设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1149|回复: 3
打印 上一主题 下一主题

[科技前沿] 突然想到让deepseek来解释一下递归

[复制链接]
  • TA的每日心情
    开心
    2025-9-8 05:08
  • 签到天数: 3 天

    [LV.2]筑基

    跳转到指定楼层
    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 & |; \2 _3 u$ K8 H

    1 L, X3 P: n: u, W4 a$ r% n7 U解释的不错% n) h0 d0 r$ A4 w0 m% {8 U

    0 N* u5 g- t8 ~  U  K8 A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。$ L0 ~5 o! U& i. w% O* A
    ( I4 I% R* |; l1 ?2 [" b5 w8 d- L* M
    关键要素+ J+ Z: i: o$ `/ a8 D; L1 d( b
    1. **基线条件(Base Case)**
    8 I/ A% H3 @% p9 @  `2 ^   - 递归终止的条件,防止无限循环9 H1 C& U/ d9 n" e$ b6 m0 ~& O" S0 ^
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    * K2 p4 I( l$ r" j" W; J; G6 O- F4 F3 g! f$ E+ C) i
    2. **递归条件(Recursive Case)**% Y0 @6 v" _- w
       - 将原问题分解为更小的子问题% `' `+ t! B8 D) @) I
       - 例如:n! = n × (n-1)!  l- S- H8 ?' Z+ }" U3 F9 Z# X

    # m" z! R1 ]  A" {9 @ 经典示例:计算阶乘
    9 V, e6 N+ Z: O/ g, ?python
    4 b+ o* g8 m' i6 C% H. W4 ~def factorial(n):! U0 i( q/ E- A9 A( r1 Q
        if n == 0:        # 基线条件; T' U; D. }- W( m
            return 1! F4 \. _/ ~7 q- ?
        else:             # 递归条件
    ; ?9 s" x4 l. ~, W2 h        return n * factorial(n-1)4 a9 q2 v: t" {4 k$ z
    执行过程(以计算 3! 为例):2 ]* W1 K7 d/ k3 S. h
    factorial(3)7 \% N, U# ?5 z8 x+ ]
    3 * factorial(2)
    # f/ V/ c% k, s1 B3 * (2 * factorial(1))
    # ~: p4 R5 l& a1 X1 }3 * (2 * (1 * factorial(0)))
    " h' _; j: K) x+ `( a3 U3 * (2 * (1 * 1)) = 6
    7 o( |/ a4 |' _: r8 E5 R) h4 \, o& l' n! o; {  A9 |
    递归思维要点0 D/ Y1 ~7 ^* ~* c! [
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 [: E8 n* m5 Z. N
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    + i* U; o( e0 L& X% o. u6 B- i4 ^3. **递推过程**:不断向下分解问题(递): \5 d8 h1 N4 U' L8 s3 h/ w
    4. **回溯过程**:组合子问题结果返回(归)
    : r3 i% U2 ^1 `2 S8 J( l9 x$ s7 v" }1 A: O! y; D+ g
    注意事项
    7 u/ |1 E# o7 J5 U- E必须要有终止条件4 Q0 j& v: C8 D: ~# G
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    + g9 |, V( S' T7 j( f( T7 C某些问题用递归更直观(如树遍历),但效率可能不如迭代! y. [1 |$ B* N6 k
    尾递归优化可以提升效率(但Python不支持)
    - a1 ~7 u2 U0 c2 ?  H: K7 y' u, i
    1 N& t. K( M/ Y: [. K 递归 vs 迭代, b" x# ]# D: F* C
    |          | 递归                          | 迭代               |
    ; Q  f' w% y# p6 t+ a: `3 s|----------|-----------------------------|------------------|
    3 e  ?& {0 O- F8 \0 b5 h6 H$ Q3 s| 实现方式    | 函数自调用                        | 循环结构            |# w1 w1 c9 \* c5 U) s  z( i# T
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 G1 Q6 s- s3 I" V7 I
    | 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         |! B& }3 Q& d* z9 y% Q
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |7 s* y* Z2 p& b1 Y! ~' u& U" X

    : U* H( M. H( `: Y4 c6 u7 n/ ~ 经典递归应用场景0 c* _/ }9 F6 ]3 p6 w3 s
    1. 文件系统遍历(目录树结构)$ K, b! E% \' _2 }3 v1 B; \
    2. 快速排序/归并排序算法
    ! y$ k* \  L8 [  I: z% r3 y3. 汉诺塔问题
      ?% t$ I/ @/ P: y4. 二叉树遍历(前序/中序/后序)
    2 B# a  X& A$ R5. 生成所有可能的组合(回溯算法)7 q' e( y% P8 H& \, T2 A6 f
    ( m/ T2 z$ I" ~# B% i# ^
    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。

