设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1926|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
    - D- B4 o7 V& M: K' n. a6 d$ Z1 \* Q5 b4 j* E! L4 E& P
    解释的不错  p' M/ r0 U7 d$ A
    : P: f* s, @0 V: P) F, m7 f
    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。
    # t1 f; _; ^6 E7 S' j3 q5 T6 [: M9 Y* G
    # |% W: E' o  g0 }: ~2 Q4 A 关键要素3 z  F3 }* z# d0 P
    1. **基线条件(Base Case)**) _0 G6 N6 E/ v* F6 [
       - 递归终止的条件,防止无限循环
    3 u( V- p- G; B% P% R   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    : Q2 Q3 D2 j1 J5 b2 ?) s( j5 f/ b* @# I; [2 F4 Y3 Y5 J
    2. **递归条件(Recursive Case)**
    0 k( [; Q5 x: X# A9 q: {0 Q   - 将原问题分解为更小的子问题) T& R7 n- v/ O% C1 C; y
       - 例如:n! = n × (n-1)!
    " T- W: A* A. A: {' R3 M+ s/ l6 y' m% k1 v7 ^+ R1 p
    经典示例:计算阶乘+ D3 O- u9 R9 t
    python9 J* n& {5 z6 l8 E7 L9 e2 @0 A
    def factorial(n):
    ) T& p! E4 t4 d) M    if n == 0:        # 基线条件; v! R$ p6 x% z
            return 1
    % p) [. O- t7 w! [$ V( u) n    else:             # 递归条件; d, R) d( _0 C1 Y
            return n * factorial(n-1), m9 Q$ e& X% Y# @4 i; I3 z/ z( c
    执行过程(以计算 3! 为例):
    1 o8 o8 J- b" I, h5 f6 Z: {9 d9 bfactorial(3)
    9 o. d9 p" `) ^. o; x3 * factorial(2), }3 _% Q5 T4 ?9 A- V
    3 * (2 * factorial(1))
    4 w6 i& n5 m- Y% U  |3 * (2 * (1 * factorial(0)))
    0 N3 ?6 ~( c' I8 d! q' b9 P3 * (2 * (1 * 1)) = 6+ v2 E! j) B; @, {
    + Q1 h& ~- g; n4 w
    递归思维要点2 Q0 f+ d, Y( x
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑, B' z9 y) n$ v' S
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    & Q- ]/ o  ~4 v1 D! X3. **递推过程**:不断向下分解问题(递)( n( }# }, ~2 ?9 r
    4. **回溯过程**:组合子问题结果返回(归); q" @7 g* r! ~- N
    & E3 F4 `5 u: d4 g
    注意事项# |' C8 @& T$ j- F
    必须要有终止条件; N( E2 ]% t/ S* n* ^
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)1 w3 g7 v; i- Y. f+ Z( Y
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    5 z% z; N$ H; N4 Y) t尾递归优化可以提升效率(但Python不支持)
    " {$ y. S; A9 ^/ Y" r6 f5 W/ @6 a1 U9 q
    递归 vs 迭代
      b' s0 X4 g4 ^1 e! {  d. x( l$ t|          | 递归                          | 迭代               |6 z. e, x2 J5 q1 I! k6 d
    |----------|-----------------------------|------------------|
    ! O' G* B9 o8 n; h7 u6 Z| 实现方式    | 函数自调用                        | 循环结构            |" ~' _4 [: M  |$ R
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |, ^( f# z3 l% F
    | 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         |
    * U" J5 d6 T& M* F( p$ W+ H| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |; t1 n* B2 h- {. `7 l% h

