设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1799|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 $ }, x$ }* S* o

    7 t+ p8 p. {0 I$ |2 p& I解释的不错6 S( C+ w9 x$ }+ M/ e' a

    4 S% L) Q) [- W6 U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。
      ]! n* ~4 J1 S) d# X( R, R
    & Q; _4 X1 Z* N 关键要素2 i, Y3 {* d* c+ p4 ]
    1. **基线条件(Base Case)**$ I3 D. L2 w5 \2 c5 F1 R0 ^! R' U
       - 递归终止的条件,防止无限循环: z, r  J, i' p% h% n/ P7 v2 G" V/ J
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 12 ?# k3 C5 Z% p. j; w& T  y

    4 B, w0 \- W0 d0 g, }) X2. **递归条件(Recursive Case)**1 ]0 d$ w) H* n  g' T5 F; l1 z
       - 将原问题分解为更小的子问题. r) m' J. \$ M  v6 ]
       - 例如:n! = n × (n-1)!" T% h% l% S; l2 \% m0 k% \! R

    ; t1 J7 ~# F# K# \# S5 B+ Q 经典示例:计算阶乘
    $ R/ J8 o7 ^  }5 G# x0 o. @1 zpython0 k+ H$ Z) i' _# V- G
    def factorial(n):8 M9 Q: K+ ^9 u& N
        if n == 0:        # 基线条件! }% `5 I7 z3 t# P1 O
            return 1# X- y: t0 e. g/ z/ _! v
        else:             # 递归条件
    + Z. ~+ S% {. Z9 V  I$ x        return n * factorial(n-1). k0 q+ o  O  y; G9 O1 W- k* B
    执行过程(以计算 3! 为例):
    3 b& h* d3 o8 b9 Y7 f0 ^factorial(3)% L$ R0 r* N$ [* Z
    3 * factorial(2)2 a2 P' k/ \! y, U0 U% H" h: t
    3 * (2 * factorial(1))% v7 S2 p" t9 G  D6 b2 W/ t: C
    3 * (2 * (1 * factorial(0)))
    ) ]1 z% \7 M' G1 X9 ~3 * (2 * (1 * 1)) = 6- A. q9 L: g- K3 f

    $ A8 [+ e: f9 f% ?  j' [! o 递归思维要点
    ) Q9 b, j2 d) K' z# r( D1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    - Q4 g3 ~3 x# t/ \  w# k2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    5 Y/ ]  r3 }1 _  O( A9 q6 o$ S3. **递推过程**:不断向下分解问题(递)5 _; i  @4 |5 R; i1 k
    4. **回溯过程**:组合子问题结果返回(归): C( F' M! [0 d$ K+ l4 q
    : d$ F! j7 t: X' k* W
    注意事项
    # I( l& A1 ~8 M% q- t  `必须要有终止条件$ ~! {4 T$ V, |2 |5 Z6 O
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)5 w* x2 C- O: F/ q) L
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ! v; z/ ]* x9 u' v# {7 b尾递归优化可以提升效率(但Python不支持)
    6 e1 n! ~6 ~* C3 h5 u
    " ~) _  r+ d0 D' v! |7 H! C8 V 递归 vs 迭代
    ) J8 U, E, Z7 S; }) d|          | 递归                          | 迭代               |
    + ^" k% p) p9 r# b0 a( b' b( L- l|----------|-----------------------------|------------------|
    # Q3 D2 N, ]$ T| 实现方式    | 函数自调用                        | 循环结构            |% a5 F4 @. A7 n: Q3 l5 @8 M
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |) A6 z1 \% R- l! ~
    | 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         |7 O: W7 ^1 n. |# w) I
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |9 A2 L. M5 I9 M

    6 N) G1 [6 ?9 K7 Z$ V! C2 t/ R 经典递归应用场景
    ) `6 G0 {/ m" J* ~/ X3 R3 m1. 文件系统遍历(目录树结构)
    / u& c6 X+ N( }" y; u: R8 d+ N4 `2. 快速排序/归并排序算法
    , ~  B$ [$ \4 N1 w1 c- A( u9 F3. 汉诺塔问题
    ( M( {8 R# p1 s" m. }/ s; d" `: f4. 二叉树遍历(前序/中序/后序): p( S- `$ f9 R& p- J. E
    5. 生成所有可能的组合(回溯算法)
      ]) ?8 R0 D0 b0 Q, E6 Q. @9 z+ F2 ]. a/ 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
    13 小时前
  • 签到天数: 3144 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,
    3 d$ b! W7 `  h% y, M# @9 b我推理机的核心算法应该是二叉树遍历的变种。# i- H5 n4 A; q1 S
    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:# ]+ m7 L  c# ~: O2 h& v9 [
    Key Idea of Recursion+ y! a1 n) Q5 F2 b) L7 W

