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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 + s% Z; o/ w. z1 v
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    解释的不错
    ( }9 K, o* ]2 O& b9 F5 [: ?
    ) T9 G4 Q7 C- i4 t' H: P; w. S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。  i( }4 ]8 C& b. `' w2 y
    & ]- T! Q* m" L) A+ \
    关键要素! P  N& m5 o2 F) L1 X
    1. **基线条件(Base Case)**
    ) U, q; \! e$ G2 H8 t   - 递归终止的条件,防止无限循环- u* ^: \" m$ W  t3 w& |5 ]5 ]8 p
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    5 f4 H" j, D0 m6 P6 S0 z7 h# h: J
    # b0 T1 x( d8 ]+ h( t! A1 R7 a2. **递归条件(Recursive Case)**
    ! h, {/ a, ~) T8 }   - 将原问题分解为更小的子问题
    * @1 {3 Q6 C# I1 m: D   - 例如:n! = n × (n-1)!
    8 X- f2 Y+ L& G# L; ^7 u4 l
      y6 n: i9 n! u& ] 经典示例:计算阶乘% f; s9 k0 ]$ s' g7 \
    python# I$ d9 e! {" V! ]7 l; L. r+ x
    def factorial(n):
    / M1 h( L1 [' K    if n == 0:        # 基线条件
    ! u4 \- e& h3 d" d        return 1% a; _4 O4 j8 L( V0 U( R+ k5 z# S
        else:             # 递归条件1 u7 M7 M& d( b2 y* @8 e$ T
            return n * factorial(n-1)6 L6 d/ {" H* i2 ]/ e
    执行过程(以计算 3! 为例):+ F6 S& t( q/ ^; q  E+ }" C/ `# r
    factorial(3)
    7 A4 [1 u+ D7 v$ H+ E3 * factorial(2)
    & n4 K' t9 d) y. |1 v8 K% B3 * (2 * factorial(1))
    & |, N& R& {' ^0 P  T  ]# l3 * (2 * (1 * factorial(0)))
    & H/ F4 y3 ]+ D1 T3 * (2 * (1 * 1)) = 6$ ], t, h# m5 j, {% j5 N

    , F! `' h8 k% Q5 P) s' B 递归思维要点
    5 t+ ?* V- ^: |. {1. **信任递归**:假设子问题已经解决,专注当前层逻辑3 m' d/ m( a5 Z( v
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ' h+ {) G" b( S: ?9 s6 H3. **递推过程**:不断向下分解问题(递)9 |& Y  X# S. \* c' l. @" E
    4. **回溯过程**:组合子问题结果返回(归)4 w7 A+ i( _/ A$ v4 b

    2 L& @* r: Y" f2 t 注意事项
    7 k" Q, u: q# n6 ?( L7 E' D) K( @, G* d+ G必须要有终止条件
    " i3 ]# `; Y! I% q) M3 S递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    . b1 ~( Z+ {+ N$ A0 e* l某些问题用递归更直观(如树遍历),但效率可能不如迭代
    + G, W( o& G4 O' L尾递归优化可以提升效率(但Python不支持)
    $ ^2 _  _; T% b) h+ w4 S4 m
    7 N# F: C" |$ Z: l- b8 u0 j! }& U 递归 vs 迭代
    ; k! a% i, ^; `# C7 c! s  U|          | 递归                          | 迭代               |
    0 \# \2 {( @4 b|----------|-----------------------------|------------------|4 y8 G" D' _, |/ S, Q- {
    | 实现方式    | 函数自调用                        | 循环结构            |
    : q: T7 a' g0 X% ]6 M| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |; T/ D- D- ^9 m2 N2 J2 v9 L
    | 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         |; v$ q' g# N: C4 Y4 s! O, L
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    . k7 s9 e. e! c8 s+ b- e6 N7 h( \
      R4 Y: n5 [" ^0 |$ ? 经典递归应用场景( O" W! U2 P4 w: G
    1. 文件系统遍历(目录树结构)' I& E' g0 V+ h4 f
    2. 快速排序/归并排序算法
    ' ^' T) o$ F$ z4 S. B, G3. 汉诺塔问题: l% G4 {, c4 A
    4. 二叉树遍历(前序/中序/后序)
    - i3 Y3 k1 o; a0 n' H* N/ B0 R5. 生成所有可能的组合(回溯算法)3 X8 H$ y1 S3 z2 J- y% i
    4 J) |7 d+ c( t
    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,
    ) D) f( c) l" ~我推理机的核心算法应该是二叉树遍历的变种。
    % K" J6 A$ @, T: i  {% i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:
    6 t" `  z6 Z$ r  MKey Idea of Recursion
    3 B$ U  B' Q" @$ O7 v) ~
    $ i* n: C  u) y: gA recursive function solves a problem by:
    ' s4 m+ N3 `& ^' j" w
    5 ?, @/ n/ G+ w" M, o5 w% B    Breaking the problem into smaller instances of the same problem./ ^; n0 M( \) D) C+ W+ P
    % K3 h; D$ T7 v4 u. A& n8 f
        Solving the smallest instance directly (base case).9 M1 `8 A4 q( L/ `) [, s1 u

