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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 " R! m$ f  K6 j& S2 P
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    解释的不错
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    : R7 W- d7 v, B- N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。
    9 a- {# Q1 b- M; Z9 o  ]4 A& J
    $ C* I7 n" k3 |# r6 M 关键要素& m+ A: S* O* ^3 p+ O) Q! k) d* j
    1. **基线条件(Base Case)**
    $ Z# g) x8 G: ^% h- e   - 递归终止的条件,防止无限循环
    # l6 [5 r5 P# I4 l. L   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    9 I: B* [9 }1 k% z0 n$ f0 t  [' E  a" [) D$ S
    2. **递归条件(Recursive Case)**5 u) u6 [* g9 m& t
       - 将原问题分解为更小的子问题& R% a. m# Y( G' ]
       - 例如:n! = n × (n-1)!# c0 O* `( n: {

    & q% k, G% X/ F' @, K' @5 y 经典示例:计算阶乘
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    def factorial(n):; W" V# ~; F* \* u
        if n == 0:        # 基线条件
    8 q5 w/ J5 K2 W) a! d# e/ {; X        return 1
    ; i, d" Q2 W# I. B    else:             # 递归条件( _# t+ W' k  o: r& {) c* g0 d3 w- g
            return n * factorial(n-1)
    8 q  r1 M7 ]* d) B* `执行过程(以计算 3! 为例):
    $ z# F( v( A; c( i, W$ @factorial(3). o- H0 u: C' ~" Q
    3 * factorial(2)
    : x- z" _0 `( Y3 * (2 * factorial(1))
    2 p4 Q; i7 a- Y5 {- d2 Z; i( n3 * (2 * (1 * factorial(0)))
    9 b) P4 @1 [' K& V+ s% H& e3 * (2 * (1 * 1)) = 6
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    & {( k! H1 t7 ^0 x: x8 B/ f' J 递归思维要点
    ! Z+ x6 k0 L6 c0 Y1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 l7 ]' k# c2 \# O
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ( D2 X0 f8 ~/ |) i3. **递推过程**:不断向下分解问题(递)
    . R4 y: N! R, E4. **回溯过程**:组合子问题结果返回(归)2 A! k+ `9 w- B1 [2 Z  l; [7 N
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    注意事项
    % q0 T# [, _4 T4 z$ P必须要有终止条件
    0 X( Y) ~" c( h: T: F. Z: }递归深度过大可能导致栈溢出(Python默认递归深度约1000层)* g: w/ t' Z, @2 G% E! t+ M
    某些问题用递归更直观(如树遍历),但效率可能不如迭代' C4 D' X$ Q$ a& G6 K. @9 B
    尾递归优化可以提升效率(但Python不支持): q( G5 c, c9 U/ e; L9 c0 g
    8 r* X0 N. o& ^: H
    递归 vs 迭代
    . E& B4 d; K1 n|          | 递归                          | 迭代               |
    4 T- e8 v/ ]+ L& a' [3 U|----------|-----------------------------|------------------|
    / V; o# ~' \- V4 Z) w5 N+ F| 实现方式    | 函数自调用                        | 循环结构            |
    * [0 [: n' V- C: d| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    . t5 h" X3 {$ E0 r| 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         |3 j( h0 B7 |8 P
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |# q+ T. R  t# ^& h+ K$ [6 B
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    经典递归应用场景% g2 b, X  e% F3 Y0 E
    1. 文件系统遍历(目录树结构)* B, A% N( z  Y8 M# T, ^
    2. 快速排序/归并排序算法
    % f5 `4 E" e8 J0 g, f3. 汉诺塔问题3 K0 Y4 @" B: P+ u6 F
    4. 二叉树遍历(前序/中序/后序)% B5 h; p8 \; s/ `: W
    5. 生成所有可能的组合(回溯算法)
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    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,
    ( g4 Z0 x3 c8 S5 D我推理机的核心算法应该是二叉树遍历的变种。
    ; g# P& T. l, l7 x$ K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:
    # t4 i6 m1 y# P5 M: YKey Idea of Recursion
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    # e( x& K$ o9 ~A recursive function solves a problem by:
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    , b4 X' b! r/ I4 C  d. @    Breaking the problem into smaller instances of the same problem.
    : y1 I8 c% l5 W' Z
    8 N& ~9 U3 N: Z. N5 S    Solving the smallest instance directly (base case).9 M' l3 U  t- Y  d) \' {

    ; g* }3 @4 U' J; U2 o4 m2 E% \2 s    Combining the results of smaller instances to solve the larger problem.
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    1 I8 k: }1 T; g! V* t" tComponents of a Recursive Function- |/ E& y) r6 o  I# c) a& V' U" n$ T& j
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        Base Case:
    , W+ N' O( N8 t: C) D. A
    ' a. [# Y1 m4 x! ~7 h& P: ]1 ?        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    ' o/ Z( l8 l# Z& [        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.
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        Recursive Case:$ q2 X' m: F8 m3 J$ s

    : _. v! g+ @, e2 W! w1 f) y        This is where the function calls itself with a smaller or simpler version of the problem.7 E& ^$ \2 G4 A; m8 m3 n

    8 ?: S. q3 W% m2 n        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 S% D# S( x+ Y+ }; m/ J

    8 ]& a( N8 ~" u1 F3 kExample: Factorial Calculation
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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:. a1 s, h( g3 U3 a8 d