    8 D& K% W9 x1 z/ p2 N  [! s 经典递归应用场景
    3 w/ d8 S( M9 \  f5 s2 |( j% i4 x, \4 Y1. 文件系统遍历(目录树结构)9 P8 x) l3 l- e$ H# D5 ?0 I
    2. 快速排序/归并排序算法  \. n9 U9 m( F5 R5 S6 r2 T" e
    3. 汉诺塔问题
    & S* O+ u/ |0 @' p6 U4. 二叉树遍历(前序/中序/后序)6 p4 o/ v" r( G+ H, u! W
    5. 生成所有可能的组合(回溯算法)
    ; C) C: O6 |! ?$ v5 D
    - A. w/ r0 x2 W& }8 N4 d5 v& i8 e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
    4 小时前
  • 签到天数: 3152 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,0 ^) \, e- z3 X& ~; }
    我推理机的核心算法应该是二叉树遍历的变种。
    ' P1 Z; W- R/ `* n' k6 y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:
    ( B" Y" t1 g3 {Key Idea of Recursion/ V3 n* m% y; O
    # H0 j: s( \- R. Q* t0 M9 K
    A recursive function solves a problem by:! R0 @, @: L) h6 F1 ]. ]$ e
    2 V. s6 A& K9 ]7 ~, B* k- ~6 N
        Breaking the problem into smaller instances of the same problem.$ G( e. M1 x/ v0 B; A& d
    0 t! F3 E' B9 b% ^
        Solving the smallest instance directly (base case).
    ) `+ b$ a( J: L' K! G) u2 A8 P4 o
    : E% S8 ?( j' ?' Z9 o+ d5 u. z* l    Combining the results of smaller instances to solve the larger problem.
    3 t3 s3 H8 }, |8 [3 b  w, Q+ U  ~7 C: A/ M: c' G' f( O+ h
    Components of a Recursive Function! i) E2 u+ y% T' A& }( i8 B% V
    * v$ n* K, A4 k
        Base Case:% v2 R6 h5 O1 T' ~

    ' b& }9 i% X1 T* v, p/ K        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    % f  g1 f( v( Z- R; w. ^" F1 |; s, Q: ^! R4 r
            It acts as the stopping condition to prevent infinite recursion.
    - V' {. }2 n  s) E+ i+ m/ \6 q
    9 X0 s% x7 R  L9 w* w( K% ?2 P        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ u5 s4 s- K* \6 Z( C4 f* k9 p
    3 L" Q. ?6 _* U- m( A- `3 C7 L
        Recursive Case:+ B6 {3 C( T" X5 l' Q2 I
    % U, W7 p" t3 T( p; f
            This is where the function calls itself with a smaller or simpler version of the problem.1 u3 w3 ]2 M3 e: c, k7 ]; _
    4 ?! _, d1 e3 J% Q8 [* r# u5 v1 Z4 O
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).$ }. Y& S7 w: \& n: C" e0 T  B$ m

    ' W4 C& B# l5 c. P+ ^6 z4 g. l2 EExample: Factorial Calculation
    $ q9 p' Y* ~/ ]( q+ ~( b
    ( F* {' _* n  Z/ kThe 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:8 g8 I' T. {9 N) j

    ; p8 }: i) P, {% N3 ~! [, P    Base case: 0! = 11 c0 x6 E9 Q  [! f  q1 l
    9 G+ i+ ]7 X  y
        Recursive case: n! = n * (n-1)!
    $ v( D' r3 x+ Q* ]3 j; |6 s0 g5 l4 f8 P4 o
    Here’s how it looks in code (Python):
    ; p/ d( d: a, Y4 {3 c1 a' vpython
    % [9 t" Y; W% V1 n! o
    ; @0 Y" E# i1 c9 k8 w" G$ E
    ( U9 q. p/ N! x: m3 f/ Cdef factorial(n):! u  r* \1 m" \/ c7 D
        # Base case5 C7 Z  b3 |; q+ t# E% ]
        if n == 0:
    ( |9 d7 z$ N4 F        return 1
    + m9 v/ W; p% h# Q3 h) J    # Recursive case
    " M4 s# c* J5 x, u9 {    else:
    0 G+ ~. V" s% T8 {3 C" N        return n * factorial(n - 1)
    % t; I+ e1 h- ]& F& r  ]8 G) k8 t1 \9 ^# K& X
    # Example usage2 Y4 G# B/ o( V8 b4 c' @. S
    print(factorial(5))  # Output: 120
    ; c6 S; L( W( |. J6 Y# c8 w+ `' }0 c) e1 b9 A
    How Recursion Works
    & s8 w1 C' f% ^! o. g! j% r% I# M- t
        The function keeps calling itself with smaller inputs until it reaches the base case./ \# c& E3 \# O8 U/ A9 D% P; R
    4 F8 ]" G" F' e# H% p( _' r. N0 s
        Once the base case is reached, the function starts returning values back up the call stack.
    ) X" {! n6 Y5 ]/ O/ i  B% T; `# i8 Q1 f+ u: G, Z& _
        These returned values are combined to produce the final result.
    - U- n( U" D$ S# ]8 {, {7 q* W
    9 N- O) l% {3 f4 F4 I; S3 JFor factorial(5):- n: i! G! V6 D, |* e, e" n5 P

