设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1425|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 2 {6 E- ?; c- f' D: y! v/ u
    - s+ W) F3 z& c5 i
    解释的不错! b7 t$ ~2 m4 J# `

    & M7 E% y& z1 {9 k; e1 |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。* r' G9 ~9 m, F
    . \2 e. O8 O! f% S, Q
    关键要素
    5 I) A; s* L+ Y, I# t, _1. **基线条件(Base Case)**
    - Q* X+ ?" A. h9 b. O) ]   - 递归终止的条件,防止无限循环* F+ c5 p4 ]# d! I( T2 P
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    2 k, X8 k- }0 J, o9 Z% f) ?% t  a- L* B" a+ D) x
    2. **递归条件(Recursive Case)**
    2 V/ a: g" r9 }: C9 s   - 将原问题分解为更小的子问题4 _; B/ z" K. J) q7 J5 b. K
       - 例如:n! = n × (n-1)!. s9 Z5 T2 Q! h* h

    6 R- ?4 }( ?: ^7 b 经典示例:计算阶乘
    ! B% n4 V2 T9 Z1 J* f# Z" |python
    4 @4 o& d6 E% y% c8 qdef factorial(n):9 D- [5 b1 j4 M! n4 g9 W, M* s& O' h
        if n == 0:        # 基线条件- |; p9 B- r0 S$ m2 M; n
            return 15 `0 ]  Y$ G/ W
        else:             # 递归条件' f5 c! y# [4 i# d
            return n * factorial(n-1)) v5 |) M3 b! q" X7 \0 H' `
    执行过程(以计算 3! 为例):
    : _/ d9 {7 g2 Q8 z  y" rfactorial(3); |% s, T7 K! E) @
    3 * factorial(2)
    - N. r9 f/ j$ h- v4 a  e3 * (2 * factorial(1)), ?0 l, i/ `' w8 f
    3 * (2 * (1 * factorial(0)))7 R) V8 `0 B) z( \, E
    3 * (2 * (1 * 1)) = 6
    2 r4 }( @$ {7 H" p; H
    / S% _9 K5 X' X8 B. t0 m 递归思维要点/ ?9 J) Y7 J5 f
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑+ o1 f, Z8 T1 w8 N( ]1 B! f
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    / ]4 j- x5 U. G( P2 N3. **递推过程**:不断向下分解问题(递)
    ( H: x2 }' {5 F5 r0 c4 l$ X7 T4 a4. **回溯过程**:组合子问题结果返回(归)( T0 ~9 n/ c/ |5 F3 B

    ; ~: B: X8 a* _4 ~ 注意事项+ m/ Y. K* ^9 q$ x5 c, e
    必须要有终止条件
    6 x' T2 n0 H( b* e1 y1 M$ D递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    1 o; K) W) e, \( {某些问题用递归更直观(如树遍历),但效率可能不如迭代8 [' p% E( Y1 {6 U8 E" `
    尾递归优化可以提升效率(但Python不支持)
    ) U. V0 A7 W! d7 F$ q1 N
    ! g3 P2 p8 t6 x 递归 vs 迭代2 D' ^" r" k( q+ {# e0 S  M2 M. {
    |          | 递归                          | 迭代               |
    + e# U7 S, O' F/ G0 {|----------|-----------------------------|------------------|% L" \( f$ ^1 a
    | 实现方式    | 函数自调用                        | 循环结构            |" T& @  m% x/ D' ]# p4 I6 _, O/ R
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    8 A; [, K% p, p! y1 r+ x! c5 z8 [, J| 代码可读性  | 对符合递归思维的问题更直观                | 线性流程更直接         |
    6 q+ F+ U1 ?6 @: U/ m. \9 ^| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    : ^( A9 A; s8 ]7 R; C4 p* N5 E  s% t( P& P0 o4 {& F1 c
    经典递归应用场景1 l7 k4 I! O) h9 a5 t/ b$ ]$ f
    1. 文件系统遍历(目录树结构)0 v& @4 t4 f5 @
    2. 快速排序/归并排序算法
    9 a- u9 t' A2 Z3. 汉诺塔问题9 Z: h# `/ B2 y3 D# f
    4. 二叉树遍历(前序/中序/后序)8 q+ l' j% |' |9 D* y/ \) h& B
    5. 生成所有可能的组合(回溯算法)
      G/ `. ]2 x0 g( N: T/ ?+ I( ^/ Q) \5 x6 Y
    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

