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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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  • TA的每日心情
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    2025-9-8 05:08
  • 签到天数: 3 天

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    楼主
     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 y! l- a5 }" E/ |& l, |
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    解释的不错1 N! k, m2 h) \' l0 k, i2 g) X
    9 [1 ]. H; q2 B. j. u
    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。, u3 N% m+ D- B/ F& U

    5 E) k9 d) o" W( n- E3 P0 @! V 关键要素
    0 E8 }' W6 ~2 d5 S. M4 k4 X1. **基线条件(Base Case)**% y- _% a: P, l* E# D* G5 [
       - 递归终止的条件,防止无限循环
    3 u) c% t. ?$ |0 L7 a8 S) l% L   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% O8 m' B! B9 d$ w1 p
    & x. [5 z2 g2 p5 L, C. y( @& A
    2. **递归条件(Recursive Case)**
    $ ~2 I! u' n( ?3 f) {   - 将原问题分解为更小的子问题
    ( c: d+ [' D/ I& c- I* P( r2 z   - 例如:n! = n × (n-1)!0 C9 s) l, ?2 k6 @2 a
    ( I9 S0 p1 v3 @( q* F9 K
    经典示例:计算阶乘
    7 z0 @( v! r% k& B: u5 c) e& b5 J" r; Apython
    ; ~+ k% n$ v# B% Cdef factorial(n):
    * u, D, B  B; y5 ~9 ]    if n == 0:        # 基线条件
    2 c, `" w; x4 ~' k/ r        return 1
    ; h# \+ \" v1 u. L1 e2 ~. Y6 _    else:             # 递归条件/ Q: C5 d' G  ?* z2 i
            return n * factorial(n-1)
    $ F. y, u/ `+ {& d8 E" l: Z执行过程(以计算 3! 为例):, v9 i9 V$ a6 g* N$ Z+ z
    factorial(3)6 J' n: V6 T  c
    3 * factorial(2)) W6 B1 b- `$ N4 q
    3 * (2 * factorial(1))
    * ~: g5 \, G  v' B7 H3 * (2 * (1 * factorial(0)))% e* Y/ M" B! P8 w; v; g) w0 I2 m
    3 * (2 * (1 * 1)) = 6
    6 R% |' }( \* L" {1 T$ z* T# u0 g& X8 F2 x) }( U# W) G
    递归思维要点
    / U* x) F2 t. X0 }9 e0 K3 z' S1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 D0 \8 s0 u) q: ?/ d  l
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ; R  H* _$ c/ x; A3. **递推过程**:不断向下分解问题(递)' [! e. Y( V/ N3 X; x
    4. **回溯过程**:组合子问题结果返回(归)
      c1 k" q) f2 ?
    ) t2 l! x6 I, \ 注意事项( l7 O5 d  P( D: v, E! l
    必须要有终止条件
    8 h% u; g, ?& J$ j  L8 s$ ]0 n递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( Q: K4 T0 I/ T. y5 C
    某些问题用递归更直观(如树遍历),但效率可能不如迭代$ I+ a8 v% \: O/ P! y
    尾递归优化可以提升效率(但Python不支持)" s) Q& T8 F- P5 g( O2 H

    & _+ w0 d7 ^. A# q 递归 vs 迭代
    . B2 D% c7 x5 w9 b. o|          | 递归                          | 迭代               |
    9 A& u4 R' |4 v|----------|-----------------------------|------------------|
    " K: u1 J# L5 p2 X" f( z) _| 实现方式    | 函数自调用                        | 循环结构            |/ V2 u4 m5 o5 A0 y7 j& u- K
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    : C7 Y4 a- Y; X' M0 s| 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         |) m+ p( d: k% U  c$ V
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |. @4 k' ~" d/ O6 N& q9 G- H5 `* V' x

    ( m, t( t4 x3 q+ m+ l; b: t 经典递归应用场景
    2 N% o% K) H2 y+ r" M( |1. 文件系统遍历(目录树结构)
    5 f; M$ [/ ?( X: `2. 快速排序/归并排序算法% m0 E1 @8 o+ P7 B7 _
    3. 汉诺塔问题
    ; m/ z4 r, ^6 b8 x0 K4. 二叉树遍历(前序/中序/后序)
    4 I8 l5 W" [+ v# B7 }5. 生成所有可能的组合(回溯算法)
    1 f( P, v0 N: N' K7 n
    9 {! l6 v; ~( e4 W8 j: A! z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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    奋斗
    昨天 14:17
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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,' C( b8 ~1 ]( q/ b5 C) ]
    我推理机的核心算法应该是二叉树遍历的变种。
    ' F" _$ C6 O8 h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:; w5 i  n6 \2 z. J7 y( L
    Key Idea of Recursion
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    # y2 j2 `; |/ w# L, x0 H7 EA recursive function solves a problem by:5 [, l0 w# [$ w; K' d1 X0 R

