设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2389|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 . m# T# E' D2 [% [: ]  Z

    5 k; Z  ~* `8 j5 D3 n9 `& Y解释的不错, ?7 Z' ?# L# V; D' w

    / @! k4 D: h+ 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。* k9 g& M1 D* b8 x" W# _! g

    ' ~/ @, Y6 N8 q! s2 p' j" a  V 关键要素! g; Y6 D1 a5 C# ]6 `: A/ L6 V
    1. **基线条件(Base Case)**
    $ G5 `9 e/ W! i0 b8 {   - 递归终止的条件,防止无限循环
    : k1 u! c1 A% t6 u0 {/ _   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ' i. R3 u" G$ \' _2 H
    : \1 \1 E5 c0 z2 K: Y2. **递归条件(Recursive Case)**$ _0 \3 c+ U0 Z- d6 R2 U
       - 将原问题分解为更小的子问题5 x( Z" g, c$ d: J6 S
       - 例如:n! = n × (n-1)!* a" V) v- \7 Q; U- v1 }# H

    & H- p" F; v- A4 h 经典示例:计算阶乘
    2 ^" W& o4 e' Z$ X: S1 ^/ Jpython$ @: x3 H* [% {$ i4 N# W
    def factorial(n):
    1 J7 t" @+ L5 {. W6 Z( {    if n == 0:        # 基线条件# l& l4 \7 C2 o$ `/ u
            return 1! l4 P7 M  X/ {  A! Y6 @* A2 x
        else:             # 递归条件
    : o/ t+ ?) c+ ]$ Y  h% O- C        return n * factorial(n-1)+ ]( p) T9 A0 U' y- V" a, J7 H
    执行过程(以计算 3! 为例):
    9 @$ ~: L% R; Q& w7 ^/ mfactorial(3)
    6 y0 F3 o2 U- N; z0 _/ Y- U; d1 p/ a3 * factorial(2)1 u2 L% z1 m0 }) s2 b
    3 * (2 * factorial(1))
    0 c8 L. |9 L! P7 o8 r. K, R3 * (2 * (1 * factorial(0))), m5 m% i" d" |9 T7 E( b
    3 * (2 * (1 * 1)) = 6
    + o& ~* q' s/ @6 {
    9 Y% b5 }; s. k- @ 递归思维要点$ K! Z2 t0 J7 B; g1 M0 Q' ^
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % B/ e6 B+ P: {2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    1 ?& ?# e2 U3 z1 s* T3. **递推过程**:不断向下分解问题(递)! i$ M( ~* j# p3 Q6 k
    4. **回溯过程**:组合子问题结果返回(归)/ K4 u+ i* Y) s9 G3 c) _& N  n

    9 T! h0 F4 |& i. b  s& N: W 注意事项
    ! i7 w8 A  X+ f6 T, h0 P必须要有终止条件
    2 G3 k8 v! ?3 O6 H$ `0 A递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ! n: {" L8 G  H9 b某些问题用递归更直观(如树遍历),但效率可能不如迭代& U& E; ?; S6 [  e6 D$ c
    尾递归优化可以提升效率(但Python不支持)
    7 H! l, }# A% W0 a" }6 E$ s' j
    ! L# W3 Y/ v3 Z! v- C" \/ d! P 递归 vs 迭代
    ' Y; _* \6 m. E0 @  K7 r1 j|          | 递归                          | 迭代               |
    ' A- d3 h9 Y. W$ t  x4 @% ||----------|-----------------------------|------------------|. A- p& u# |/ f
    | 实现方式    | 函数自调用                        | 循环结构            |# F* q" Y0 _' X4 {+ N& ~
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# K% Z0 Z; R9 q% 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         |
    / {. H, F* ~9 c- X| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    % U  @4 d4 |: j7 a5 r( \9 O+ n" W
    7 R- P' V5 J& s 经典递归应用场景
    . b  T) Y! C. b4 E& c1. 文件系统遍历(目录树结构)0 h1 }; E  Z+ l( b
    2. 快速排序/归并排序算法# H2 f9 I9 O* \/ v+ |
    3. 汉诺塔问题* E& {3 u9 w  C
    4. 二叉树遍历(前序/中序/后序)
    ( T/ P5 @& t7 p+ P5. 生成所有可能的组合(回溯算法)& J1 A6 a- p2 @/ s  \0 q