    9 ?4 \9 \0 n" P$ c$ A! \* M    Base case: 0! = 1
    6 p7 R* |1 u$ Q& \8 N" ^+ M, Z6 Z" [7 h/ k# x$ ~; H
        Recursive case: n! = n * (n-1)!  L; ]" A1 M) f! d/ }

    8 D  [' E) H' V6 R# Q% X4 aHere’s how it looks in code (Python):8 \0 ~  }! X& M. b4 m2 l+ l7 H
    python
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    def factorial(n):( F! ?% S5 v$ t* {, u
        # Base case
    . x0 D* J1 q5 X    if n == 0:
    4 ?8 n. c  o" o) i- V/ C; N        return 1
    1 C7 n) f. O' h5 V    # Recursive case6 Z3 s2 f- d% i( w; H0 H
        else:- S! c  M: K6 U" [$ ?" q
            return n * factorial(n - 1)  q- n8 z" k0 d: {

    . O' ^7 [, n6 i" V# Example usage7 Y& D% O/ {( O2 Q
    print(factorial(5))  # Output: 120+ @  F5 U5 F5 H

    1 p8 l. Q" z2 \$ W8 E& C! lHow Recursion Works
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    1 F' P: S, m; R9 i0 T  |    The function keeps calling itself with smaller inputs until it reaches the base case.. w% }# l' V, v; R/ w" I# `& m

    ! V3 \! \- `; C9 m7 W    Once the base case is reached, the function starts returning values back up the call stack.1 U' |  \' x8 a4 i  v6 T
    ( d6 h- w" ^1 v' P+ w
        These returned values are combined to produce the final result.! x( W3 ?3 x9 E

    0 a9 n) H6 \1 J' @% M$ \3 V' hFor factorial(5):5 _% a  b# O/ |( S
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    factorial(5) = 5 * factorial(4): y/ l* Z; s/ M* C' O$ L, v
    factorial(4) = 4 * factorial(3)5 B& \1 S6 U+ C3 \# K6 Q
    factorial(3) = 3 * factorial(2)  [( H' F+ W6 b  o
    factorial(2) = 2 * factorial(1)) T( ~' w: m, f' r" j2 j
    factorial(1) = 1 * factorial(0)" W3 `: t9 w; x% u  G
    factorial(0) = 1  # Base case
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    # a; h  \" V1 E; e3 SThen, the results are combined:
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    ! i+ C& r- ~9 I7 A5 Wfactorial(1) = 1 * 1 = 1* o: j! B% Q$ R5 b* M* T+ H
    factorial(2) = 2 * 1 = 2
    ! H* F; \+ v* l7 E: F0 `# Q  _factorial(3) = 3 * 2 = 61 J9 l+ A- f" b
    factorial(4) = 4 * 6 = 249 w/ y( X4 I) ]  |( P. _4 t, t
    factorial(5) = 5 * 24 = 120* h- P+ C& X4 |' |8 J9 `1 ^
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    Advantages of Recursion) J& d5 m* {- ~! g2 X: S; c) ~( y
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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).6 @' e- j2 K2 t5 C9 H0 A, o( w
    , ?4 o5 ~0 @- u
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    / A. l$ n7 T! m7 ^& l2 \1 W% ]2 ?. Y3 s' A
    Disadvantages of Recursion
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    4 R  C  Z4 v1 O3 E- 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.- D$ O: `3 t4 t
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        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).5 N  W- X6 x6 d; U. p9 t$ i7 a
    ' l0 h. C% C0 a1 `* h* }7 m' E
    When to Use Recursion
    + I% H( Y; a& z1 p
    . |* L6 L- G( R    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).- _9 P" J+ R& }6 n4 ]; x* I/ l0 Y
    % i0 Y) b% J& @, v+ H
        Problems with a clear base case and recursive case.
    5 K/ e; S: \' Z( g' u' P$ x# M7 o. b8 s' {% h* N/ E
    Example: Fibonacci Sequence
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    / b' R( a7 E1 J7 Q9 ~The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    7 l; O( z' @7 Z! j' t4 F* I( w! n; `* ]
        Base case: fib(0) = 0, fib(1) = 1' y6 Q, y1 J% q3 S

    0 s8 F) \; \! J2 D* ~8 _    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python
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    0 z+ ^& @# w/ k" A9 Tdef fibonacci(n):
    0 ]% C+ U  i: A4 x, \# D    # Base cases6 q6 h+ |( R" H( G: y( F) ?# Y
        if n == 0:8 [8 `+ Q, O2 C- d* y7 |* ]
            return 07 C+ w+ b' o7 z6 x5 A: r+ ?
        elif n == 1:& z; r: k# Q+ z' @& J
            return 1$ ^( m" E/ s0 Y* V' D
        # Recursive case1 u& v7 O3 d( {. e' @3 L7 T6 e
        else:
    ; N1 z8 k3 P* D0 j        return fibonacci(n - 1) + fibonacci(n - 2)* P2 \4 Z  }! B4 N( K! |

    " D. Z" W$ j" c3 s& H# Example usage
    ( x  x4 I& ?# O% n: qprint(fibonacci(6))  # Output: 8
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    Tail Recursion
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    1 |, ~2 a0 _) HTail 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).
    ' Q6 j# b& @$ M1 |* s+ U
    1 L6 M, t/ l" S1 Z1 iIn 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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