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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 ! [' O. p* m9 X- v6 O  A4 O3 z& w
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    解释的不错
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    ) L/ D, @% o4 S. T" v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。4 t( m' o" q: s5 a  Z. W7 f
    - F- L1 N9 }7 i, W- ^
    关键要素
    * ?' B' w) h: j9 N: R1. **基线条件(Base Case)**
    + k7 H5 L* x% q; Q8 P. B   - 递归终止的条件,防止无限循环% |- B5 x, `4 u* E% e3 [9 P
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    2. **递归条件(Recursive Case)**2 S" t: x, {' o$ d; p: K+ X5 [
       - 将原问题分解为更小的子问题
    . V4 e7 M, p* b# {- Q   - 例如:n! = n × (n-1)!5 V! A& H' P6 r( W& i# D" d- z( v
    % @* v/ q+ {5 Q' Z4 N* z
    经典示例:计算阶乘% \! A. E! C7 p4 \
    python4 w8 j2 }) G) Y3 r; @* R( s
    def factorial(n):
    7 B; P* a9 l; ^. i4 h1 T4 p    if n == 0:        # 基线条件3 T' q! A3 x% \3 P  ]
            return 1
    ! k( l' T7 p0 e5 L( o8 K( Q    else:             # 递归条件' A, B# d! w% Y: a
            return n * factorial(n-1)
    1 c9 [; ^# l8 u- o' `3 t8 e; h9 V执行过程(以计算 3! 为例):5 {* F; T% t" z. z  n/ V
    factorial(3)2 J2 `; a' p6 j- L. h/ E, K6 u6 _; M
    3 * factorial(2)
    + F1 x3 i9 }1 W5 I) O" B3 * (2 * factorial(1))* F+ L/ u- T0 M2 d$ r7 k0 S5 m
    3 * (2 * (1 * factorial(0)))
    ' m4 l0 O, a4 |) e3 * (2 * (1 * 1)) = 6/ y, x3 |$ h4 n6 [( b

    ! H- }! ~+ f; m! u" |) v 递归思维要点
    6 _) K+ J* X  L8 I( V1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : @- y' O: q4 ^5 K, V$ o2. **栈结构**:每次调用都会创建新的栈帧(内存空间)5 q+ K7 m' C2 g9 W0 w3 S6 N
    3. **递推过程**:不断向下分解问题(递)* e9 N! ^8 h: r% Y& m% Z* w
    4. **回溯过程**:组合子问题结果返回(归)
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    2 q( ?( N. y7 H, @% J* U, C( A$ ^6 h  m 注意事项
    # }( m5 R  H: D# V# p% l必须要有终止条件
    # L& o  V0 T% R3 z递归深度过大可能导致栈溢出(Python默认递归深度约1000层)5 M" p; V  s# Q
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    & S% f' P- |" L4 t: n尾递归优化可以提升效率(但Python不支持)
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    递归 vs 迭代
    9 b8 K. L  D: @* D, W2 e|          | 递归                          | 迭代               |6 H+ |. C! T" m. X* h' C" O
    |----------|-----------------------------|------------------|
    $ d3 J0 c6 }7 k+ C  R| 实现方式    | 函数自调用                        | 循环结构            |. @- Z. w: i8 J3 T9 M' m( l, [
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    " a( ^/ S& ?( Q! t& }+ ?! [| 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         |2 ~4 `) }) \$ I
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
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    经典递归应用场景
    ! G& |1 i" e3 [# P, t. ]9 O& {+ s1. 文件系统遍历(目录树结构)+ b# |8 q1 K/ x' a/ v
    2. 快速排序/归并排序算法
    ; C. {" E* ?9 t3 P, n/ {: i8 `3. 汉诺塔问题
    9 l1 y; X# k3 u+ v. e* o0 _1 M0 Y4. 二叉树遍历(前序/中序/后序)5 p& P1 B$ t& `  R
    5. 生成所有可能的组合(回溯算法)
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    3 v6 s) Q8 N& a6 o/ 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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    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 x- N" i* b- @6 A8 x
    我推理机的核心算法应该是二叉树遍历的变种。" h" K8 G8 |0 \# 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:3 P" o% E- X. M5 e4 x( f- }
    Key Idea of Recursion
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    A recursive function solves a problem by:
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        Breaking the problem into smaller instances of the same problem.
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    8 P7 O! }  s; Z6 L, U5 ~  @    Solving the smallest instance directly (base case).  J! r, R" A& v/ U, g. c7 p
    3 g: X$ V2 e7 p7 m1 S  l5 N& u6 T" B
        Combining the results of smaller instances to solve the larger problem.1 u4 n  O4 o& Q& }2 D5 x1 X

