设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1974|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 $ C4 N* V% V. T/ V% O9 X

    ) U/ F+ r' J) y7 u) B解释的不错/ o5 `4 t4 G# L
    ! k, e8 I0 V" R& |4 A+ Z2 }) i
    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。: }. Y4 t! N/ V$ b. y

    ' r, `9 k8 e0 Q, d 关键要素: z6 [+ p. T8 |) |( |( e- [; t2 c
    1. **基线条件(Base Case)**- V. `$ S! k. K; r5 Q
       - 递归终止的条件,防止无限循环
    9 S- h4 z$ j, }( d8 G8 H8 x  v   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      c2 f8 W9 ^. ?
    9 X4 L& I1 T+ |: x8 H* U6 L2. **递归条件(Recursive Case)**, x% I4 J  A; @
       - 将原问题分解为更小的子问题
    1 e0 L. i. o; b, ^, n5 j8 `! O8 O   - 例如:n! = n × (n-1)!! ~/ A6 w3 Q+ P( _: n7 ^# b
    . p# y7 Y$ r- g  w
    经典示例:计算阶乘/ s4 h: ?3 C" t7 z
    python
    5 c4 N+ k' o7 Z; @  D. a1 U  R" mdef factorial(n):
    1 U( a- n9 V) w    if n == 0:        # 基线条件
    9 T  {% q% N) M  J6 R& `9 u        return 1
    6 O  r- c! O9 D3 v! m, x    else:             # 递归条件( h, j* r* e$ o% k5 o! p
            return n * factorial(n-1), ?; S8 t6 {& c: X
    执行过程(以计算 3! 为例):
    % \: C4 Z% f* H' p4 |+ A- \factorial(3)
    . c/ j& P" W" `1 w% S% ~3 {3 * factorial(2)
    & G8 ~1 [- T+ q$ ]1 P8 Z% H1 K# y3 * (2 * factorial(1))
    0 M9 z) X/ s3 s3 w( ?3 * (2 * (1 * factorial(0)))# z! J! ]& Z$ k# z( {
    3 * (2 * (1 * 1)) = 6
    7 C3 ]1 F! F  ?6 L" X( L, I* w& _0 O) U4 w+ D& G1 ?% t  e# a
    递归思维要点) G. M8 a$ g( g" h4 e, p/ R* Y
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % \2 Q0 E+ e% x# u* N3 Z5 d2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    2 O* n( B/ _0 L7 \. \5 ]; j3. **递推过程**:不断向下分解问题(递): q/ G* j: f* a: Z) s
    4. **回溯过程**:组合子问题结果返回(归)
    : n( s6 z, h% ^, e/ ^& ?; b( `4 m! ^: P3 I5 G7 f
    注意事项
    7 c7 p& l# J' Y% B% \% R3 g/ {必须要有终止条件
    6 n' V, w& Y6 Y, O6 C; [递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& Y1 Q8 C0 r$ I4 v$ ^
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    8 R; o5 S& g8 x' }尾递归优化可以提升效率(但Python不支持)
    5 a) j2 {: @0 K
    8 u. K' [. @# v1 \* M( x# Q 递归 vs 迭代3 q, h1 {, {+ z5 d
    |          | 递归                          | 迭代               |
    . M. N8 v' \1 A+ i* ?0 B) b$ M( e|----------|-----------------------------|------------------|' I9 Y1 J+ h  d
    | 实现方式    | 函数自调用                        | 循环结构            |
    5 M2 f" J" a& G+ r! h| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ) n) w8 b9 {7 z9 Q1 g0 k| 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         |
    2 f# z6 @  W! ~; D  @| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |3 E3 b( Z- E% f1 x- V
    4 h. j# G1 I4 c
    经典递归应用场景; U6 f$ @  r& p
    1. 文件系统遍历(目录树结构)
    4 d' f' I5 v" a6 T$ k2. 快速排序/归并排序算法
    4 R8 Q5 q; L8 s" J/ ~. }$ J8 Z3. 汉诺塔问题$ c8 H; N( C* `- h
    4. 二叉树遍历(前序/中序/后序)# G" i7 A2 d% k9 T/ {# m$ R+ B
    5. 生成所有可能的组合(回溯算法): l% n) v8 |( |) c" t6 ?3 D
    . \/ I5 V+ R$ i: t1 _1 Y$ E' \+ p: A
    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