    5 小时前
  • 签到天数: 3113 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,
    5 N1 T; u( K. V1 u我推理机的核心算法应该是二叉树遍历的变种。& a1 M1 \- E6 z$ |+ Q7 j
    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:
    / ~! S& l) v- U; I( ?" wKey Idea of Recursion( z  p5 \! \( Z3 {# H1 I- m$ j# o4 p
    ' W: M+ A" Q. J/ d0 m
    A recursive function solves a problem by:( `" x7 \2 z- I" I3 I7 Q

    ) l8 [) Y1 U# I2 P" r    Breaking the problem into smaller instances of the same problem.( o* d8 r6 w9 p6 m  m6 s

    + h3 h. H% P5 O7 c: v    Solving the smallest instance directly (base case).
    : {- E$ g, f$ l! D' {4 I6 B. D
    0 q+ c( {2 O7 Q8 N' T) U5 n$ j    Combining the results of smaller instances to solve the larger problem.
    ( S  n- X' w7 k* w: `: C
    + S( ?/ b" B" n, f& ^Components of a Recursive Function# b$ h0 {+ Z5 @# @5 E
    3 L( H9 V( F4 F- o/ s
        Base Case:0 }9 i7 [+ j: [; }
    2 s  b  [0 M/ _' Z: O, g
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; f& J# x6 ]* C0 Y& Y2 x. `1 z/ M9 W4 j3 @% Y: a
            It acts as the stopping condition to prevent infinite recursion.# B" l3 R& G2 X7 J/ C* s

    ! S# ]6 q% M) b+ o& c5 J8 r5 ^        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.1 `7 v5 n  Y% S1 W

    ) K, ^, W6 t; y( H    Recursive Case:
    " C/ H3 g6 k7 z; ~1 J" A  ?$ G! Y% Q2 b" y
            This is where the function calls itself with a smaller or simpler version of the problem.
    , J4 G8 N( s* s! P" J7 s. p6 \: Q" E  a7 D
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    6 ]7 y$ [, z: C3 D; E3 e0 r; s9 \$ M" F
    Example: Factorial Calculation0 e% J2 x6 O5 ^2 U2 Z9 w0 J
    # V' P9 {) G2 l$ B" `
    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:
    ( }/ ^! f6 ?2 s3 [
    5 `" m4 V: m# K! Y8 K) u    Base case: 0! = 1+ ]% V. m( C7 y% `" d1 @% M( k
    6 J6 A4 _& g$ U4 m, B0 A. R8 b
        Recursive case: n! = n * (n-1)!5 |; Q- H0 Y; Q5 m% c0 R* u3 i
    * |- ^. i3 p+ c6 S* y8 R0 K/ i# J
    Here’s how it looks in code (Python):
    3 k+ k" {' {( L0 V5 g) }5 ^) mpython8 a/ Z6 r! X! Q- N) g+ y

    ! A9 `3 h/ v  J
    3 v/ V, T! f; r/ [7 l, odef factorial(n):
    * P3 o8 I' \, w    # Base case8 B1 a8 z  ~0 D
        if n == 0:
    $ [  h( D6 p2 Z5 A: r! g( `        return 1
    * H1 Z  O/ C- V2 g    # Recursive case
    1 {7 l2 Z" W+ ?! s+ y5 {    else:
    - [: T& w# N' ~        return n * factorial(n - 1)
    9 \4 k5 P! U4 N2 a8 s/ q- w& Q; V) b
    # Example usage
    + Y1 J  D% w2 j1 e3 Hprint(factorial(5))  # Output: 120
    6 ~- Q, |7 A  h* p0 t+ O8 L  `  i8 v& [' h* x# q& R
    How Recursion Works% `  I, o3 J: i

