设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1319|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
    % o" `7 ]5 t/ p0 X( Q0 s% f6 \% Q6 K$ {
    解释的不错
    ; c( {" z  u9 J- Y3 [9 e+ p2 B, q9 x1 e: \* y4 ?1 E7 r
    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。
    * k! E  D3 ~0 `( G- k
    2 L( ]/ D/ v7 U 关键要素% s, D+ z0 T4 f% [& u/ F
    1. **基线条件(Base Case)**8 m" G& g2 K+ L
       - 递归终止的条件,防止无限循环
    : R) D% r1 R( Y1 {; l! I* i7 i" u   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1. i1 b+ j  f2 U+ ~& a0 c4 g

    / F( V" U3 }8 G- e1 X4 [% @2. **递归条件(Recursive Case)**
    . I, |: Q5 @3 q+ B. T" e! I3 @1 a   - 将原问题分解为更小的子问题
      y& m: w, n' J) O& `8 K   - 例如:n! = n × (n-1)!+ L) @7 C; I( Y3 L1 T: i3 h
    1 p* F$ l5 }7 k/ @
    经典示例:计算阶乘! c2 {4 M- |- h; ]
    python
    ' X7 R) P  Z$ N% D5 sdef factorial(n):
    - n1 r. G# K0 U0 Y2 y  o) q    if n == 0:        # 基线条件( a6 z2 ]4 Y8 u$ g
            return 1" ]1 I. a7 @* G! W( V# u$ s
        else:             # 递归条件
    2 g9 M( W% q- D        return n * factorial(n-1)
    5 `, M2 m5 @2 I, \0 y执行过程(以计算 3! 为例):5 c; g4 {  a& {1 c
    factorial(3)- h: p  {/ }8 ]. J' ~1 d
    3 * factorial(2)3 E  X% G5 d& h
    3 * (2 * factorial(1))$ M$ E* `) m) t" ?: ~- \# _
    3 * (2 * (1 * factorial(0))). ?0 B8 ^) w9 e* v3 E
    3 * (2 * (1 * 1)) = 6, I" E0 f1 i( ]; L# Z% g1 J

    2 k4 W6 V5 p6 b  } 递归思维要点/ k+ ]$ y% t2 r* G
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    1 g! M4 r" c( C0 Q2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( r2 w1 [2 s0 H' K& d- F& w% N
    3. **递推过程**:不断向下分解问题(递)
    % u1 I5 m" i# n  d4. **回溯过程**:组合子问题结果返回(归)
    9 S. G. u8 q3 |8 b4 R8 n; p: u! g
    注意事项
    ; b  a+ n7 F9 J$ B! I& {. g9 K: J必须要有终止条件
    ' y9 z& O7 z; N0 ^) R$ E7 c递归深度过大可能导致栈溢出(Python默认递归深度约1000层)8 i+ R  [" c6 B2 a+ i: L
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    6 B; J% m8 v* m( F; Q( q$ O! s尾递归优化可以提升效率(但Python不支持)3 v6 y& V1 G- y  w0 T6 }# c  |7 G

    3 G/ a, ^% [9 _; B: f4 `" a: | 递归 vs 迭代5 U) [! K+ _+ e9 \0 H- g+ o
    |          | 递归                          | 迭代               |
    # C7 z: b  l9 k7 k" V0 @% n5 O|----------|-----------------------------|------------------|
    ! |3 S5 U( a% s/ ~; D0 g' N| 实现方式    | 函数自调用                        | 循环结构            |2 [* W. m% x$ l1 q
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & d& M- ]9 ^0 S5 b1 W: k+ [7 }| 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         |
    - ^% K6 x) l0 r3 W7 N& T1 o6 s. O| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |! {: O$ F% x$ D4 `4 M7 n+ g9 n

    : L7 r8 c6 U) o( n 经典递归应用场景6 K5 w8 X: ?2 v1 |
    1. 文件系统遍历(目录树结构)
    ! j( W$ B) _( B* R5 r1 c2. 快速排序/归并排序算法
    - p; v$ c: z: h1 m3. 汉诺塔问题
    ) \& V4 O- m! u" L3 p! ~0 k) b  G4. 二叉树遍历(前序/中序/后序)
      g: @3 l; t3 {2 y* ?! H7 j2 b5. 生成所有可能的组合(回溯算法), L3 R' z3 l3 v* \; W) j

