设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1517|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 % b1 b3 k0 r2 u1 B$ a
    5 z$ A# E' b, X0 V9 X% E
    解释的不错: b" f& _3 }( z% U+ y

    $ V! x; z1 `& ?! z9 b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。
    1 m' i# I0 h4 F9 f. L! b/ R! L) Q5 \- k, ?  M
    关键要素
    0 i; z* r, Y) U1. **基线条件(Base Case)**
    " l$ ~( z+ ]! c: c+ c   - 递归终止的条件,防止无限循环! y+ S1 H! f$ w! R, T4 k
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    , s$ d3 u% e3 y* k* U: b$ E9 I$ O/ x# Y- c1 b
    2. **递归条件(Recursive Case)**2 ^& U. C0 @& @2 f! o9 {
       - 将原问题分解为更小的子问题
    / S- ]; s; h6 w& p) J9 l$ K% t   - 例如:n! = n × (n-1)!
    * p7 f8 Y7 ~0 `& I. n% @& t6 B' N: p. |! g1 V7 S6 F! U, `) P% B1 p
    经典示例:计算阶乘
    # Q; G7 [9 i+ B% U9 T8 V- ~python( S! F8 |7 ^4 j/ V- o6 m& n( P$ S" M
    def factorial(n):% f$ d! G; o* H  h
        if n == 0:        # 基线条件
    " G& H: K" ]4 s" C& s4 k        return 1
      m/ ^+ c! c. G, k# p  p    else:             # 递归条件
    8 e/ t* r; w3 h4 N        return n * factorial(n-1)( w, o7 [7 L+ E9 k
    执行过程(以计算 3! 为例):+ f* p! E1 j# d$ }, o' a+ T) ^
    factorial(3)
    / b& T. K4 {; S4 x/ I  T7 s1 F9 ?3 * factorial(2)
    0 ?  s" v8 e3 Y* i; w3 * (2 * factorial(1)); d* m4 c4 T5 g9 X( v! E+ |1 q9 C# h
    3 * (2 * (1 * factorial(0)))3 ~! L5 {  B+ Z) f- _: z/ Y
    3 * (2 * (1 * 1)) = 6* w: n% H4 P" y* n  r$ N* I

    ; P/ K5 S! H, c9 f 递归思维要点
    ; S- }* P. r4 F: U  T1. **信任递归**:假设子问题已经解决,专注当前层逻辑% ]4 H4 n: T* Z; F7 Q0 A2 m  \4 w
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)" K2 N  E' X' w3 [8 S. d
    3. **递推过程**:不断向下分解问题(递)
    1 I$ w& ], G3 m' d$ t8 Z& E4. **回溯过程**:组合子问题结果返回(归). b. d5 o# a/ n6 C) B  [

    , F( B3 v6 ?; d  m, S) S 注意事项
    " m! x2 R6 Y: x0 _% I; {1 _1 X必须要有终止条件
    0 u/ W) o7 F* ?! s递归深度过大可能导致栈溢出(Python默认递归深度约1000层)) l! D' G) X; c4 M" L
    某些问题用递归更直观(如树遍历),但效率可能不如迭代# U* \  t& x, l# A
    尾递归优化可以提升效率(但Python不支持)4 {; ]" o' ]2 B, j0 V: j1 T
    / P' F- N2 a+ ~" U$ X- E: o
    递归 vs 迭代
    + U* _- B- b, T1 U* z( G) ||          | 递归                          | 迭代               |
    , J8 `* Z9 ?* z|----------|-----------------------------|------------------|# b7 L- ^) e9 d; p* H
    | 实现方式    | 函数自调用                        | 循环结构            |, i7 l: B; u0 b( C* k1 |( Z
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    7 S* r/ i: ?5 ^. r2 G| 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         |. [) _  h5 ]6 Q5 m% y! I! k
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ) \. X* X* j; F) L8 L  }
    ( s2 i. Y7 U7 h' X# m 经典递归应用场景
    3 B: v; N; S* [& e7 R7 U+ C1. 文件系统遍历(目录树结构)
      o+ p+ c7 N8 e5 f% W5 R2. 快速排序/归并排序算法$ ~# I* S4 K" ?% Z9 I
    3. 汉诺塔问题
    " `- v! Q. Z" A( T  m% N2 g4. 二叉树遍历(前序/中序/后序)
    ' |  U4 ?0 O6 c; g5 G  N5. 生成所有可能的组合(回溯算法)
    + L. A" R. K  r
    7 w! ~! S: E5 Y! z6 |$ h' z/ 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
    无聊
    昨天 07:32
  • 签到天数: 3117 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,2 [3 q) h3 k% M: g) {4 ]
    我推理机的核心算法应该是二叉树遍历的变种。1 }/ g% a3 j, B! t* H
    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:# R# @6 ]: Q! n9 m% M6 l
    Key Idea of Recursion0 k/ g4 z/ n: Q  K, P) `! w2 H