    4 }! L  ~. a+ 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
    开心
    昨天 06:54
  • 签到天数: 3210 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,
    1 m& L: W9 k6 d% X( X我推理机的核心算法应该是二叉树遍历的变种。
    4 s) N. T) j2 O' |0 C* ]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:
    9 g- a/ k# c9 r( ~! B* y& JKey Idea of Recursion
    & P" _* f. G) {/ V- L* o: X4 r
    * P  J. w' u& R/ ^3 E! j. xA recursive function solves a problem by:+ ?+ O% P  [( o# U" b; e
    9 k$ v9 g6 H4 T6 U! v! a! h! M% _
        Breaking the problem into smaller instances of the same problem.
    1 c& i/ l5 p0 @6 ?* z& V7 h8 b* A' b, {% t, H; c: \2 _) C+ Z
        Solving the smallest instance directly (base case).
    ! O9 C8 y  ~: g+ B9 L/ q. r3 S( I+ d8 F. M* V: l' x" `
        Combining the results of smaller instances to solve the larger problem.6 g4 j0 w5 |9 l* Q0 r

    2 [9 m; q5 U( XComponents of a Recursive Function
    3 x+ f3 l; e4 M+ S. V8 Z' n
    # a5 p: S- }1 x  g, g' Z    Base Case:
    / r6 ^9 i, d& m% a: R5 o9 y
    ( ]* q# k3 o# ]' C5 F7 P! k( U* p        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.2 G! m; \: ^" {6 ]" v
    9 X4 W& y$ T/ |% Y5 I, ?: a
            It acts as the stopping condition to prevent infinite recursion.3 \& |# }* j+ j* T5 Z
    7 S! T" h1 I8 S1 b( }) N
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ! b6 p- K" d1 x5 y; [7 M/ R* D2 \2 j
    8 O: ^! _0 d8 {1 N- i    Recursive Case:
    3 Y- i5 u1 P/ m. }1 T
    2 i% U9 R, ]8 x$ I        This is where the function calls itself with a smaller or simpler version of the problem.& a% r4 T$ z. @4 {6 r* h8 k$ P
    , P6 l' \6 N# t* T; w7 x
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ( Y; ~& [7 [) \2 |5 j1 d9 y7 n8 x
    2 L& ?+ E# k5 n# UExample: Factorial Calculation
    ! y& t; p( ]$ Y
    0 ]: p5 l; N9 k2 ~$ Q9 a1 lThe 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:
    7 c% E$ b3 \( L  u
    * d: D' A) h1 U7 ~9 U    Base case: 0! = 12 e$ C4 j; o/ [/ s  l! t' U

    6 D# `) Z5 l3 Y, r    Recursive case: n! = n * (n-1)!
    ! }. |/ F- [/ C0 ?+ @; p+ R& J% m( @9 R
    Here’s how it looks in code (Python):
      f( [3 M* b5 B) ^. }, E) p# ypython) _: F/ x# O' X  N2 ^" b* Y6 n+ B
    0 u* O& G  f9 X' E

    / q* X5 P% p5 Q# s* wdef factorial(n):- ]4 g9 A, C. C: O8 O' Q) w! G# m
        # Base case$ i& x' |& w) }& S7 U
        if n == 0:
    , f, N& H, Y) t2 }' [; \        return 1+ W0 h) v" \' X+ C; K2 z
        # Recursive case
    $ j. A! ]- u1 c    else:- }; _! m4 T7 K0 B4 D
            return n * factorial(n - 1), C7 l1 n" b/ E* V! U# P
    , ]1 ~: G0 @  v  c5 N8 l" ]  w
    # Example usage
    3 v2 A7 g4 G4 }/ F; b) f& Mprint(factorial(5))  # Output: 120
    + X& P* M, I2 m# y' v1 y9 ^; ?2 f. ^; `( m8 a) @. [0 B
    How Recursion Works$ P% d  N4 d, [! e. N

    4 e. _, e; M: r1 V$ N    The function keeps calling itself with smaller inputs until it reaches the base case.
    2 [% g$ q0 q8 I
    & g* p0 |' b+ ?8 m# a! U    Once the base case is reached, the function starts returning values back up the call stack." s$ y. J5 Y! a6 X5 i; s2 u* k6 k

    ) r& M: `5 K* O9 ~: k6 l    These returned values are combined to produce the final result.
    6 I8 s" G5 V% X2 |2 r) n% D' G. _/ T$ v: s" r/ V
    For factorial(5):: p. k# K3 H& y* I2 m" ~: ?

