设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2129|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
    % j0 k& ~4 I. g( V
    , f9 y( D# A3 p& n( ^7 K解释的不错
    ! ?2 Q: N( m1 Q1 T5 r( z% g5 A* k, V; r! b! Y+ m7 K
    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。3 k0 }* `5 y8 ]  b

    : R& @- W/ w' {+ |3 n" h 关键要素  g' I; ^0 i" T
    1. **基线条件(Base Case)**$ o7 ~6 {/ K: J; n
       - 递归终止的条件,防止无限循环  `  f$ @6 ~8 i  x: [
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ; i2 A$ F0 e- c2 U, r. c' ~3 n  T  e) [) \5 D/ ]" N
    2. **递归条件(Recursive Case)**/ I- w( ^0 [2 ~' d0 u
       - 将原问题分解为更小的子问题2 Q0 w9 Q4 B. L+ X
       - 例如:n! = n × (n-1)!* C- i* }& j3 ?1 A" o; R4 x& b
    9 S& `- Q6 l& A( b/ G7 H
    经典示例:计算阶乘
    2 w! B' E$ i# t. D6 @python
    8 S; w8 j0 K# t$ }def factorial(n):' M6 |8 [; Q# ~, u* T( W
        if n == 0:        # 基线条件
    ! O3 K3 v% q, }+ R, u$ o3 G9 V        return 1
    4 [" s2 S! q' t% N9 p9 ~& {    else:             # 递归条件% W/ \2 ]3 y; A' y1 H
            return n * factorial(n-1)
    % J- B. p/ q4 i2 d- r1 R执行过程(以计算 3! 为例):" k, O5 G1 ]+ \+ y
    factorial(3)/ l3 p/ w' _3 v0 W& `
    3 * factorial(2)
    0 C& G- G) l  G3 * (2 * factorial(1))7 v# @6 x$ z9 F9 z
    3 * (2 * (1 * factorial(0)))8 [. d# @  s1 E) |0 A
    3 * (2 * (1 * 1)) = 6
    - ?! ^. C3 W; Y5 t7 b
    6 G. j: `/ u1 o% u4 R% n+ N+ Z; C5 d: C 递归思维要点
    * Y" }% a/ F4 I: _7 t4 J1. **信任递归**:假设子问题已经解决,专注当前层逻辑  ~! c! U. P5 D
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)) H5 Z+ b+ p8 \0 M- \& m
    3. **递推过程**:不断向下分解问题(递)
    ' c% B: e: F4 m% z4. **回溯过程**:组合子问题结果返回(归)
    . R$ i5 k2 a1 w& E# B9 ^; t1 M. P8 l/ W
    注意事项
    2 R( D/ o3 ?2 S% I必须要有终止条件: ^1 g! U# g# t) Y1 T! l
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ; x" [- S& x1 K  @3 w某些问题用递归更直观(如树遍历),但效率可能不如迭代' o- y4 l- c9 p8 F4 t
    尾递归优化可以提升效率(但Python不支持)+ P3 k/ }3 z+ i8 a7 o7 f

    , G9 @0 ?+ d# u1 @4 R3 Q 递归 vs 迭代, |( `3 ~% D3 `/ E9 X# \7 w* ^
    |          | 递归                          | 迭代               |/ U: D: y" }/ s
    |----------|-----------------------------|------------------|8 x8 k; e, q3 _/ c& u% {
    | 实现方式    | 函数自调用                        | 循环结构            |, ?% I+ [' Z! c( D7 B
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |: O5 ^+ K1 I$ `
    | 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         |/ v% I1 e1 q! c2 L+ T
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |8 M. T+ f4 J% m+ O  ?- V+ w
    2 W4 s8 D/ ^$ {( M
    经典递归应用场景
    / Q1 ]# `( ^2 q1. 文件系统遍历(目录树结构)& \; Z( J; e7 u& W8 V% ?. I9 J
    2. 快速排序/归并排序算法
    , b0 l3 M+ x, j& A' S7 {( Y, w" S3. 汉诺塔问题
    1 u, r. n8 V4 \3 @- L2 H* m/ F4. 二叉树遍历(前序/中序/后序)" a# U2 w- M2 ^' ]8 \0 `
    5. 生成所有可能的组合(回溯算法)' D+ c6 V/ s* e# ?! T1 b! F
    7 h! L2 w! y9 S' ^) X
    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
    难过
    7 小时前
  • 签到天数: 3181 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,
    ; N1 E  D, u" O2 F我推理机的核心算法应该是二叉树遍历的变种。, c2 ]( Y; R' l
    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:; U; k+ O/ U2 E0 g8 T
    Key Idea of Recursion
    / l$ a5 C+ L6 _. Z/ b1 G1 C6 M6 m4 E$ c/ c; N. \7 h
    A recursive function solves a problem by:8 c& R& @% {4 R, |
    1 h6 a( e  O) i8 c5 o, ~! m5 C
        Breaking the problem into smaller instances of the same problem.
      S3 i' b$ {8 U4 `6 G, v% O3 v, c- B) S; K$ v- X  \
        Solving the smallest instance directly (base case).7 F% H1 ^, O* ^4 O
    & N: f& D6 _) e3 d% k9 o. ]- C# A
        Combining the results of smaller instances to solve the larger problem.  E+ c! \8 t3 c) d9 H9 j( z& P" }2 a
    8 r5 w8 ~3 Y7 M4 I1 H$ m) c$ M" @
    Components of a Recursive Function1 g' w% n6 W1 _" Y9 t

