设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1418|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
    5 F7 S, G3 x, j
    * D: x0 e+ x: `/ t解释的不错' \2 D4 g- Z9 x9 _; O

    2 V3 T" A. u; e, \, B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。1 k, h2 s1 `- _  J0 K+ v: k1 S; }
    8 z/ r% q# n+ s, @+ L' G
    关键要素7 \* m/ W& a& D" b
    1. **基线条件(Base Case)**
    1 S( n4 n4 a) a/ G9 J. J( \   - 递归终止的条件,防止无限循环  B# ~  i( U. ]4 M3 R3 j4 C7 w
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& q2 E8 `( _6 d/ H" [

    3 X8 ]. p& r* V% _8 w+ |2. **递归条件(Recursive Case)**
    , m6 p/ x  W( D8 C- E8 |7 v   - 将原问题分解为更小的子问题
    . U4 t; g2 Y6 Q( z9 h0 j   - 例如:n! = n × (n-1)!# w+ L7 [4 ]- `, r
    & t$ r$ c  l5 H2 ~* y: j
    经典示例:计算阶乘
    & k; i) e9 c' \% o( s! U: J2 D! @python
    , O; Q% c, Y' z3 G4 {8 O! |. |5 v8 udef factorial(n):
    . D5 F9 U- A- A1 C    if n == 0:        # 基线条件/ K$ W1 P% Y) n$ O
            return 1- Y1 g' g/ y6 D$ b4 Z
        else:             # 递归条件
    + \, F; s# |# I3 }! Q        return n * factorial(n-1)& `: _$ P2 X* h$ \/ e2 o0 G
    执行过程(以计算 3! 为例):: N$ H: T; g+ D3 k  q: {
    factorial(3); s0 R8 Y" W1 J* P" L/ i
    3 * factorial(2)4 ]& K7 H$ i: z& R; W6 }- J
    3 * (2 * factorial(1))0 Z. _) D* f) _' {# |. g# K/ G
    3 * (2 * (1 * factorial(0)))8 e( z$ j7 X" {8 v# S
    3 * (2 * (1 * 1)) = 65 Q7 ?9 A9 T! p) k

    ( W' _+ B/ d8 E1 ?2 G0 G4 f 递归思维要点9 k0 `5 j" O3 G) t1 u
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑! P$ z, b0 s5 T( l4 G0 N
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- C, r: y$ d% z7 }- e7 E
    3. **递推过程**:不断向下分解问题(递)
    9 s* ~7 b! Q$ H$ R4 S) ?4. **回溯过程**:组合子问题结果返回(归)
    5 }7 q7 {& u9 R1 H- T, Q. S& D* v2 _" \6 O3 G5 N
    注意事项: M2 X) r' v0 R$ G1 |# _9 {! b& G
    必须要有终止条件/ E7 }6 a( o! }; {6 t
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ) }& g& F8 n, y9 Y$ `, _; E某些问题用递归更直观(如树遍历),但效率可能不如迭代% \! |/ q2 ~/ k% F3 `3 t- p
    尾递归优化可以提升效率(但Python不支持)% ~& N- [! I' L/ x; S  D4 y

    / T& A9 A- `' U: _5 G 递归 vs 迭代
    4 y: d( I3 v- N8 N|          | 递归                          | 迭代               |
    % e* {( B2 `- W5 X0 Z) K. v4 z5 H|----------|-----------------------------|------------------|
    , L1 c7 |0 @! }8 F| 实现方式    | 函数自调用                        | 循环结构            |% W5 h2 v- I8 _, J8 I0 u
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & ]0 [2 Z( Z8 M* y( S+ M5 d8 i4 V" I8 m| 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         |% g: U, }! L: G. k
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |+ O. `% {' ^; Z$ u9 ^# M2 T

    9 U' Y5 }! M  |7 c8 s 经典递归应用场景  }: S& y: H7 l5 h3 H
    1. 文件系统遍历(目录树结构)
    ; Y& P/ i2 W. c+ \$ \5 x: \, J2. 快速排序/归并排序算法
    ( e( a) Z  D. T: |+ V1 V7 w3. 汉诺塔问题' f  H7 @+ S/ ]+ p  M& I6 J" v
    4. 二叉树遍历(前序/中序/后序)
    / q2 p) y4 \9 S3 E2 ?: w, H8 ?5. 生成所有可能的组合(回溯算法)
    3 O1 X% R) x4 J  Y! K/ i7 G: N- w, b9 M# F% l  ?2 _( R$ 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。