    6 z" q2 Q  f1 E* a( T( V% O9 i& T2 @, v
    factorial(5) = 5 * factorial(4)# |. Q: s0 g' }+ _. r" u. T2 ^
    factorial(4) = 4 * factorial(3)1 L$ e+ h" @) i0 c4 N; O, R
    factorial(3) = 3 * factorial(2)& O* V6 T9 T& x) y
    factorial(2) = 2 * factorial(1)
    1 b! i3 d  r( B* N  rfactorial(1) = 1 * factorial(0)) m+ m0 P+ P  ^; ?; {0 ~& {
    factorial(0) = 1  # Base case  e& t* g" K3 R( t, q# Q
      G; h$ o# v) Z$ e# u1 p/ t) i* g
    Then, the results are combined:/ {' \6 A" Y' b

    1 E" F/ Z( d1 `
    ' x( r" |. u  a0 x; ofactorial(1) = 1 * 1 = 1
    6 @- g- ]  N' Zfactorial(2) = 2 * 1 = 20 m' X! y  }) S% p/ _% G3 n9 Z: {6 U
    factorial(3) = 3 * 2 = 6
      y9 d! X" S1 n9 s' S3 J" o+ Mfactorial(4) = 4 * 6 = 24
    9 H) m: A1 w* S# U9 ~factorial(5) = 5 * 24 = 120
    4 f8 Y* v5 s0 ?: @9 a6 L2 K4 B  r' M+ ?, t0 [7 S( B  v
    Advantages of Recursion
    3 h0 E& T. J9 M- A# G" y1 z. g/ `0 `# P/ O7 n" k0 g  M. c5 X4 {/ t8 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).
    6 L5 S( p3 s# x( V! v
    6 v5 D- B3 I, g+ s! y6 o( |6 v$ M    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    0 h( E1 Z& B& x0 T, T9 i/ Y6 `% X0 `) B  ~& n" z! L
    Disadvantages of Recursion
    7 I6 g8 S" U1 y: \8 ?; {! U/ X  V- U7 ~) [& J; d6 f
        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./ K" a+ d; t' O% \
    ( U! f* u3 V$ J# u1 P# q8 m
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ( Q# R+ X2 g6 P; P/ l+ {4 y* D6 \: n. f% ^% q
    When to Use Recursion
    % \: |5 n4 t* y  q5 Y* w- Z) \2 P% [/ k5 ^" P) [1 _: I+ q
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    2 e5 P7 f3 \" l. L3 C9 b7 J0 ]* |7 o6 F6 F. l! G9 g, N
        Problems with a clear base case and recursive case.4 n4 X3 w. N2 \

    * F. ?5 w5 P! w* M! u! MExample: Fibonacci Sequence% Y3 Q# `$ H( @3 _( j+ W: K
    / }& v* Q& C6 n6 G( ~; N+ B  u3 b+ c
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:8 e7 [8 L4 i4 s  K
    / i% }8 \9 c0 _) m9 [! _+ u7 f
        Base case: fib(0) = 0, fib(1) = 12 M* j: [9 K1 C0 ?; ]0 p
    + h4 }8 R0 J, B) N% H  ^  e
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    1 ?& Q6 Z& o/ Y% `# V
    $ d1 X- K$ ~" E4 N+ {& J# lpython
    : I0 |( C0 b, t/ t) _3 d8 H3 I" ]5 V5 K1 m6 E, H

    " ^+ O# f- o8 P8 c8 t( w% Xdef fibonacci(n):
    & y8 y; p" ]. ~7 v) S    # Base cases
    " e' @8 x+ r) p/ Z    if n == 0:7 P9 X6 B* |! [0 G7 l" [
            return 0* k- e6 d6 J( ]  p7 \0 W7 s+ g* W9 S
        elif n == 1:
    8 m7 s. C4 e3 z2 B$ y* k2 S& s        return 1
    ( ^9 f0 T1 I7 l- j9 C    # Recursive case
    $ c% u4 x( E( ?    else:+ i% R! T' Q4 ~" t/ T& N: L
            return fibonacci(n - 1) + fibonacci(n - 2)
    * |/ P# G' f; c, Z
    # [. k2 F* ?% X* ~, p5 F8 o* W& h# Example usage
    ( W. |1 N. {2 ~. Y  Sprint(fibonacci(6))  # Output: 8
    * e* l6 P% O/ e# o
    % R: W2 c9 s6 ?+ B5 F3 uTail Recursion9 W& C3 `3 Q$ S, X

    & K" f* ?. [) kTail 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).3 H7 m' j, \" Q, G
    ) b* l& d' i. T8 I& x
    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, 2026-1-23 11:57 , Processed in 0.059668 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