    3 g' @. \5 L" _0 g0 y) R8 ], v+ g    Breaking the problem into smaller instances of the same problem.
    % ?. E/ f% A: X9 _
    1 d- Y3 m) Z9 G. T2 Y    Solving the smallest instance directly (base case).
    . p5 f) X9 M0 X4 Z" g! Y3 Y& f4 \. m- v3 {8 q
        Combining the results of smaller instances to solve the larger problem.
    * q: C: a0 Z8 \% g, {2 \  ~+ K5 V
    Components of a Recursive Function
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        Base Case:
    ; T& N. H2 J# L1 `. I$ `% \# Z: q+ @4 n" `/ S# l8 x5 u
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.  V. R: s9 I2 Z% ~# I& y! X

    1 F* E. {7 u( b- R* [/ h* ?        It acts as the stopping condition to prevent infinite recursion.
    ; O3 F$ S0 x( B' B* w- |# y8 P1 C! P1 Y
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ L( J# t. K  c8 {7 H9 ]3 x
    7 u$ Q! A& s3 K% q  u1 [/ C
        Recursive Case:% h9 ^7 N. n7 J
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            This is where the function calls itself with a smaller or simpler version of the problem.
    5 ?% d7 c- p' B6 ?0 E+ \- J9 I* X! A6 F( A" ~
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).8 ^5 ]' X$ E- O, F( Q! ]# q5 {
    ( C" w/ a$ `$ R4 ^3 o2 e) ~2 I
    Example: Factorial Calculation
    3 x' b! f/ M. W, B8 v! O4 H" b. ]
    " u: W. V# P$ ^; p2 YThe 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 \+ S0 C+ O' f5 t
    7 H" N( o$ ~( w( i/ n( u
        Base case: 0! = 18 u/ ~4 V& g0 r3 P

    5 I1 N/ n1 _" z3 p/ l    Recursive case: n! = n * (n-1)!
    0 _* F! E3 C5 O0 S! v  k6 x; B) \# p' r2 ?/ \. i$ h; F. d
    Here’s how it looks in code (Python):! [9 [  F7 l( L- S# a& B
    python, B6 w" @8 h% z+ e/ C! A) L% h
    , Z2 P* \' x- X, b  V

    % {4 y4 s# S1 D7 f) Mdef factorial(n):7 m  C& \! n( |* y3 k
        # Base case; h9 i: H! A( R
        if n == 0:
    7 Q) E7 {5 Y: e/ t+ _        return 16 l* j# S& v' Y3 }! {
        # Recursive case) I* S9 v  v5 s* Q9 }! N
        else:$ N/ C! K* I$ b3 T* o7 T. x
            return n * factorial(n - 1)9 m7 \; ~0 c4 A/ x4 {6 C. `

    1 B# l1 l6 V+ x3 |9 q# Example usage
    . H- {" O* {" ^9 {# jprint(factorial(5))  # Output: 120
    # ^# h7 u/ M! c3 y, Z- A+ n; w) {" g9 T! r
    How Recursion Works/ F* X9 T( E- g/ H
    $ A, d/ _* z& K" E2 M% k
        The function keeps calling itself with smaller inputs until it reaches the base case.( a6 j5 c7 p$ e' j
    ; Y( s, _8 f1 \5 M
        Once the base case is reached, the function starts returning values back up the call stack.6 m3 `5 K" M) _" ~

    $ T; a+ v! A  r; K$ H    These returned values are combined to produce the final result.: F. E! u7 [/ x$ n5 w- W. I

    1 q1 }" K; R8 h4 O+ Y2 [/ o; GFor factorial(5):- g* t7 F+ g! t2 }7 A. |9 Y( `0 ?