    : \% I- m$ E$ ~% K, c) t( H7 y    Combining the results of smaller instances to solve the larger problem.
    : J- _3 F$ O7 ~: D; s9 Z, ~" P; h1 G# M+ `. N
    Components of a Recursive Function3 b3 q: M6 x& R6 p: T
    1 g) P* v7 O  d$ U
        Base Case:
    0 z+ W; Y. D) s$ }0 w) W: {! T0 u- R: f7 r, `
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    7 Y8 @( }' }9 K% `( j+ u6 e
    + X' Y# o5 \+ Z+ D3 `/ p7 j1 K        It acts as the stopping condition to prevent infinite recursion.
    ; _) {; X6 i7 s0 Y2 \7 E" e) y" N% r: K8 e! a1 p; J/ e4 I
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.8 `+ `/ g  t& |

    # q' g. K8 D$ c2 u" M    Recursive Case:9 v( Z+ ~5 S9 M3 n; J
    8 e) C& ?' W, O7 b) E' \% m5 o
            This is where the function calls itself with a smaller or simpler version of the problem.; f3 T! s$ B1 a+ b; o& S! y
    / w) c$ [3 z1 |" S3 y; B
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).4 M7 ^8 y$ N. }! x

    2 b% R' `, f9 J! @2 _' U3 qExample: Factorial Calculation1 t7 w0 z+ {) l6 p& c
    / ?- i4 D' T; q$ o+ N0 H
    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:
    % @% a, E  R/ ?; l
    1 |6 E8 ]- x7 ^; [7 x    Base case: 0! = 1
    % w& T; J7 i1 j& X, l$ T# E; N2 y9 O" G3 S2 L' w
        Recursive case: n! = n * (n-1)!) o* S+ ^1 }, |7 n1 H

    % d5 S) M$ f4 o& \# @$ j9 VHere’s how it looks in code (Python):2 f- V1 e- f1 f" H& }8 }. r
    python: e1 y6 W1 }; N" b# h" k

    + i: V7 P9 a; ?: J6 s* s
    " T& l6 X/ q1 X+ K& G! {def factorial(n):
    9 T6 }1 k# c. j# ?6 o9 D& a3 }    # Base case% [! @1 T$ c* w3 i. M& e
        if n == 0:7 a4 @9 i1 q# H' N7 w3 r4 I
            return 1
    ; l" j7 @& P1 {$ A6 N    # Recursive case
    - E+ r5 x  C; c5 R& D    else:
    ' O4 S/ S. A( ]8 z( i& a% Z        return n * factorial(n - 1)
    & D2 J( z/ r% T- J3 x4 j7 t- P  C4 b" [6 z8 r: B2 L/ ]
    # Example usage- R* t0 W  A- u' v$ `) U4 U
    print(factorial(5))  # Output: 120! L4 h0 r! J8 f7 V- C
    1 q* u* j' T- ]2 d2 l0 K
    How Recursion Works0 K: j' ?  p3 r; b# R
    * ?5 `: R; J; O9 i# T
        The function keeps calling itself with smaller inputs until it reaches the base case.6 T+ z% G  Y$ S9 k( N9 x

    - F' j8 E7 s9 A( r% }8 v$ m    Once the base case is reached, the function starts returning values back up the call stack.5 K  J& {9 F) F

    ' h0 x6 i/ u3 z; _    These returned values are combined to produce the final result.1 A- }  @. g  V# N* y

