设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1919|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
    % I$ H# a1 y9 V: |3 Y2 l; f  L# S6 N
    解释的不错
    ) k% f6 K1 {; k; g& Z2 T
    - }3 `% z# D2 ^5 S6 o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。
    ( Y2 a  K5 x8 h3 U( k+ \
    $ D2 x9 K3 e- G: C 关键要素
    3 L- S& k) a7 N% V# O9 a1. **基线条件(Base Case)**
    " _2 k2 ~, F1 H   - 递归终止的条件,防止无限循环1 X0 K9 V7 E" D( {9 k$ S4 w* [4 Z
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 19 R" {. o) e+ _( T' u0 g0 T
    " F& }7 n( C+ @8 b: g: k1 X
    2. **递归条件(Recursive Case)**
    1 r6 N8 F- |$ g! \9 h( i   - 将原问题分解为更小的子问题
    . f5 Y. Q9 k8 T$ o6 i( y   - 例如:n! = n × (n-1)!  y) t& P: H; r3 x' O
    " y! C# B" x+ r; D' D
    经典示例:计算阶乘
    , c2 M  Y: D1 d. J( ?python0 P4 H% o( L; r9 k
    def factorial(n):! l7 o; I" J& _: N: m: F" q. e+ P
        if n == 0:        # 基线条件7 D7 l0 D& o6 W6 D
            return 1  q3 l3 f' t! Q) O7 M
        else:             # 递归条件
    $ q( s+ P4 i7 M# n        return n * factorial(n-1)
    1 _% c+ @+ S7 u  R* c4 }执行过程(以计算 3! 为例):
    - u  v2 H& @$ O$ Qfactorial(3)) u% B: W$ T. @& u5 k/ c
    3 * factorial(2)4 T" F  {+ ]) w- s+ x: W; s
    3 * (2 * factorial(1))6 U8 M. k! G) @9 e- p9 @! [6 o  Y
    3 * (2 * (1 * factorial(0)))
    7 Z$ Y, K0 l, e3 * (2 * (1 * 1)) = 6
    * ?( {5 y5 R" T( ]$ q& M" S. G/ e+ M( d! n" N. i
    递归思维要点
    6 D6 Z6 u( i7 a1. **信任递归**:假设子问题已经解决,专注当前层逻辑) ^: ^; K4 _1 x/ y6 t1 W6 h- \
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 c( M- ?8 k" q1 z- k3 p! Y
    3. **递推过程**:不断向下分解问题(递)
    ! q' l4 [! p$ j0 _7 n4. **回溯过程**:组合子问题结果返回(归)9 E+ y$ N# j0 y/ h0 x1 b

    5 s: R( L, q) h9 b 注意事项
    ! p" L- Q/ }: p0 R  o必须要有终止条件- m0 P1 `1 F; H) p7 @
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层), u8 G* F* W( ~; c6 b
    某些问题用递归更直观(如树遍历),但效率可能不如迭代! n' [$ G& d" N
    尾递归优化可以提升效率(但Python不支持)
    $ L& }' x; H  F! h% E8 S' K
    * N9 M! p% W# H9 `- u 递归 vs 迭代# P7 N9 v: E5 d) d% c1 O
    |          | 递归                          | 迭代               |
    2 Q4 b- l! H7 L* Z5 Z5 K|----------|-----------------------------|------------------|' v4 S- v' d* I; d
    | 实现方式    | 函数自调用                        | 循环结构            |
    , T# N0 i: q: |' `* S. S! e0 P, b| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |- @( y& I2 s/ Q6 }- w
    | 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( b/ P6 S! T| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |; T& }4 s/ O. ?/ S; t- g

