设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1837|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   M8 J2 e: G, `8 P. e& f# |, @

    " A/ X+ F$ x, h3 j7 w, o解释的不错& l9 M2 W3 g# A% R
    ' l9 [% S+ |$ I  c" G
    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。
    . V* x8 W2 {  h, }# c
    2 M' W# M( F* | 关键要素
    " [* }0 `8 u' B1. **基线条件(Base Case)**
    / \  [% W2 g3 y4 t   - 递归终止的条件,防止无限循环% N2 c  [1 B5 x, O. i. t; P
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 11 ]& m  n; a8 b% y5 [; F! Z# ?
    8 T" {( c7 i9 d2 m# @- |; p
    2. **递归条件(Recursive Case)**3 A# t+ t0 `' Q* f: v
       - 将原问题分解为更小的子问题
    2 `; E9 h5 Y4 D6 A! R( E   - 例如:n! = n × (n-1)!
    , s9 A7 x" e' S, s8 r( _9 i* K* e) O
    经典示例:计算阶乘
    4 }& G' ~3 J: gpython# v" q* p1 T# e3 Z
    def factorial(n):- M$ n( w) m, v3 U6 D$ v& s: @
        if n == 0:        # 基线条件
    + c2 c% L& M9 Y  W/ ]        return 16 a* a( M; |/ d, F8 x$ I- b- Y# r8 X
        else:             # 递归条件
    , d( i: ~  g3 x6 T; ~        return n * factorial(n-1)
    & p2 P8 d4 _+ J执行过程(以计算 3! 为例):" y% }* h9 {& R5 T/ \5 A
    factorial(3)
    " ]: L) \" }" G1 W4 Z% {3 * factorial(2)
    * r9 ^1 S' a6 F0 O3 m3 * (2 * factorial(1))1 i! b6 a1 N2 L6 |2 v8 A
    3 * (2 * (1 * factorial(0)))
    0 v4 y1 C2 \- k* p4 V3 * (2 * (1 * 1)) = 6
      _$ G$ `6 T( O* ~. z
    & Z% \* M9 o' R! e7 Y) A 递归思维要点# `/ l% u. E6 J. D# [" E
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    5 H9 H& W& `$ D; O. Z5 S2. **栈结构**:每次调用都会创建新的栈帧(内存空间)$ R6 N, R, Q' Q. k2 q
    3. **递推过程**:不断向下分解问题(递)
    + I6 G* x0 W' A. C( B  G: K3 u  ^4. **回溯过程**:组合子问题结果返回(归)
    * X) c; ^$ n+ j5 H! X- P' v$ _' U  ^; f% s9 p4 ?* V. [' L5 T
    注意事项; R3 P. ]- |6 G4 R( j5 [4 s1 l
    必须要有终止条件+ _$ \& p/ p, R+ ~& z1 n
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 ~4 m; q( O' W" m5 Z2 l: W% m
    某些问题用递归更直观(如树遍历),但效率可能不如迭代) }3 b" r. _9 o& Z5 J
    尾递归优化可以提升效率(但Python不支持)
    % |: K  Z3 t- g6 Q- g/ D. u) ]% t/ S" ]2 f( S! Z: _
    递归 vs 迭代
    ( r( z" A0 r% N8 I. c|          | 递归                          | 迭代               |. L/ U( z- R, y+ t3 b
    |----------|-----------------------------|------------------|# w  H/ h; C" ~
    | 实现方式    | 函数自调用                        | 循环结构            |
    ( b2 `3 s$ M: n3 J! i% ~' d; \| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    # z. |3 o5 Y4 Q2 V' B# v| 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         |
      f& f+ a7 H' p, O| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |, E7 E9 g0 q6 S! V

    6 u, }1 k1 H. \: c3 X 经典递归应用场景. q( {, S+ ~3 u7 j- e
    1. 文件系统遍历(目录树结构)
    3 c" H) E: a& x. C2 j+ G! U3 ]3 z2. 快速排序/归并排序算法
    - `( ~* k9 u+ c5 q. o3. 汉诺塔问题3 R/ l/ D/ u" M' C. P
    4. 二叉树遍历(前序/中序/后序)8 v) q$ f8 c4 G- e
    5. 生成所有可能的组合(回溯算法)
    6 d) x+ }3 n9 N+ @' T: R% v7 b
    3 a9 O" B5 v8 A9 D# ~0 e( J4 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
    无聊
    昨天 06:31
  • 签到天数: 3146 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,
    ( G/ l5 k" q# R. r我推理机的核心算法应该是二叉树遍历的变种。
    2 ~! i. a4 A3 N/ G) e  W( K4 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:$ J. l4 ]5 w5 z% W, N
    Key Idea of Recursion0 c$ T# W/ ?" @' V6 G' R, X