    10 小时前
  • 签到天数: 3157 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,
    6 y2 Z8 \5 j% @/ K/ P- {& A- i我推理机的核心算法应该是二叉树遍历的变种。
    . m9 Y$ _1 N6 A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:  F- E/ d/ w) k! O; ~8 v
    Key Idea of Recursion/ G: Y" w: K' k) o+ B6 T! E

    3 ^9 r  z7 ^" @/ t# x" J% ?+ i; HA recursive function solves a problem by:
    4 ]  v, O9 m: u. i( q
    5 i" c" U* U' m8 a9 _2 c    Breaking the problem into smaller instances of the same problem.
      z9 o5 {/ u) K6 t8 o2 D2 o6 t) G5 O7 ]7 X
        Solving the smallest instance directly (base case).7 t8 V* a1 r" g$ f7 P9 Z
    & t. U- G1 U7 |9 s  ^6 R
        Combining the results of smaller instances to solve the larger problem.
    ) L  }" r) o, R3 [. ?# V( G
    ' B/ k- i( _' |, r0 XComponents of a Recursive Function3 @2 h' u$ ^( [1 A2 ~
      N- }9 O8 T" \- y$ t7 I$ V
        Base Case:; U. |& `. R! f8 K

    * n5 B1 M9 N2 d        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    * q& Q( ^, m+ H, G* j$ |. m3 ]" a& t
            It acts as the stopping condition to prevent infinite recursion.
    6 R4 |3 q' T# a3 {" g" o6 ~
    : V4 N6 B. g7 w8 q# O9 Y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.# p, t! L( u0 |: c

    7 h9 {& Q/ e9 S2 c    Recursive Case:8 r: o- _9 @, ?! K9 d5 [
    * I0 B, k: X3 f7 H: B) y4 [) r
            This is where the function calls itself with a smaller or simpler version of the problem.
    : e2 L# z, J" j& L, L6 L- ~& d5 U+ B/ A% C
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    * N5 a7 R/ Y  X# Y# e% F& [% m+ u5 M9 [
    Example: Factorial Calculation
    9 b1 Q# R8 y! o3 ?: L+ _+ Y0 ?
    " z6 S8 u# B0 Q! w& }# Z# ^0 VThe 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:3 b5 u7 O; U3 l# T
    ' y- {  d* ~% o" L6 O  ?
        Base case: 0! = 10 i& G% w% ~* ]3 X( `

    7 D& r' r" d! H5 X2 ]2 ^" e' {    Recursive case: n! = n * (n-1)!9 T% C1 O6 y" _& ?. X2 [9 m

    + r/ ?, i9 B1 R& rHere’s how it looks in code (Python):
    5 o0 \( L% E. B1 Vpython
    & {% b" E" O7 b% i! b
    1 G/ C# v4 p/ w
    3 \' n0 n2 n; P! {4 Bdef factorial(n):
    3 a" K' Y$ I! `' s! i1 N# q    # Base case
    / f6 |! m# n# Y* t8 ~8 k8 u# U( U    if n == 0:
    & \! D8 J+ ?5 I# c  J        return 10 y2 e( [5 G9 H
        # Recursive case
    9 ^1 J( ~  L4 y5 l    else:
    4 L8 Y) Q+ a1 p6 ]* T        return n * factorial(n - 1)
    2 U, ?1 n4 O: U* w/ o) Q6 q  d" \! c$ [/ W& P& C- y) v% Z1 I+ \4 V
    # Example usage
    9 b5 T3 {  C, E- ^5 s; Lprint(factorial(5))  # Output: 120( t4 Q" f/ C) w* j6 n1 b
    4 _, G2 }6 T9 a, f) d! x+ s
    How Recursion Works* Q! i0 ~' q. K8 N
    5 H1 R1 s5 V2 u. R- _3 `
        The function keeps calling itself with smaller inputs until it reaches the base case.
    . Q7 ]7 x1 R, t. \. ]" _% V( P9 w; T# r8 N/ l+ q
        Once the base case is reached, the function starts returning values back up the call stack.% P$ e" I  q2 K6 ~8 \1 t! q

