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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
     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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    关键要素* C( H/ O, b& U& I, q4 P
    1. **基线条件(Base Case)**4 Q5 ]3 d8 z8 N& F
       - 递归终止的条件,防止无限循环! v$ a+ H: a& C& p7 G
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# d- j# P4 M: t
    * ?! b$ |1 Y  q4 ^
    2. **递归条件(Recursive Case)**1 {, B3 D7 y1 x+ o
       - 将原问题分解为更小的子问题
    , s1 q4 m7 {  R( X   - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘# r' S' G& U/ q" B0 B5 ~
    python) P5 D+ L& a, |
    def factorial(n):
    6 @+ k: b, G5 W    if n == 0:        # 基线条件
    + y9 O9 c* Z; d7 e7 Y/ V        return 1
    0 U/ D* O+ u0 N# `3 V+ f) ~    else:             # 递归条件
    $ v! L0 _" X) O( G- h( B4 v/ s        return n * factorial(n-1)
    * @+ K% |5 Y' s1 \% F7 o执行过程(以计算 3! 为例):
    % [/ ~1 L2 |- z  {& s" tfactorial(3)7 i+ z- Y$ O+ L9 a7 H- i! h6 O
    3 * factorial(2)3 Y! W8 j# l0 L7 |! u. ~
    3 * (2 * factorial(1))! u) y" Q* G9 m; W/ A
    3 * (2 * (1 * factorial(0)))
    ! [0 \$ d: e5 n8 W6 g3 * (2 * (1 * 1)) = 65 |/ g5 _/ D" W; Y, E
    2 ]4 N- f& v- U* F9 y" ]
    递归思维要点
    5 X) V* S; `' N1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ) k! }/ y, i) I: }. {  x) U2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 m9 ]5 q, J/ L' e) o* z" W+ |
    3. **递推过程**:不断向下分解问题(递)
    . a! i  `7 |9 y+ j( u* G* M7 }4. **回溯过程**:组合子问题结果返回(归)
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    + O& C9 o% @6 J' c) C* ?" i 注意事项- ~* P: ]# q+ K& Z6 o: ?" s
    必须要有终止条件
    % @; q9 N2 m  O2 M递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    $ e8 c# B2 K: k某些问题用递归更直观(如树遍历),但效率可能不如迭代; B8 {5 t) d+ F1 P; [' I; M
    尾递归优化可以提升效率(但Python不支持)$ L! ], f' a/ s8 u- J& [
    " {/ u( d& [/ q3 `7 q0 h2 K( m# F( c
    递归 vs 迭代6 P/ U- C0 \  ?
    |          | 递归                          | 迭代               |
    # o( l8 P! `4 Z. \3 N3 V3 m8 u|----------|-----------------------------|------------------|
    8 x0 g2 {4 k7 M( K7 l. T( H. q' Q| 实现方式    | 函数自调用                        | 循环结构            |
    / }2 w+ Y0 K  ?9 X| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    / Q% m( S+ P9 t) s, w| 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         |8 R( U4 x' e4 M) d. ~  z" R
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |" p8 F( s+ i1 z! J

    6 Y; c8 V1 u9 @1 F9 j 经典递归应用场景
    / T' x, v/ E2 @8 W; v1. 文件系统遍历(目录树结构)& F) O; e- K+ R+ K$ c! s. k. e
    2. 快速排序/归并排序算法$ y& O: `& i) w1 j8 R( G8 h1 |
    3. 汉诺塔问题. v& l! ^/ W: {+ {
    4. 二叉树遍历(前序/中序/后序)$ G8 B/ J! }1 P6 j; Y  W+ h5 t
    5. 生成所有可能的组合(回溯算法)9 [6 ^4 N# \- s
    * E8 r/ V, I8 t0 {& Q3 d
    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 f6 z- ^' z$ p5 S- R我推理机的核心算法应该是二叉树遍历的变种。
    : l; Z& ?, A2 G: ?. H/ b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:
    1 f& M+ F5 c% o. W5 F# tKey Idea of Recursion
    4 \1 [- u" g" a3 b0 P  n0 Q; ]- ^. D- B7 g0 K; c* q, Y
    A recursive function solves a problem by:
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    6 ?& i' n& v. Z6 k% ^    Breaking the problem into smaller instances of the same problem.' W8 U: \4 C5 w* q7 ~, L" x