    : S  e( m+ o4 V4 v: C2 j7 rA recursive function solves a problem by:
    & b" T$ ]- o# i7 `0 H4 }
    ! m: @/ W- C- ?4 C  M- K' D/ w& o    Breaking the problem into smaller instances of the same problem.3 J/ @8 ~; I$ [$ C2 ^* B& h% @# s

    + }/ C) }# U- N2 g# {' \) }    Solving the smallest instance directly (base case).
    / f" d* N9 `3 V1 J% ]* ?, m7 i
    8 V4 }4 F- w5 O2 @1 H  x    Combining the results of smaller instances to solve the larger problem.1 T0 m2 l; i: {( `5 Q# @+ Q

    8 \" Y! j, \( w  \Components of a Recursive Function
    - b& v+ N7 Q9 J/ j# g* A4 v5 v6 X/ S
        Base Case:0 q# \1 j' e5 i( i' d; Q1 L
    % a+ P" G! Z% k$ K7 s
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    1 l; ~# B3 a. z% e1 b# \0 [% u
    " z& x( @! ~; g. v0 K        It acts as the stopping condition to prevent infinite recursion.$ ~5 n9 L* E: w3 i" Z/ u) L" d( \

    ' |0 r. t- t5 [2 `" Y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( I! n1 T* D: K* e* s1 G2 s6 |8 M4 X
    * D& Q- J& [- y: V* I4 K% l2 f
        Recursive Case:
    % W# N# [- Y; m4 Y* T0 z$ e! E. L8 k" \
            This is where the function calls itself with a smaller or simpler version of the problem./ |( n) L0 p% R/ Z
    - P3 _9 e0 v% T* s! D$ V$ ?
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 }- z. ^2 V7 F/ N; H  }
    ' P+ L  [* K) o' C* n7 V: }. r1 V: h9 V
    Example: Factorial Calculation, k' j  L/ z$ f) g

    $ L8 ~1 |3 u8 N! X) c$ o+ {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:
    ' M% E7 Z. |( A9 [# Q: Q; j5 X- M6 F, J$ Z
        Base case: 0! = 1
      \% U; g- i8 {1 |  [" J+ M/ h; L6 \1 |
        Recursive case: n! = n * (n-1)!
    2 k5 y6 H5 ?" v, @* H9 T
    # e& v1 R1 B9 ]! M  ~3 O- w! L& BHere’s how it looks in code (Python):
    ( S3 {5 t/ g6 o; a& ?# I, ?python& H2 w+ R! Q! `* S
    + j& Z$ N* z/ H: h/ H! D

    " y) T* v) u/ F3 F- x2 ~5 }def factorial(n):
    4 o1 W) W/ F0 c# A- S! n    # Base case
    $ M! ~; G2 U- i; H& J( Z6 S9 ~    if n == 0:
    # h1 ~. `% L9 [        return 1$ Z. X4 Z" K$ S
        # Recursive case
    2 b; r: ~, T! f* e- K- _3 d3 r9 O% A/ Z! e    else:
    & e* ]" S, v8 n( L% X        return n * factorial(n - 1)
    5 G/ q) r1 y4 r& ~7 [; R. q: p* W* j! J  L: k
    # Example usage
      t8 i9 h! b6 c- r! ]7 l9 Wprint(factorial(5))  # Output: 120, w) w4 N7 ~# ?5 {9 U8 A

    0 Z6 c/ G6 L' B0 X4 ?How Recursion Works  K# n* U5 M' p$ I% b) }5 R7 l
    " _& ?6 A  P. M
        The function keeps calling itself with smaller inputs until it reaches the base case.
    : p- Q. s; g- h- r! K# B" M/ i3 l+ \. M' M$ M+ O
        Once the base case is reached, the function starts returning values back up the call stack.3 I! `4 x6 v& q

    1 v) y+ x7 |( ]8 j2 ^2 M    These returned values are combined to produce the final result." h  ?* ?, y* C0 T* J

    * B; Q7 m  w' Y4 y& WFor factorial(5):+ s% i( j  p: E' h/ F
    / B) G' E0 }' l! t