    评分

    参与人数 3爱元 +26 收起 理由
    pcb + 4
    老票 + 16 涨姿势
    住在乡下 + 6 给力

    查看全部评分

  • TA的每日心情
    无聊
    11 小时前
  • 签到天数: 3101 天

    [LV.Master]无

    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,
    , t- X1 A, }3 L  |$ B我推理机的核心算法应该是二叉树遍历的变种。, f9 ?; s1 ~9 G* R
    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。
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    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:$ j% O5 c/ Z0 F. w
    Key Idea of Recursion
    * ^- H0 d8 `7 t0 E
    & ]9 E$ A. h, L/ f$ FA recursive function solves a problem by:% l- w5 x2 I( n- o
    9 ^6 P5 m, X( m: H  W
        Breaking the problem into smaller instances of the same problem.
    $ T8 p; s; q! J: D0 X( ~: q
    ' N' [+ B* o9 y' }1 g    Solving the smallest instance directly (base case).
    % _: R4 R- n- y2 Y: C$ g6 |8 c2 i  o) G) A
        Combining the results of smaller instances to solve the larger problem.
    $ m- ^+ b9 l. U7 D( e8 k+ O7 Z9 P, `, ]
    Components of a Recursive Function( o: Y8 R0 ?  h1 N6 r% B0 u  Z7 D
    $ ?- f4 n2 L/ q1 k- h4 A
        Base Case:
    9 K, b5 Z$ r8 f6 a$ X2 P$ Z' e, h. t+ N8 a0 K$ G# D2 d
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    : V3 q1 k# U' K4 i
    ) ]/ Y$ e/ w% }* x% j- v0 u$ K0 C- A. |        It acts as the stopping condition to prevent infinite recursion.) L( p" }1 K4 O
    , n: w2 a4 e/ z5 l/ ^# W
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.& ]" u8 x& I% y# G- g& p4 W
    % I) ^' A4 r1 K" W) V0 g7 t
        Recursive Case:
    8 ?" ^- R$ W' a. j* w* w' T& u8 V$ {$ H$ V9 Y
            This is where the function calls itself with a smaller or simpler version of the problem.
    0 L8 d: U  z& c. u8 P
    ! @; o# Y- C+ x2 D$ c        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    * ~/ R# w3 ^9 V) p- Z
    7 T- e- V, J0 H+ ^1 W8 _7 a& x: PExample: Factorial Calculation
    & h$ z  I8 H8 ^2 L; [0 R! G6 m8 }7 D! h5 x! B
    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:
    8 t( T9 _+ A: l7 W( T
    ' f8 W, m% G2 i( i    Base case: 0! = 1
    4 k+ H2 n& y2 j
    + [9 i& ?3 ?1 f* b$ E    Recursive case: n! = n * (n-1)!
    3 X( S3 F0 Z& W& E5 Z2 e6 t6 }; w4 H
    Here’s how it looks in code (Python):
    2 a! g5 @& l0 w9 ^  G5 [python
    / X/ r4 j; K; X/ t5 H( g2 B6 K
    % O  T0 @* G+ E( g# s7 `- j/ _" `  @; \3 _# D: K! z4 i; b  X
    def factorial(n):- n( ]; c& M8 a" E  _$ f
        # Base case
    . A0 F8 L& U- ?8 k' ]& l! _+ o2 X    if n == 0:
    ; a3 Z3 w8 \1 ^        return 12 c' N  o( r! v" t* P
        # Recursive case. ^' l1 Z/ h2 @6 _4 ^) @. |- |7 f
        else:* v: ?& Y/ I" k+ u& w
            return n * factorial(n - 1)
    - M* J5 e: X" P9 }) d
      f) `* I: T2 G* N# Example usage
    4 I/ Q: U' y0 q; r$ h' G4 Eprint(factorial(5))  # Output: 120
    ' P% S2 l+ A+ s7 ~% [; A3 D; U3 N5 N
    % [& A7 j8 G% |  dHow Recursion Works7 _- v8 |! |& {6 i& A" b* t  z
    ( E& d& Z5 X$ }# M" z
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ! t4 e- E4 ^+ P3 X7 \" x$ T) p/ q
    # z7 e7 ^- m; H0 d# n" g: m. g    Once the base case is reached, the function starts returning values back up the call stack.
    ) K0 {4 u2 S. ]7 `
    4 _# G7 _* C: d% R! c  a; d( Q    These returned values are combined to produce the final result.
    & l+ p& D) E; h  \/ C' i
    5 A, B: ?2 H4 O1 {For factorial(5):8 R: L, j) ?* T7 E- }8 w" |

