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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   ?4 D# i8 a9 q  K6 |7 S" F

      s+ q0 V+ [" N( U解释的不错$ R# L5 o1 _8 A2 `) v
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    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/ L/ E, J" t/ W" z. V
    5 n* P5 ^% ?7 [  ~% F  j  {
    关键要素
    0 u- @- K( x/ q* K, |8 t1. **基线条件(Base Case)**9 M7 X" A/ r8 b7 v) y
       - 递归终止的条件,防止无限循环
    0 Q9 e/ E1 v, n: x; C4 C$ k* t   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1/ g" O- Z0 ~* i8 P( ?3 ]

    * v2 S. R& I/ R$ D5 B  @% }3 s4 |2. **递归条件(Recursive Case)**1 q; W. M8 @1 d: K4 E2 W
       - 将原问题分解为更小的子问题, x# ~) t' z" J! P' u( Q# \. l
       - 例如:n! = n × (n-1)!9 g" P/ @, H7 D, D

    0 U6 r5 L$ @5 ^4 F; v5 S3 H, \ 经典示例:计算阶乘! Q+ I$ ^4 i+ V0 T  l5 @6 y% F
    python
      H& T  K6 q6 z0 V) Idef factorial(n):& b$ E& d! p/ V& u. r( P: ^, ~
        if n == 0:        # 基线条件, d: w9 }  [& N( `: J2 J
            return 16 e0 `& i9 O9 ^8 O) h) ]# r
        else:             # 递归条件* G, u8 L7 @) e# ?, I* f( {
            return n * factorial(n-1)3 z4 l$ f" Q" t' |+ N+ R8 x' m
    执行过程(以计算 3! 为例):
    4 R& n# p% _" g7 `' wfactorial(3): D% P+ P  q0 s, ?  Z
    3 * factorial(2); `) f. Z( ~. O3 l8 d
    3 * (2 * factorial(1))
    - |$ [" G, Y& Z8 s. i3 * (2 * (1 * factorial(0)))" D' ^* t4 `; q7 }+ I  i
    3 * (2 * (1 * 1)) = 6- K) U: t/ }  B+ S' s
    2 x3 ]( }1 ^+ O: o/ x( \: P* I
    递归思维要点0 |/ H9 d3 L$ p8 u
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑1 q" F6 M" `0 J9 ], {, a
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)) S' t+ e9 g. h8 V
    3. **递推过程**:不断向下分解问题(递)5 G9 c4 V* v1 t1 q
    4. **回溯过程**:组合子问题结果返回(归)
    7 f: U& b/ H% K! [+ L
    & q( ?7 Q; q8 r6 J 注意事项
    # ^3 @  r( k+ W. R必须要有终止条件
    " F: E8 W6 |; [' W6 P0 ?% _0 J. T递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    " V5 k" c8 m5 q某些问题用递归更直观(如树遍历),但效率可能不如迭代, Q/ K' f. m8 B: G4 J0 l+ [% `
    尾递归优化可以提升效率(但Python不支持)% ?9 s, I  s* R2 t" J5 |6 v( R
    4 e- H) t% E7 q9 O5 @* K/ T
    递归 vs 迭代
    ! A  {& V. [  v" j|          | 递归                          | 迭代               |) }4 |8 m6 h3 L1 f# y! q' W" s
    |----------|-----------------------------|------------------|
    # C) @/ H2 U4 D: c% g6 ~| 实现方式    | 函数自调用                        | 循环结构            |
    4 v' u2 V7 A. h3 G0 V7 || 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    5 {; l6 \8 S1 }  c8 K| 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         |/ h# U+ e# j* L9 D$ b
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |" \1 A: X5 I) V% |( C
    7 u. a* [5 m' H9 Y) a9 h8 D
    经典递归应用场景
    + b, _' f2 n! W/ W: u7 d1. 文件系统遍历(目录树结构)
    + J) S2 e0 d0 d2 P( U5 h* k2. 快速排序/归并排序算法, Q- ?7 K1 H& m4 e( u
    3. 汉诺塔问题
    2 M/ Y* O& s* d' R4. 二叉树遍历(前序/中序/后序)
    7 k' ]  Y$ ~1 _; ?# g! w# @: l* p5. 生成所有可能的组合(回溯算法)
    & E( }7 q# i4 X, a
    2 O. F8 o7 x4 D. a+ ^5 t; m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,1 W- Z- Z: ~2 `6 R& B& j
    我推理机的核心算法应该是二叉树遍历的变种。
    8 ^2 W" o& ]: @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:' q7 N9 [+ `6 e* l# B- M# l* G
    Key Idea of Recursion
      X; A6 u! y+ E+ K$ s. M+ L5 N5 f" Z# h6 ~
    A recursive function solves a problem by:, m: G% n+ L$ y: n2 F
    $ @) i4 I; P- m2 y
        Breaking the problem into smaller instances of the same problem.
    ' b8 a& s* U7 E" e4 ?
    8 x3 z8 d# a  M& j+ q, k4 w& |    Solving the smallest instance directly (base case).+ [: \( I0 h" P" b: ]. t
    + q9 X- x. [4 Q) N5 M
        Combining the results of smaller instances to solve the larger problem.
    . i+ J2 e0 N% Q1 P: t3 Z) k: C9 `- y- h8 W
    Components of a Recursive Function: T- {: C( K3 J2 v+ Z0 h

