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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
     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- v/ r1 J/ ]* O5 {4 T6 E
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    解释的不错
    4 I; A" T& b6 {$ T4 X% w4 r8 F8 Q: g5 B6 {) x; B; V5 f
    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。$ q; d9 m  b0 ~! b1 h" P4 _( q

    " D) J+ f* B& K3 _! T 关键要素
    & a% D/ f  z- E, J9 B! X% Z7 Q1. **基线条件(Base Case)**
    4 w; g2 c9 l! E5 A" d3 c   - 递归终止的条件,防止无限循环- ]0 d/ ?& z& l9 B
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1+ }6 s' p9 G7 @. P) }$ p' R- J4 ~

    3 J7 z* D4 |5 j" H1 o( |! }9 e; N# x) s2. **递归条件(Recursive Case)**8 w5 U1 a% N4 Y8 O* S" R
       - 将原问题分解为更小的子问题  \9 F  `$ u  ^4 T. u, q! S/ q
       - 例如:n! = n × (n-1)!
    ( X% {; q: `" q* s" h. f3 m' \. m! R" i4 ]6 g, L
    经典示例:计算阶乘
    . r3 L' [: Y6 r6 k  Ipython
    & B& c( y% P4 M  N/ udef factorial(n):
    9 t" u  W9 p; y/ A7 U' ^; |    if n == 0:        # 基线条件0 h4 Q" s  [- B6 Q/ c) ?
            return 1, i, @1 a. k: ]  \# b
        else:             # 递归条件
    9 m7 z8 }& U" q4 U# X, f        return n * factorial(n-1)9 p( B$ o% S. ~7 @9 E% Q# P4 y
    执行过程(以计算 3! 为例):
    $ L5 J: Z" N& A0 b% ~factorial(3)
    5 b2 ]9 W- i& J3 `! a3 * factorial(2)
    & P- Z+ a' M1 ^3 * (2 * factorial(1))
    9 i/ {% i; @  Y" G: P3 * (2 * (1 * factorial(0)))
    ; l; }, J7 W4 l3 * (2 * (1 * 1)) = 62 Y6 T( z/ W& [: A1 t  W9 t

    - n9 L5 ^/ k- K 递归思维要点' I6 N6 ]- p0 R6 d3 d
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑, `  V6 `+ t9 m0 X7 V0 r: |
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ) o& i2 I$ J% k8 ^3 C3. **递推过程**:不断向下分解问题(递)' Z/ M! g& E8 g# s# N; d
    4. **回溯过程**:组合子问题结果返回(归). h6 }# {: s1 t: n

      B3 r8 i& O1 a$ ^$ ]' n  @9 T4 ^8 S 注意事项  z. y  f& G  x! \. p
    必须要有终止条件
    0 J( T. M4 }8 Z* y递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    $ ]# p: K( c! P) J某些问题用递归更直观(如树遍历),但效率可能不如迭代
    4 I! {; }6 O2 f" S3 e; I* H, V尾递归优化可以提升效率(但Python不支持)
    / n) w) z# \. `& b! D. \& n! x  R3 m! C% p
    递归 vs 迭代
    0 `5 a# {6 U4 c6 T4 F|          | 递归                          | 迭代               |+ r$ b4 a. [" e  {
    |----------|-----------------------------|------------------|
    ; n% g6 H# m/ i| 实现方式    | 函数自调用                        | 循环结构            |
    5 }! {5 e0 _& `2 s2 K0 o| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |4 {2 A$ \7 A2 H9 y, |5 j6 H
    | 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         |% b6 O: i  P2 F6 w$ f! a: S: \; x
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |# o. U/ `2 v9 e) D$ w3 E) P$ O! g
    3 m0 k1 U4 z* a8 [2 p
    经典递归应用场景/ m% B  u4 k: w0 Q. C/ X3 v- _' }6 W
    1. 文件系统遍历(目录树结构)
    : L- u& ~! [# d# ?6 x2. 快速排序/归并排序算法
    ) d+ Y2 W; l9 J9 p/ W+ D3. 汉诺塔问题
    3 S3 [! {! m4 W9 u4. 二叉树遍历(前序/中序/后序)8 P: p9 H( P5 J5 B7 k
    5. 生成所有可能的组合(回溯算法)
    # R1 Z0 j9 R+ p* d8 V2 |5 A# y
    ! N: b9 s; u# U: v3 a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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  • TA的每日心情

