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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
  • 签到天数: 3 天

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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
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    解释的不错
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    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。
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    6 d( a2 g7 R& g; F  O+ L 关键要素! |; z& s: ?1 X8 y) n, m
    1. **基线条件(Base Case)**1 L; o: t! J) m4 Y4 K+ j. w6 o
       - 递归终止的条件,防止无限循环# l3 m7 ?+ S$ Y' e, ?7 ]7 {, O
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    $ Y# D) g; N% ~/ B( Q. ~( J
    5 t  D5 y3 D( C  Z2. **递归条件(Recursive Case)**" B7 `7 Y$ U/ R: n: S) A$ Q
       - 将原问题分解为更小的子问题1 q' S7 I) [8 Q. ~! b, u% |% V) y
       - 例如:n! = n × (n-1)!+ c: b& W$ i7 [
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    经典示例:计算阶乘2 w% R' r  {# ?, m3 |
    python
    ) h/ e- }5 ~$ A  {! n2 L4 i( U( ?def factorial(n):7 C9 M9 _! u9 c* g% V
        if n == 0:        # 基线条件
    $ R: d3 L- r# Z2 C        return 1/ J: k/ U2 C+ Q# ~! E7 C
        else:             # 递归条件5 J3 @! U; V( h
            return n * factorial(n-1)
    6 I8 a/ Y( ^9 l9 k& B4 I) t执行过程(以计算 3! 为例):9 K4 J7 O/ r( |. ]" W% j
    factorial(3)
    & Z! X4 I& ]( N9 `: k) M8 |+ T2 B; k3 * factorial(2)2 `( m1 w) C" m2 L7 \) ]
    3 * (2 * factorial(1))0 a6 k; ~0 K, s" w/ ^6 A6 N0 U) o
    3 * (2 * (1 * factorial(0))); `3 x  T, d% v1 d
    3 * (2 * (1 * 1)) = 67 y9 p& o1 x: ?7 f
    % I4 ]* Q; w4 ^
    递归思维要点# l- j; r4 P$ A; k
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑1 L- _0 ^, d; d  V% w7 {8 t; W' W3 Y
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
      {: i& W8 D6 p& u. Q* W9 m" H3. **递推过程**:不断向下分解问题(递)
    4 A( B# }+ g3 g" j) I3 Z1 Y' ~4 d4. **回溯过程**:组合子问题结果返回(归)
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    注意事项
    0 K( a1 ~) {! E: q必须要有终止条件# V; f" A3 n# G2 f7 p2 d  d
    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+ _- ]- u+ u: `5 X某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ! `* L. W! ]. x( E尾递归优化可以提升效率(但Python不支持)
    # i2 Q3 |4 Y. q1 f  w
    " s6 [% d- Y7 T8 Q7 h5 A/ F 递归 vs 迭代
    ' P- k: O9 c! y) p$ a1 x$ W|          | 递归                          | 迭代               |: p: X, t: F2 p3 A4 e& l3 h0 y
    |----------|-----------------------------|------------------|# p9 `; X) Y) K: f0 C" E) V) K
    | 实现方式    | 函数自调用                        | 循环结构            |; T0 f; W3 a7 j( v6 K1 {8 R' V2 k' k
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    0 V7 T7 L; D0 i7 U6 T0 W# V| 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         |9 a9 ?- O( [! ]
    | 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+ m, l. R& j* _7 P
    1. 文件系统遍历(目录树结构)% e5 P$ N7 w- S: v8 b5 m
    2. 快速排序/归并排序算法' _7 O% i2 M6 r
    3. 汉诺塔问题" i! F9 ?% N& }+ ?% p
    4. 二叉树遍历(前序/中序/后序)* H2 d/ ^; ^+ S6 s+ n  g
    5. 生成所有可能的组合(回溯算法)9 c9 x0 p% @8 E, z: ]" n" D

    ; w; K. r2 ~/ a' B( o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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    昨天 11:23
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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,
    8 h' l- l" X2 A( p2 l. `7 ?我推理机的核心算法应该是二叉树遍历的变种。- ]' }/ C" T  E! S) s
    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:
    9 E( J$ d* y3 u( E% N' `- j- nKey Idea of Recursion3 d! c, n6 ^* {) C( t0 _) r0 y

    : P+ F3 O0 M& a$ dA recursive function solves a problem by:
    : Y& y9 L2 z: A- Z& F' p$ u$ x9 i$ C
    * [$ b" g7 {7 W+ r) ~! u    Breaking the problem into smaller instances of the same problem.
    % ]1 H! S$ B$ x) O
    # Q& N. c) c( {' D    Solving the smallest instance directly (base case).
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    $ h: ^0 o6 m3 s/ `/ u. c4 J- j    Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function
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        Base Case:( u, Y) N+ B$ w( }' a9 g

    . i; P4 r) M4 N- O        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.  m) p* X# y% u

    & F5 U1 m  J( ^        It acts as the stopping condition to prevent infinite recursion.
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    ) ?0 b& N" t+ j3 ~6 U1 k        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:
      o$ x, G7 c" E& G  |) H
    # S* x/ _6 z! e, N! [$ F        This is where the function calls itself with a smaller or simpler version of the problem.
    2 P. h% E) S/ C6 c$ J
    & _: b. Z3 C" T+ Y8 P$ T; {) w! {7 D        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation1 D/ T' z: Z* s, m% g. S6 K1 d
    3 _" H+ x# X9 Y6 ~3 f8 T, _% ~
    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:
    3 \" d) _! }: }* f7 u2 L! S" C: h! ~! _: ^% {- u2 V' t
        Base case: 0! = 1& r; r' K+ L. j( Z6 x# n

