
    " ^" B9 ?- |3 x  l        return 1
    " E  z: s! Q/ P, t! J8 H    # Recursive case6 L$ }' y7 m8 a- C! C4 V) W4 b" P/ ^
        else:
    9 ?# ~! d8 l0 J& P' S. s" A        return fibonacci(n - 1) + fibonacci(n - 2)
    ; D5 J3 e4 O! l6 @' y. ]# _! [: S3 t0 Y. Z9 k
    # Example usage
    + w+ Y. o. ^2 }) n) a4 Rprint(fibonacci(6))  # Output: 8% V+ ~5 ^6 ]7 o) O
    7 V- y( A; t, d: b8 U) Z& G
    Tail Recursion( y% r, l8 D, E5 e5 u! [

    ! F' F7 O/ @! z% A2 ETail 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).& s3 P' _0 a# Z: P
    : b) y! _! Q: P# c1 y: g
    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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