    4 P1 P/ W0 {: m. PA recursive function solves a problem by:
      G$ [' F4 Q! a1 H, S3 M" H4 w5 |$ |+ m8 i
        Breaking the problem into smaller instances of the same problem.+ Q3 ^* k2 m& m- i8 S

    0 z+ f3 Z& g6 }4 w& y    Solving the smallest instance directly (base case).
    - B) J* |! r" O& @0 ~" P' _+ H1 U6 H; F8 C% F& V4 r$ I
        Combining the results of smaller instances to solve the larger problem.
    & u: {2 ?( e( k# w3 y4 ]* `$ D$ k
      w1 j! |; g- v  w/ UComponents of a Recursive Function4 b5 t; U' _2 g- u; q
    1 T7 ~0 f# r: p' e! H! F
        Base Case:
    $ s" a0 O* |, l+ e  F0 T) u* A- w! U2 a( i
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. t4 z# u. l. O2 L# c# R
    4 g# |# Z, P/ r4 H$ B& j
            It acts as the stopping condition to prevent infinite recursion.
    - z# W+ s0 C4 r5 W
    ; I" ^6 n. c. z/ G0 {! q        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    * p$ S/ A6 k8 t+ b3 [4 _5 p& C3 n% L& s, u
        Recursive Case:1 }+ W" j. R7 B" S) d

    2 R, Q0 {+ i5 m; c. S5 y        This is where the function calls itself with a smaller or simpler version of the problem.
    " j8 x7 b9 I: V$ o: j" X( [, z& n
    7 ~2 ]: I6 D) p/ k: E8 I        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* [5 |/ R; N( h( d+ E( `: R" T9 l4 x

    5 Q2 K& v  Q! d0 Z2 K0 B* AExample: Factorial Calculation! j$ }7 Z, [& `+ R  V
    , s: x- b! Z  W% L- m5 @4 W! [1 \% q
    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:
    # t9 a1 m% Z. A2 ^- y* n- z
    ) \1 l7 d) G' P. K    Base case: 0! = 1
    ' \- x$ P1 J' M: v9 B2 O! W) o/ f: |* i2 l
        Recursive case: n! = n * (n-1)!! l  r$ t% I9 N9 K5 S
    2 @3 P* Y5 ]9 U
    Here’s how it looks in code (Python):
    3 t: Q$ G7 V/ f! D: S  ^python* ?5 }2 W4 q4 P2 D4 m- B2 e9 o

    / }. J1 F& d: F6 j( I$ D
    1 G1 d# ~, \3 D0 odef factorial(n):
    3 r" m* v: K: ]9 W) N/ K0 j    # Base case
    # o! r# l2 ^4 h    if n == 0:- N# k% Y8 |2 [" g& ]; Z
            return 1
    5 e9 V; s+ E0 ~- Q- X4 B/ O( Y    # Recursive case9 N) Y& J3 i2 W. Z/ K; u5 l3 {
        else:
    6 h4 U) R+ ]8 c- Q. D        return n * factorial(n - 1)3 h+ H' u! H9 i4 ^' w/ a, X

    6 a& F  ?8 S# t: x  I# Example usage
    : g! r# o& n2 p, I0 p+ I' Q" n7 Dprint(factorial(5))  # Output: 120/ n, d# L6 O, B$ j+ r1 h9 u
    5 E# h, u$ Y5 p* f- E$ S! t
    How Recursion Works
    5 p. k0 E: ?  P. ~4 U6 G* w7 {  l- h7 L& g8 a6 P- _2 W
        The function keeps calling itself with smaller inputs until it reaches the base case.& d  P! {* S5 ?6 }' x+ J+ {2 Q
    6 x. k. N% V; ~
        Once the base case is reached, the function starts returning values back up the call stack.  @8 f' |0 W+ j0 A* L, |5 s
    & A$ c7 ?& f+ U5 m, o
        These returned values are combined to produce the final result.
    / p' `- ]; q9 h3 _6 |, Z5 r& I
    7 O4 X! c3 [2 OFor factorial(5):# q9 u) d& I" @  }: w
    5 m- H+ l9 F: t1 r% k. D- I; w3 M
      S' K& N# O) u% F  t5 Z
    factorial(5) = 5 * factorial(4)9 U& m: q, L! d
    factorial(4) = 4 * factorial(3)
    " |  V3 Q/ D7 r" X, zfactorial(3) = 3 * factorial(2)  m: N" v9 L$ J$ x8 ~0 {" K
    factorial(2) = 2 * factorial(1)+ f  i+ _" N' q. i
    factorial(1) = 1 * factorial(0)
    9 T. J5 H4 f* j2 cfactorial(0) = 1  # Base case
    4 |6 n2 B4 Z. ~: u( F: @6 W* X- V; g0 }  r7 E5 }
    Then, the results are combined:. t( Q7 ]' J4 E; J3 t( e( g
    % s/ C" W* ]2 M. E9 d5 q