    $ o; V% l! b4 ~3 C- R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
    郁闷
    昨天 12:19
  • 签到天数: 3107 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,
    9 {" c$ y# i! d6 j, l2 u. W6 P我推理机的核心算法应该是二叉树遍历的变种。' M$ C; D  T, E, A  x& m4 S% p
    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:
    . h  g' J8 I6 B, `4 K5 \# L& eKey Idea of Recursion/ U: j- q; V( `
    * N  y3 x3 ]1 v+ T& z1 a
    A recursive function solves a problem by:
    5 d' x7 J# O. N- y; _- D+ W: d2 O' X: d
        Breaking the problem into smaller instances of the same problem./ \7 `0 C; v. d. t
    & R9 P6 R) H; r; b
        Solving the smallest instance directly (base case).
    . f  l. x) Z- q
    " @, S1 r+ R/ m. ^1 Q) y    Combining the results of smaller instances to solve the larger problem.
    ; ?  ?+ O8 |4 r4 d5 Z% p; e, J: @3 ^, K! e4 U1 Z9 B
    Components of a Recursive Function7 g6 ~; k& O/ ]) \
    % E6 k& q2 T2 h% Z
        Base Case:
    ) G( U- e! }7 h/ ^. l; X
    . z; d. {1 e' o" I/ _( ^5 }        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    + S' @+ }9 v( `7 K. k. ~6 v1 J/ y# _2 `5 Y
            It acts as the stopping condition to prevent infinite recursion.4 h' f6 A( I" O3 i2 ?  U8 f

    * c+ x" o; w2 U! h6 n% {$ L        Example: In calculating the factorial of a number, the base case is factorial(0) = 1., }* k/ A3 |7 o, k; R

    ! h# q. T! ?! d* J' e0 T  j    Recursive Case:8 G6 _* {/ c  z8 X& r
    ) g, r5 x* |) K' K+ |, F% S7 h
            This is where the function calls itself with a smaller or simpler version of the problem.
    8 Z0 K8 J3 E, \4 ~7 ]8 N9 n1 f! f3 [
    " O. b8 j5 a4 Y& N        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
      j  z* X0 I8 ^' H3 f* ^& p  \  t9 q. X, O' F( e, R. W
    Example: Factorial Calculation
    1 J& f3 x1 u3 m* @7 D( z" R8 _+ h: V( j7 E
    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:
    # A5 D" s9 X; s, O+ Z" P3 l; N/ H% r3 r: G0 a9 D) _2 K
        Base case: 0! = 1
    ; s2 d) D, n! b* d$ e
    + \; f. ~8 `7 z1 X$ @/ t. E    Recursive case: n! = n * (n-1)!4 p1 ]+ ^1 R" R! }0 k

    4 K8 \/ i7 x3 u' t' U+ y2 eHere’s how it looks in code (Python):$ ?; B2 {/ W* o7 J$ K( V
    python
    . o4 [" B( Q8 m- Z
    3 T5 G+ }% s6 Z5 Y+ F) I4 F& R1 E. L* b% L, C3 E% u  v1 c
    def factorial(n):
    4 K! y" E+ |+ Q+ t& \3 G' {' B5 R$ ]    # Base case4 H, \& J& l' z
        if n == 0:( H' r" I: J( i# @6 {% C% L/ V
            return 1
    8 F/ n: _8 R. o0 M    # Recursive case
    " T5 n2 S8 ?1 P7 j    else:
    . O/ p6 P+ n* ^+ E7 \7 t        return n * factorial(n - 1)
    0 `: _5 Z! n) k7 A3 {7 {9 o/ V6 j& B5 R4 f
    # Example usage/ A5 D+ Q5 ?" g9 {0 V
    print(factorial(5))  # Output: 120
    % t5 i3 b6 Z2 h8 E# ]* m) R9 C
    . {* h1 k8 v( d% f* V- h, i3 ?How Recursion Works
    + ]" V* Z0 R, [8 e, J3 O1 l0 z) _3 C+ z) p6 A: G9 V- \! @
        The function keeps calling itself with smaller inputs until it reaches the base case.9 d8 n3 n" e1 Y% G2 t9 c4 y
    8 C& Z) w' E% U+ U# O; Z; Z* U; @
        Once the base case is reached, the function starts returning values back up the call stack.* [, L$ d6 ]3 a; V& S* F
    + [: U3 _. z  z% E4 V3 ~& P
        These returned values are combined to produce the final result.
    : Q5 R! Z1 }0 Y' w0 E* f' H* A6 I; G6 |1 n  l- Q
    For factorial(5):
    ' I1 r  X* D( F
    ( N7 R3 a1 c' Q* N) A5 e+ k! O6 z5 B, ^7 Z$ _: F& a! S
    factorial(5) = 5 * factorial(4)
    9 @8 m8 Y6 m4 {- R0 E) g, hfactorial(4) = 4 * factorial(3)
    ( X0 I1 x9 j( f- qfactorial(3) = 3 * factorial(2)5 w& `0 D2 _3 Z/ U  T! [* {- |
    factorial(2) = 2 * factorial(1)8 p. @; N3 \( d. J
    factorial(1) = 1 * factorial(0)) Z) {* a9 V$ J" L3 f
    factorial(0) = 1  # Base case, b' H: I1 `3 z1 O' U7 |9 p. m7 P
    4 {0 V- f* F# w
    Then, the results are combined:
    5 m- ?+ o, m* \
    8 f" W9 a1 H1 B( a0 u& y4 a4 B) `4 m2 u" G% ~! @" g! I
    factorial(1) = 1 * 1 = 1" _9 N5 V; ^1 G; ]2 f2 o  ?# z" u6 r
    factorial(2) = 2 * 1 = 2
    - ?: s3 l" |' s0 l# G" vfactorial(3) = 3 * 2 = 6
    ) P* Q0 [( {9 b# Ofactorial(4) = 4 * 6 = 24
    9 S: k* ?0 _$ A- D) t. Gfactorial(5) = 5 * 24 = 120. ~0 n' L4 r" q: s1 J