    6 t* l+ {  m0 Y    These returned values are combined to produce the final result.
    * Z6 J* k' i) \" \+ [9 P# M, I( B* j6 w, K9 u6 ^* s6 C
    For factorial(5):
    3 G) L4 h0 @, @: d( w; T1 W& k6 v+ a' E# b
    & ]" O! T0 [# J! _
    factorial(5) = 5 * factorial(4)
    9 p! T$ n' B' Q4 p6 Q( E% Qfactorial(4) = 4 * factorial(3)2 m- J3 O$ T  Y: U
    factorial(3) = 3 * factorial(2)
    4 T1 J4 \) ]: B! T1 bfactorial(2) = 2 * factorial(1)
    % M% }$ y7 P) C& Y) lfactorial(1) = 1 * factorial(0)' z  Z' l! S7 W; `* {" _3 ~
    factorial(0) = 1  # Base case
    4 A& F! I- J, j; o- g  q) v% ?" b4 j. {- ~/ e( s
    Then, the results are combined:
    0 y$ C; c# Y/ }, y7 [* ~/ i% v2 m3 o( L4 I8 I' q! U
    ' s9 v- i2 _  J* Z" B% v, A
    factorial(1) = 1 * 1 = 1! L% }) a: W$ \- L; `8 `
    factorial(2) = 2 * 1 = 2
    8 I$ E5 A7 ^" ufactorial(3) = 3 * 2 = 64 C4 F, N8 t* F7 [, V( {
    factorial(4) = 4 * 6 = 24
      H$ S4 b% @0 ?# t9 h' g7 @factorial(5) = 5 * 24 = 120
    , c9 P5 }- x# h* L( `" @
    & J) [* O4 p# |% `% u0 [Advantages of Recursion
    ' o! w1 V& [3 u/ u$ Z& T% k+ r0 I2 C, [" z' l/ `$ R9 M) h% O
        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)./ W9 Y6 n) w; q+ e! e5 [

    - _. u  {: Z! H1 F" t1 C: M" B7 R    Readability: Recursive code can be more readable and concise compared to iterative solutions.% L8 F4 Q( a/ C7 P1 S' T, o) z: X
    2 b! c  r+ p+ \9 w5 \1 n6 G7 j& S
    Disadvantages of Recursion
    3 j; G# [6 W) @+ K* H) j1 L8 Z( c+ ]3 G& D1 w8 d( q1 F  Z: p! R1 W
        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.+ y; D$ ?4 I3 V2 N8 _) f

    3 Y& M- i. O  ?7 c3 f+ {    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    " J' g& H4 l3 y- s0 K/ s
    3 x& S  S# W( g  w1 ?% a/ uWhen to Use Recursion
    + ~' ^# a5 V# b# Y# Y4 P+ l) X( h) B4 P& I+ D/ y% X3 G
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).+ i" \1 {) Y7 o- U5 W) C

    5 I1 b2 I6 \& [9 O+ k    Problems with a clear base case and recursive case.4 x5 O7 Q# X8 Q+ f- J( M
    $ g& w4 X. \( ]: r! W8 d
    Example: Fibonacci Sequence) y( Z$ i6 l7 n; p% q8 z

    7 z  c; E! M2 e  eThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    + ^7 W5 ?" t: Y! V- D) b, u; w( F. {3 }. K5 \( R9 E
        Base case: fib(0) = 0, fib(1) = 1
    4 j1 p5 Q' A1 k$ L' f6 V* H* s6 K" {  D1 B3 I/ T, x5 v
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    0 ?, ~& l2 Z; y% v; j; z& A- Z9 ?) g5 F" [4 {) t
    python
    % k* j& r! l8 ]& l7 J) J6 b" h5 S
    ' U/ ?" ]# [/ L6 b4 C7 F8 K# Z. I% {. P  n
    def fibonacci(n):5 }' d5 \  g4 h3 ?
        # Base cases1 Y5 j/ h# k* ]
        if n == 0:% l3 C) i4 R+ f0 V6 [  m% f
            return 0
    5 p0 T/ b, ^; J+ R0 `- i: m    elif n == 1:# J9 k7 @7 N/ o( y1 u" x  B
            return 1
    ( n) S5 ?* I9 P. p    # Recursive case2 d; z; ^) B# m6 P
        else:0 a! N8 J6 S  l' S
            return fibonacci(n - 1) + fibonacci(n - 2)
    ) G& w: p$ Y7 T8 y
    7 L. N6 ?3 B( m3 y* j2 X! l# Example usage
    ; ~, @8 j# J$ h9 \: Iprint(fibonacci(6))  # Output: 8( a. d# b- k' f( s* Z7 V
    ! \" I2 }5 A! ^- h" c2 _  U
    Tail Recursion, K7 e0 Z1 \4 E# [% t
    2 @' T2 [( e0 [% u. j8 L1 o* Q
    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).
    / I" G* n3 I. e1 g2 O+ g% E8 O& w' u: q" a: B3 t1 r
    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, 2026-1-29 16:24 , Processed in 0.069931 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