    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
    慵懒
    昨天 07:13
  • 签到天数: 3112 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,
    $ P) \* p2 C0 A我推理机的核心算法应该是二叉树遍历的变种。0 M% O' ]% `2 ^
    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:6 w9 n% o/ b4 P2 k9 V6 R: B
    Key Idea of Recursion% L  W, T5 e7 E- Y

    ' Y& ]  j5 F6 e1 Z4 C+ yA recursive function solves a problem by:
    3 w$ o& M$ t; m4 v. _+ _* @% `+ w% u. N4 }5 M9 g
        Breaking the problem into smaller instances of the same problem.. b. j+ y3 K- B( h, Z: X7 m

    8 b* p: J% `5 |' I+ E3 A; J- o    Solving the smallest instance directly (base case)./ W! C% I) w! O5 U6 Y0 r, b

    , H7 d. v$ U7 W4 Z+ M9 S( R' y: i    Combining the results of smaller instances to solve the larger problem.) }& x9 X' `2 f* H3 q8 b5 r
    3 x8 ]7 \  {2 i- K$ n* b. @/ u4 f* M
    Components of a Recursive Function
    2 ~( `- O$ X$ K8 Z
    , @# U) J( S) @    Base Case:4 q/ l! f3 m5 n- L! G% H( h; t
    ; H, ], |1 K3 `- j3 m7 v
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    7 u8 g* k8 n! {. U: F* C" b8 w6 a6 L4 ]0 Z# F6 W
            It acts as the stopping condition to prevent infinite recursion., b3 `2 N2 w- p4 u; }* ]+ X% a+ o

    4 W* f! `6 [5 y6 K8 o7 Y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
      R- ^$ Y  z4 Z& |% U3 K, {! `2 O0 L
        Recursive Case:$ u( I) M# \% h2 F

    1 c5 p( z. e' s* S: f        This is where the function calls itself with a smaller or simpler version of the problem.
    ' m- T3 r9 h! R1 q# ^. [7 g1 p( g) h+ ?
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).) s1 w9 m' Q1 ~- a" E- q4 k

    . F: q% X. t* d  B- [+ f: NExample: Factorial Calculation/ e" K& X; B6 O2 p

    0 t% o' e2 q; @( JThe 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:( w8 T, s; [9 o8 C$ w9 O
    9 h9 Z, ?- {- [/ M4 I$ F7 y
        Base case: 0! = 1
    ( C( ^) F. h* c% H: ?% N& c6 O8 p" L3 t" M
        Recursive case: n! = n * (n-1)!1 V( Q/ ]) C  @) k" y

    * @8 n" j! V8 ~9 O7 [( AHere’s how it looks in code (Python):
    9 t5 u& k; F3 A+ J  Opython
      e! k9 C( O2 p0 c, k: E
    * R2 l/ j5 w. D! ~' k/ @  w- j6 Z; l% Y
    . q" Z4 A- T3 }1 l/ e1 K: |! P8 ndef factorial(n):! F# r5 a5 [" D% g6 J8 t& A' Q
        # Base case
    , m3 b; T( f9 ~* @0 @2 u    if n == 0:
    + l: S6 V3 u/ {* V& i        return 1
    2 X2 d0 u4 L4 [    # Recursive case
    , v4 U0 E8 g, U# H    else:
    : u. p( p( Z1 w        return n * factorial(n - 1)
    . @9 e: R1 i. C8 ?) u: k; |' n9 m3 C- g0 X$ `! g
    # Example usage2 S5 S- q% h5 \, C2 t/ h
    print(factorial(5))  # Output: 120
    ' V+ k  i" }# N: o8 r) f8 g' ^- Y3 z* f, S) H
    How Recursion Works
    : _8 Q9 H7 k6 X% W1 N' @; T
    / U- e/ o1 f8 I7 d/ h+ D    The function keeps calling itself with smaller inputs until it reaches the base case.
    5 D  ]! {/ w8 ~3 A: T; D! V0 M8 H/ s' O. n. g% q, m
        Once the base case is reached, the function starts returning values back up the call stack.
    + A; C5 Q5 H+ p" b/ @7 E  k( Z$ F- l, s
        These returned values are combined to produce the final result.
    / w6 K( G7 f* f7 ?( U3 \' g  P# Z1 W' [3 U7 \6 o
    For factorial(5):
    % D/ o' ?$ p2 y2 B) W5 l7 u  ~1 a% ~
    ' m* m! J+ |! V: x( C% q
    factorial(5) = 5 * factorial(4)
    , n% D+ U! {3 k7 h9 Sfactorial(4) = 4 * factorial(3)
    # I0 D' \( H# lfactorial(3) = 3 * factorial(2)
    8 T! g+ V$ q: ^6 U& Tfactorial(2) = 2 * factorial(1)
    ! A# \$ P4 L% L) A3 Gfactorial(1) = 1 * factorial(0)
    - @7 |! S4 g4 M1 vfactorial(0) = 1  # Base case
    , b4 T, i3 F4 M/ i2 L) R& z- \* Z) o5 O4 Z
    Then, the results are combined:8 M, H# n8 _/ k1 ?