    % Z% L9 n8 y2 J) l) _5 m$ R    Base Case:" x8 p* _  H. p+ v  u

    ; r0 f& q: x" S1 v        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.6 m8 p8 p' ]% k
    3 Z2 G2 z0 j/ s: i+ t7 o
            It acts as the stopping condition to prevent infinite recursion.& k% c' L  s8 q) W$ ^+ j  U4 K+ M
    7 T9 Y4 W4 Q5 t$ y) B5 d4 @, Z$ l
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.7 y( g# ]7 l# a' L
    : |, e/ P) s* e3 `
        Recursive Case:
    9 @$ X+ e" c  n  \6 K$ R5 w' }4 g* n/ g2 A2 O* i' P3 h
            This is where the function calls itself with a smaller or simpler version of the problem.
    5 r5 s: R$ h- k; p3 Z: z+ u
    ' g( \: E5 ^4 r* L        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% @1 ]2 ]  A! `$ ?% o4 \% H

    3 h  i/ L. _. `5 k9 gExample: Factorial Calculation
    4 l7 Z* _2 W/ q+ o" X
    6 G% w- ~% u9 }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:
    ; ~6 P7 j- V  ?
    # x( U1 {8 V  i9 V+ d& {    Base case: 0! = 19 @5 T! O1 [1 m; k+ z8 P: O( u( B
    % A* z6 U$ i/ M, s1 g+ B' @4 N  d
        Recursive case: n! = n * (n-1)!
    $ H! K" `5 w; t; y! }8 f9 J% y1 L% v% C' }! x$ O0 r, R# d
    Here’s how it looks in code (Python):
    ( I6 H% }/ v4 j7 \2 T6 Xpython
    5 Q. v$ g" v% s* ~! a7 c" Y; s/ g! n4 ^: e5 N
    & ]. \8 m$ M3 `
    def factorial(n):/ s6 O6 N$ s7 }: O2 D% G
        # Base case
    ! i) ^: Y$ j& j/ T* f    if n == 0:
    " M5 x6 G. v7 H        return 1" ~7 j3 p! B; A4 C+ C/ l
        # Recursive case
    7 F, _  n$ t6 i' l3 C% Z/ h    else:+ y* |3 |8 Y  q, ~0 M7 O
            return n * factorial(n - 1)
    , Y0 e! V" I9 m9 h# g! V/ ]$ \( \* X3 S1 e5 X
    # Example usage
    + t5 W, v, l, X% U) O9 d) Gprint(factorial(5))  # Output: 1208 ?0 P2 E4 U& z" H2 I7 u

      F+ k; Y; g  q  l! J. F, @How Recursion Works% E5 v4 y: H+ n

    * `+ h: e/ h" Z! b* @* }( L    The function keeps calling itself with smaller inputs until it reaches the base case., n$ t% I3 L+ L1 Y; G: \+ p9 s