    7 D) B1 B: e$ Z9 [% \5 \    The function keeps calling itself with smaller inputs until it reaches the base case.
    6 I6 N; [; g) r4 Q1 {6 S1 P" T0 Y- f0 U7 j" S6 O# e
        Once the base case is reached, the function starts returning values back up the call stack.8 Q- Y6 Q  x. e; G
      p5 p+ {+ d9 e% d
        These returned values are combined to produce the final result.8 _8 z, G# `, w$ C7 N9 E
    : G( O" ]% A  K. a- B5 P, N
    For factorial(5):
    ( z8 T3 u% m* v
    + t5 k* N" m5 n
    ! w5 U! k) j4 A/ T6 Q: ~( Ffactorial(5) = 5 * factorial(4)) |0 N2 z' R; K$ e
    factorial(4) = 4 * factorial(3)
    : K" D9 r* r. M7 q; b) c$ efactorial(3) = 3 * factorial(2); l9 {+ A' l, C* Z4 k  d2 |
    factorial(2) = 2 * factorial(1)7 p& W# H$ d$ Y3 Z# C
    factorial(1) = 1 * factorial(0). Q% o: \0 r. B0 r
    factorial(0) = 1  # Base case
    - H( |8 X& j; ^" v$ u
    + W) Y+ W$ q1 {' N0 \- D+ oThen, the results are combined:
    1 r0 k& C8 w- s! ^% G
    - [8 p3 T5 \# q2 j! Y9 t2 T+ Z: `1 s( s/ v% |3 Q
    factorial(1) = 1 * 1 = 1+ _. O7 ^' d$ ^( O8 ^
    factorial(2) = 2 * 1 = 2
      l# n& g4 {& M5 nfactorial(3) = 3 * 2 = 67 \& t  g% O! j: @, y( Y$ k
    factorial(4) = 4 * 6 = 249 U4 g, w% P. H1 V! G4 S
    factorial(5) = 5 * 24 = 120; Q! A, ]/ x& H4 A4 @/ D

    : m4 t' y3 w" j( t3 eAdvantages of Recursion: y- U8 ]9 b; {+ h$ P
    ( z3 C8 f! o8 A: W5 B) J6 N9 _" F
        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).- V+ ]3 R; a0 x- U* f: H

    ( ^/ A) d6 T. }$ ^/ ?    Readability: Recursive code can be more readable and concise compared to iterative solutions.7 |$ u  V. P3 {* m" {! K
    $ h4 r7 c5 C2 H" n3 ^
    Disadvantages of Recursion
    5 n+ y( x$ I# s9 ?  r: G8 Z+ Q7 f% L/ U3 P# J6 Z% q) 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.  ^' s( ]8 {$ J/ f9 t1 r
    , @) G1 Y6 H# h3 z& p# F- P
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)." c  V+ @7 q, c4 T/ C" L/ _

    , c) V# z) F: b; PWhen to Use Recursion! \, ]% [! P4 O6 @' G4 i$ V/ M) ]% d
    $ a) \7 C+ u# Q- m/ C
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 r' r. G6 i3 k/ G7 f/ u$ n
    % [! R  b! ?- y3 E
        Problems with a clear base case and recursive case.8 b9 T4 [  P+ v6 x6 `

    - K/ ]( P% m3 h( v3 [0 NExample: Fibonacci Sequence
    1 v5 ]! C. O) L. R' v$ k0 o2 P/ D1 a2 l. Q
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ) r% q% e( z- w; [1 y
    2 x$ v% c3 \/ C% z8 V, t: s( }    Base case: fib(0) = 0, fib(1) = 1
    / T! O- M  _3 x0 F: F
    0 j2 Y& r2 }9 _" M    Recursive case: fib(n) = fib(n-1) + fib(n-2), f& l1 w- {" Q# T2 x% Q" b

    % y) W3 C$ w2 k( [: Wpython
    2 f' c+ C5 l5 Z$ o  g. R/ c4 u. e, \1 D5 H& b- M0 y+ Y
    - d7 X$ }5 U; y- `; y) Z+ i8 `& h
    def fibonacci(n):
    ; m4 m% z! }* ~6 r$ y9 E1 D# G* H% `    # Base cases9 [$ F9 q, D" J6 r9 e
        if n == 0:
    # h- s( g2 C, K8 h4 I        return 0
    ( p, |; x% X( q8 {    elif n == 1:
    / L5 y2 e: \- X/ r. Z0 T: Q' g' ]$ r        return 1
    " e5 D% s1 a. c* I    # Recursive case
    " k$ O) I: [, o1 [6 X    else:
    - Y, q$ A+ G. |/ n, Z8 V. G$ o+ z        return fibonacci(n - 1) + fibonacci(n - 2)1 u8 c& y" C/ s" U1 M: @

    * `3 J, R4 X5 ^; Y. }# Example usage8 P* A- T2 @' Z, F4 `6 \
    print(fibonacci(6))  # Output: 8; v- P' i. Z" L, k! ~; C! h

    8 g3 e9 c/ F# d5 ]6 W* H$ C" FTail Recursion$ j8 ?6 a" }. s* U! S6 Z
    % a8 L8 U5 `2 C8 i: q; R
    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).
    6 P6 m5 ?. r3 H0 n
    2 ~  T- p" [5 S( ]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, 2025-12-9 13:03 , Processed in 0.039652 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