    2 i8 v& `) r- V6 h' O1 \A recursive function solves a problem by:
    8 t8 e# s6 P! Q, @% Z# h8 G  S$ n- I! U9 L; z' B4 P' f2 T) v# k+ u
        Breaking the problem into smaller instances of the same problem.
    ' {; A% H' `. ]9 a; l  l
    " |# [( X( W* Y5 A+ a" y    Solving the smallest instance directly (base case).7 I7 E$ i. t2 o4 R/ @( R; f
    % Q# [3 B% m% ^  K9 m3 A* @" f
        Combining the results of smaller instances to solve the larger problem.
    0 j: ?" I1 D5 `0 z
    / e( ?- d$ R' r% i! K$ O. u* CComponents of a Recursive Function# `0 `9 I9 K7 L1 D

    6 q6 R% W1 ]' Y0 I( L' ]* G2 E    Base Case:. l" S( S; |! W( q% E+ o" n

      a* i3 W7 j4 k: \* F* c        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.- H; ?( L, Y7 ^0 T% E- @

    ( v3 h1 W2 a. {$ M- n        It acts as the stopping condition to prevent infinite recursion.8 x( `* S! M, E# b0 b
      [2 c! o) Z& m2 [% _$ A+ ~' q" V( L
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.7 |2 q  F5 i$ i$ T6 t1 B4 v) ^5 ~

    4 n& `2 T3 f2 b* C9 F# v    Recursive Case:4 U5 J# G3 e4 v6 z/ R
    ! p% I3 o$ }2 q: r8 A
            This is where the function calls itself with a smaller or simpler version of the problem.
    2 P2 W8 V8 }! v. b; N: b$ H6 L5 @. W: ?, B8 n7 {' c& K
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." Q# P4 j. b4 {8 Y% H1 T+ K
    0 U* e: E2 a+ N+ e9 j6 L' O
    Example: Factorial Calculation
    " [3 c; V- h3 [! `" ~5 K& }
    9 n. Q9 X$ L( w4 j, `( i' v" MThe 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:
    " r; F) @7 s' Y8 v$ [. @  Q( i% n' j% X1 t/ p
        Base case: 0! = 1% Y/ ^, Y# |3 V% K& k1 H- j  J
    ; V: f1 f) R. X9 R
        Recursive case: n! = n * (n-1)!
    7 V- w4 y+ s' {0 M& t- \# w
    % T, X- }; y5 M: J) UHere’s how it looks in code (Python):! A! t2 i& b/ M- }4 e% j1 W) p
    python
    2 B) K6 i0 p. |3 U2 D* m1 e( A7 Z" X: i  d5 S; T; O

    ( D; }. h/ h0 o, qdef factorial(n):
    ( l' j" Z7 {; L+ e% E' N    # Base case
    6 L8 V! O( c) ^, w4 w$ S    if n == 0:4 t4 n5 R# i+ `% G0 l/ Z
            return 1
    , k* |" L) C* S7 G* z    # Recursive case
    3 L3 U4 e# O6 a0 A* L' i1 j8 g4 L    else:  W8 \0 [1 J0 I
            return n * factorial(n - 1)8 w; V1 Y" I- X% b7 W8 O( e
    8 z) C! |4 h: h8 g0 d7 M
    # Example usage
    9 w8 T; y4 q. u6 e6 m! Oprint(factorial(5))  # Output: 120
    $ Z- p0 Y7 b1 N: Y$ j- M2 a8 l9 Q
    7 y& P" V+ H5 E3 M* OHow Recursion Works7 @3 x0 a' i# z1 m0 |! _
    8 f3 y2 W$ J  Y
        The function keeps calling itself with smaller inputs until it reaches the base case.; I1 v! r% |" d/ H* N

    . {1 T" h0 M, Q. i! A! O    Once the base case is reached, the function starts returning values back up the call stack.
    ) ]9 r7 i8 p8 o  q6 a. u! F' Y" P$ A$ T2 N/ `" F" h
        These returned values are combined to produce the final result.
    : O8 S" ]; n* ^3 R' c& i- o$ p% `% E' }6 d3 `# q4 R/ o: [0 S" r6 u
    For factorial(5):! j6 t1 ?6 X7 N& F( \

