设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2170|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 6 W/ ~* ^8 o: L- m. B' ]2 F9 z2 F

    - K9 o1 m6 q# O$ j/ T2 a1 `解释的不错$ r1 S0 G- P) ^9 U
    9 W2 G/ f5 d1 U+ P( S; s9 B. T" v
    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。
    ' t3 I8 h3 u- n$ d7 J# k. |" ]9 p. f
    关键要素+ E* T/ u* ?8 Z0 W; Q2 j9 m1 r' Z& O
    1. **基线条件(Base Case)**
    8 }0 G2 p: ?, Y6 ^3 C   - 递归终止的条件,防止无限循环. ^# U+ ]5 D2 F+ g$ {/ C+ \
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: F8 P. Z" g, P- x
    * J6 c4 q) H2 W
    2. **递归条件(Recursive Case)**
    7 W8 ^* e! X' R/ t5 Y% T) O. e   - 将原问题分解为更小的子问题8 K* I& r: D* }* }: r
       - 例如:n! = n × (n-1)!" p0 A3 F& ^+ U- @9 V
    3 i5 w% l) e( H- f8 S% K& R: c9 T
    经典示例:计算阶乘
    ' M0 I+ R+ T; u& }& cpython5 L% O- j( m1 ~1 L% N* `; a8 e' x
    def factorial(n):
    6 f* q% o% ~( w% \) |5 m& N( H: i    if n == 0:        # 基线条件
    & y9 I3 F* G' D! l# n        return 1
    " [4 r; h( r3 d    else:             # 递归条件
    ' i1 K* ^* d, j' l. |9 j        return n * factorial(n-1)1 P+ [7 K3 o6 _
    执行过程(以计算 3! 为例):
    ; a% L2 t+ t. Hfactorial(3)
    / w+ ?, l+ C  |% {. O- n9 R9 c3 * factorial(2)# ~4 Y& c+ D! P1 M- d
    3 * (2 * factorial(1))8 ?4 }5 |; @" d6 |( w0 ~5 U1 f
    3 * (2 * (1 * factorial(0)))3 |1 n. X$ z0 `3 g$ Y
    3 * (2 * (1 * 1)) = 60 U3 h& ^4 [; u/ p
    + {* `" c6 m5 s5 M- r1 }
    递归思维要点/ Q! Q/ Q) u+ w3 V' |* k
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑; m3 q# i% E, Y' B7 h  D
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ( W  Z* b1 K$ Z8 H( s3. **递推过程**:不断向下分解问题(递)! a+ Y$ a# }( D9 f
    4. **回溯过程**:组合子问题结果返回(归)
      g7 R) ~5 ?: `  G/ U2 ?- c
    & x0 D3 Y* t3 Q& P 注意事项7 H- F! A/ h- Q0 R
    必须要有终止条件
    + w$ A$ p* E( }- l2 H+ N递归深度过大可能导致栈溢出(Python默认递归深度约1000层), w7 S% d8 \7 ?9 E/ e% \8 H7 D2 m6 O
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    , o9 o) ^; ]4 M6 c) p% Q2 A! b尾递归优化可以提升效率(但Python不支持)7 _0 @; j; g% e, J. F4 {* Y+ z

    7 a7 c( {: |! }- |( f+ P 递归 vs 迭代
    9 G; ]! A1 q/ a, Y* V3 Q|          | 递归                          | 迭代               |! _  z& L; a+ G5 C5 D% H
    |----------|-----------------------------|------------------|- U  y, G: n# c5 r
    | 实现方式    | 函数自调用                        | 循环结构            |2 |  ?/ u+ G- m4 u
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |/ L$ E4 N. f) Z; @6 {$ y4 g' \" R
    | 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         |( p7 i. K1 z- q, o' {
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    - \! J$ g; b0 h. U- n# I# J( k' N( _2 R7 Z* G; a8 ?" v+ V' _, t1 m7 c
    经典递归应用场景
      t' H# \9 O5 W  G" ?1. 文件系统遍历(目录树结构)' ]. ^: D6 t1 ~6 h5 M
    2. 快速排序/归并排序算法3 v) i  G' |7 @9 u, g
    3. 汉诺塔问题( Z# X/ U& G: y" b# x& l: O% v& }* j
    4. 二叉树遍历(前序/中序/后序)
    8 T7 {- `+ _9 J& p- ^. L5 t5. 生成所有可能的组合(回溯算法)+ ^/ W3 c  x0 T0 g