    , U6 `& p! g; ^2 g8 V$ l. G1 @; _7 h8 B) U
    factorial(1) = 1 * 1 = 1
    + ]$ }! A: f7 G, k8 ]( |; h; r( |1 ]factorial(2) = 2 * 1 = 2
    5 G) B3 E9 n2 N+ C! dfactorial(3) = 3 * 2 = 68 j3 ~3 p1 J7 D# @# u5 D" t- z
    factorial(4) = 4 * 6 = 24  p& D6 a8 j. D
    factorial(5) = 5 * 24 = 120+ R3 k0 G( p: ^# s+ F: H8 G1 G

    4 o1 O) q$ a* G. ^. vAdvantages of Recursion  q- A% m- m! x2 Z1 Q" V; X

    ; H2 e8 @8 G4 m2 h( y) O' c    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).. S0 c+ v# t7 j- s
    ( z' z" @; |' Z# J! ^
        Readability: Recursive code can be more readable and concise compared to iterative solutions.- j6 o) W) c. m+ F

    7 n- `6 }6 Q- K3 _+ JDisadvantages of Recursion
    % u) C7 ?2 b% Q9 n' ]$ E( t3 d3 }! }
        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.
    4 O) S) `5 B7 N+ M
    & q) ^/ y: ^8 f7 M5 e    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)./ G6 Q( r+ y, p' U' o6 y
    % T0 e& y" n0 i# _0 ]2 k! t& m6 I
    When to Use Recursion
    ) l  A3 O4 j- F7 S! Y$ r
    9 _  \+ l7 o4 ]    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).% S( T2 d0 r% k- W: n  _$ m

    0 z0 `3 J* E5 {' n+ _    Problems with a clear base case and recursive case.! s5 c& F" m6 K( g* _4 q5 \
    # j4 I  Q* S( J
    Example: Fibonacci Sequence
    : F# q& v3 {( }+ o3 Y. U3 x2 w7 Z0 E  Y. F$ r5 F! d. K1 Y
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    / t, E) @; F4 a1 e# v# B, d
    7 x5 |7 D8 x: M' Z+ K9 O$ y5 r    Base case: fib(0) = 0, fib(1) = 14 M( \& O, f" A0 E1 H/ n9 k
    / c  c! i3 ^5 i( b5 O; t
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    6 G8 S/ V: G/ y7 X+ s0 U4 n
    $ `: F( g% R/ B/ Bpython! y' A  F8 j( t- j3 x* O

    2 L7 ]5 D: a0 f! o% q' K2 ?' Y% Y! @& `: K1 R; v
    def fibonacci(n):6 U( o& ^7 \, n$ ~; Z( t
        # Base cases( J& K2 s9 z( d5 ]$ k  U1 i
        if n == 0:3 r& L, `+ W. j) B" _; B+ a1 d
            return 0, P6 F& v0 O3 v. @" [
        elif n == 1:
    * Y2 U' a( {1 |: C        return 1
    6 t; e+ R# I& u0 M: d( @( r1 \  N    # Recursive case
    , d+ V; \: ?0 F    else:) i# D' b2 l& b7 J
            return fibonacci(n - 1) + fibonacci(n - 2)* O: G- \5 O' Y) `1 F: u/ W

    # g- m6 H5 @: p% r1 R/ _. \# Example usage
    # L( D% i1 @2 o5 z% a$ h# t, |0 pprint(fibonacci(6))  # Output: 8
    / }: v2 `1 S+ B8 I: R. n8 d0 C- n: O! o1 i* H: g7 H
    Tail Recursion
    9 A0 m/ E9 z+ q, c
    ! |7 X. F1 u% z. V3 z6 ^3 M" CTail 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).5 S( |) F$ Q0 K

    0 R& ]8 B& k% t) x* OIn 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, 2025-12-9 00:26 , Processed in 0.046216 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