    1 O# u6 ^! b) s! l  ]1 a
    : a6 m% W$ d) x6 V$ c) i5 o2 m6 Mfactorial(5) = 5 * factorial(4)
    8 b; w% i8 e6 p( R) |* j; Sfactorial(4) = 4 * factorial(3)0 U3 `$ w+ s) J
    factorial(3) = 3 * factorial(2)
    ; D  q# f0 X  E0 B" s, ^factorial(2) = 2 * factorial(1)
    $ S& \, l% [- m# Zfactorial(1) = 1 * factorial(0)% X7 h! q. B& }% b& u1 ]
    factorial(0) = 1  # Base case4 y+ x) l9 Q; S/ V
    4 i/ d# e3 N+ x; w& l' g: A. Y: O
    Then, the results are combined:
    ' d% _3 |3 m  J: _/ A% P
    5 f: v7 B, V! \2 H
    3 r% g7 w' b( l7 [2 A4 kfactorial(1) = 1 * 1 = 1
    6 L* q7 s9 i  s% n& \. ufactorial(2) = 2 * 1 = 2
    ) D5 S" f6 N' f* c" J, Nfactorial(3) = 3 * 2 = 6
    , Y+ g2 r8 m" e8 {+ q7 xfactorial(4) = 4 * 6 = 24( R* V1 ~. @2 F  P5 M' ^0 I8 }, B
    factorial(5) = 5 * 24 = 120
    0 x3 q+ @# w  v; A
    & `' k" O# C/ ~$ w6 BAdvantages of Recursion
    / J- G  H& l6 l3 m& O0 x5 c5 u9 e% L
    " c, O8 w* V5 {* i; U) ~    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).$ _2 p$ t3 w; [: p
    6 ]7 Q6 h" k1 F
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    + E2 g% `9 ^8 G6 R" t% l; r8 s% U% W  b
    Disadvantages of Recursion
    0 X; d4 ?$ [* N0 g8 m% [! v8 S- C& M* K& P
        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.3 X7 r5 g- J. O( k4 G

    & ?  ~2 G& w/ U    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    / C; L% {  _& Z! V. e; \- m8 |! R/ l+ g
    When to Use Recursion
    ( L# {) H3 H; U& G  k4 [6 h. F% J" J8 W/ }$ Y7 Z9 O) H* y# \6 N
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)./ d" F6 |+ g/ l% n8 w3 W6 m7 q$ K

    ( {& U* P2 U) p, f& I( {1 J    Problems with a clear base case and recursive case.# h( l4 J* P6 Q' I3 m
    & m' D% `- [1 I2 N. a7 I9 n
    Example: Fibonacci Sequence' k0 S0 X/ k4 T% n0 w

    3 U3 v) F) O/ I+ s- fThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    : v5 d  P9 ^6 X; P- z+ t
    , c/ O! e5 {" L    Base case: fib(0) = 0, fib(1) = 1) B0 I6 u; a/ c" M+ v

    , |( o; T* B2 b, ]    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    8 P! V1 p+ q$ j2 w/ c* }, l- x' {1 J0 k9 J1 g1 i7 Y1 V: N
    python
    - h  b, J, A- G
    ! W  S4 L. p8 e) u# I0 K5 r# y
    : ~, T. {/ E' ]& C5 K/ \def fibonacci(n):/ ]0 S1 M' i3 |% M
        # Base cases. u4 v) F# G3 V5 j
        if n == 0:
    " \# K: a% h+ W! ~- N        return 0
    + X  W8 e- z+ G8 Q    elif n == 1:
    : ^7 C8 D$ r+ H3 i$ A1 g& n- ]        return 1/ n& Q; D! |4 ~5 T" @
        # Recursive case
    3 U+ q2 |- C% b    else:
    1 m" G$ b3 G0 C        return fibonacci(n - 1) + fibonacci(n - 2); Q( L- U7 \8 F' X4 }; O% V5 c
    ) D- e) H) ^2 `# z! W! V
    # Example usage: L: y( l$ ^: _) I$ k
    print(fibonacci(6))  # Output: 8! x$ c) X: b+ w% `
    $ q0 i7 [7 x2 @5 t. P) u4 j% W
    Tail Recursion
    . s5 V# s1 ^+ g* a* q0 j3 T! H1 [6 P: V7 |4 L! a
    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).) a9 d0 @2 N  J  J  [- T1 r
    # M3 F- a1 A3 d  B: I( |/ k
    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, 2026-4-1 04:31 , Processed in 0.055383 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