    昨天 07:03
  • 签到天数: 3118 天

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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,
    " a6 }/ @/ S' |. v0 Q4 e我推理机的核心算法应该是二叉树遍历的变种。( w, C* W0 e1 w% D
    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:
    % x2 I% A  v$ m$ ]/ Y) F1 c. BKey Idea of Recursion( h8 F9 D6 |8 x1 z; h
    ! Y. e3 X( @6 y
    A recursive function solves a problem by:
    6 C  l7 B4 k4 B9 o% Y7 ^0 `% N! ]% d3 ]% S5 X0 x0 c
        Breaking the problem into smaller instances of the same problem., T0 P# q5 F6 x. C' L
    % w/ _5 Z, a2 ^; E8 N
        Solving the smallest instance directly (base case).
    ' q5 q$ D* h% I8 ?; {. |- L5 s  e  R0 U
        Combining the results of smaller instances to solve the larger problem.
    8 f4 n, x+ P/ j9 b& @# d. {! c% {; n9 B% N9 x
    Components of a Recursive Function
      {6 M/ ^  S0 y! L0 Y/ |$ s4 Q1 d9 x, j2 e
        Base Case:9 v1 N( R4 P/ i& K$ k
    4 E: \8 F1 n# D: s1 s/ a# ~
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion." @$ N$ f2 |# Z0 D; I" I; A" c+ d. {

    ; e6 @' L, u' R8 s        It acts as the stopping condition to prevent infinite recursion.3 ^8 {9 B/ L  A5 J2 P
    & i& c  f9 v( ?/ p  G7 J
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; q$ Y- x: ]+ x3 b* r* Q- n8 U+ A

    8 C! E1 g- \) o& }) q1 D/ A    Recursive Case:+ g% h6 Z5 b, e, X/ K. ^" K

    $ X9 _; V; F# H# v& F: _        This is where the function calls itself with a smaller or simpler version of the problem.
    , |. C, _1 A( t  [3 d; n" {# x6 b8 k9 b5 X2 e
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    + l9 A4 R" W, S: T- {! Y5 M$ u: ?2 R
    Example: Factorial Calculation3 x4 m4 c" u) w+ P: x  o! T# D/ l2 \7 F
    $ u0 N' X; N0 e3 F, S
    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:* F! y2 b; v/ K, P; u3 j

    2 S: n0 B! }% t8 x+ d    Base case: 0! = 1( b' d1 Q+ a% e; ?+ O( y# I. M2 P2 ]
    . V& E2 m1 d% D2 P
        Recursive case: n! = n * (n-1)!" ^3 r1 A# P# E' V( k. Q9 N
    ( H. F7 J' U: ~4 a& m- P
    Here’s how it looks in code (Python):2 X: B# d7 B; l% B& F
    python8 n) ]1 M! _1 R5 ?* n2 X1 ?% y
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    1 ]% D1 q# J) c/ @$ Z: I
    def factorial(n):4 n6 G; i# u) q/ F* c  a( P
        # Base case' n/ j- Y; A- s% @" ]
        if n == 0:( M. V% ]+ _7 O' W" y5 e4 z
            return 1* V7 B" f% Y1 z8 T  ]
        # Recursive case
    # \! H: Z4 x, r7 z    else:
    * u2 o. T8 R5 `- I  l        return n * factorial(n - 1)
    0 s# r) o: S6 s, L0 ^/ m0 q/ x5 K5 h1 i* }
    # Example usage
    " i# _8 |( k7 S1 bprint(factorial(5))  # Output: 120- h2 {; j( X# @4 U5 L: X

    9 m+ i2 w; q4 x' h7 QHow Recursion Works
    8 I0 W5 ]: ]( N) U$ A% ^; W
    8 D, N% H4 T6 [6 ?, V    The function keeps calling itself with smaller inputs until it reaches the base case.5 l9 B$ ~$ l# U. K0 Z

    8 H  b/ W6 }& k0 ?- p9 o6 B# M    Once the base case is reached, the function starts returning values back up the call stack.# K  H$ \  f: b' U* v' ?
    - I2 x( \+ G# X1 x
        These returned values are combined to produce the final result.: r; `3 z- n& ?* a4 S2 Q, Y5 V# F
    0 j1 x! O* _1 {. }' o/ O$ ?" E
    For factorial(5):! o" O; D  L8 n% C2 {
    ' u4 v" V' h' b$ s, d# n

