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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 / q; f, }- I5 O. r# N3 f& [

    + N! S5 j6 r6 W5 H4 z解释的不错
    ! T2 Y9 @, Q9 M2 M$ ^
    ! M% w9 h. G2 D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。
    - L+ e9 ]) u- \! T% Z; f& N! Q1 c. h9 E& r$ ]+ }) E
    关键要素+ Q0 e1 x4 Z/ }$ P' N
    1. **基线条件(Base Case)**! e( J) W2 U# y3 H: ~
       - 递归终止的条件,防止无限循环4 t4 {1 t- S+ @+ F3 \* w: [4 m
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    3 H+ ^. L( k$ N+ A# k9 P
    . Y& D* i2 w) {- m- i' l2 W5 O2. **递归条件(Recursive Case)**
    % c$ O6 e! s0 E9 T- ]" j   - 将原问题分解为更小的子问题
    2 I1 A0 ?& U8 p; L- x   - 例如:n! = n × (n-1)!9 K+ ]: p! I. N0 M" F9 s5 Q9 L
    % U. _" t* r4 I# }: L
    经典示例:计算阶乘0 r! B! Q/ i* J5 B# X+ ?
    python  O  _# J5 Q- r' f" e
    def factorial(n):/ G1 p: e! ]& _. `0 Q- N
        if n == 0:        # 基线条件, a6 w% b; n0 ]" c" I8 F1 {2 h7 o
            return 1; x; q. n! r# b" S
        else:             # 递归条件$ D# X7 k5 m% \* I. F) ^1 v
            return n * factorial(n-1)1 C7 Q- u4 s/ t. S5 H4 j+ e  X
    执行过程(以计算 3! 为例):
    % c3 ~$ a$ q: A7 U% f" `" r9 s# Pfactorial(3)- e* v: G- `9 A, B; S/ Y
    3 * factorial(2)" ?# V9 e/ _; _4 L- @' Y  c5 A
    3 * (2 * factorial(1))
    8 E! ]& E6 U0 y4 B% v: B; f3 * (2 * (1 * factorial(0)))
      I4 ~+ c& u- x" `3 H8 e* U! m3 * (2 * (1 * 1)) = 6! _  e5 a9 r1 ~! m# t! m) L

    + y- g" E3 T2 g  A. G$ ^ 递归思维要点3 `6 X/ x; u* ^9 P. T
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑% c9 b3 }. l9 f" z8 o! l. S7 F
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    2 w" p" R0 z* u4 V3. **递推过程**:不断向下分解问题(递)5 n) w5 M6 ?2 J6 y7 s( [  e
    4. **回溯过程**:组合子问题结果返回(归)" c5 p2 Y; w' P2 M. P! v+ j, V

    3 t; i& w0 W- A5 A 注意事项
    9 M! ~; h5 A0 k5 X+ Z必须要有终止条件
    : t" J7 ?6 [0 a. f, B递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& \, h$ ?2 I! \  r4 c8 F
    某些问题用递归更直观(如树遍历),但效率可能不如迭代# N" r  E3 \: Y
    尾递归优化可以提升效率(但Python不支持): S/ x  E# @& u# N# x9 R/ A1 U6 k
    + J6 u' p3 j+ T- L- _
    递归 vs 迭代
    9 {0 N+ k" Y2 e. ?  B, `; C! f|          | 递归                          | 迭代               |
    5 O. I: D& O" V( g3 I% a6 p% o/ ^|----------|-----------------------------|------------------|/ R1 ~. B  U( ^; j* V- x
    | 实现方式    | 函数自调用                        | 循环结构            |
    , w" k9 O8 N& X1 `$ h; z2 z$ ~! W| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    4 N: N7 k, S4 v4 ]: k4 Y/ ~| 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         |
    3 `5 S: g+ L  j% c6 e+ x3 n| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |* k5 p+ q: E: e7 j
    8 N- O$ C' @: E
    经典递归应用场景6 |- S, D/ g3 M
    1. 文件系统遍历(目录树结构)
    $ A6 T6 |' G) l8 @  P: _2 i! Q2. 快速排序/归并排序算法/ _, E9 d- V7 K1 V
    3. 汉诺塔问题/ S& K3 [! G/ N2 m
    4. 二叉树遍历(前序/中序/后序)
    % G& d% @( D4 A# w! v8 o: l& d, g5. 生成所有可能的组合(回溯算法)( _- o8 G! b0 V: Q' |: c( z
    : i1 u. ~  {9 Q5 R, ?( f
    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,
    ) T1 Z9 k/ |, Z( H我推理机的核心算法应该是二叉树遍历的变种。7 a, X7 c2 M! J+ |4 Q. c" \
    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:
    / R1 i- Q& @. b8 MKey Idea of Recursion
    1 M! P; x" _7 t7 P( C: H
    - v# F! b+ Q& f0 {  S) p7 }9 o+ kA recursive function solves a problem by:
    4 k; h7 c* v& G! |) p$ G: v5 w
    4 @$ v1 a, F1 n! C, W: u4 L# E. r    Breaking the problem into smaller instances of the same problem.+ C$ N5 W) n* I" b; M7 J* x