    ! Q1 U. i2 X0 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。

    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
    奋斗
    8 小时前
  • 签到天数: 3186 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,) |7 @; ~2 s  r5 _
    我推理机的核心算法应该是二叉树遍历的变种。, V) c0 o, a- 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:3 m. h5 X; v% _1 m2 N0 K& a$ |
    Key Idea of Recursion/ Q' |) ^+ t/ P' p
    # Q6 t9 Y. y$ G+ Q7 c* b# C
    A recursive function solves a problem by:
    + O/ |$ h5 u: ]% ?& V0 R2 g. v3 J' E
        Breaking the problem into smaller instances of the same problem.* M8 S, a+ [+ P2 E2 M  y5 ]

    ' L, N& X+ J' m2 L6 u0 i/ }9 K    Solving the smallest instance directly (base case).
    0 g: W. k" D( a4 m& p; V( N
    * B" k8 N& y( S3 Q/ e% x    Combining the results of smaller instances to solve the larger problem.$ ]9 ?; C. T& _3 D; b: ~2 W2 t

    $ b# \& \' r; ^8 K# W( {' w) J' s" E* ]Components of a Recursive Function# Q0 d2 R  E, S: H8 U( a

    . l, j* q3 h( [; J    Base Case:
    0 L) S5 Z4 j4 r4 E
    % s* s# P& d0 r. _  l" h        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    , _* g7 A( c$ I7 m8 W( [; A; o4 z
    5 r8 m5 i7 a/ L7 f3 [3 R4 d* j. H        It acts as the stopping condition to prevent infinite recursion.7 c- M' m& o. d  s( O' f4 y6 }& H
    + u2 |. D" {# i7 q
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.' R& x/ w" A# r  S& Y( Y0 r8 N

    / R/ P* ]; j- u! |' ~0 v1 I7 h    Recursive Case:
    ) D9 z4 V( v' k7 I1 e
    2 W& t" @; K; z# J* U# p4 b# Y/ k0 E4 U        This is where the function calls itself with a smaller or simpler version of the problem.4 z* c4 `9 q' z, Q& p

    4 n7 [$ \2 j5 I& g        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).& _# v0 z2 h" v! L% X$ [$ v/ L
    7 I2 i* a& ]7 y  S7 p# q$ H2 u
    Example: Factorial Calculation# L$ ^2 i, c; C/ O! g

    * `- R1 Y- m: v, B& R/ BThe 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:5 C9 t, ]' d5 g" x& f/ h$ |' ]
    , x; G# X5 L) Q5 B- Z
        Base case: 0! = 19 A9 N, [" J. L) r
    1 g7 {5 E, l  Q- h; P
        Recursive case: n! = n * (n-1)!/ L+ F3 `: W. Q

    ) b/ W( V3 O0 v1 J5 }. \* JHere’s how it looks in code (Python):+ T0 X! B- d+ \) o6 G* Y. Q
    python. P6 M; f  A+ {6 P( w9 e

    8 I& f0 _7 l" i) y1 }# h: f
    & j: _+ j! ^9 @& g% {/ H5 r) F) rdef factorial(n):' P( ^; r. w/ c" E! c, g
        # Base case
    1 X) _& N/ C5 W7 F$ ^' a    if n == 0:
    5 a6 |6 z( A; }" I        return 13 E) [  `, Y% g( c2 U- [+ W( O+ c
        # Recursive case
    7 u1 Y0 E" O( s$ B4 p0 x5 f, W+ H9 j; \    else:. Y9 Z' F4 J: a" B( X6 @) _5 S
            return n * factorial(n - 1)
    3 _! N- `- u" V  c, c% z, p1 F6 r" f, h  z
    # Example usage
    + n1 D' m0 i3 u9 b$ _print(factorial(5))  # Output: 120
    9 [( S' T8 I9 T8 W' V3 {) w8 k6 a0 H! v
    How Recursion Works  c0 f0 x6 M! R* Q7 ~% m8 s, F