    1 ]' q3 i% }9 b6 m    Base Case:
    6 j/ N: |: u2 ~$ k' G2 |- D: Z; J
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ( i: @  ^; @; k# p" k- X: ~( ^; [
            It acts as the stopping condition to prevent infinite recursion.5 v7 t" r. |/ b3 b; e) X

    1 |6 F4 x5 E  f/ \        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ) ~  l& f5 {, }! E0 p" q
    % C% A4 f! n8 g& E) q" a$ p4 T. d    Recursive Case:8 u6 Y5 `4 o9 ?0 y3 V
    / Q8 D0 V8 p6 ?4 W" \& `7 y6 b
            This is where the function calls itself with a smaller or simpler version of the problem.
    1 z$ O3 x6 _; T( \/ O( p0 @
    ) T, L2 @9 ]: f/ I8 g9 ?        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
      ]. @' i3 Q2 F1 [5 \# o
    , J' z5 M6 G; C7 Q& B" |Example: Factorial Calculation
    , w3 A, U& P! f2 L* b7 m& j2 p+ q" g$ ^. E  e
    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:1 Y4 C7 H4 g* ?2 `% D/ F
    8 f% I0 ]! W6 G4 U6 |9 z
        Base case: 0! = 1* X0 B$ ?; Q1 D/ |
    " ~! ^' q# X$ |1 S8 I, j- T( u# P
        Recursive case: n! = n * (n-1)!
    ( L: y- b# n* a
    4 [, y$ V6 U# E& b- gHere’s how it looks in code (Python):
      [! o7 v, _& F, Hpython
    7 Y3 b7 W1 ]3 }7 t0 c/ c( C; @/ g; B) _; i2 {$ }; Z9 y# z
    7 I* j3 b3 j$ a. \* l& [
    def factorial(n):
    5 q% L6 X8 I7 V: \: c4 Y6 x6 p2 t    # Base case. P: r# Q6 q' A# j4 ^, W
        if n == 0:
    8 m% n& Y7 ]' w4 G        return 1
    * Z. [! T- o8 G  K# s7 f* D    # Recursive case( c0 Z7 G* z0 c  ^
        else:" }6 `6 V& p; R! V" K
            return n * factorial(n - 1)( e1 O  u( L% @" Z& v

    1 j& l) I. s2 L! d# Example usage
    7 H8 V; l, |7 W- S' I+ B1 Sprint(factorial(5))  # Output: 120
    ! `* H7 t) b" G% J+ k7 T4 [3 }. ]# ~% E' d2 u
    How Recursion Works: T; n9 N: N) F/ X! _9 z
    1 W$ ?" q( ], {
        The function keeps calling itself with smaller inputs until it reaches the base case., \: W$ ~+ B* ~3 u+ \! _
    ! H# v6 R: u' X7 g
        Once the base case is reached, the function starts returning values back up the call stack.4 U- M/ c3 V8 s; `& E. v) A. A

