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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
    # E- f2 o( s, e6 H5 p5 R# h7 }6 z/ w" Z  B3 o
    解释的不错
    % K' v8 G  @6 h4 X% z7 f  V) g6 T! P' F' J. \7 F8 Y, c9 g: ?
    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。
    ! x% x4 W/ F9 D  ?4 @3 x
    ' q% A6 g( w1 e4 d: c' ^- x 关键要素7 y+ r$ E8 M0 x* J$ M
    1. **基线条件(Base Case)*** O/ Q! ~' b% A9 S& k, ^6 q
       - 递归终止的条件,防止无限循环' r% e0 l- Y$ Y* r3 y9 K
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1  N: `% U1 S% I6 K" v1 G/ G  d  W  G

    7 s! w0 j# S. @# O" y% x2. **递归条件(Recursive Case)**7 o$ Z8 W9 G2 j; M
       - 将原问题分解为更小的子问题
    8 \% U- i* l' \: {   - 例如:n! = n × (n-1)!
    4 g3 N# t5 Z. b
    2 H/ d1 q% V0 I" Z9 V1 t 经典示例:计算阶乘
    ) S; T5 K% x% p; v5 W" X+ Y5 ppython: w& r* z# J2 m) J# x9 G1 [
    def factorial(n):
      @8 a: e( t% \) Y% }4 i8 T    if n == 0:        # 基线条件
    % D( T0 m5 m6 B% Y        return 1
    : G' E0 E& v4 o. C, Q8 P0 M    else:             # 递归条件( X/ p2 m5 O4 R
            return n * factorial(n-1)
    , N2 k, X' L& G. ~6 R" E, ]$ T执行过程(以计算 3! 为例):/ ?3 |9 x4 N3 b9 t" {& S4 ^( |2 h
    factorial(3)7 V3 @0 _, y. q& ^  l4 E
    3 * factorial(2)# f- s/ b7 y; N7 v' X7 y
    3 * (2 * factorial(1))) r# Z3 ^- F/ C& I
    3 * (2 * (1 * factorial(0)))6 l* f2 @9 B1 Q. k: `3 V" M1 ^
    3 * (2 * (1 * 1)) = 6) K! c! q4 v0 `3 w

    5 U+ s4 N5 \$ a0 z: F 递归思维要点! `; W7 L6 F) e5 c2 e& o
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑5 c8 p% _7 ~+ s" Q, X8 L+ t4 j
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)7 N6 s3 a) e* k; X7 G
    3. **递推过程**:不断向下分解问题(递)- U( N* d! j) z
    4. **回溯过程**:组合子问题结果返回(归)
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    $ o( L" L6 x& w0 r2 Z/ s1 v 注意事项
    $ Z- c9 Y5 A2 U% c6 J必须要有终止条件" x0 W6 w" m8 z1 C3 g. f) E$ v7 G
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
      p, {5 z' V: J: W! {8 E4 k某些问题用递归更直观(如树遍历),但效率可能不如迭代
    : O5 u$ F( U4 D$ w+ u尾递归优化可以提升效率(但Python不支持)3 q/ a% e: Y$ c; i& E2 r! ]) i& Y% \