    ; L( E; K6 C  g4 O- t' V& n    The function keeps calling itself with smaller inputs until it reaches the base case./ i% I7 l1 K" L: [) V) J5 M2 j
    $ t1 o. c. o$ v4 F* A! B
        Once the base case is reached, the function starts returning values back up the call stack.
    9 u) e; ?$ W9 s6 ^6 P1 e3 A' [6 {! W) G( V- }' z
        These returned values are combined to produce the final result./ Q* C4 h' P7 B3 n8 s2 h
    6 y; r2 |( Q; E# c
    For factorial(5):
    # q+ D7 X. _* r9 ]4 O& N# ], Y, S7 O! `! t# q- a
    / t, p3 O2 `; U; }+ C* n# C
    factorial(5) = 5 * factorial(4)6 @7 Y' \/ b6 Z& K/ U
    factorial(4) = 4 * factorial(3)5 W0 j5 u( R  |- K! |
    factorial(3) = 3 * factorial(2)
    7 @" v$ j0 E6 h7 w6 Ofactorial(2) = 2 * factorial(1)8 r4 ?9 k: C8 U
    factorial(1) = 1 * factorial(0)2 V7 E0 L: M7 M
    factorial(0) = 1  # Base case
    ! Z. F/ D2 P8 ^- P( v$ z7 P
    6 S* E9 x5 s* x( ^Then, the results are combined:
    * Y( d/ T9 j9 n# v; j0 U- H1 n, f. U
    7 u' t! F/ F0 C+ v5 d7 L+ {# A6 g, J( s  K8 l7 ]
    factorial(1) = 1 * 1 = 11 i  K6 I2 v- s+ q( q" `4 `+ @
    factorial(2) = 2 * 1 = 2$ [8 \+ `  R# s# {' N0 X4 l
    factorial(3) = 3 * 2 = 6
    ; U6 f/ U% ]5 ?/ h3 P5 R/ X2 ?9 ^factorial(4) = 4 * 6 = 24
    " G! |$ O& Y6 n7 T$ k2 zfactorial(5) = 5 * 24 = 120% u3 \5 u1 ]" ^( d6 y
    / p( K; j, k5 P( j$ J
    Advantages of Recursion  k/ P5 b  Q2 _) `: D
    7 W4 f1 Z2 B- f3 K& D) G& ?
        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).  n9 n2 S3 {* O! u4 T! M9 \

    . A& }& |* m: R) Q4 i) X& u+ Y$ o    Readability: Recursive code can be more readable and concise compared to iterative solutions.: v& ^$ n6 w4 H! J" K: \3 h

    : v. _( E/ Y/ |( e$ pDisadvantages of Recursion
    ; l' z0 p5 j# [8 D5 b0 I$ e- g" g3 t. u5 y# z$ U
        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.
    # C6 _/ U4 L& A. a' Y- Q
    ( V4 d* Z* N. e( |9 ^4 t( B- c    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)., u8 p0 @8 ~3 x6 T: n
    $ y: x2 c- U( A& A
    When to Use Recursion
    / S" k* [7 ]0 ~1 B" W% Y+ W  N  H' S% A8 J6 T
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
      U# [5 z2 e4 u$ y, ?  P  M9 O: j3 z1 c; W/ e4 A
        Problems with a clear base case and recursive case.0 Q* p2 q( R. N8 ]" A7 R, O
    * [' {/ c0 b# ?" K9 n% J+ L5 N- K
    Example: Fibonacci Sequence' R) f# A3 {" D* Z" r- W

    7 k( h# I9 C- BThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:+ ~0 p. I  x2 Z2 ?+ k9 k- E3 _

    + @1 H: g3 L. N/ _5 C$ {$ T    Base case: fib(0) = 0, fib(1) = 1
    . \8 V+ U, u3 f, l7 x+ z1 `7 e, o3 ~3 x9 c0 \( M" ?
        Recursive case: fib(n) = fib(n-1) + fib(n-2): b7 _/ b$ {; h9 p7 S* \2 M
    ) K5 }0 {, b6 R3 x0 t/ S
    python6 j/ H* r! ?5 m& r7 [
    8 K8 ~1 u% X- ?- g

    ' H$ ~4 T$ ^+ q- d: x* Mdef fibonacci(n):
    " i, w' K  l( f: X- j5 U0 T/ B" t( U    # Base cases
    3 ~/ E2 b" o' j6 n    if n == 0:, j$ e4 t) A5 L
            return 07 y# L2 P6 U7 k3 f
        elif n == 1:; Y8 u3 ?+ o; S  H- N6 m; b
            return 1
    $ D: R( S/ k7 d* Y) F! @    # Recursive case
    ) R1 m3 X) l+ Y, D. t3 e$ ?    else:) Z6 _( o/ V: `$ q& p
            return fibonacci(n - 1) + fibonacci(n - 2)
    5 O3 E$ f# k0 ]# S' l& H) f0 e: ^' D# M1 P, k1 d0 a5 |
    # Example usage$ Q- c0 h' c. Y: a1 h% j5 |+ r$ ^2 [7 T
    print(fibonacci(6))  # Output: 8
    4 ]& t" m+ Z9 Z) c$ a1 B7 _6 d4 M% x! o
    Tail Recursion$ l- f3 M* V9 ^
    3 V8 t( y& f" i# n, {
    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).
    ' R; C: S, o0 v) a/ _) x% E( T3 [3 C9 p7 s) g! M+ M
    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, 2026-2-27 15:15 , Processed in 0.055688 second(s), 18 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表