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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
    ) F( a# Q7 ~* i( [# I" A" y" y, A$ X" y/ ?. u6 K
    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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    关键要素0 S+ s" Y! J- D
    1. **基线条件(Base Case)**
      t. H  X) H3 J3 a  ~& X   - 递归终止的条件,防止无限循环
    - Y1 V/ C7 e5 [, N7 C: z  ?9 L$ f2 L   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1* ]7 _; l3 j! n% w4 D0 x% W3 P1 f! k

    , w8 V0 z  q- b: u2. **递归条件(Recursive Case)**  e2 H! y4 G& O( z. j
       - 将原问题分解为更小的子问题5 l  [5 R5 x1 Q% K; X  z1 n. q
       - 例如:n! = n × (n-1)!
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    ; E) {8 i; q1 y. U8 Y1 V* U- r: a 经典示例:计算阶乘
    / A# s$ }1 _8 ?& A- w! Y- |) V% Npython
    2 ?0 }! {* j8 }+ ?7 j" Y( u  {def factorial(n):$ `; `: w- H! z/ T' B
        if n == 0:        # 基线条件
    # u* N3 {, @' h- j  W+ k        return 1
    0 H- y* W1 A- f( g; G    else:             # 递归条件
    ' i! g+ F; X# k        return n * factorial(n-1)+ z2 {5 `/ d9 h0 w& L' c% c' }+ W9 p
    执行过程(以计算 3! 为例):2 m$ n2 M' z' Q: g1 i' C) c
    factorial(3)
    ; x" w$ s% o' f3 o' W3 * factorial(2)
    ! Q% h' b/ k+ L' a' d6 _) ]/ L) s/ c3 * (2 * factorial(1))
    $ H  p. ?- P# N& I7 Q0 m! t# c3 * (2 * (1 * factorial(0)))! j3 m. m0 ?' R# g! v9 }; k
    3 * (2 * (1 * 1)) = 6, G; P( _: i% B1 W
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    递归思维要点
    9 S5 f0 z+ k$ v5 w2 G1 U, ?$ w1. **信任递归**:假设子问题已经解决,专注当前层逻辑1 d! r# T6 Z' b, h+ D1 }+ `+ j
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    # u# M" d' }2 L9 w9 ]  b; ]' V3. **递推过程**:不断向下分解问题(递). X" H' z, {# \+ }, r$ m3 A' F! E0 W
    4. **回溯过程**:组合子问题结果返回(归)1 P* z# \! a! `: V9 J. u

    8 |0 r7 I* `. ~5 v+ Z 注意事项
    - U0 k; d- c; r2 \6 A必须要有终止条件6 D& |% N3 u; w2 {, w, `9 Q7 ]* I
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层); z' |& S9 h' u: D$ B
    某些问题用递归更直观(如树遍历),但效率可能不如迭代- _( Q: I9 s) f; s# i6 }( c& G
    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# z! S  X$ C6 U* m2 }. G
    |          | 递归                          | 迭代               |
    5 q$ [& H1 P$ c9 ~5 a|----------|-----------------------------|------------------|1 i" R, W: A8 n3 ]  w
    | 实现方式    | 函数自调用                        | 循环结构            |
    4 \* l0 @3 k; D  x( S7 [| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    + J# @0 a4 O, m9 M: N& T4 n" c* i, n| 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         |) J$ s- _4 D1 l* ?# A
    | 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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    # @) \0 P' k7 v; { 经典递归应用场景
    # h4 B' V9 x0 c0 {  ^1. 文件系统遍历(目录树结构)
    & C! G3 X: ?/ B! A: \2. 快速排序/归并排序算法
    ! F8 x6 K* M" R4 F1 U1 I' u3. 汉诺塔问题/ v6 r; o/ Q9 K- x9 z* S
    4. 二叉树遍历(前序/中序/后序)
      r9 P1 T$ ^0 k$ `* E4 n$ O5. 生成所有可能的组合(回溯算法)- @' h5 ]5 l9 c" I- n' `
    ) E2 a% b3 O; \7 q- W
    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,
    4 q; W' T  Q4 [# u7 M& P2 V我推理机的核心算法应该是二叉树遍历的变种。
    " O, v5 ^& S7 }. a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:
    . P' K9 _7 N; xKey Idea of Recursion6 h% S4 O4 L& ^, X  F

    9 ]$ d6 T6 Y5 N% BA recursive function solves a problem by:$ @. q3 U8 J. t; j2 l
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        Breaking the problem into smaller instances of the same problem.
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        Solving the smallest instance directly (base case).- U* b5 k8 X8 O* G8 L/ V

    3 K$ K9 p6 K( ?/ q) J    Combining the results of smaller instances to solve the larger problem.
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    ' [5 D. o, R, MComponents of a Recursive Function( ^0 k7 c( P- t" n
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        Base Case:
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    $ b4 P, y/ s6 z+ g& ~        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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      l, {- p3 j7 r; }7 ]1 J        It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.' _5 P4 _: i4 G. h; b
    ) z+ x/ o: k0 S& }
        Recursive Case:7 J3 D, w9 o) J
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            This is where the function calls itself with a smaller or simpler version of the problem.6 l) l" K9 @$ \! H! v& W
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            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* P5 g# n* K$ }) E
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    Example: Factorial Calculation
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    ; E9 l2 N+ `% ~6 \1 ~% ]6 sThe 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:% n+ O" d' [( f