    ) c- C& U! O% \8 p) V! K    These returned values are combined to produce the final result.
    4 H- X* Q' B% m: I) u* {
    . m" Q$ p$ M+ r5 iFor factorial(5):
    ( w; D# a2 F( w8 a
    . U" U! s& Y1 n4 T9 Q4 N" h0 ]( @3 c# m: q( v5 P5 P, J0 a4 x& E9 {
    factorial(5) = 5 * factorial(4)  }0 O, G5 L3 l) e+ r
    factorial(4) = 4 * factorial(3)
    . I( b; P# d3 f7 S9 b3 u3 n# ifactorial(3) = 3 * factorial(2)! |1 P3 r9 Y6 W1 g: K$ i
    factorial(2) = 2 * factorial(1)- s9 H# |9 F$ G; S- a
    factorial(1) = 1 * factorial(0)
    " d, Y) \  L( L) P8 D. C! afactorial(0) = 1  # Base case) S9 {: @8 Y0 r: _
    # G/ ?/ M* p! f* Z
    Then, the results are combined:( V( Z+ p) H+ D+ ]
    3 N% d. B6 w$ D1 l1 S

    4 ~) n4 y: S! _; sfactorial(1) = 1 * 1 = 1
    & P6 L/ b/ w; m. h3 i8 [* b' Dfactorial(2) = 2 * 1 = 2& C$ ?+ ^- c8 J" }, v
    factorial(3) = 3 * 2 = 6% Z8 v- i% w3 B2 j0 ^
    factorial(4) = 4 * 6 = 24  M% M3 ~7 X7 G- k  e
    factorial(5) = 5 * 24 = 120
    7 R0 Z+ I+ l1 `2 i* ^/ ~9 c: H+ o- V
    Advantages of Recursion. I6 j' Z, _5 }" c0 q4 ~
    + d+ C2 C( c# Z# F- h2 H
        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).' u! h, s9 V- j& E
    ! u( m7 z) y4 o: j/ h
        Readability: Recursive code can be more readable and concise compared to iterative solutions.3 e: S9 i1 V. n# Z7 A7 A
    9 A5 `! s9 E0 ?- {: H
    Disadvantages of Recursion
    7 z! V2 d; n) y  ?, B
    $ p0 x) r; o8 k" m  H- C" G    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.2 x" L" \  ?2 x- i: t2 `7 K

    6 |. F' N- h# {7 k, {    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    5 _+ {* c4 l5 h" a, E% L% i
    4 `7 E% E' K, `9 g$ e0 q& N, D0 oWhen to Use Recursion
      J4 a0 @8 x3 o
    ; d( o, |# o5 ^5 n    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)./ x' H, d0 O2 U  \6 C4 `

    0 O: G8 q& ~, t- }  V' U4 |    Problems with a clear base case and recursive case.
    8 h; V4 ^8 u- r+ n
    - v0 S$ n5 S" @. Q$ _Example: Fibonacci Sequence6 k0 ?7 n# q* j$ ~$ w+ v9 [+ v

    1 e- k  t" |9 x+ gThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    $ d7 A) w/ _2 a
    2 O( o/ F! c  {- v0 ]4 L2 N    Base case: fib(0) = 0, fib(1) = 1
    % y9 q. _; g4 h7 c- n! U) S) ?/ @5 Z) J6 y6 K0 P! k$ i' M5 I, K
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    5 `  ~, o* C- R$ [0 ^1 V
    % k: `; h/ ~. U9 Q4 v  p$ zpython3 x7 ^" l' W- u6 H/ N9 G) d% `8 _/ T
    3 P/ q# o  F) p& k3 ?
    5 U  a8 T- L; y* \; n% I
    def fibonacci(n):
    6 u  n5 V- b8 c    # Base cases
    6 M& O$ Z; |2 L% b, ?& Z! R0 j    if n == 0:
    , [8 m8 h2 h6 {* g" y- e4 r7 b/ j        return 0+ r% m) {9 q6 v& |' ]0 b9 w7 {
        elif n == 1:/ K: V3 a0 {/ E) \  Z7 e
            return 14 `1 N/ Z* y/ [9 Q5 N
        # Recursive case2 L# a: W5 A1 A4 {" j
        else:
    4 C. ~! o4 ^6 O. {9 l5 A. z        return fibonacci(n - 1) + fibonacci(n - 2)6 r9 j2 ~0 ?+ M$ _) \9 T
    3 N2 d2 k, x7 l# \* H8 v
    # Example usage
    9 y6 F$ L' z- W+ c( [' w' u( d+ |print(fibonacci(6))  # Output: 8* \. y' w, D; L
    & u/ c5 r) T! A
    Tail Recursion2 h' r8 N* ?- G( `- J

    / V$ U1 W. M6 c3 tTail 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).
      h1 i& `  R+ h3 R. q
    8 h/ J0 \3 _3 n/ R3 a7 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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