    3 H  Y& j% u! r, W
    ' `9 @, E5 w" l1 ]& B' a: Ufactorial(5) = 5 * factorial(4)
    # o/ l1 v4 [& g; n, Tfactorial(4) = 4 * factorial(3)
    1 R8 y8 }* n8 h( _. v" afactorial(3) = 3 * factorial(2)
    4 t+ I- q9 d- c9 n9 q5 Q/ ^factorial(2) = 2 * factorial(1)6 }: f! ^! O3 D
    factorial(1) = 1 * factorial(0)
    9 o9 t4 C' v. D! m/ Bfactorial(0) = 1  # Base case
    ( F8 `- U$ s: `1 @5 s5 Z2 Q. D; U' O+ t1 ~) F
    Then, the results are combined:' r0 T" f+ B" A" s+ d

      \+ O  C1 f) b' o+ v6 n6 f2 {- p/ n( h9 D' s; q
    factorial(1) = 1 * 1 = 1
    9 i  C" q5 ~" @factorial(2) = 2 * 1 = 2
    - X9 W, Q/ T$ N/ mfactorial(3) = 3 * 2 = 63 [* S2 s6 ^/ X- U
    factorial(4) = 4 * 6 = 24, O! o1 S) |7 g- s: Y0 a- V6 B
    factorial(5) = 5 * 24 = 120
    5 X) Z( N; [9 W, O" @0 v$ o" ?
    7 K! N7 K  u! E8 L4 pAdvantages of Recursion1 C. m& R7 X( h* O

    % C1 {; i4 @- L8 d    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).
    * x" H. V7 f' Y
    & v, F) M1 k) I1 T4 g    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ! b' z8 I5 U: z: k9 h
      v; C5 D" ?1 A9 r# L. q- ~) ^Disadvantages of Recursion/ u6 M$ {- B8 e) ?" F2 G

    8 a! W% v9 I+ D% V& P    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.
    & r: _) }0 }3 g8 F( i4 l8 V: E# j$ U/ a# m9 q  @5 y6 h( W8 [
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).! _+ f& B) Q- N1 t) |" T; ~5 T8 C

    - q+ T% D$ g+ m! m5 fWhen to Use Recursion
    6 Q7 K: r+ V8 ?  ^
    2 K9 [* s  Q8 e7 z    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    : s5 z' W) A2 q$ l
    7 v$ p: m8 C0 y& P4 i$ D    Problems with a clear base case and recursive case.! X- l; T0 F3 k0 q- w2 _
    & q! M. f5 [) ?! i$ ]2 {
    Example: Fibonacci Sequence. e+ d4 }' t2 {" e1 A/ u
    + R3 j. q8 Y% g: E: T2 b2 q, [
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:$ A- v( T* N2 e/ H  J9 G' N. E

    2 [  H/ y- q5 \7 ]0 c' O/ |4 D    Base case: fib(0) = 0, fib(1) = 16 E% P. T$ x9 g8 {1 ]  p

    ; H, E( U7 U. W5 h7 J! ?    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! R# ]1 s- U# N% v
    ) t7 r. b: t. E! J! S4 ], Apython6 ?% k- M% M, T! m
    $ ]/ n" @6 A' r3 K0 M
    * S" m; \- ]6 B# n
    def fibonacci(n):
    $ b9 m0 q' I. H' J2 m    # Base cases0 ^( T, @3 K, R
        if n == 0:6 l3 x) b3 m1 y- l
            return 0
    9 v7 s; L* i" I  T! @' B& c: i    elif n == 1:
    7 N0 Q0 k6 l7 V2 C4 F' D        return 1! `3 f4 V3 K, L- f* M
        # Recursive case' E* ?0 ]( ?# g
        else:
    ) k, D8 ]" F( J3 @" K        return fibonacci(n - 1) + fibonacci(n - 2)8 ]) @$ T, }- J4 o) a$ n$ f8 a( f( b

    ' S! Q1 v9 ?7 b* n% C# Example usage& i0 g; S& V) R9 y" V
    print(fibonacci(6))  # Output: 8, c  \5 a4 R: _+ A9 V
    ( G/ c7 ?$ ?5 v6 i- a0 c8 U
    Tail Recursion' i( H# X1 C; \1 k& n/ k! k: o

    2 W; ^9 J2 H* g# Q$ U. NTail 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 X. |5 ~& A4 o' L) s, w9 r0 U' P4 R2 |2 T
    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.
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    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。
    回复 支持 反对

    使用道具 举报

    手机版|小黑屋|Archiver|网站错误报告|爱吱声   

    GMT+8, 2025-11-24 18:43 , Processed in 0.030590 second(s), 19 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表