    3 s. R4 y& N5 X    Recursive case: n! = n * (n-1)!+ q# G/ t& Y4 T
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    Here’s how it looks in code (Python):
    2 K& f( E3 u- |! ^  ?1 b. Ppython
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    def factorial(n):+ z% y5 N7 ]  b- g6 Z: o, j
        # Base case8 I* e" E# h9 w; ?9 q
        if n == 0:9 D& E4 X# S/ K, R$ G! x: `
            return 12 e4 X; D/ S4 G, N5 y8 t
        # Recursive case
    3 O, d' k) e8 P3 h, x8 B0 D  l    else:
    7 ^" H  Z) U% ~! y7 c        return n * factorial(n - 1). e" r7 K" X. y/ Z- V5 k

    . l7 a; O/ ^; [0 n5 N8 [( @% _# Example usage* h& ^3 N" Y0 q
    print(factorial(5))  # Output: 120
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    6 I0 H# p; h! o  V, F7 LHow Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case." ~6 O0 v* L3 r( A, o/ t

    ) s* }) W( c9 O5 ]    Once the base case is reached, the function starts returning values back up the call stack.
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        These returned values are combined to produce the final result.7 `% p: F3 T4 Z: Q5 D  f  O

    / I6 T, m% D5 h8 hFor factorial(5):. b4 J  H( K1 M  {

    2 x5 N. ~( ^7 w. s8 R* w+ i( ?# K3 G% y. _% {- `% o& q
    factorial(5) = 5 * factorial(4)
    " U" N) Q% h( ]0 t, N4 Yfactorial(4) = 4 * factorial(3)
    6 `7 J) [3 j8 C" [# s. E4 }factorial(3) = 3 * factorial(2)
    2 F- d7 h1 O" d" M* ?& b) cfactorial(2) = 2 * factorial(1)1 L& [# d; p4 l/ D1 ~$ q2 d0 T
    factorial(1) = 1 * factorial(0)
    " F8 v6 [% ^2 A5 p1 mfactorial(0) = 1  # Base case7 \/ j1 ?8 ?, x! X/ s6 s/ O9 u3 R  e
    3 l+ D. b3 x  S! i6 V
    Then, the results are combined:: i1 b$ H$ i9 ]7 M# y+ L

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    factorial(1) = 1 * 1 = 18 `# S. h, u* @1 F( W7 b( G
    factorial(2) = 2 * 1 = 2
    0 n7 O( T- f/ b0 s  e" x" }factorial(3) = 3 * 2 = 6
    , M# N& K+ ?/ p8 T. ^) H$ ~factorial(4) = 4 * 6 = 24
    " v) `1 r; }5 ^/ U8 h* y8 nfactorial(5) = 5 * 24 = 120
    9 [* P0 m5 H  d8 P2 R( k0 t/ l% s5 n1 R8 L( }
    Advantages of Recursion
    0 M, J" n) Q# X8 X
    : f3 l" E# `7 g    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).! v# i' b- m# @

    4 X/ R0 ~# e  W9 V( a    Readability: Recursive code can be more readable and concise compared to iterative solutions.4 V; n+ N  L: D  ^, ?0 \
    8 W/ T; G& r& d( r9 T& q2 i
    Disadvantages of Recursion
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        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.
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    2 d/ C8 c. X/ x& p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    / _3 M3 R" X5 n' M7 D, ?- ^9 r! o( I2 \  Y0 m  e
    When to Use Recursion
    5 c, g) T+ }- w. k  t
    ; |) G1 j! A0 s; s    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ' `7 `' S- |2 v3 p6 N' x. w( q1 k! t7 @
        Problems with a clear base case and recursive case.' i* F+ o% E( m: s/ O# K
    / @+ P/ t3 {( H0 X2 ~3 X1 a" A
    Example: Fibonacci Sequence1 b. ?% U4 Y1 q7 t; ^# i
    - I6 i+ |% z& B# s
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . v0 @2 m2 O0 C/ c, v& `
    5 i% B4 @$ `; a3 t2 F7 _    Base case: fib(0) = 0, fib(1) = 1" O0 p( Z0 K; l9 H& \
    . `0 ?) R2 C# M* O& c
        Recursive case: fib(n) = fib(n-1) + fib(n-2): @) u% R/ [- V* {, f$ j

    ) B/ \/ B# H8 b3 a0 xpython
    / R! v7 l" J  [+ {+ K7 H. P  ]1 t. H0 x( O

    8 \2 \$ |" P) Y7 [7 [2 Ldef fibonacci(n):
    4 b: u) k( Q+ J- B0 a    # Base cases
    . O" t( n" q# l4 s, F3 s3 g    if n == 0:7 N; J% q$ H/ z, J6 j
            return 0
    5 |, L: S/ U8 }" y0 g3 q2 m3 T; Z3 N    elif n == 1:
    - \* c( U; z7 ~# p% G" w; I/ J        return 1$ x( ~# W4 `* [& i) Q2 F
        # Recursive case
    1 f. G# M" F' I9 ?5 P! N. w    else:
    ) O# C3 v6 G7 |; A0 [: Z6 N- X        return fibonacci(n - 1) + fibonacci(n - 2)
    ( p; s: f1 v. j3 N) F6 ?9 b* }6 K9 S  g8 W
    # Example usage: p9 _& m5 u6 g3 |
    print(fibonacci(6))  # Output: 8" G  a1 V" I/ @2 G+ \  o
    ' k' U8 Y/ l, J5 i5 _- f9 f
    Tail Recursion1 C) p8 V2 U; F1 o+ S
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    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).* u9 Q  D7 \  t1 R# V9 Z
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    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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