设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2113|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
    & ~3 T  H3 s* Z# s( R' b
    $ \9 n: v: ^5 J  }9 m解释的不错6 j5 f  f' d: P5 [2 H+ ?; I

    1 {4 R3 D3 Q: ?( f7 Y: l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。
      \" Z# z9 H+ [3 @# \9 e4 T5 _2 P# j7 F" @0 `6 T2 |+ G* G
    关键要素
    ) B# |) m' `$ N: _! H" f1. **基线条件(Base Case)**+ s/ a4 ~4 ^3 L$ \( J+ u& x$ g
       - 递归终止的条件,防止无限循环
    7 C; x" x* O0 X/ g( ]. ]6 ~4 L3 L   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    & Z4 Q8 F: P6 U! c0 r
    " {) V0 h, Q# H2. **递归条件(Recursive Case)**" O. j. i/ x# t: J/ x; Q
       - 将原问题分解为更小的子问题
    ) k8 a! u: P5 w2 \$ M   - 例如:n! = n × (n-1)!- \6 x1 ?2 L$ E( ?

    9 y8 I0 I. G2 \$ I 经典示例:计算阶乘
    ; G9 q# K. K6 m7 S7 gpython
    4 |# a+ I2 p" W& A8 ?def factorial(n):- Q7 H. i! O' M) U# [' c
        if n == 0:        # 基线条件
    + }! n/ J% v  g! y        return 1+ C! u/ H2 c! v9 [
        else:             # 递归条件  i3 |. M5 j, d
            return n * factorial(n-1)- q8 V7 J* T6 s5 p; ]; X
    执行过程(以计算 3! 为例):
    ' {3 ?/ a: f; p0 m0 C5 e& Y1 Y6 Ffactorial(3)
    " r1 b1 p8 h9 |1 A, e% t7 S& n/ ^3 * factorial(2)
    0 i5 T2 ?& v& M4 \" V: S3 * (2 * factorial(1))
    ; ^7 [+ R1 x; }0 {+ v. z& y3 Q+ w% _3 * (2 * (1 * factorial(0)))/ U: e( e, C4 [" R' s
    3 * (2 * (1 * 1)) = 6; c5 G6 I: Y8 D+ N# y3 [

    3 O3 x) r( u% B  h! w: C% \ 递归思维要点
    4 k. }6 W' Q" u# U0 M" R* Y1 t4 F8 {1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    # q7 U3 F8 F* A) k' A+ s2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    * B. }6 ?( d9 _3. **递推过程**:不断向下分解问题(递)
      ?" n& B7 K. [" H4. **回溯过程**:组合子问题结果返回(归)
    7 s1 [* e2 ^! w" j$ B
    , a6 }$ o. q% [# s- L5 J 注意事项# E& k* R' Z6 l; }2 x9 f  @% c( C
    必须要有终止条件
    + `0 A/ {. M: _4 T递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    - U5 x! x. ~7 ^  K某些问题用递归更直观(如树遍历),但效率可能不如迭代( r) R+ f& w5 f
    尾递归优化可以提升效率(但Python不支持)
    ' Z& K( ]7 K/ ^7 A" C/ y' E
    * C7 ~' G& X4 ~9 \ 递归 vs 迭代
    4 t, S6 Y* c7 h: V; _( i|          | 递归                          | 迭代               |
    ; ]1 A' J. Q5 O- R. O. W: f|----------|-----------------------------|------------------|; |; N1 S/ M" m$ m3 [- }
    | 实现方式    | 函数自调用                        | 循环结构            |
    ( P( h8 X% Q$ f| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ) t' }' X7 `- u  g; V5 K3 d' 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         |( g7 b+ L! N: I! N  c- h
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ' [/ F0 N* b# P2 i" j9 N9 i7 ?& |( {
    $ g- L$ m# o3 {  S0 y. R0 x 经典递归应用场景6 b! G5 r, C4 v+ ^$ |
    1. 文件系统遍历(目录树结构)) e& s- h) m" X
    2. 快速排序/归并排序算法* V) a* D4 ]1 |9 `* [
    3. 汉诺塔问题% y/ ?# J2 d3 Y, j1 E
    4. 二叉树遍历(前序/中序/后序)6 D7 j& k! h/ }3 o/ r
    5. 生成所有可能的组合(回溯算法)
    - @4 K" G9 Q. J8 ~  U! a3 r: E8 {4 k- |/ T
    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