    1 g& x6 ]/ X8 K" p9 x1 r) {    Solving the smallest instance directly (base case).
    ' Z& A' I; ~" S+ N8 j' r. v4 I( X3 a2 J" G  `* W4 P* R% o
        Combining the results of smaller instances to solve the larger problem.8 q' d  n5 Z' Y- ^; p
    & p4 v+ J+ H7 T: _7 a" H! t( X
    Components of a Recursive Function; n2 B# q, ]: a6 l6 h, ]2 R' N
      T9 n& k6 W* o9 T0 Z1 z9 k% x7 I
        Base Case:
    , N& P( C2 t  L& E+ o9 @
    $ t) v  ~% V; g7 p- ~  f        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. V6 S$ h* _0 F9 e

    0 R0 ~* g, I; r2 d1 y% q        It acts as the stopping condition to prevent infinite recursion.
    / {* L# d* J+ N/ Y$ c
    # F: Z/ U2 b. G2 u7 f( ?. W: ]5 F        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 V- w# s; t  Q# I4 o3 z
    6 l( z, w7 \2 x: r
        Recursive Case:3 R) F9 B! M1 n% h* C; p/ K

    4 c& e. I9 U5 u        This is where the function calls itself with a smaller or simpler version of the problem.
    6 u6 R& `* s; k8 Z; c
    1 u; N5 O# F; F# M. \- x        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    3 \0 g/ j9 F- A8 y& [. W8 F" k/ j1 w7 P% o
    Example: Factorial Calculation
    / O- O' F7 D- i1 f  U
    ! z1 L& ?* Q7 z9 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:
    " @3 E2 u; X" ?. Z8 l
    , ]- q7 p# z" z6 U2 g    Base case: 0! = 1
    / J. k4 K& `  S; S' {
    & H; p; g! Q/ m# q; m- r    Recursive case: n! = n * (n-1)!/ @- D! s' c# j

    4 q# m# k) `( d2 N& m6 y3 SHere’s how it looks in code (Python):  L/ g2 Y) w9 @3 y1 H
    python4 W9 C+ o8 m( H% k5 E
    ' i# k% o# ~0 c4 a. Q
    " v- r% Y1 z0 a
    def factorial(n):( Y- |1 J( t7 E& }. G  k: K  b
        # Base case% C  J3 k4 j( T* m- j; J1 e
        if n == 0:
    5 R% D5 }% \, n0 e0 t, I        return 1
    6 G7 S7 X" G, n0 S) w8 L+ t' {. ]    # Recursive case
    & V0 }' n$ G2 w% T1 q0 v, T9 w    else:
    / T! P. j$ Z2 M1 h: v        return n * factorial(n - 1)
    8 R' {/ ]$ [0 y1 H  e& F
    ! T% U1 E6 ^* e% P# Example usage8 n1 s" j4 z; g& G- `* \. C
    print(factorial(5))  # Output: 1207 v7 M. j3 N/ l3 a7 l

    ' u: r" K2 a# z) I# I# ?" wHow Recursion Works
    ! O3 ]0 }# k* z0 W5 @5 q- Y3 c5 j% s1 j' c) L+ Q
        The function keeps calling itself with smaller inputs until it reaches the base case.
    8 o8 H* J- u4 C2 N7 y0 j( I7 `
    0 r  }/ b( S/ D$ N    Once the base case is reached, the function starts returning values back up the call stack.
    3 b5 V! c' I0 A' ^3 f, [' f+ P5 r$ U# a2 |9 _/ K
        These returned values are combined to produce the final result.
      p: p& O2 G0 X( v. w3 S3 [) C' u9 D7 r
    For factorial(5):
    ' C1 A+ B5 N3 M8 k: E+ z& B. y, I9 u$ C7 J
      Z" v3 p2 s8 T5 M- O( P  ^9 Z
    factorial(5) = 5 * factorial(4)" n- Y; P1 z0 v0 I/ a
    factorial(4) = 4 * factorial(3), I8 G- Z; s. E  K! ]
    factorial(3) = 3 * factorial(2)
    : b2 z) f$ s0 L1 G9 S1 ifactorial(2) = 2 * factorial(1)
    9 Z3 |3 O; G' r* Ffactorial(1) = 1 * factorial(0)1 c: x/ @) _+ Z) F
    factorial(0) = 1  # Base case" j/ t1 k  b8 ^8 M' X