    , T$ f/ Y9 q" R$ Q    Solving the smallest instance directly (base case).- p- z, w6 d! N  A, T" F- G/ y" B

    3 A3 I0 b* o4 G! o/ G- M    Combining the results of smaller instances to solve the larger problem.2 w7 F4 e/ g$ L3 ]% a0 {
    8 q  q3 {$ x4 ?/ y
    Components of a Recursive Function
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        Base Case:5 i2 U0 B+ D; h9 \4 m3 G

    # C$ K; i/ Y0 F* }7 c        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.6 k3 V8 m- _& Q$ U; O

    , Y" i6 V% N1 u6 ?9 s& T        It acts as the stopping condition to prevent infinite recursion.( O" v4 T+ `  \& m

      F( Y) S, S0 S( c/ B: F        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:4 J* ~. U; i6 \9 h5 v% A2 B6 h
    . U6 ~+ X7 t) B" |' J  ^8 h' _
            This is where the function calls itself with a smaller or simpler version of the problem.7 `/ \/ d1 O$ y# p5 q
    ' Z3 D6 P8 U& g& C' V5 G! Y- H
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).  Y$ T& c5 c' n, H! Q3 U+ l
    % \1 R3 a+ i2 j; y$ b2 h
    Example: Factorial Calculation
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    / S% O" m+ \0 b, E! [* mThe 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:" s1 }1 o3 U" o+ z

    3 ^; l6 R4 ^  W+ z+ R& c4 d    Base case: 0! = 1% p2 M3 i2 O& u
    1 p6 d! G$ g; g' O5 U& g' a
        Recursive case: n! = n * (n-1)!
    8 R: G" i2 C" o% v! E: \1 o+ Z" S  k5 m* D
    Here’s how it looks in code (Python):
    ! k) n: ]' J+ ]7 o2 u0 apython
    & E9 `9 u6 t, s7 |' B& J8 r+ a8 n" u% P

    ' P  ]+ r* O- C* |9 Y8 t! gdef factorial(n):
    7 m0 N: C+ m! W. J, O    # Base case
    ! C% Y, g: W9 n# \* ^+ u# W6 Q    if n == 0:/ Z. {) [; P! {' q9 _& A( ^
            return 1# g8 Q1 f) R9 ~+ l; k% G4 ]
        # Recursive case
    + b$ G9 G6 Y7 @/ w# n' G' f    else:
    7 D9 _" C  I2 t. Z/ p        return n * factorial(n - 1)
    ) r/ r2 s6 p/ Z4 w) i6 I
    8 g4 g- J2 P, w& A  E- [# Example usage
    " X* b: X8 Y8 g& x9 @print(factorial(5))  # Output: 1205 _8 v2 I3 p& I: m3 F; u

    / K$ {9 r: C7 ?+ z- U* D0 h6 W2 Z& |6 HHow Recursion Works, l% S$ @' q8 {* y8 H; x* p2 @0 z

    , ?! |$ h: b' R" y    The function keeps calling itself with smaller inputs until it reaches the base case.* n! l6 O, Y% Z+ B, _& T8 r3 X
    ' ^$ [" F8 K6 N  y! `9 L# K4 X
        Once the base case is reached, the function starts returning values back up the call stack.' _8 Q2 t" h" _) |
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        These returned values are combined to produce the final result./ S; b9 ?6 P: \: ~5 o6 q
    6 A3 m* J* \: d% a6 l. V, j4 b
    For factorial(5):) K2 ~* q  x. |7 T5 {

