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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 5 r: \0 r, Z5 Y& A' _
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    解释的不错
    ( F  K3 [) M" \) E0 z; ~. K' `2 M8 I- P6 R
    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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    9 s, C# U# S8 _( X4 O1 a 关键要素: V, A- A7 @9 w& W
    1. **基线条件(Base Case)*** s) c: A$ G" H' m# T9 k. o+ e
       - 递归终止的条件,防止无限循环
      d6 c- }) I+ J   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# c0 `4 N; S3 c0 m8 z

    6 |+ r! W: {% I2. **递归条件(Recursive Case)**
    . Q3 D' [4 W. y4 l% D! \   - 将原问题分解为更小的子问题
    / z+ h+ r# C& @& V   - 例如:n! = n × (n-1)!
    ' X* K6 a9 J" W. |  _
    * E, Y8 z  ^& x 经典示例:计算阶乘
    " w0 ]1 w$ t# z4 Tpython
    7 ?! J! e6 s2 H, R0 H" D6 @def factorial(n):' a: q! U, y/ w; K/ x0 W) z
        if n == 0:        # 基线条件6 D; p5 y1 C" u, k9 a
            return 1: V' P" @% ^/ g9 G) X. x
        else:             # 递归条件6 c* F" l, d1 \: m$ c1 C: i4 E
            return n * factorial(n-1)6 @4 R8 y2 ?. n  S
    执行过程(以计算 3! 为例):
    ! d4 b0 {9 P1 m0 [& K# h1 Efactorial(3)
    ! l* z1 x  o: f& X+ i3 * factorial(2)
    7 ^+ q1 K* B8 j8 {8 K+ O$ |3 * (2 * factorial(1))
    ! W! i+ ]& _" Q: R0 G8 c3 * (2 * (1 * factorial(0)))% g) \3 k0 }+ Y: G- W+ O
    3 * (2 * (1 * 1)) = 6
    ( ]% c7 a, O4 J4 T2 L; R
    7 i" J, B7 P; r9 R3 d6 B 递归思维要点1 o" ?5 N; x0 r! e
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑; g7 v7 ]# _: `, j7 f
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    $ ?/ u) R9 y8 g3. **递推过程**:不断向下分解问题(递)
    . M5 e7 f6 T( R4. **回溯过程**:组合子问题结果返回(归)- e8 @! G0 w; Z8 ?
    3 Y3 Q8 F, y0 E1 e1 H
    注意事项3 L; m# r% d9 \- k1 \
    必须要有终止条件
    - p  y1 B# k/ [( @! }  v% ~递归深度过大可能导致栈溢出(Python默认递归深度约1000层)) `6 O: Q3 [" e- R& ^* G
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    0 v- R- o! v" J尾递归优化可以提升效率(但Python不支持)
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    , {6 x! a+ W1 D  n6 y! m 递归 vs 迭代
    ! ]4 j* P7 f: z2 f) J|          | 递归                          | 迭代               |
    ' f& M" H' C% \$ G|----------|-----------------------------|------------------|
    1 o8 _* f4 b! S. z- b2 x| 实现方式    | 函数自调用                        | 循环结构            |
    ; N" \, ^1 j; Y) `2 }| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |0 K8 z3 \. O8 |! G! T3 n
    | 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         |
    6 [0 K- k4 r" q| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |) ]: n; ]2 \+ k2 j9 P% S9 p
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    经典递归应用场景
    & g! P/ D* W  n+ c1. 文件系统遍历(目录树结构)8 o: T8 b7 Z2 f% E7 Y
    2. 快速排序/归并排序算法
      C0 D/ Y+ F& A; \  q3. 汉诺塔问题; @$ @9 ]" @/ a& a4 A
    4. 二叉树遍历(前序/中序/后序)
    $ S( W/ F7 M4 ~5. 生成所有可能的组合(回溯算法)3 C* |/ a4 I" m5 k* l