    * C( k9 K5 c& h6 u7 U. D1 @  o, S# S6 v! {9 P* P
    factorial(5) = 5 * factorial(4), R/ ^' Y& ?! f; `& k, A
    factorial(4) = 4 * factorial(3)' z' b% }" O6 e; L
    factorial(3) = 3 * factorial(2). y: Z7 ?3 i% Y  q0 v3 _
    factorial(2) = 2 * factorial(1)( g" V8 T) n# T9 F$ B* B
    factorial(1) = 1 * factorial(0)
    4 [7 U: H) u1 m" w7 i! g/ v& bfactorial(0) = 1  # Base case
    4 Y* m! t1 E* R+ }( L1 g$ N0 ?5 ?' Q, f5 a6 K4 b
    Then, the results are combined:
    ) S- N% \$ z9 d6 f
    ( ~* _! m' e0 D5 g
    1 t$ z0 O; u: s2 _: yfactorial(1) = 1 * 1 = 1
    + ], Z4 W8 `# u. Pfactorial(2) = 2 * 1 = 2/ s* m: |$ c* ?# f& A6 X
    factorial(3) = 3 * 2 = 6* w/ A$ a; I2 b& U
    factorial(4) = 4 * 6 = 24
    ' d; W/ y' H6 b* Q. ~  g" t" ifactorial(5) = 5 * 24 = 120
    . T+ Y$ n" K5 U' m. ]$ b  r9 [) r6 [) {3 ?$ N
    Advantages of Recursion
    5 Z6 u" i' K, w1 }" P0 {) \$ C* L/ D$ L1 M$ R3 r3 J/ ^
        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).
    4 J: d, j1 w) a1 g/ V: w4 r% K
    % X: t& {/ D5 R8 w: U    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    : y7 l3 w  g; |7 T0 b
    " V% U- B$ B. k- P* n( r8 }" e( K$ wDisadvantages of Recursion
    ( R* d( ^6 K5 Z$ Z$ f0 Y* j) S. S4 S8 R/ S0 C
        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.1 I% {9 [1 F6 Y( o! `

    ; y! c# |. u2 i5 F, W    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 B; a- r) W4 U2 @: X' T7 `) c% l

    ( @, a. z$ L: JWhen to Use Recursion
    ( p7 \% F! j! f* O$ L) F: j. X
    7 z1 d; i# ]/ d    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ( k) a2 F! @9 Q+ Q# M1 e
    3 S6 Y9 [4 X$ i6 w    Problems with a clear base case and recursive case.
    ! H9 Z! n' Z# }
    ' v$ U& V; C& F' SExample: Fibonacci Sequence( O" W- O. A( R' r* p

    . t0 `- ~0 A! q' k; cThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:& d/ r; T8 F5 g" S& P1 }

    6 k# D4 i8 @% S) O! H) m4 E    Base case: fib(0) = 0, fib(1) = 1$ L: W% O/ j% c5 J

    3 B8 V. I& s: c0 x9 c    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # \. J5 N: d8 b1 G" i; H) u  w; }: K7 T3 H% h. s2 K3 }
    python0 c4 `/ j3 D: R
    8 x- I5 p0 }9 A8 W" D  V
    * J4 U+ k9 ~5 R; {$ o, @! p
    def fibonacci(n):
    5 w4 O+ J7 W9 K  ?, t# i    # Base cases0 S6 q3 t, ?4 n6 Y+ W. W  E7 g
        if n == 0:
    7 O" b/ e. q5 ^, B        return 0
    3 A" r1 |% l* Y5 v; y& a$ _    elif n == 1:
    6 D8 `& |  w; R# k6 k0 W: d) ~$ i        return 1
    + j' O4 F. z# n8 l& O) K    # Recursive case5 G( q0 i2 X) r$ ^: w' q/ d
        else:
    ; x- {5 Q3 k: i+ u$ v+ D2 r. u        return fibonacci(n - 1) + fibonacci(n - 2)
    & V1 k0 b% o9 H+ I% x; ]) R- I, n" U9 {3 N; X0 v( i* z) D
    # Example usage5 U3 l2 P9 C. D$ G" ]9 {$ o7 C5 Y
    print(fibonacci(6))  # Output: 85 i$ z0 y' a0 G) c% l$ ?

    8 N$ d4 X9 x9 l$ Y- [! H5 }Tail Recursion
    9 R! u1 U+ }' d0 i- }# o9 B
    + a6 g: j+ N" a+ G  o! LTail 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).
    / |7 t) A8 q. T/ h6 o
    " q5 b& }* [2 l6 ^$ m# y7 EIn 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, 2026-1-12 20:47 , Processed in 0.030074 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