    & A, J3 y& s, r. b$ f) O" WComponents of a Recursive Function
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    ; g- m2 W0 j. N# n: {3 k    Base Case:) y% t8 _3 H) {
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.2 O# a1 P8 p$ ?" @8 y

    0 g; j$ `* r; U- w0 y; g        It acts as the stopping condition to prevent infinite recursion.7 v2 g) a3 Z! c7 }+ p/ r1 a, Z: o

    3 p) K, W# \6 y$ i0 p3 d0 B! w        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 `) x0 c- \! _- n, p+ B7 r
    * L) L1 Q) u7 V8 F2 }
        Recursive Case:
    ; s0 g3 a$ W1 h1 x1 n# [! [2 }/ X: n. r6 j9 Z% s3 A
            This is where the function calls itself with a smaller or simpler version of the problem.
      X+ Q8 K, f% C# y; d0 d# H! n) }9 V( J# l/ l8 {. n
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    5 Y' U4 o: @0 U, u& L' j( b' Z! O; f4 o' r: [" s. G- b
    Example: Factorial Calculation( \- @9 C: d& I/ j0 e& u. L! g& O
    # h) p: T# o) q: b
    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:5 }8 n6 K, I1 C8 F& r
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        Base case: 0! = 16 U. I* Z0 W3 z% I# n* r( M

    7 u4 E5 \1 [" V  }$ M' O    Recursive case: n! = n * (n-1)!
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    . q# f) z% f- O, PHere’s how it looks in code (Python):* s0 L) @  _. H
    python3 t: T8 I1 Q( @' U5 R/ U+ _

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      c! {, R( \4 x2 }' _) S+ k7 d$ ~def factorial(n):$ k6 M( U9 i5 t) L# _
        # Base case( I/ }7 H  Q0 R" @3 _7 Y' w( {
        if n == 0:$ d' g+ N+ v+ d, a. z' `
            return 1! t# Y- K) V* c- U
        # Recursive case9 h, V! Y# _, |, r7 A: G
        else:
    8 ?6 _$ K' @. q, c% g        return n * factorial(n - 1)
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    2 k# r( e! A$ q7 c- C2 k# Example usage
    . f6 O0 ]4 f8 U9 s" b. Yprint(factorial(5))  # Output: 120
    # O3 `, k9 Q5 w
    1 S+ ^! Y8 F; r2 C+ _3 x7 Q" pHow Recursion Works
    9 C% k( _& a6 c  g4 Z6 h* N  D0 a# r: ?" @* [5 `
        The function keeps calling itself with smaller inputs until it reaches the base case.- ^6 E9 @+ ^) \$ F- ^

    3 }4 A2 C% n4 a- r, N! m    Once the base case is reached, the function starts returning values back up the call stack.. Q9 t' X; _' l% R) N0 F1 x  X9 T

    4 M2 N' Y1 H# L& L3 q0 k' ^. V0 R    These returned values are combined to produce the final result.
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    , @; a- L) W! O9 v( X0 j5 JFor factorial(5):9 r. J0 i& x) h. }! V0 Q

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    factorial(5) = 5 * factorial(4)/ q5 B9 P  G9 g; X# M+ p
    factorial(4) = 4 * factorial(3)
    ' _7 D2 H) Q/ g; O  k1 r2 Kfactorial(3) = 3 * factorial(2)4 X0 l6 j# j# Q/ N# j
    factorial(2) = 2 * factorial(1)
    ! @" p/ J, S+ r( E3 Ffactorial(1) = 1 * factorial(0)
      q. I: w0 j7 }factorial(0) = 1  # Base case
    9 e$ w! _  M1 ?4 ^. u/ r$ V6 o! R3 y- E/ J4 r6 N5 U2 z: n
    Then, the results are combined:
    4 O  x+ E+ ^' m4 T" H. Q( `1 }1 h. `' I0 R( u

