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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
    9 ^- V. U" `: d, x# y
    $ ~% k1 k0 ]7 O8 }! Q- {解释的不错' y$ ?1 b. d6 ?* T1 O

    ( b) w) I7 u8 E0 W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。% a* _$ r9 z  \% F( o; f1 Z

    ) n0 ]( \; z# a; Z0 i, {8 Y# V 关键要素+ U5 d1 U3 G( ]6 a
    1. **基线条件(Base Case)**' D$ w. g5 }- Q( P8 Y& R
       - 递归终止的条件,防止无限循环. q; @7 S( ]* H! w4 }9 b" s0 i
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    % K; h5 H$ t$ M! f& I& y" g' K. k! e' l, j3 b4 `0 W" m
    2. **递归条件(Recursive Case)**
    * g" g% G8 ?4 o, u9 g; Z9 y$ z" _; R   - 将原问题分解为更小的子问题* ?- T0 y& M# c; K6 {" u, D$ H
       - 例如:n! = n × (n-1)!; G- m* l" H6 y6 e: K

      Y. ~$ N4 l5 S+ j$ W4 Y: W 经典示例:计算阶乘
    ; A8 A& S' _; }8 upython9 u, V$ ~0 N$ n0 B
    def factorial(n):
    3 l! @; u. q6 |    if n == 0:        # 基线条件
    + d/ i. D1 p# [5 s6 n; ?" c        return 1
    ! m8 |+ o( Q. ^" |    else:             # 递归条件
    ; B/ K# q/ M% N6 M        return n * factorial(n-1)
    2 W( q) a" X' C7 ^  P执行过程(以计算 3! 为例):3 u; |' D- p# g
    factorial(3)
    * g+ ]7 o6 p, Q/ D/ u- c# M3 * factorial(2); E  ~! f% {0 e# e) A/ L
    3 * (2 * factorial(1))+ F& ]. O  p3 g! C
    3 * (2 * (1 * factorial(0)))" @0 D1 T2 q( V/ d: s0 |8 S
    3 * (2 * (1 * 1)) = 68 P+ X3 g, Q4 r6 ]
    2 N4 m! |2 A- }" C
    递归思维要点) G& F9 z- O1 X9 Q1 p2 i
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    7 @: u4 `8 o# M) ?1 |9 x. S: q2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- g) }/ H7 M  O/ y  H# U" Z
    3. **递推过程**:不断向下分解问题(递)
    . P' ]1 _3 m! V. ?" G4. **回溯过程**:组合子问题结果返回(归)% \3 v5 p3 Y4 S. \$ G

    * a( n$ g' R* y, R 注意事项
    ! z  _+ s. x* {/ V( J必须要有终止条件
    , J( U5 f/ v: z2 m( H" n& T递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  |- H1 K! i9 G* `5 L
    某些问题用递归更直观(如树遍历),但效率可能不如迭代/ ^+ p2 y- {1 O. a2 ~
    尾递归优化可以提升效率(但Python不支持)
    , _9 j8 q' ?3 y& O' p# @! X/ u% M( T: a2 F
    递归 vs 迭代
    & f4 P/ g, O+ [6 i|          | 递归                          | 迭代               |
    3 [, O  V* F, z8 V|----------|-----------------------------|------------------|( {+ u1 n& `% n  S
    | 实现方式    | 函数自调用                        | 循环结构            |# Y) ]. ^8 i; i: y' E( k/ i
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    2 H! l  d: }" @1 ]1 \| 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         |
    0 m& q7 e& p/ ~* B| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    + v4 s+ i1 b1 s' v) c; s8 F7 k  s6 x7 N+ D* {6 M6 P; t
    经典递归应用场景. `$ e3 h% Q- w# P' ?2 w
    1. 文件系统遍历(目录树结构). v$ U! H" P( Q
    2. 快速排序/归并排序算法' P# g9 X* |6 ?8 W- f3 Z1 N
    3. 汉诺塔问题
    + h0 E  O: O  h4 a* m4. 二叉树遍历(前序/中序/后序)
    ) _- v1 ]8 {( t6 t5. 生成所有可能的组合(回溯算法)" |6 x& _- V! {6 Q

    ! x* \' q) q: `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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    昨天 09:10
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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,
    3 Q2 k% U7 G; F  D我推理机的核心算法应该是二叉树遍历的变种。
    ; g# _- n/ |* e# p3 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:
    + A. C# {; B8 g* D# I; tKey Idea of Recursion) l$ H" |. g9 ~( Y4 M