    $ y1 u8 {3 }8 h 经典递归应用场景
      F3 z; L& {: Q' W' D% J3 Y% d1. 文件系统遍历(目录树结构)4 @" E4 X  Y/ d7 i* Z) P
    2. 快速排序/归并排序算法
      F% h; y/ ], b" C/ T- E- t3. 汉诺塔问题* l( s) y# F! K0 _( u- p7 g( Y
    4. 二叉树遍历(前序/中序/后序)
    : B# Q# l4 X6 H) t- }6 d2 g: D5. 生成所有可能的组合(回溯算法)
    ' Z% f( T* Y4 P+ U1 v3 B) b! |7 z1 a' B# i) W0 s! e
    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
    难过
    昨天 10:21
  • 签到天数: 3151 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,
    8 D8 S$ V. C. b  S我推理机的核心算法应该是二叉树遍历的变种。
    % }+ m% T+ m0 \& B. G" d. r; c& p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:' [# x8 w$ ?" J
    Key Idea of Recursion
      v( P% q; M* d' K! u6 P0 [
    3 t. t8 G# D4 ZA recursive function solves a problem by:- o1 @8 e/ D$ W8 Z& Y  ], B
    + c" n4 P% H/ M. @  f2 `& _
        Breaking the problem into smaller instances of the same problem.2 K/ C; s* `; W
    : e; U* A: H+ E/ b$ k& X
        Solving the smallest instance directly (base case).
    1 P9 ~9 i( K$ K, b8 v
    * h! M$ ]; n4 [5 m0 o# o/ L    Combining the results of smaller instances to solve the larger problem.# a& E* J+ Q8 z* n. G2 m
    & s7 Q! A/ A7 x+ Z
    Components of a Recursive Function3 C  k" c1 S# j- \, u7 a

    7 C" @1 Z' E1 k" m4 F    Base Case:
    * y: M& b3 e1 V# }8 y' S+ r
    ! g! A. A' Z9 E+ i5 T3 y        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. H/ L* g, u% U

    & |' l( Y! A- F5 b' M        It acts as the stopping condition to prevent infinite recursion.
    # o& g8 v! }8 n: I
    6 d7 ^6 Y% P2 E# g& D2 V        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    , W8 p, r7 b2 E2 f  c& A9 l' `+ u+ G7 p% s+ x* e
        Recursive Case:3 i2 E" b$ ]  ]) v3 h( s
    + k" C/ Q$ K5 ?4 G+ z
            This is where the function calls itself with a smaller or simpler version of the problem.
    # c+ m( C) W* n7 c/ \- m; O% z- u8 s) H
    / ]1 ]+ M7 |) W+ h/ G        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' D7 m8 t0 G2 A0 w
    - q% ^  c" z1 E3 ^, m4 Q, s1 z
    Example: Factorial Calculation
    - P2 ^; I2 k9 x! D/ ^& k0 z
    # v6 X6 C% ^* s) j- L/ I, }" QThe 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:( g  c* H# v1 J$ h

    ( g/ y2 \' c6 |$ R! I$ H    Base case: 0! = 1
    ( b* N* U# n) l# B7 j2 \  \% o$ e3 W9 `1 ^, J1 L* R
        Recursive case: n! = n * (n-1)!
    ; d3 U. G0 S0 W+ L8 R/ T$ p+ F) K$ D' X8 g" z) ^
    Here’s how it looks in code (Python):8 {% q, o) p/ ]* R; z5 m
    python
    9 w: C2 r, y# y3 V* [- s6 m: u
    4 _8 a" r' ?; Q( A+ k  T8 @' h) J3 W  |( B+ j2 k! A- J4 Q
    def factorial(n):- J+ e8 G% N! n2 C1 B
        # Base case
    $ t  z: w# q; @& [    if n == 0:
    : t& l) N, T. h( P  ?( R4 y        return 1
    ; B/ n8 I# y& P    # Recursive case$ N. d' o( y$ H- q% V
        else:" x. M) C* O, g5 N
            return n * factorial(n - 1)9 e5 J  L+ T/ `/ G8 `

      U; l0 [( w& {' A# Example usage
    6 x, Z$ H- o' [5 F5 eprint(factorial(5))  # Output: 120
    ) t* }6 v1 x7 W0 {% z4 I# b
    $ e1 G! Y' ^* N+ Q  ^" zHow Recursion Works
    2 s- A: G5 i4 r6 U4 w- \& m! j5 O7 E$ d, ]2 D! n, A
        The function keeps calling itself with smaller inputs until it reaches the base case.( e! @8 M! Z$ r- e' h8 v6 R9 R

    " M1 c9 l4 q0 @( b  y    Once the base case is reached, the function starts returning values back up the call stack.# e! m- R( J# M3 k7 C
    4 c$ M! H$ c$ t  D
        These returned values are combined to produce the final result.
    1 }# W* ]* h. x! k5 C6 f- Q1 Q& T  @7 B3 L5 W3 P& T
    For factorial(5):. `6 X8 o4 g2 X, ]