    4 X( ?5 v# F; Ofactorial(1) = 1 * 1 = 1" A6 i7 h" e- o
    factorial(2) = 2 * 1 = 2
    2 p9 I5 t0 `- f8 Afactorial(3) = 3 * 2 = 6* s9 k/ D1 O$ f, g3 e
    factorial(4) = 4 * 6 = 24
    & f2 c; M0 O2 ~9 w5 t0 P4 `4 |9 Gfactorial(5) = 5 * 24 = 120, `( O2 F" M$ x! e  l9 V- p

    5 r$ R+ U/ a5 R$ T5 x& w- Z, L) X+ ]Advantages of Recursion, I7 E$ I1 T6 A4 s) j
    : F8 i. p# W6 l, H! k0 M; V7 e
        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).
    : b- q. U. k6 ]- u2 S/ [' m9 J* }5 {# O8 i( e4 B7 n; D
        Readability: Recursive code can be more readable and concise compared to iterative solutions." U4 k9 E. W  T8 u+ E. P6 g0 a8 B
    ; q, ?- d: h1 W
    Disadvantages of Recursion- N" n: g$ S6 Y: g7 d; J* `; g. h

    8 G% y" g$ y, a$ E2 g$ Q: }    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.% f$ {4 d+ G5 k3 Q

    ! z3 k/ c! J. M. f    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# m& ~8 s: w3 C! ^

    $ J/ c4 f; W2 |& qWhen to Use Recursion
    8 F" w' L% T  Y8 f( g; T4 t2 M# N$ f; V4 j: l* r
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 j- [; Y8 Y) t' \1 x# I& [: h

    8 S7 J4 T+ A# S    Problems with a clear base case and recursive case.
    8 B( E. X' h8 u# Z: a6 C1 {+ }& Y4 \: g" x
    Example: Fibonacci Sequence$ Y( \8 X! P: j% V# {0 v% N% y) X" H; N

    0 _4 t, K; ~0 u/ }8 oThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    4 J  l* m$ W$ j# N: E
    8 C7 X: x# v3 I+ \& B4 k0 ]$ j7 D    Base case: fib(0) = 0, fib(1) = 1; Y% q) D* ]* s- r, ^1 n# R: U
    " \$ R! O" f6 K
        Recursive case: fib(n) = fib(n-1) + fib(n-2)( [; B6 N  f+ B5 o/ C% V

    / \6 [1 j8 N$ X- W# t5 e8 lpython
    3 B+ z5 I" G: D, Z, }. O. z7 Q- |6 _9 X

    6 N7 ]/ `4 x3 w- j5 j! _def fibonacci(n):
    0 s+ O$ U1 @& D; b! j. J* l5 w    # Base cases
    * i1 ?% Y/ `5 i$ }( K    if n == 0:- X3 R7 k1 v& B5 G+ }6 E+ w) M
            return 07 o0 W6 j- n8 q5 O
        elif n == 1:
      E% a0 t& `! T5 i8 X$ i        return 1; o, a" i- s; E5 b) n0 z4 T4 d
        # Recursive case
    ( b) R: Q" h* d$ h! C/ I    else:
    7 M9 O: `, O6 x7 B' q1 n8 I$ d3 U        return fibonacci(n - 1) + fibonacci(n - 2)
    - q) z& S" h. h' g, H
    # p8 h0 o: \7 O* `2 b# Example usage& @+ A4 ~9 D% w! q8 Z
    print(fibonacci(6))  # Output: 8) s. ]" F. b* D9 P+ s9 v; o
    . ]% n: @4 [0 f
    Tail Recursion
    % z" A3 x5 ^5 C* a
    $ @/ G; ?# T2 P/ x# u( |0 J9 K7 w' U- CTail 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).+ U' v" d6 y  l8 N" u
    ' v! z" O( N( w
    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, 2025-12-15 05:50 , Processed in 0.032199 second(s), 21 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表