    6 p! U& r( j$ `) zA recursive function solves a problem by:& q1 N; x$ M- Q8 _0 I6 ~

    : J" s! R: L5 ~9 k& p    Breaking the problem into smaller instances of the same problem.
    + @* m4 {' o. P- O4 l5 G0 D( [$ Y. f9 }6 m) D0 m
        Solving the smallest instance directly (base case).
    8 b! ]5 [9 @% \; s( k$ R
    & d4 A5 m2 h: ^$ A! Y    Combining the results of smaller instances to solve the larger problem.+ U8 z$ K* A5 \1 f! c# L4 M

    ' j2 B; G9 ?" o# GComponents of a Recursive Function* w/ u; B5 W6 m; W" o

    , B# n1 R  y, c, @; y) c5 b; X6 |    Base Case:4 B4 A2 V  p8 }( `, U
    3 o; b# A$ n, j8 s
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.' S  D$ i3 @5 F& ~: ?: c

    : @$ o% B, w: |. K. i  q        It acts as the stopping condition to prevent infinite recursion.
    ; z- }4 z& Q7 m7 n0 A- k; z$ j6 S" r, C3 t5 w7 t# B
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.& ^  |+ g4 F3 F

    1 U# ~4 a) Y* c- b9 C    Recursive Case:
    " |" P: P5 t0 H1 F: J. K: z6 M
      x. |3 I& s6 w( d$ x1 `' I  l        This is where the function calls itself with a smaller or simpler version of the problem./ M! T: M) [. r+ k( g/ M! w. V6 D

    2 C" K) @# z' b8 S9 L3 S8 [0 m        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% \6 t+ A7 m5 b) e# }" {
    % {) u# u$ P3 ~# {* K
    Example: Factorial Calculation1 I& K4 V! k' U

    0 |* R0 a: \# B( j# \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:: s/ y& v3 Q' U: K! y( E. h

    # v; G. g  ]  }: y    Base case: 0! = 1
    ! H0 ^/ L+ Y+ k# `: q# N/ o
    8 I0 ~1 C( {5 k; t3 u5 G    Recursive case: n! = n * (n-1)!
    ( q3 d6 ~; R4 C( ]6 _5 a
    . o. d+ E. o$ o8 e. [7 PHere’s how it looks in code (Python):9 B6 Z" ^" k6 S# {( w9 c
    python
    4 S+ G5 B, A9 O$ W' S# b/ q8 U; L$ D5 g; K: \+ h, `

    # Y, O7 Y2 Q  }def factorial(n):# W+ Q5 P* q! O; B7 X" R( D0 q
        # Base case
    " k: e! B1 h' z! W! O+ x    if n == 0:) Z' P) H: }2 |, |* B
            return 1
    . P6 A9 T) E3 \  j9 H# s8 i    # Recursive case
    9 N: L: P3 \2 F, Q4 g; ^    else:
    , M7 [! q  y5 M: t" Y        return n * factorial(n - 1)
    2 Q$ F% `: C  y4 u/ r
    $ y5 Z9 p2 _" G) d+ f1 H5 v# Example usage* P: x6 j9 r+ V: H, X% M/ F
    print(factorial(5))  # Output: 120
    $ c7 H1 S4 A1 e5 ]8 k
    ' s7 Y) B- I7 l4 l( X4 p0 ~How Recursion Works
    , E) ?' ?; g  t. G0 w( Q5 ^1 Z5 Q: ]2 N- V+ z, E6 N5 t4 x; `. ?
        The function keeps calling itself with smaller inputs until it reaches the base case.
    1 F- `4 i$ o% ~. d1 a! C9 l1 M
    7 e: z8 P$ P% V* Q! Y( B    Once the base case is reached, the function starts returning values back up the call stack.
    ; e  V" Q: @, Q. y4 E
    2 n/ {& P# j# ~1 \. H    These returned values are combined to produce the final result.
    * Q" `0 p4 l! j/ q+ C6 P9 p; u: P) x6 r* |) C* o% Y
    For factorial(5):% g2 s! u& U  A+ _; T
    2 u: b" M! g. G4 e* J
    4 I3 z) V" d7 l5 `
    factorial(5) = 5 * factorial(4)% u6 B5 G& J$ Z8 l
    factorial(4) = 4 * factorial(3)- O7 n9 Z! l+ i
    factorial(3) = 3 * factorial(2)4 F% J$ ~# L5 k: K
    factorial(2) = 2 * factorial(1)% R- D! ^( b9 ]; d, C
    factorial(1) = 1 * factorial(0)6 W! W6 M- {+ b' ^* n4 b9 ?
    factorial(0) = 1  # Base case9 u4 d/ W# f2 H5 O8 e
    - L& B/ L: y9 M  P% D
    Then, the results are combined:9 j& {& |* |! O+ H2 V
    , V# _9 A6 t* M$ H3 e