      v0 a2 o, E' `- p# P* U: \" s' i 递归 vs 迭代
    7 }; y5 K7 V* t" E4 o6 B8 h|          | 递归                          | 迭代               |/ G& W* g+ R6 B! l3 u8 X. b
    |----------|-----------------------------|------------------|1 o. c8 T% y) N
    | 实现方式    | 函数自调用                        | 循环结构            |( [- K7 C: w4 J' y
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    , Q2 X/ l9 \! {5 W. X; O| 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         |
    + u" U6 V! I" P. k' }8 v- g' Z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    9 l: o" |0 _  M+ a  k& q1 ^
    3 U, V( j9 c$ Y* I  Z, }' L 经典递归应用场景
    $ \# [+ Z' C" \5 h/ `2 n) ~) T1. 文件系统遍历(目录树结构)
    1 k6 \7 V$ B! m( h( R+ J2. 快速排序/归并排序算法- A5 b! \0 b4 [: ~# J8 j. e
    3. 汉诺塔问题
    - O( `9 T* f: e. d' U4. 二叉树遍历(前序/中序/后序)
    2 s0 ~' l5 h4 `# l5. 生成所有可能的组合(回溯算法)
    % H6 u. _" {2 N# a1 t1 \& |9 E) C8 R
    + p& X  {( z( K) F% l% b& 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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    昨天 12:10
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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,0 k% U' @1 }0 _8 Q
    我推理机的核心算法应该是二叉树遍历的变种。
    2 R  s6 s4 ^2 A# r  B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:
    / D+ f  P- H: [+ x% H; H/ W1 v- ^3 NKey Idea of Recursion
    ( {" O; S8 |0 ~! j) D$ R' G, C* K1 w5 t. W  |+ L6 V
    A recursive function solves a problem by:
    + a7 R! Y1 P5 A: [3 R
    . G& Q( E" E6 ~: _0 G' A    Breaking the problem into smaller instances of the same problem.
    + O8 ]" M% R$ C: `( S0 n% F: x3 f. s: [8 @
        Solving the smallest instance directly (base case).
    - b+ F$ @* j. M" z) Z& Z. X
    5 l& o* W7 u- `- H9 t& d% H1 [    Combining the results of smaller instances to solve the larger problem.
    * q7 Q( `2 w2 k# ?
    / s; j0 [) w7 t" MComponents of a Recursive Function
    8 Z, ~7 y: R- t0 ]  ?& m2 ~+ T: o% d
        Base Case:/ ?" J! p& K% q) ?, B
    , y) H' _8 c3 V. O) [  A
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.: p- m- M- u- T7 N

    4 K2 \' e: o0 I6 K" E        It acts as the stopping condition to prevent infinite recursion.  s- f; l" [) Q, P* D
    4 z8 o, U8 r' ?3 `% a1 a
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:
      T: u8 F3 |6 \8 X2 V0 Q
    9 _6 d- T* }3 z0 N. Y- R        This is where the function calls itself with a smaller or simpler version of the problem.8 J" G4 K0 J6 Q" K+ M
    1 ~& E) v& l6 W2 w5 H. E( I- W/ w
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' a1 P& `; u+ u/ ^( S
      E3 I3 g8 V7 b
    Example: Factorial Calculation/ }$ p+ T! Y. ^# }4 x! @% @
    / d4 h1 I2 e: J: ]; C
    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:" U, v0 O* U2 ^3 X1 _7 m
    ) g/ ~5 @, ?  t3 {
        Base case: 0! = 1* ^, L! c: r9 D9 ^# j& Y  ]! _! P
    . b- B9 K; d5 D6 j
        Recursive case: n! = n * (n-1)!7 @- `$ W2 u3 o( l

    . V4 y, [- ?! H  P( I$ OHere’s how it looks in code (Python):( K2 J# _4 h5 H- W
    python! r0 _" }9 Q: m4 m
    * ], r( M5 m& P8 d

    . [4 n! M6 L4 T( Zdef factorial(n):% ^, W  _' w5 g. a: Y. L
        # Base case5 N) m9 P: p2 E/ K( r8 p
        if n == 0:2 Y$ A' S6 J8 U' Y
            return 1# f& n; C+ f, z7 F) @1 M
        # Recursive case
      s! c7 o. V0 N4 i; C4 M    else:
    $ A# `5 ~% s2 c1 `  X+ c! U  ^        return n * factorial(n - 1)' X1 ]8 p7 j. \2 g
    + b3 w1 i0 w( a' \
    # Example usage7 |' O8 v$ d1 {0 A
    print(factorial(5))  # Output: 1205 `0 l7 U8 g. i! |* t% l% W- h

    8 y) P5 f/ F$ L  DHow Recursion Works& u- S! U. d  m1 p3 I
    7 @% w/ y- g1 z" [$ o
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ' j0 _: N+ z/ z+ e7 D: r3 T  X2 v1 W. K" i: M
        Once the base case is reached, the function starts returning values back up the call stack.$ X4 E& t3 A7 j1 s
    4 Z8 W3 Q' D( M+ `+ R5 o  E6 q
        These returned values are combined to produce the final result.
    & x$ u) L) p* v' ~
    3 m7 m& r6 t* p/ i* O0 w9 s$ XFor factorial(5):
    0 i0 j  ?5 c0 M% \1 o6 ~+ k' B- R, o6 c7 q