    6 小时前
  • 签到天数: 3178 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,
      l8 ^; K4 R8 a- p, M1 r- j% C7 J我推理机的核心算法应该是二叉树遍历的变种。# s) o4 M9 t! M( V, E
    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:+ Z  ~3 \$ }2 D
    Key Idea of Recursion
    ) \, ^/ O0 ]1 l: U0 M2 I# v$ p6 d! c5 d& ?7 A- q
    A recursive function solves a problem by:
    1 M7 ]! g& o4 p8 E
    # }& B: B/ k3 |% N    Breaking the problem into smaller instances of the same problem.9 j/ q  J  H1 [) e
    + `$ |! H: |+ B1 u  |# N
        Solving the smallest instance directly (base case).5 P* Q4 o4 q  n- X% x, m
    ; e3 \6 r. w$ f" z5 g- q- _3 k
        Combining the results of smaller instances to solve the larger problem.& r. h* |7 L2 v3 J' m

      d/ D; _4 }5 j% c# V/ f* R1 M, vComponents of a Recursive Function
    & |4 K) J* U! [! q* p! I  }% y. b
    9 y/ w3 O- o" _; D    Base Case:
    , B( m, ]& w, `$ m! r- J
    & p: A( u; w6 k        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.8 h- p9 o& P3 Q/ G. M7 m# y; Y
    2 [+ m* b/ x' M; ^% {* B
            It acts as the stopping condition to prevent infinite recursion.  l& o9 \# x9 E& b1 g
    & A# N, F; T: @& w
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    5 ]% O8 u4 J. H+ V0 q) c7 j+ a0 W
    , W. F5 G; K% G" P5 c+ E* ]$ S    Recursive Case:
    2 s: @! j5 {* E- k5 b, R
    : F: J) D/ e$ E( ]/ k  Y        This is where the function calls itself with a smaller or simpler version of the problem.5 O! _  e% ^* X7 e, `1 Z, u
    ' j4 L: q* _( g* L
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* E) ^% g% h, d6 J

    7 X, R9 f2 w& a  d5 d# w  qExample: Factorial Calculation
    7 Q3 U) P$ G; L7 A$ t  O. U/ T. i. `* y8 L2 ~4 `
    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 p( M; u8 w3 a  e9 f9 [# l8 `9 s1 _* I4 v- S
        Base case: 0! = 1/ Y& _6 G7 |: `: Z) n
    9 V  B9 Q$ |; i: m2 D* X) M2 n
        Recursive case: n! = n * (n-1)!9 n9 l' p9 ^; Y) u
    3 M7 I* I1 p( f. N
    Here’s how it looks in code (Python):) ]1 B: X3 S4 C
    python
    2 I2 F. O: }" u9 D& H
    * n" M5 Y$ j) Y/ u+ S7 C# k' H: s2 `) d! M6 d) R* F; s- ^& ]
    def factorial(n):
    ; _" j6 f4 e+ [( Y- z3 P. i    # Base case7 T% R7 J0 ~8 b9 d
        if n == 0:$ ^' \$ e- h5 s+ V# E! E
            return 1
    6 h" C( }$ \7 U9 y: l; _  v    # Recursive case
    5 d; j) L) m7 i* y3 A+ I. f    else:
    / O% Y& j7 [- Q! k2 N$ d        return n * factorial(n - 1)
    1 P1 w: U- e8 r) [+ u) t
    % p4 c/ J6 u! e/ o# Example usage
    5 x8 U) m5 c2 E# eprint(factorial(5))  # Output: 120% M0 X5 h, P4 ]& N
    2 A- L0 q# Y% e, X3 D2 z
    How Recursion Works
    ! X7 H6 K$ T& m; D% d' M2 ^
    ; Z# [* W- }6 l+ P6 u    The function keeps calling itself with smaller inputs until it reaches the base case.
    7 a( g8 Y# C1 }6 b+ J+ C+ p. `) f0 X4 v4 Q- l( L& a  v5 t
        Once the base case is reached, the function starts returning values back up the call stack.0 e" I0 x) [& n$ Q6 ^3 Z4 e: u" S
    " a+ Z5 m, @" F% F& L+ c
        These returned values are combined to produce the final result.
    # ?3 p$ O# [1 N3 v8 I8 b3 Z  n  R
    For factorial(5):
    + e4 `! @1 d2 ~* d2 h2 y! \' u+ m0 |! r' L# y# ~* q* i& D5 b" ]/ u# N& P
    + Z% N9 m$ B  Y! n/ g( K
    factorial(5) = 5 * factorial(4)
    , Y$ r2 N' ]$ J, z! L3 Xfactorial(4) = 4 * factorial(3)
    9 }" e! L. {: |- l/ ofactorial(3) = 3 * factorial(2)
    : c! \: \8 o. S$ Dfactorial(2) = 2 * factorial(1)* V. K. d% ~' ]! y0 s
    factorial(1) = 1 * factorial(0)3 o0 I# L) S4 g# b: G
    factorial(0) = 1  # Base case
    / h+ Y5 f. Y+ ]4 u+ L' _0 i8 K4 |5 T# L: t1 ^- l; n
    Then, the results are combined:
    5 T2 c% t9 O2 }4 O6 ]9 K
    7 K8 Y4 Q: w3 Z! J: [. j; ^
    & j1 K9 S" w: T' Dfactorial(1) = 1 * 1 = 1
    + x9 u6 q5 D% |9 L- \factorial(2) = 2 * 1 = 2
    * }5 A5 c6 j9 |factorial(3) = 3 * 2 = 6! ?/ @0 W/ K5 j6 B% \! ^( W
    factorial(4) = 4 * 6 = 24
    # H7 M/ ?* @  {factorial(5) = 5 * 24 = 120+ I+ a8 `+ N+ {* P' f6 O% m% R