      \# a5 K/ O$ ufactorial(1) = 1 * 1 = 1
    1 Y8 t7 g; |' _) X) \factorial(2) = 2 * 1 = 2
    ' C  x5 {- J, i9 \9 t" Vfactorial(3) = 3 * 2 = 6
    ) N5 R: k* I: _) q  sfactorial(4) = 4 * 6 = 24" H2 T" a8 G. m$ y1 ?' \
    factorial(5) = 5 * 24 = 120( M0 u/ S  p$ R! a1 A. ^$ b  `
    4 `8 T7 n2 U4 `
    Advantages of Recursion0 }) K8 R) R: C" F5 J7 g( F

    6 ]3 _! c: p+ x& I. l! R8 E$ y    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).5 o/ _- n. s. C" O
    # p3 H  i! T' z  v/ l. l) J) y0 ?- `
        Readability: Recursive code can be more readable and concise compared to iterative solutions.5 ]( |  X2 \$ m% s2 P) w6 c
      b+ N+ @3 Z1 Y% `6 `
    Disadvantages of Recursion
    4 z) h, w$ O: @, J: Z0 B
    ' ?1 r7 ^$ X8 `- R% k4 v( h8 E    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.
    . l  M1 j+ x! k% y& U: V; m1 }# T' n# @5 r( V
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    # m+ \% w4 {! d7 K/ Z& @3 k9 Q% F1 ^3 n/ ^
    When to Use Recursion
    1 e8 k& F4 K$ u# i% X
    ! [) O: U9 R$ x; s4 W. Y; `    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    4 _8 o9 @% u% P( o; r( P# l: [5 I. t) h, a# _# b  t! N9 U) s
        Problems with a clear base case and recursive case.# A. T# G8 V! L" \

    ( V6 b; r6 ^! g" M+ I. HExample: Fibonacci Sequence5 B3 S* k1 R1 t: l2 M  H& z3 M' [/ \
    * A5 d/ W7 L4 i9 b: m  X& Z
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    1 U% L& y; R, y9 t) \" c- i# a7 }* c+ E" T5 O" ?; u. j9 c, S( F. C5 ]
        Base case: fib(0) = 0, fib(1) = 1) }7 g4 S8 [7 F2 `

    8 [& }0 f7 F+ u4 I/ H6 v    Recursive case: fib(n) = fib(n-1) + fib(n-2)% q5 O" i, l/ n+ P4 \0 c; [

    4 L/ \7 S- K. c) p( Q' Zpython% V# y) W2 R% K* w$ R

    : w6 E% k- z' |  M7 W, X+ |
    7 m) T; d# K$ Adef fibonacci(n):
    : O2 i& o. _- w    # Base cases# N% }( q$ w1 O- [
        if n == 0:
    , l- V3 ?& x, ]+ e2 s8 U0 V. Y        return 0
    + I( b. m& ]5 g/ B: B" ]    elif n == 1:7 `' m% f7 v! t% P
            return 1
    # C1 O; b* M6 f0 z. q8 W1 E    # Recursive case) O! w0 [8 |0 |& J/ ^5 A9 O
        else:
    " p2 N8 d# s  b        return fibonacci(n - 1) + fibonacci(n - 2)! Q# s* B& {0 e4 Q

    * v, k; O5 ?, v7 h2 u# Example usage
    2 Q+ N* C" H3 p$ a$ I! c" c. Vprint(fibonacci(6))  # Output: 8- e( {' W1 Q6 i0 u

    + G0 U( n( M; H2 m0 g. y8 ATail Recursion
    , c4 q& W. a1 t6 j4 F" b2 E
    . Q0 y0 n* m" l. ^1 a( yTail 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).
    & k; i/ p- F+ [% l4 O8 h" c0 g: j& l8 I% |
    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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