    . H  ^! t- W; K6 w$ w- q; R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,
    " K% o, [3 n$ l3 @$ X/ `/ }0 \我推理机的核心算法应该是二叉树遍历的变种。
      b2 S4 h; i/ L: F% b& F" @! 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:4 z! r9 R9 \9 h- ^* v& Z
    Key Idea of Recursion0 U* V7 r- F0 S9 Q

    ( Z% p2 P+ O% y% }A recursive function solves a problem by:
    9 T/ d$ e, I# R$ c9 R( l6 B/ Y3 n3 ~( w9 d# K0 J/ {8 L" d/ o. I  _3 Y  _
        Breaking the problem into smaller instances of the same problem.  r2 q5 i" D7 g2 I

    9 e5 Z$ z2 D9 k+ V' r    Solving the smallest instance directly (base case).! X$ X4 [$ n% z1 p2 ?% X  L
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        Combining the results of smaller instances to solve the larger problem., T, F7 @" t) A  b$ Q
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    Components of a Recursive Function4 i6 j! g5 p( H# D1 r% z, s

    2 J  k: ^1 ]% W+ f5 o' w- @$ h    Base Case:
    ! s1 a7 I6 Q+ ?/ I7 @8 U, {7 N3 T* U& c/ R$ G# D$ B
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            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:8 g; p/ s$ M( @/ X. I6 a
    : \/ F1 V, C  O) t
            This is where the function calls itself with a smaller or simpler version of the problem.
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    ( W" c: ]" t. d        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' {4 D& v  C5 V, J- t% r+ {
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    Example: Factorial Calculation; M" X5 }6 D; \+ w9 t

    0 k) s/ r" Q! r) |$ n. Z" h" P# nThe 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:
    + o; f( C# K/ v! J3 G# A$ r
    1 z+ r0 w! F  }( S% }6 R    Base case: 0! = 1
    $ Z% t5 B9 J# y3 ]5 Q6 |! H3 q2 J* F: n
    9 l0 u6 |7 m2 f    Recursive case: n! = n * (n-1)!
    : ^$ q) V+ U7 ^+ b
    ) a- w5 {! H5 f( yHere’s how it looks in code (Python):2 u9 K6 Z5 ~  t7 Q2 s
    python
    4 ~: I# l% G, j' l6 I( i+ Y. H8 ]+ m8 V' V! e' k
    / C6 X/ A- o2 J- j- x
    def factorial(n):' T! y( r+ x9 G7 o3 N( O
        # Base case3 C* o( a" z# {
        if n == 0:
    # ~1 H9 v7 w; F        return 1
    5 ?: V  W" F4 A4 F- h, N    # Recursive case
    ! }8 X4 A3 T6 [5 H    else:
    $ B( g# g3 g4 t6 P0 o# }        return n * factorial(n - 1)
    ; E6 S1 g6 }- B7 g1 J7 f$ v+ o  O" I, h/ A% ]3 v2 R3 F7 o! E: g; P
    # Example usage
    , d  @2 D& N/ i! p! [print(factorial(5))  # Output: 120
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    How Recursion Works) x. u" Q+ L' s9 U

    8 A5 X4 G) U& T6 O. F    The function keeps calling itself with smaller inputs until it reaches the base case.
    ; c+ }' _0 U) M4 @3 |% A8 a
    " N0 _' w$ ?3 T0 m' b8 b( Y+ R+ s    Once the base case is reached, the function starts returning values back up the call stack.
    / D' p* U2 y0 j8 P, k$ X0 |5 |# @+ t, J. ~# @2 V/ t  {
        These returned values are combined to produce the final result.% }) G, B" o2 a* E
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    For factorial(5):1 Y8 o" d  J9 z( q% ?; x