    / n' a$ X, b5 M& T' p# _/ l7 Pfactorial(5) = 5 * factorial(4). z2 W9 E9 j& J7 m+ p1 s/ p; _
    factorial(4) = 4 * factorial(3)
      t9 s# w0 `' s$ o6 ~8 A+ M* tfactorial(3) = 3 * factorial(2)
    9 r- P) y3 b" bfactorial(2) = 2 * factorial(1)- W  [8 f$ o: L! _, C7 H/ M% p
    factorial(1) = 1 * factorial(0)/ f* p: d, }& U6 o* J  |
    factorial(0) = 1  # Base case( D' S* c' x+ }7 {( k+ t

    % {  k& x+ v' w. @8 QThen, the results are combined:
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    # V4 A- j8 @  _3 P) G& d) g) `/ q" E$ c8 X6 `$ f  ]5 V: L, n
    factorial(1) = 1 * 1 = 1
    6 f. [' [( m* Efactorial(2) = 2 * 1 = 2& p5 L3 R1 E8 ]; m4 q+ R
    factorial(3) = 3 * 2 = 6
    & L( i/ e3 X% o! m! c2 a( u4 |0 Mfactorial(4) = 4 * 6 = 24& V' `# ?  x- V+ e8 `. l/ ]
    factorial(5) = 5 * 24 = 120
    $ K# Z8 C$ e5 V/ K% {' W4 f- X5 ~; I% r
    Advantages of Recursion
    2 q3 [8 e5 e: t- Q
    . z9 Z- G0 J- R  s! J    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).- d$ L" ?! k) o# d. U
    ' X  Y; {- k4 ?
        Readability: Recursive code can be more readable and concise compared to iterative solutions.9 ~2 e+ r. q$ ~5 X

    , F7 h( j% V4 N1 h4 YDisadvantages of Recursion
    : r, }$ [- m6 T! K8 L1 g- B5 ?# E+ t1 R, P1 o: S
        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.8 v5 s8 ^0 d1 @- B8 m2 w" s. k

    # |$ `& q  k; T: _# t    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).6 C) z6 N1 A+ G+ J" v/ p

    ' f8 H4 T0 ?& a* v2 \" pWhen to Use Recursion! n1 v/ u% w9 x" V  j& K

    3 `/ S- h% ]0 R2 [$ [" q    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& M5 x0 M+ v- Y; Y
    7 n1 X2 {9 Z3 A
        Problems with a clear base case and recursive case.6 [& K) m3 C' B% F; p. S4 q$ E
    ; s; i+ j9 I& i1 I% i! Y4 M) i
    Example: Fibonacci Sequence' Z2 h8 _* o, X3 y7 |

    ) y2 u( o' j4 PThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ' U' ]9 K# w  d7 T( r9 L3 c$ [, u5 |& O# Y& k; B) U  `0 S
        Base case: fib(0) = 0, fib(1) = 10 S* f- t$ z! O5 }" Y3 A. X
      Q# ?) J9 d& B! d* K
        Recursive case: fib(n) = fib(n-1) + fib(n-2)1 J" Y* I* ?  O# z  J) b& A) G
    4 {$ {: f! C0 a2 v6 s9 i: N
    python  u7 v. b7 S6 _+ A

    % l. S. I1 O0 s& B9 c; N( G/ v1 l% L: q2 W9 `4 @
    def fibonacci(n):1 N0 E4 y1 i* [+ a. \
        # Base cases+ k8 p$ e0 s9 o( ~
        if n == 0:! q% J. ?1 {+ t1 c0 ]
            return 0
    * ?! n( Z3 a1 d8 ~    elif n == 1:
    ) g# g7 L7 R# L) B8 G* w& ]" N        return 1
    , |; c- Y# u% z# o. ^7 N    # Recursive case
    9 q' g+ m8 M# F( T4 R    else:
    4 ]' D7 L, f. j        return fibonacci(n - 1) + fibonacci(n - 2)
    + m' U7 ^2 T) K
    * L2 T" H% p5 V5 Q; P! X+ w# Example usage% W, _* J- J5 X; s
    print(fibonacci(6))  # Output: 8
    ( ?+ l7 Y) F0 g( }. g* }) G
    ; p% V/ m5 }3 @" V; ^" zTail Recursion! a" G& [7 N% W

    0 m' I$ C& ?0 ITail 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).
    2 h5 Q* X: \: I+ Q* }# i
    ) D0 O7 u$ |! O4 q. |& |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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