    ) G/ l& e6 c5 u; x& X
    ' |1 L# R; g+ @factorial(5) = 5 * factorial(4)( b9 Y0 R1 D0 b4 {( `  j" s
    factorial(4) = 4 * factorial(3)1 s6 I# q, p6 t- u+ H/ M4 j( X; Z9 B
    factorial(3) = 3 * factorial(2); N$ f+ w+ x3 g( W; e( l( d
    factorial(2) = 2 * factorial(1)- z$ W7 }: y$ c# g$ t
    factorial(1) = 1 * factorial(0)
    # r5 q" Q+ @3 j0 f+ g6 v. ofactorial(0) = 1  # Base case
    7 C3 p6 A! m2 c$ E6 u5 R2 u
    0 G5 Q! D  b2 T( N! k. R# mThen, the results are combined:
    & x0 m. L- P" ?# `+ W  V4 e/ j  Q, U9 ]: m/ g" x

    7 o7 `4 m' j1 I" Ufactorial(1) = 1 * 1 = 1+ I! K; K7 N* G  G
    factorial(2) = 2 * 1 = 2; L/ o( o  C7 g6 d
    factorial(3) = 3 * 2 = 60 O( I: [, V6 m
    factorial(4) = 4 * 6 = 24
    & y% I& ]! d+ _# n. p2 }factorial(5) = 5 * 24 = 120* ]' v: Y) b( Z

    0 [) ~  g& o5 x5 i7 F1 C- W& J* mAdvantages of Recursion* D: k. M& D8 e$ l5 F# C- O) @; `

    7 T+ s' V! X7 V$ m    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).
    / C8 E2 @. V1 ]4 U+ q) v, h7 x% b/ r4 p: o3 J5 C. @- w7 v& B
        Readability: Recursive code can be more readable and concise compared to iterative solutions.7 R! K+ X" f# Q9 P

    * R3 M. l: L: jDisadvantages of Recursion  {- Z) w5 p  t8 X0 Y

    9 M. R0 Q2 T$ @% j* I    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.
    1 Q  e4 P$ `+ F) x" s+ Q6 F- C  Q$ r# _& t' d* K0 n
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    + _9 B" Y6 m0 J+ e5 ^, u' g! m# F# ~9 a$ ?8 G. z
    When to Use Recursion3 |6 k5 A! j# m" [4 T
    + ]. G7 B/ H4 y5 w% d
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 B+ P; R- j5 n. e/ _: G

    . N; _& G8 k! M$ [3 k8 |    Problems with a clear base case and recursive case.
    & E8 q0 X2 S0 y) i& T1 }5 U% E/ _1 s/ J, N" A# L. j
    Example: Fibonacci Sequence* ]5 N$ H: J/ o3 `

    3 i- v4 c& [3 Z; L. k: H9 Q/ V% BThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    7 n8 K) f' R- A4 n
    / Y7 K9 n$ m  p% h5 H& t    Base case: fib(0) = 0, fib(1) = 1& G+ ^; u: u% t, @+ m# G7 u, I! o+ T

    + H+ D- ~% Z0 W6 D    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    , h+ V3 a2 e6 [) y4 v  ~1 E2 D
    ( P. B: h, n( _python* d8 ]4 `2 A% v' J

    5 |0 O) X0 F& `+ }! N1 t8 D+ Y- C' s7 D1 L. v/ B
    def fibonacci(n):9 s$ l! P; Q7 y! f6 V7 {3 p# ]+ X
        # Base cases
    5 a9 }! w2 G* V! N5 Z3 H! j6 e9 O    if n == 0:
    : j  g7 D! {/ x! [, \6 _        return 0
    0 ~6 j/ t! s% e- ~6 M" q, Q( `7 m( ?    elif n == 1:$ w9 U4 O! `; f  `% E' m, B) B
            return 1
    - O9 T  h  T( H) m4 D  e- d    # Recursive case# H+ d/ X& ^& r9 ^( U1 l
        else:, S& i2 x3 I8 u4 c
            return fibonacci(n - 1) + fibonacci(n - 2)
    + ?0 C9 A- M! P4 H& o! G' m! I' J" j4 P% l, P
    # Example usage5 v* m) A! c/ v9 L: [$ |. b- y
    print(fibonacci(6))  # Output: 85 k( S/ n9 g; v* S
    " A% o' y8 Q4 G6 @6 q
    Tail Recursion
    & e/ a, E# g% @  Z+ Z2 s! O6 b) Z' n' x4 x. m# z
    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).
    9 V  A$ H. R. {- k5 f
    - v# Y( p  ?  X; LIn 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, 2026-1-23 00:26 , Processed in 0.055842 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