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    5 U1 M' u( j& v4 f+ Ifactorial(5) = 5 * factorial(4)
    ) i  M& A1 Q( N8 j" X, Q: ?: ~" rfactorial(4) = 4 * factorial(3)
    + F0 R5 i9 |5 m) v2 S% dfactorial(3) = 3 * factorial(2)
    - d( X, z! J  K  K& r1 Zfactorial(2) = 2 * factorial(1)
    ' a3 [7 l5 Z+ _* Sfactorial(1) = 1 * factorial(0)
    ; n3 C/ O$ }% t  qfactorial(0) = 1  # Base case5 P3 ]- S9 F- k3 w0 _% x3 L

    5 @0 N5 H1 r8 B4 Z5 oThen, the results are combined:  x3 k% j0 G! M, S  [

    : U9 ]! X+ |( F! ?6 i
    ( @0 v" y* `% L- |" R$ E% o* F; Z" `factorial(1) = 1 * 1 = 1/ c8 F: x4 y: O! I$ n0 D! l
    factorial(2) = 2 * 1 = 2
    5 Q9 ]" p: e* V4 U& Bfactorial(3) = 3 * 2 = 63 c8 w: b6 s, s: z$ _# @
    factorial(4) = 4 * 6 = 24( W* m% Q0 |( j- P# E
    factorial(5) = 5 * 24 = 120& B$ [4 I2 r' }, T

    $ d$ I/ A. K/ R" Q% CAdvantages of Recursion, S4 |+ \0 t* p9 p& X
    4 f" m8 F- c. K) |# O' `# a& Q
        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).
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    % O& t; w/ @! |1 \/ S1 @$ w    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion
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    ! H' n; D- ^- y6 u    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.
    , X! _) `# d% U0 S# p" Y8 i" J0 `2 k; z% Q1 e1 s; ~( `0 A
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 X- O+ W7 g$ o4 e6 k+ M0 H

    : y& c; G( }% W: H/ p! f" R3 x3 SWhen to Use Recursion, t7 O, ~4 j; B, w0 e& q

    6 b  ^3 M$ g4 B( O2 V    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . I. O" B% s; k# t5 G% i; [' w( z5 e
        Problems with a clear base case and recursive case.- z. }) ^4 D- I- w) d* Q. W+ H/ V
    9 f5 v" h& v' c
    Example: Fibonacci Sequence
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    : l5 ~3 a# I7 h  T) b4 ]9 S( l+ ^3 @The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . e* a! j' i  C3 l) x' u
    6 g0 Z$ z" `+ g5 F    Base case: fib(0) = 0, fib(1) = 1
    7 m" b4 D  p6 Y  q6 {' B
    8 _" |. k" Q: x, k0 `    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    9 G+ y! [2 t" d7 i! \4 G' k0 m! C# F5 e$ W
    python$ m) s$ Z& [- C  D: Y/ p

    9 K2 c  L8 H3 T0 C; @4 ^& t5 G
    def fibonacci(n):% u5 U  [8 g/ H4 w$ ]. n4 d4 ]
        # Base cases
    2 C! S* R7 e, e+ |1 W    if n == 0:% n$ g) m3 X0 @+ s+ J' X" k
            return 0
    & b& h% P  L' j  T  ^. |% s    elif n == 1:& B" v% N) {- t
            return 1
    % B2 _8 j1 {: A+ k- n. J( M( Z    # Recursive case
    : o, J3 `( ?- {    else:
    ' T& |9 J) }& {        return fibonacci(n - 1) + fibonacci(n - 2)
    ) |' W0 Q" _2 k
    ; \4 H2 P+ Y1 Z# Example usage; \9 J. p; s2 G; K5 |
    print(fibonacci(6))  # Output: 8
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    Tail Recursion
    2 y$ \9 T/ ?% q/ V! d* S
      K# f3 L) c- y+ r8 y8 D/ \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).
    7 q+ u  m0 f* K1 P! C
    6 ]; R* B8 h# m* o) m; OIn 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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