    ; F4 |+ @7 N3 U: {8 d7 u8 F7 ~Then, the results are combined:
    ; ~1 u$ X8 P) N/ }. V# K
    8 I, A# h9 [1 G7 [& X: w: V9 k0 v( h9 p( h- s7 ?( `
    factorial(1) = 1 * 1 = 1
    * D8 D3 b0 R( Q! S6 Dfactorial(2) = 2 * 1 = 2
    " ?% G: w' K" C- K# u" R+ ~  Ffactorial(3) = 3 * 2 = 63 f# ^. y" Z. U: p: ]: E$ e0 K. c
    factorial(4) = 4 * 6 = 24- [5 ^, Y- \$ f) [8 X3 O5 C) I
    factorial(5) = 5 * 24 = 120
    # L) S- i$ H/ O8 K' F" j/ F# l9 z: x' d# m) Y& E; O/ b5 Z# u
    Advantages of Recursion5 k, G3 S( ?3 z) `/ k3 D9 b

    : G4 }$ D% D( i9 g" e$ X( s    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 `4 e9 o! ^5 F

    * ^$ R5 C# e0 }    Readability: Recursive code can be more readable and concise compared to iterative solutions.% ~' G! B1 F/ z) }0 o1 x0 |9 @- w+ z
    ' v* W" S" Y2 _. q2 `' b5 q, J
    Disadvantages of Recursion) F9 w# u: x, i, k% C, |* B# b
    - ?, m3 l3 z4 T6 \" L
        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.0 n, t2 ^% o0 H% `) @

    $ I  u* U0 N% M' z6 C+ j/ h% E    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    6 ^' h4 @$ a& u2 k3 i' }( ?
    5 y5 d7 R8 {- ?When to Use Recursion* F/ j0 x% s. d

    8 a6 M; C8 t$ X0 q; N. w5 B7 G    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).. R! e" J$ ?: `+ d: N5 I1 M

    . t: r6 g9 u$ b    Problems with a clear base case and recursive case.# F/ C1 h. O! f# v+ f; a2 t

    6 |# M% ^6 d2 r; k1 bExample: Fibonacci Sequence7 h1 p" @) t8 d0 ?8 C1 j7 j. C! @

    1 v% a6 u, `7 \9 l, zThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    # b" `/ X- j: E, ~, ~4 p
    8 N) U0 s0 k8 r3 e    Base case: fib(0) = 0, fib(1) = 1( p) O. M2 j* w, m  x

    1 r7 _7 p, y) @4 o& ]8 [& D    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ; S* ]% Q  T: ?  ~5 I5 d' s0 U* n$ m! _' ^. e
    python' K( O" i, ^8 D$ t4 e
    # Y% q3 h) e4 C  a7 W  g

    # }* _3 u  i) |$ K' qdef fibonacci(n):
    7 r$ C# P6 N4 _6 D, g4 n    # Base cases
    " r0 v; h- X; X$ z$ U* r    if n == 0:/ C9 K% h# `: O: k7 i
            return 02 f3 N* B% }2 @! M% J6 V
        elif n == 1:$ n: w3 W+ B. b# ~
            return 1
    / q7 b- ~. f3 n! w5 e: b  Q& h& t    # Recursive case) P, o6 i2 u* c. j  X4 I
        else:
    2 b( x( z% J& T( z% c        return fibonacci(n - 1) + fibonacci(n - 2)3 t; Z3 Z3 B: Z/ m! ^( ]1 h

    ) z8 s/ H7 c: ?- H# O( x# Example usage
    $ W3 }3 p9 Q* ?- b2 ]print(fibonacci(6))  # Output: 8
    5 I$ d% Y  l& {) k5 r
    $ f& b$ H0 G; Z3 l5 [1 GTail Recursion) D, R4 s$ V9 h* u" q

    + I* P8 x0 b$ Q( w7 JTail 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).
    1 W, q$ ~* }: ?+ a7 @8 m' J3 K* L
    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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