    8 H* V" x. h7 ^$ X    Once the base case is reached, the function starts returning values back up the call stack.! U0 e* J6 q# o' x. z- J/ {
    & s" Z+ q  W2 a" g( _
        These returned values are combined to produce the final result.8 |6 r6 `- |! D( S: H
    $ @+ [; x8 U- T4 |1 ?
    For factorial(5):! g2 y% C2 H5 s
    5 _4 C2 f! B: K+ W
    ! W2 C4 i9 Q9 M- @" Y
    factorial(5) = 5 * factorial(4)7 e5 x3 E& _; y$ s3 d% f! L
    factorial(4) = 4 * factorial(3)
    5 N9 C& ]- V/ U& y1 `/ s0 v6 b& u1 Lfactorial(3) = 3 * factorial(2)" |: t3 |# }# d  O" U+ o
    factorial(2) = 2 * factorial(1)) F1 P- y+ e9 Z' J* C! v! U# L
    factorial(1) = 1 * factorial(0)
    3 I$ j) u' b+ x( K, U- `factorial(0) = 1  # Base case
    5 _3 z* e3 `- S# e# u* w( \' M, w  X8 X3 k9 T; O
    Then, the results are combined:/ h+ X7 ^& n- b& X5 N2 P3 d
    & Z5 ^% y4 D  z& Q& C

    # a& Q5 k6 ?+ [% Rfactorial(1) = 1 * 1 = 1
    0 g) ?7 j0 i* L5 _/ j& F' dfactorial(2) = 2 * 1 = 2
    & `; Z- ^( Z  f) cfactorial(3) = 3 * 2 = 6% z+ n% ^6 I; E/ o( H
    factorial(4) = 4 * 6 = 24
    5 t+ |8 _  X% B4 b* Z% ?9 Ufactorial(5) = 5 * 24 = 120
    ; j9 \2 n8 X, {. G% x6 C( ~: I# \- }. H
    Advantages of Recursion
    ; F9 t2 f3 \* P
    4 [% G) Y# J1 [9 s    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).
    , o0 ?4 K2 J# p2 D1 t, i' G7 H  n3 A
        Readability: Recursive code can be more readable and concise compared to iterative solutions.4 o- X1 ~7 Q6 @% G- x$ d

    ' {; j5 k8 X, uDisadvantages of Recursion
    ( r; n' b7 C0 v& N+ _( W& _& I3 Z# P+ N: v
        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.
    - Z3 y* m, B! D8 r7 s- W
    - s0 J% u6 d7 G8 H- R- N7 s    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    8 C% h  f: I# t; z) A" p, h! F- L% ]4 I# X. a
    When to Use Recursion
    3 ]5 T4 A  l% q! N9 V/ _
    0 t: e8 a* B/ l9 w& [# w    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).3 `0 ^  p; O5 h! G" L- H
    % e: n7 S9 \+ @3 ~' }+ t; ~+ R7 d
        Problems with a clear base case and recursive case.  g0 D* m$ c+ V: l! f2 ~$ e
    7 h3 x, W- y+ d3 u+ K
    Example: Fibonacci Sequence
    7 K! P! H) i" f4 o$ H1 F% ?2 g, A+ n( M8 H
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:! O, C) q% [5 B+ m% j# \
    / G2 j( t& H, ~: c; ]5 A* |
        Base case: fib(0) = 0, fib(1) = 1- v1 {/ d" O' j
    : D8 t  n7 C, D0 B: b0 D
        Recursive case: fib(n) = fib(n-1) + fib(n-2)4 J+ ?& D- a# Q8 @* k/ i4 J; l* R

    , a# J! b3 d( z& v$ q  d6 Bpython* X1 }* g) \) R3 s( d8 h

    ! {! c$ r$ R5 x, j7 {; O4 q5 j, }4 L/ G: j
    def fibonacci(n):
    5 I0 Y9 |, l  g- l7 V    # Base cases
    $ p) M- }2 x# U# E7 y% H8 x) m    if n == 0:$ N" e/ Y/ R. e0 N/ e7 T
            return 04 `3 R5 v( a) U$ ]' p% j3 D7 q
        elif n == 1:
    ) I7 ~- ]4 H- G! O/ R        return 1
    * r# N/ [$ i' x4 b& J: I    # Recursive case2 S8 }& Z/ c9 @6 _! q( H* g
        else:  ]  z4 u1 d7 G& F6 t9 b
            return fibonacci(n - 1) + fibonacci(n - 2)# Q1 g3 R, V! C( T9 t  {0 t

    5 b! v" F0 ]8 q8 n# Example usage7 v* s( H( f9 g1 z+ w& M7 D
    print(fibonacci(6))  # Output: 8: z: E) p6 h1 v
    7 q, M: j" U0 B. O8 {8 `4 F" X
    Tail Recursion( D1 m# n- b* f4 u6 Z$ K

    0 D) o# w/ c6 k4 j' Z( i+ Q  T4 DTail 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).
    / Z4 `0 d, S+ \0 s1 I; A1 E, @; e5 \, C  h" ~. [* \  C
    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, 2026-2-22 14:11 , Processed in 0.056984 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