    & U7 d, P& j* {$ }& aAdvantages of Recursion3 y$ O2 j- H$ i8 N* f! w; K

    $ t; L# H' X; C4 A: r! j    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).% X1 h+ q9 f) ]7 u
    6 [1 }, E% K  B- j; ^9 a
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    / \* b/ ~7 v6 V/ h0 d7 K' d; ^( z
    % a* z6 @0 u$ f8 U; V$ K+ t& H3 KDisadvantages of Recursion
    7 q* Z; F' C# }' V2 \  \7 ^. `% k8 j1 H/ ^
        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.& y5 {4 ^# S  W# W+ m/ l
    : J0 D. \# R5 n! a
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ; U5 b. ]- u- w" G) b. f3 o& ]6 P2 L& T* K6 u  d
    When to Use Recursion! O8 x7 `1 S+ }/ i7 ?& z

    ) ?" v' ^' G1 [9 L9 S: \/ Y    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 @# M# ^1 c, b9 T% o( R$ E0 H
    ( {9 s" Z( z$ [  u
        Problems with a clear base case and recursive case.7 U2 M# G- a  z* P: G
    : U) _( l9 p: T' u
    Example: Fibonacci Sequence
    ' }+ M/ R5 \" N2 A
    5 R5 ]" Z' h) ~4 E; T, pThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:4 A9 ^, g! ]3 z* O' d8 j7 {: |; u

    7 z6 u  u* g1 a1 q& z    Base case: fib(0) = 0, fib(1) = 1
    * L9 [: M7 H  @( O5 z* D7 k2 S" p
    & H* c! i; C# U. a3 M    Recursive case: fib(n) = fib(n-1) + fib(n-2)+ z. M) C3 d6 c7 F0 a
    4 C/ u  U; y; k; V
    python. b" E* O0 u0 H+ q, O% I. k
    ' z  \& K  l+ W( P! r. x6 z
    3 V1 J& a( x/ z0 ]- \7 M5 F# Y4 W0 y
    def fibonacci(n):: D" X9 N# J, J2 \
        # Base cases) z' ]- h* r- X* p& ^6 ~  o0 Q
        if n == 0:( T5 f0 Y% d( v% v, m( i6 n" k
            return 0
    ' I7 n( O4 Z) N9 Y    elif n == 1:
    - {; O/ v2 h2 l& G; i2 M5 m        return 1
    ' J$ a% m% q3 j5 e& T( d. p8 \    # Recursive case$ P% e6 A. [3 j2 K1 k
        else:5 `3 L) n* x$ I# b
            return fibonacci(n - 1) + fibonacci(n - 2)  L( C6 F5 C% _- m5 [
    ! _7 e: ]# S; V+ ?7 D; q5 N
    # Example usage
    ! t8 Q- `- `8 V% f& {print(fibonacci(6))  # Output: 8$ Z  I& |/ J1 ?3 i6 {

    3 Q7 f" S% ~/ a1 V) }' ~Tail Recursion4 ~1 u2 A0 E! \" s- j  w& T
    0 P& I! n3 z6 _" }# F4 o7 H* b
    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).$ a$ O! T! f2 R

    ! H. X& A3 b0 ~4 [: v9 _  BIn 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, 2026-2-19 13:22 , Processed in 0.060244 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