    5 I* \* S2 f# Z; R/ Z2 z3 O8 B    Base case: 0! = 1
    1 h0 x6 O/ t7 ], u+ W; M2 w1 ~# j1 G
    : q2 ~  H; [+ G3 v9 L" U    Recursive case: n! = n * (n-1)!
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    Here’s how it looks in code (Python):
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    : ^& x4 K5 {1 _. |
    5 {9 {% A4 G, U  W: zdef factorial(n):* G1 K% |. j& z1 U
        # Base case) ?' y, s, m, U) Q6 J% p
        if n == 0:7 `& J5 z4 t! {$ J9 u
            return 1% t, y+ U$ {& d3 f) C4 z* n
        # Recursive case
    , `% A# i1 }* s' s' n% M, y    else:
    + W: Z. U* g8 Q' n6 v0 \3 n/ W        return n * factorial(n - 1)
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    4 a; P* h$ W1 j" @3 i9 H# Example usage5 }+ ~2 A4 P8 u6 W3 ^/ m0 |
    print(factorial(5))  # Output: 120; }6 Z9 k, S* U/ P( j' S

    % s0 N% |. U/ Z9 x7 O4 ~% v$ dHow Recursion Works- C3 s" A. P8 P* {& H
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        The function keeps calling itself with smaller inputs until it reaches the base case.
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    5 w8 K- ^/ g1 g  `4 {0 q+ Y3 z    Once the base case is reached, the function starts returning values back up the call stack.# B6 f. [: M$ _# ~6 p7 f5 V

    . D4 f$ ^# w: f2 T3 A1 A    These returned values are combined to produce the final result.
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    For factorial(5):
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    + K. T! n" |8 t8 o; v; dfactorial(5) = 5 * factorial(4)
    5 h1 c. y; P+ v+ r7 v. z/ w& Rfactorial(4) = 4 * factorial(3); j4 L# R4 W# V
    factorial(3) = 3 * factorial(2)
    & i4 U& S2 B, y" S. s! Z* wfactorial(2) = 2 * factorial(1): e- g' Z- I* z$ i$ J7 |7 K! N* J
    factorial(1) = 1 * factorial(0)0 ?! V; u3 w! f4 w) h' f
    factorial(0) = 1  # Base case, E0 Y) T4 [9 Y* o- Z0 e

    ; d% d% N& D6 U* XThen, the results are combined:
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    ; S2 ?9 R$ n+ z' t
    5 ?" s1 v# d5 i; P/ nfactorial(1) = 1 * 1 = 19 r' j' z0 Y0 W; x1 H
    factorial(2) = 2 * 1 = 2
    . l) y- Z! ~# E6 w+ `1 gfactorial(3) = 3 * 2 = 6
    7 Y. E9 Y. `& g2 A2 Ifactorial(4) = 4 * 6 = 24
    & @3 x4 l* p3 V' t3 [: y1 d% ufactorial(5) = 5 * 24 = 120
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    Advantages of Recursion
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        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).
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    / o# U) E: A+ s9 \    Readability: Recursive code can be more readable and concise compared to iterative solutions.) s" u- [0 e% S% Y% l& p

    $ l& B3 Q. i4 `Disadvantages of Recursion
    " R: X3 E6 m% i' F4 C' {( E5 _! K$ R9 ^3 w  p; {2 s& F5 \! [
        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.* M) l0 u0 |+ C# @  z

    9 c9 L8 Z8 G+ {; [+ G3 O    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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    When to Use Recursion
    , {" u. O! Z% @' t! K) l6 Z: P- w% w% z8 n, J8 c
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' m% x7 g7 ]' l9 [& m- r5 R/ N

    8 L8 Z# |9 i, ?+ O3 q2 D" c3 U    Problems with a clear base case and recursive case.0 i. ?$ B! j6 f
      d& [: t7 T1 u. A
    Example: Fibonacci Sequence" s! b2 p$ _, M

    3 Y. t! R0 `: H5 ^8 c- zThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ; z4 V7 s/ w9 d; a8 E: m2 D/ v7 y5 Q2 N) J: J
        Base case: fib(0) = 0, fib(1) = 1
    4 m+ c, e# u' b% V
    7 U4 E) `; k6 K! W    Recursive case: fib(n) = fib(n-1) + fib(n-2)# u7 z: Q8 V; _* B/ w* t

    - k# u- W% T9 K2 ~python; g3 O+ U- H% A  e

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    " Q' L3 h$ V0 q* A: ldef fibonacci(n):) I# b+ Z7 [" Y2 ?3 t  ]8 \( x: {8 Z
        # Base cases
    : N* q0 n1 P1 `8 I5 O( W9 v) s    if n == 0:
    7 ^* C% D0 w. B" I+ f( l) D        return 0: a* l6 F4 y; u( J
        elif n == 1:
    ) z$ ^( R0 c4 T/ }# b        return 1$ s5 f# K* Q9 V* i- g4 y
        # Recursive case
    ) Y1 G) O- B# g# p7 U( U    else:( L% y- n: W  ~  W7 B, O
            return fibonacci(n - 1) + fibonacci(n - 2)# ^2 e8 a: |4 `

    ) b$ a& t0 A$ {# Example usage# `" d" _1 |/ Y( k/ a- x. {
    print(fibonacci(6))  # Output: 81 T7 W: ^2 t( [
    $ F& G9 G3 W" X! ^5 i0 n1 o/ @
    Tail Recursion
    1 j7 _3 I7 J, t0 E# k* d6 _6 o/ X5 y3 d7 K+ {% Y* I/ ~
    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).
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    0 c& O; T% b- N& S# |: W/ z% dIn 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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