    5 n( [1 c1 ?% ^& _, kfactorial(1) = 1 * 1 = 1
    : w" F$ J0 @* w  m3 R- j# [' xfactorial(2) = 2 * 1 = 2
    9 u! m: H+ X) E* f' d$ j' F+ ]factorial(3) = 3 * 2 = 6
    3 N  ~! O# f% t' e3 Xfactorial(4) = 4 * 6 = 248 a. y/ y3 V% s8 i- r4 C3 w' t
    factorial(5) = 5 * 24 = 120# L% e' [% \' M* t: }( L# }
    # S2 {- ?% v4 o* }, {
    Advantages of Recursion
    ; G( d  L7 Z; z; v+ r" X. m4 F
    6 q) z' Q4 o& I3 j; `: H8 i$ \1 F    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).
    " ?2 D7 U" c9 q8 {- S' b( I+ c- S* |/ x- F4 b5 H. j) [) y# ^
        Readability: Recursive code can be more readable and concise compared to iterative solutions.9 f6 i2 l1 ?; f7 G% ]
    # n8 Y# V; {( w( u
    Disadvantages of Recursion
    6 x! c- A: o8 P. P4 }! h8 W" B9 ^- f$ j2 T1 O9 S; u0 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.2 l2 F; t! O5 h) Y" M
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        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)." @1 q/ s1 a, m* }2 H1 @% g6 ?
    1 ~/ _  m: M5 y
    When to Use Recursion' k+ p5 n8 |( I1 M, Q
    # J, m2 H* T3 i& j7 X' E4 J. l# J( B
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    4 t1 a. \( x+ c
    ) R' I% T  Z& o% {    Problems with a clear base case and recursive case.6 j* Z8 T, u& R0 G

    ; X6 o3 R; |0 h# C% y2 `! N! {Example: Fibonacci Sequence3 E! z$ n. }* d5 o- C* ^' Y9 L
    4 W$ w4 r$ V& X/ ?
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    5 p, ]2 e$ f. i- n% G* [$ G. F8 k$ |5 i
        Base case: fib(0) = 0, fib(1) = 1
    & b+ G  C0 t6 ^9 _3 m
    9 [1 I9 Z2 N" T4 [2 @! T    Recursive case: fib(n) = fib(n-1) + fib(n-2)/ u7 V1 j4 p% B, W* @3 I  \1 c
    $ @& J, {8 u! U% A- _
    python1 K1 H$ q+ r* i$ [! T8 R

    : a3 X4 s8 F& r$ L9 @
    9 [- V" s# z5 h# gdef fibonacci(n):
    9 x. G4 G6 i1 q% \& q    # Base cases) [. v* }7 M. w8 t  u9 U2 Q
        if n == 0:; R' R  |# W9 Q' t( o6 {" F
            return 0, T( N, S. d7 h( |; V
        elif n == 1:/ T7 E5 B1 M/ Y( [, c* y; T8 O# G2 [
            return 1
    1 w# D2 v& T. `# o* N, y    # Recursive case6 N5 E& S% Z1 q" Q$ Y
        else:4 ]4 H' F$ X4 |. P$ R" z
            return fibonacci(n - 1) + fibonacci(n - 2)
    1 j( d+ E* _% U
    ' w9 Z: X% q, |- P# Example usage
    ' H8 t7 g; F9 K" E  Gprint(fibonacci(6))  # Output: 8% o8 }8 F+ R+ f2 t1 M

    " V& S2 ~- v$ J) f$ q. Z  O8 eTail Recursion
    + V! R8 Z3 ]: ~3 r' s
    ' W' _+ i. G; j( 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).
    ) u! q8 x! J3 x. S. Q# z" Z1 i6 d# a2 k! L0 e$ L- }4 T# O1 J, ]
    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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