    ! i) H8 ], O/ j/ M6 V' tfactorial(5) = 5 * factorial(4)  o' I$ n7 s) K; R' H  n
    factorial(4) = 4 * factorial(3): X' e3 C3 {# v8 u9 p/ [3 ^
    factorial(3) = 3 * factorial(2)5 |, D" n2 w" l( w, L
    factorial(2) = 2 * factorial(1)* j" G* M0 X9 ?& i2 _- K* X
    factorial(1) = 1 * factorial(0)
    6 ^" Y. ^  d7 \: H  }/ b5 Ofactorial(0) = 1  # Base case: A. u1 p% s8 o- H. f4 {9 s

    ! C# h) R3 l7 W- ^% _Then, the results are combined:
    3 F8 ^4 @, V' D
      f/ ?5 s7 l' N
    3 O2 t. s: A1 Y* m' U. nfactorial(1) = 1 * 1 = 1
    3 O( y9 x$ p( G/ Z% Nfactorial(2) = 2 * 1 = 2
    ! ^' r+ O6 \# Vfactorial(3) = 3 * 2 = 6' d( A* V# R6 z
    factorial(4) = 4 * 6 = 24
    3 ~/ E6 o( H& |3 Y; J: z6 f# nfactorial(5) = 5 * 24 = 120( }/ Y5 M- M1 k

    " Z) A5 S; J% s, b0 {. BAdvantages of Recursion: i" m3 }8 _$ D4 F. p5 M
    4 O1 h9 z& h0 ^
        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).
    * h( O7 R) R9 B0 H$ `) w' L% Y1 U. J: e
        Readability: Recursive code can be more readable and concise compared to iterative solutions.$ C1 y( y- h7 n/ }
    % M* U7 c" k# M5 a
    Disadvantages of Recursion3 k, y  s. I3 _( H2 w" Q# I5 ~
    $ ?( y1 ]5 J9 V. f7 r
        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.! F4 V4 k- V9 n1 ^4 z& h1 p6 h
    5 C5 Y4 ]8 m. H5 U: g0 Z4 ^3 O  \4 n( c
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).' ?9 C1 L/ `# D& ]" E# |# {

    & N# ~7 U& x, q, k+ uWhen to Use Recursion& G1 m9 z& w! w9 h9 }  J7 _

    4 q3 A% O9 J8 [: e( f    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . @" R5 Q/ I" ^1 k* [& M" ]9 L% O
    # v4 K; i( c6 w: l0 _    Problems with a clear base case and recursive case.
    ! X- S5 m5 X2 r: b% F/ n
    4 Z, [, R/ n* e# U4 E" M2 vExample: Fibonacci Sequence$ |9 G6 R6 x& f% R" v) d/ c

    3 B; b0 @" g9 c: R+ b, zThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * b2 q( \- O9 Z7 K1 O+ ]5 @1 o2 [
    ' X' T7 a, F+ e' {) B& P    Base case: fib(0) = 0, fib(1) = 1" Y: E" F% |' Q2 u# ^! s2 m! ^7 H

    : O! y+ R6 g4 a    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    & i# H' R# C/ V; [5 ?7 w) U2 d, w4 P! k1 f. k  F5 U1 c0 G
    python
    4 f0 Y/ _9 i: r5 L7 S% l3 }% q8 E, P
    ; c! A+ n3 U" s4 k$ b
    def fibonacci(n):
    ) S- Z" M% K( p& [6 @    # Base cases8 ]2 Z! R9 K$ a0 J9 e
        if n == 0:
    * N6 @+ H0 {8 R  `        return 0
    0 R+ m0 O4 I/ \! p    elif n == 1:
    * x! c8 S1 F; I5 d! P        return 1' f  m5 B$ j0 v8 K
        # Recursive case4 ~3 j& b3 ?% z8 h
        else:) f' I2 e0 ~0 O- E) @
            return fibonacci(n - 1) + fibonacci(n - 2)4 W- a- z# d* f; N6 r5 V2 ^; L! m

    ' p0 f- D% ]- c) N* g# Example usage& b0 a  J5 s' i- o0 C5 s+ L
    print(fibonacci(6))  # Output: 8$ N- C/ f1 Y& |7 ~- V

    ' d: i$ e1 n7 ]$ H' J. g2 |* GTail Recursion
    ! X  m! S2 j' d( N3 W/ Q7 p7 l( `! Z9 [
    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).
    / {" q3 R  O+ A5 h$ y  w! z7 y: Q1 T. L6 T8 z) Y
    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, 2026-1-15 04:58 , Processed in 0.032218 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