    . l* M: `% ?9 I
    5 {& S" J& A1 f* B. Vfactorial(5) = 5 * factorial(4)' v1 r+ l5 r* Y' j# U
    factorial(4) = 4 * factorial(3)
    6 V9 I+ E. C, S3 ^$ S- ?$ f1 ^( [# sfactorial(3) = 3 * factorial(2)
    & i# @: a- N& e5 g: Kfactorial(2) = 2 * factorial(1)* H- _2 [# |2 F" v: q& L: d" ~0 ^1 w/ c
    factorial(1) = 1 * factorial(0)& a, e3 J) P& i8 c$ P
    factorial(0) = 1  # Base case
    ) N" G* }* o. n/ a
    & E) t9 T9 d# \/ M; O& P# zThen, the results are combined:* W/ {2 {4 L! l5 |
    6 A" u3 \$ c$ V- W3 ^. o7 U" x

    ; j; k" O5 i0 ^; G/ Qfactorial(1) = 1 * 1 = 1  _6 B$ E& a; [, N
    factorial(2) = 2 * 1 = 2( }, {8 L& e) V4 K( Z: L
    factorial(3) = 3 * 2 = 6
    # o8 [- _! X0 Y: }+ e0 y- _factorial(4) = 4 * 6 = 24* {5 E+ E& C9 g, b
    factorial(5) = 5 * 24 = 120
    ( G) J+ v" o0 H) U8 u+ _( ?
    ! j! p, ?& T! V; i' S! WAdvantages of Recursion
    1 J5 \1 i5 H% Z! d. x+ a, F0 V& z% p8 m$ p3 [% n+ s) m
        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 D7 `$ v7 I$ m8 K! f

    % t% h9 B$ ~6 I  |3 f" y3 u    Readability: Recursive code can be more readable and concise compared to iterative solutions.2 V4 ?6 N  J6 _) g
    " E' c2 o* D+ _5 V6 I$ \
    Disadvantages of Recursion
    % B: v0 `& [8 C* _0 D3 ]8 g3 y
    : |2 ]% c+ O  ~$ [; @4 p    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 o2 q1 f4 V, c. Q6 b) I/ m+ A$ p& @7 J, z& L
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).+ L* W6 ?0 C* n8 N0 }
    / E- J" E# `; K5 a! X. t5 M  B
    When to Use Recursion
    % W: O8 x7 r* o; {: a! X  k2 R9 z' o' L
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).  O/ z: ~  c0 u
    ( c- V% X5 f! A! X+ l
        Problems with a clear base case and recursive case.
    * i8 X' J; P  e. [
    + G7 Q6 q5 T* z8 I! PExample: Fibonacci Sequence
    9 d: q/ K! P+ L! T/ X; Y* b5 @. x7 x) q% s1 u4 a
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:/ ~  O: K% ^$ p4 h: {$ L
    + ^  A" O+ \- k5 y4 r$ U
        Base case: fib(0) = 0, fib(1) = 1, o  R; Y# d2 S( B% U2 `
    4 M4 P1 ~: W0 q( N2 z" W- ]
        Recursive case: fib(n) = fib(n-1) + fib(n-2)1 q3 x5 e" \* P
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    python( g( p5 f: U; N- l2 V4 Q  `$ b) r  y

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    ! v4 Y' c' K6 ^/ \def fibonacci(n):
    & ]% Z7 M6 h/ X+ U, b. x" e& q    # Base cases" p& Z. [) |7 \$ r9 |( r) X. k
        if n == 0:5 d' o- y% W$ ~' |+ K
            return 06 o; i& m) |7 F" [/ `& s7 e
        elif n == 1:
    % ^6 S+ T  E  K+ o6 Z1 `        return 16 @! Y! E' j" L3 k( a
        # Recursive case
    - ~; m; w- K, b9 I    else:. q7 H$ V" [& m- o( [; T
            return fibonacci(n - 1) + fibonacci(n - 2)
    ! _+ C  j1 O6 Q) a( c2 w$ X* J+ n; }6 d
    # Example usage
    ! V8 u+ K  U4 X# Pprint(fibonacci(6))  # Output: 8$ s$ o5 r4 R! M

    9 n/ f0 X4 l  C7 I; I; m# W2 z- {  J$ bTail Recursion$ a. I. A2 d$ [2 Z0 B# r. F" }8 K
    9 |. U7 l0 G0 h
    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).9 m: {' h; ~' T" J: A3 h) w3 ^
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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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