    ( e2 ]7 m: w: y4 \2 W6 s4 I: w5 Q6 V, _/ w
    factorial(5) = 5 * factorial(4)
    . }- v1 ^* U+ \+ E" yfactorial(4) = 4 * factorial(3)* k# p9 C, M! c1 }+ h4 \6 P( Q5 f
    factorial(3) = 3 * factorial(2)2 t- n' |8 O7 G7 w" O" h( r
    factorial(2) = 2 * factorial(1): z% ]: t9 Y  c; B0 Y* o
    factorial(1) = 1 * factorial(0)
    4 C9 j! G7 y. d, O, x, [factorial(0) = 1  # Base case+ }8 W3 g; k2 ^. z+ \/ d: m0 ?
    ; I9 a" u* X+ R( ^/ z
    Then, the results are combined:0 P- q/ C$ _7 G8 }7 C1 N+ q& E
    ) w# O% e* B& q' b. x; X
    . H* `, o( }# C/ X# b
    factorial(1) = 1 * 1 = 1
    2 l7 V) g. o6 O0 p' `! w9 g" Jfactorial(2) = 2 * 1 = 26 a& d4 f; Q9 X3 f+ v0 j
    factorial(3) = 3 * 2 = 6: R& m1 r/ A/ F  K7 b1 m
    factorial(4) = 4 * 6 = 248 x5 Z/ ?/ E; h1 K* V
    factorial(5) = 5 * 24 = 120" E- ~2 S' `9 K$ z! Q# p
    ( e+ c6 a% D) ~" N, Z
    Advantages of Recursion& o- C, V4 @( c! }$ |
    2 z) Z2 Q* f/ D9 k
        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 T* z- G" ?. z0 O- l& x+ t
    8 v% l- K. ~/ Z2 [% @    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    4 H; a3 A( e9 c6 X1 O# b/ y6 f/ {
    , R& E: W0 `- j8 ~Disadvantages of Recursion8 r* ?+ Y  r! b! k) W! g

    ! Y5 C3 J! }( P1 S. F* U( }6 k    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.$ {. J5 M- p2 G( {, p4 J
    / o  f9 G& r  {' I. |0 t
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    6 t2 k$ X/ h8 U& a  ^3 h& s
    . w0 z' V0 S& b0 R+ \9 kWhen to Use Recursion% R( {" @, w9 W3 {6 r' e

    6 B+ f# N9 X1 ~5 I) I; J7 G    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ' a, U6 i1 n* Z7 f" G& ]% G& P! `6 b7 Y; Z0 d( }
        Problems with a clear base case and recursive case./ }. d5 E( s4 Y9 j; }

    0 `3 W. A4 b% F& F8 lExample: Fibonacci Sequence2 Z+ \7 r7 l- G' u  w4 e4 q

    . f( ]' [# y0 ~The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    7 i/ K/ ~. k+ a% C' }& Z, K9 m, O
    , P9 G7 S. S3 O: `    Base case: fib(0) = 0, fib(1) = 1& I. P5 Y" {5 n6 b. d$ U, @
    4 Z" z* {! ?# p) k! G
        Recursive case: fib(n) = fib(n-1) + fib(n-2)2 B' O8 q) H$ I" _

    ; Y/ ~, Y+ a( L) S8 fpython
    * [' D, I; T" N2 A) _  Y1 w$ W" t4 H4 }* j$ ^/ D

    8 x) q3 U; P; m) O  Ydef fibonacci(n):
    * q/ Q9 V) |% v5 H  j1 h    # Base cases  S1 s7 v& m. j2 F
        if n == 0:9 j6 h  u2 k6 a9 Z  O8 N
            return 0
    $ |+ N1 E; ?; `& s$ A1 n. |    elif n == 1:
    ' r$ j% y  z6 X& H        return 1+ c8 S6 O) W0 e% t1 L* ^
        # Recursive case
    6 F+ w# W0 @2 k; A3 y* x. }    else:4 o% w; Z% G, H8 _  q8 Z% T
            return fibonacci(n - 1) + fibonacci(n - 2)
    9 p& O/ @3 {6 V) A% f) O; }' S+ B4 ~' b7 g' c# k* ~
    # Example usage
    . ^* I- k( e; L7 @" p" T9 }) Vprint(fibonacci(6))  # Output: 8
    / w) l, m. |( |6 B: B' H9 M7 j( U0 j* p$ a" B
    Tail Recursion
    ! }. P3 y8 R7 ^0 n! R8 G4 U+ ?7 j" w% i$ A
    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).
    ! l7 ~+ R  b/ `# S" W+ P; E  l1 W  I2 j9 S. P+ |" b
    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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