    0 I% e/ P+ H* T# y" O- o' }factorial(5) = 5 * factorial(4)
    : t; A7 x2 u9 ]2 t1 X2 mfactorial(4) = 4 * factorial(3)
    : T2 V! Q/ _* ]! jfactorial(3) = 3 * factorial(2)8 U: t4 L% o9 ?& C$ f2 B; {
    factorial(2) = 2 * factorial(1)
    $ ^  K0 v; p) {7 G: v( c6 m/ zfactorial(1) = 1 * factorial(0)
    & {) @& k4 w1 g9 d, w# ~factorial(0) = 1  # Base case+ `8 u, T; L: R

    $ \# D. d% |/ J: x8 `6 n' \; C$ {Then, the results are combined:
    % R  n) P& t. a, K/ M& X  c
    * g- P& X: r' Z, i- n& t3 H/ I* l: ]1 D2 y
    factorial(1) = 1 * 1 = 18 o4 S; M6 q! h
    factorial(2) = 2 * 1 = 2, m4 c+ E' b0 e0 w! g" E  b( @$ m. W- Y
    factorial(3) = 3 * 2 = 6
    - d3 K& b0 D5 ofactorial(4) = 4 * 6 = 24
    # z/ E7 n' r! A3 O) ]factorial(5) = 5 * 24 = 120: C( N2 a& ]. L+ P+ V" H7 k" M+ @
    ( o/ z% a9 |. T2 Q/ G% `5 {' o
    Advantages of Recursion
    0 a5 z% S3 S$ E5 T+ X1 G1 b+ {% l6 `
        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).
    ! j8 M% c/ N% k" Z. ^* e0 d$ L/ H" W& z3 [- @* q7 a
        Readability: Recursive code can be more readable and concise compared to iterative solutions.# m! J# M+ N9 s1 {
    - y& E8 A0 C1 r7 [
    Disadvantages of Recursion3 a9 C: V7 }" `! z/ n

    / p& M3 W0 z5 F5 |- z, f    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.& Y: D5 i; v8 h9 M# A9 w- p

    $ D1 W' s* n; ?    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    - v0 P% m4 P; C: g. u& c$ B# x2 P/ B
    ' l/ c3 I: \6 B8 m# |& vWhen to Use Recursion
    / U  g7 n2 }% G! p  U6 w7 _* ]
    % N4 J6 O  G2 B/ g0 S9 l. W    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    2 V5 A' n7 ]/ ^
    1 h: e$ E: i# V5 `5 h$ `    Problems with a clear base case and recursive case.
    # ~% t  F0 W/ j& l, x
    8 t# V8 A  _, ]8 r: v5 W5 l2 v* FExample: Fibonacci Sequence
    . o6 D! N# s+ n1 ]* U% U7 R" i6 |. ~3 r8 D" i. ^3 v. x
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    : H9 }0 R& g  U; Y" r# K1 l6 C
    - l: G# e( O6 Z9 l8 ]0 }    Base case: fib(0) = 0, fib(1) = 1
    " Z2 ~& Z5 W9 Y- C4 R8 g" S5 _4 a
        Recursive case: fib(n) = fib(n-1) + fib(n-2)6 m9 h0 D% D( a$ G
    & h7 s% D  S  g" |7 R5 c
    python
    3 g: e: ?# e3 @+ M" N8 _1 T5 k( w  Q
    ' n2 G4 Q4 L# ]/ l: C* W9 Q
    6 P; t4 L* H& |( D: s# a$ u, cdef fibonacci(n):
    9 d! L$ f  @8 S# L; b4 j8 |    # Base cases
    2 m# u5 }3 O# F. @* g3 _1 N" Q% s    if n == 0:, _' C3 U6 A8 e% K2 d
            return 0
    $ s" |& L6 W" H3 z; E( `    elif n == 1:+ A2 E- q6 u8 M
            return 1
    , S6 M' r( U" m+ d' i    # Recursive case% z" h6 K: T9 f' a0 _
        else:
    ! G! f7 U2 O, J/ `        return fibonacci(n - 1) + fibonacci(n - 2)- a+ \0 r) i: \6 N2 t7 v

    * ?* Y" D4 M2 s4 t& x/ ^, K4 i# Example usage( a- ^7 R8 x! i& S6 ]& c/ g
    print(fibonacci(6))  # Output: 8
    + Q( X( R3 }0 o: e' I
    0 z& B8 l, N% p% [: xTail Recursion
    0 l, O. R; o& X) p+ L/ x9 d8 F4 p0 e& Z9 S$ Z" B
    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).8 I8 r' t% Y+ `2 l. b

    & W/ x# t6 d. X1 S, R  wIn 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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