    + l2 r5 n4 y" \+ ]For factorial(5):
    $ r9 z  V' C* b) u! O
    , @  L3 L* s* m! e1 ?, c6 e0 [! i- J$ x6 j, A
    factorial(5) = 5 * factorial(4)
    0 a/ s. @* v% \8 b" |& _factorial(4) = 4 * factorial(3)
      {  ^8 D1 L/ }. Vfactorial(3) = 3 * factorial(2)
    3 b, k$ }2 C! w# ufactorial(2) = 2 * factorial(1)3 C, y- g/ f5 S+ I  S
    factorial(1) = 1 * factorial(0)( E. y+ h; p9 l) U3 R
    factorial(0) = 1  # Base case- W0 j. \6 H* j8 M% G) S/ x
    2 E! L8 Y5 k; @1 E' ^* h$ F
    Then, the results are combined:7 b6 ^( b) E( \8 f. y5 ^
    ( d5 L1 e" p1 \+ D3 G/ H4 d- |
    / `4 `5 |1 k1 L5 u& m
    factorial(1) = 1 * 1 = 1
    6 x  J  K0 a0 v9 k+ M4 l- P5 T# L+ Gfactorial(2) = 2 * 1 = 2
    : P6 A3 B  b, m$ p6 Jfactorial(3) = 3 * 2 = 69 T- t7 @$ Q# w
    factorial(4) = 4 * 6 = 24$ G, v7 v( v2 \" k; W
    factorial(5) = 5 * 24 = 1201 o  [/ ~+ O, t1 q* `7 {
      @; o& I! U1 [+ w
    Advantages of Recursion
    4 H* z$ \9 {. `  d/ o6 I* N- h4 M1 T8 y, X
        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).
    3 G1 n8 h% X% X$ {9 _5 T! |4 A/ u# T, }: R' I. |1 l
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    6 C4 J& Z6 a5 ~. N/ w; H; N- X. W# k' }2 C+ B/ v
    Disadvantages of Recursion9 u' _! Z; @) k7 u8 h3 s5 ?/ ?% \

      i( [" O/ a. {9 W5 V/ a    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.
    9 G9 ^- |0 c! Q- r9 q3 n6 r' M$ {: Q  m! M1 ~6 r
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    / O: T, X; e6 V% E, Y, d* M) W) }# o  R) \5 `( t  W
    When to Use Recursion6 }' @* f$ n  A* @+ T! d' g7 J( `
    6 G3 H- M7 J8 k0 W& o
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)., q6 W! I, s3 @* O

    ' D- _; u/ c% v1 e+ q# e    Problems with a clear base case and recursive case.* u, x# N  u- [. W
    0 ?; F" y* V& U6 C. N# f' g
    Example: Fibonacci Sequence
      m7 `1 e4 R5 S* m
    : g2 W4 v, O. r- WThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 O, _0 ^! T9 i$ Y- q
    4 ~% u% j8 b3 P+ ]7 k6 X- K
        Base case: fib(0) = 0, fib(1) = 1: [+ C* H& e" a+ k  ]- @2 J
    4 l9 r; J# f. h
        Recursive case: fib(n) = fib(n-1) + fib(n-2)% b& b# I! h. L. v( ]
    ' I6 m* P- P5 j" u) M! [$ S6 B
    python
      S$ S5 E1 p8 r0 C5 {- f2 x2 h7 U2 y* t) ?' P/ q0 q1 Q1 D

    - z/ a* a% q8 }4 o! zdef fibonacci(n):
    6 v& o& U- u- I    # Base cases; S9 S; F' H  m- ^& n3 k
        if n == 0:
    # K+ o. S8 x8 w        return 0
    % _' o# K1 v" e- @; e7 z: J9 d    elif n == 1:
    : Z4 b' J( t+ n8 \! C6 D        return 1" C5 W/ W, s3 B
        # Recursive case
    / q8 m8 G# z" @* H, n    else:
      t$ f  Q$ e3 T! {! l        return fibonacci(n - 1) + fibonacci(n - 2)
    % b, N7 \3 P4 ?9 Y1 D) x! k7 _4 S. Q' y4 p3 {) }% o* U
    # Example usage
    . R' h2 ?3 l3 g# N& Q5 r, Rprint(fibonacci(6))  # Output: 8
    / e. ]9 y; e. g6 [5 R& _5 I9 W) T$ m  h3 m0 d
    Tail Recursion3 P* j1 w) f$ z4 w3 n8 ^

    * F4 Z, R% a! B- \+ B# {% {3 CTail 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).
    2 u* N  R" q! W, ]7 t$ z, W; q3 V. b3 E7 ?4 v' Z( d/ P# m
    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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