    . v& Q0 d+ f/ X2 D& @9 ^- R& |9 ]Advantages of Recursion
    ; }5 j2 @: Q9 j: j3 {5 B' H. F- a: ~7 h% I* E* n9 P
        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).* t# W! e; M  Q; I
    6 L5 D; f1 k) q6 ?+ @' N) Z0 S; L& B
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    # f, w+ g& h6 U7 ?+ a2 `- L
    ' e8 a* t9 m0 T" h+ xDisadvantages of Recursion
    . ]' c: t; c6 c) _. c3 y/ b8 u) B+ Z5 c  L
        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.
    , U" B! S5 U5 N$ c+ Z
    $ p0 C4 Q' l1 T) l3 ~' ^    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).5 h  i" O# _" x# M2 L

    8 N5 y% }+ C" L/ T9 `: L9 lWhen to Use Recursion* K1 _  G' J' |- E! y
    $ N" N$ M; L3 T0 M5 I% G, D* ?
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . x# ^3 X+ r" f: K3 _. ^3 s0 ]$ D/ i/ k  n0 ~
        Problems with a clear base case and recursive case.( @/ I- b" w7 n
      c/ z* P, H4 M9 `% J5 v; F  \
    Example: Fibonacci Sequence
    4 O, p, L) v4 D2 w6 V' q* b5 }0 A* \+ q" e
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    : _1 D" k0 T$ S( k
    5 t9 Z8 K8 n1 A. i    Base case: fib(0) = 0, fib(1) = 1
    . V" i# U2 d6 B4 Z8 S/ i: K6 R0 V# S2 ~
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    : z* u- J/ f! o- \$ M
    ( x! a# S  p- S8 f. ^  ^' P7 Ppython; O/ _5 S% p' b( `

    ! {7 r& e1 |6 O8 k4 Z1 Y0 y1 b$ {1 b( {/ a. j/ @- S
    def fibonacci(n):
    ( Q' x% H' ]- z0 x( v- H    # Base cases( `! a( j5 v( Y1 x9 O
        if n == 0:) a! ^# y6 u/ A7 N4 f- b
            return 0
    0 b  f- _, F/ Z0 g( A/ \% w& a, o    elif n == 1:; z6 A$ k0 Z) S
            return 1
    , O8 |3 G8 K% }& [' c    # Recursive case
      y( q7 [8 ^# n' h! c# V5 o8 m    else:
    + y1 L  I) h2 B2 n1 f0 m        return fibonacci(n - 1) + fibonacci(n - 2)5 J, p, s0 y) q

    7 t7 ?' ?/ _! m# Example usage' L' E( ~( r: ~8 [% q
    print(fibonacci(6))  # Output: 8
    , H$ h3 m7 o) |; }6 s  m/ P& i5 g9 N! e5 U- M7 n
    Tail Recursion2 ?: X. z. p' L. b. V

    7 w8 O; O  A1 \; r$ TTail 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).
    " f9 i6 D1 I+ r: g& A. E" B: z- x6 H0 W- q) d
    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, 2025-12-4 04:18 , Processed in 0.031444 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