设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2007|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 3 q* G* r4 W- C% b7 B
    : b, B% M- _/ {4 G5 ~# _
    解释的不错- N2 K4 R/ J4 l% ]* g# n

    3 }$ T( d! S) j6 j+ H( w& E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。
    $ q& I. N. V3 T9 G4 _7 N# Z  n5 M" B
    关键要素
    ' }8 L% E; ~* |* |1. **基线条件(Base Case)**: t; i- [3 f7 G. c$ l( ]
       - 递归终止的条件,防止无限循环( g1 @6 u9 H* Q# n4 z* ]1 S( C* J3 v4 _
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1) h$ w. \# x9 _/ L6 R- B5 _8 {2 `% o

    6 u* s6 d5 x8 I2 A# D# o2. **递归条件(Recursive Case)**# t1 S$ c# d& Y' c; B# f( T: N
       - 将原问题分解为更小的子问题- t3 N. T; d9 z. T  s4 B
       - 例如:n! = n × (n-1)!1 H& @% f6 L% J' o% ~5 z! Q

    + j. M4 i0 _8 ~8 j$ G5 q+ g6 N 经典示例:计算阶乘
      c$ V/ S9 S# c7 @& J1 h8 r9 jpython! [  k: p0 u+ ]( S. G
    def factorial(n):
    & k" {) Q0 y) @6 P    if n == 0:        # 基线条件  j# F3 {; W! ~5 E; n
            return 1% G( U# P* S9 o8 w/ x
        else:             # 递归条件+ h/ _. u+ P! ~
            return n * factorial(n-1)" c; U. v4 N) T$ O1 \" \# L
    执行过程(以计算 3! 为例):
    , v& s  d# q/ J( lfactorial(3)6 g0 t+ ~' e8 h( J1 d4 C
    3 * factorial(2)& C6 M7 l1 ^) o7 T
    3 * (2 * factorial(1)). u" `% w& `$ y
    3 * (2 * (1 * factorial(0)))9 O) `/ _; y  P' Q: k3 g
    3 * (2 * (1 * 1)) = 6" F, K0 f2 n) H/ \0 E8 a( [
    : F( }2 _) o" ^0 ^) _2 N2 |# ]
    递归思维要点* ]% r' J, W( r! Z5 H6 |% @
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
      j7 e! T5 _# M5 a- M2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ( ^, ]* [; [9 ~! c! Q8 ]3. **递推过程**:不断向下分解问题(递)& P1 u, l8 x5 B
    4. **回溯过程**:组合子问题结果返回(归)5 F; D. C# E/ ^$ B! B" x$ B
    ! ~) n' }' d5 Z+ G' @) d
    注意事项, b& U/ K+ p* v+ Q  c6 x9 S3 s+ l
    必须要有终止条件+ x7 s* [! U+ o/ E6 Y6 _, x
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    $ D, I% u! e& \某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ' \, @& \, Y/ V% r7 b尾递归优化可以提升效率(但Python不支持)
    7 P- \/ [( Z9 h8 q& Y8 t5 Y' [" V/ p4 s) B
    递归 vs 迭代
    ; |* x5 T& |( u: u4 T|          | 递归                          | 迭代               |6 a: W$ N; E. Z2 K- k1 Y, z
    |----------|-----------------------------|------------------|
    . X0 f/ W) p5 \, F$ h( L| 实现方式    | 函数自调用                        | 循环结构            |: Q, A5 |0 S' b
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |5 f2 _8 C7 I; ?1 `8 P4 i: R, 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         |
    ; _" T8 k# N' h9 ?2 q& c+ N| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    + I5 Y+ U% n4 I" N  ]( N
    . I1 |& o5 ?& Q0 j  o2 x 经典递归应用场景
    9 g1 e' J) z5 U/ k1. 文件系统遍历(目录树结构)- @7 e$ u* x! S4 ~
    2. 快速排序/归并排序算法
    % \. i2 k1 Q3 r8 O/ U3. 汉诺塔问题
    " r3 Z! N( ~( }7 H3 V4. 二叉树遍历(前序/中序/后序)
    $ K3 ?, V0 g) X: W1 R: q: E1 Y5. 生成所有可能的组合(回溯算法)
    1 b. e8 r+ `% E
    + Z" r& g: o( J) _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
    11 小时前
  • 签到天数: 3162 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,+ q( n0 m( E% g. T; a
    我推理机的核心算法应该是二叉树遍历的变种。7 Y. A' F6 J+ s" r  D% _( A8 @) k
    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:  X. B% `7 o4 y! d1 f9 G# X
    Key Idea of Recursion+ C- g4 G8 K4 T$ l2 A

    * c+ Q, K+ q# c2 d. b" {, UA recursive function solves a problem by:
    0 |+ y$ Q+ J, o1 g3 `8 C2 g( c1 ~2 Z1 I5 `7 B- I
        Breaking the problem into smaller instances of the same problem.
    " K1 ~! k, n; [0 u
      _" f( N0 [0 j. l& @$ T    Solving the smallest instance directly (base case).
    * P# T  A6 T: G( L9 m' T
    & y6 Q9 p! y. K* E( b- Y    Combining the results of smaller instances to solve the larger problem.
    & f& ]3 ^. q; A1 U: a+ s* P. h* f/ V- N) _0 N- M- K% W. w0 \$ Z7 G
    Components of a Recursive Function4 q1 f9 c* P% @* N0 n) t

    7 m# _$ }! p) E" z$ s# J    Base Case:
    1 L% r' y6 S3 q: c
    6 d" v- H% h( g% H: [* i& N9 z/ b. s        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; I- U- {0 X9 s: @1 Y& t
    1 ?# R2 R/ w  k: {        It acts as the stopping condition to prevent infinite recursion.$ I% C2 w/ ^) m! l& o: M& I

    ) J) G% q! d$ \7 F2 Z3 ^        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    & {+ {5 d- W) ~
    2 G, t% Z' g/ L( b4 J  S    Recursive Case:# V4 L. P( Y! O  g- C
    / _1 B; t8 h+ u
            This is where the function calls itself with a smaller or simpler version of the problem.$ I# y! K7 E2 p1 W# C) E" Y2 S; Z( \

    3 J6 L, Y. G  C- X, K5 k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    6 x) t/ u9 e- A) J4 L7 |4 s8 e: j$ _* Y' g4 X5 m" b
    Example: Factorial Calculation4 [5 {$ J6 G* }/ ]' p
    - H  P. o  W$ P* K" |" N
    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:1 Z4 N& M; B4 ?( p0 P" p0 t
    7 v  Z) g- u- o7 @* S
        Base case: 0! = 1" P! s8 A! _3 M5 I$ ]% Y1 B
    ) u5 j. T2 t; i/ d
        Recursive case: n! = n * (n-1)!4 \1 P$ [- U3 Z

    ' @3 P( K& y! o9 Z6 f( c3 F+ MHere’s how it looks in code (Python):" O9 c5 C7 C2 j9 R: o8 T$ s# Q* c7 K
    python# [: u/ ]6 u# L  d! A
    ) d# C" {$ ~1 d: J" e  g

    0 D) C$ p- b+ z# \/ ]def factorial(n):( ~" V' P) C+ h8 |
        # Base case
    & }. ^2 |/ a, X5 @3 O% v    if n == 0:# J- w* K9 p: v0 d
            return 1% R+ l, X# m, }* ]
        # Recursive case! j- q3 G8 {$ M  K& X
        else:( |, }. S( D* B8 h6 X' r
            return n * factorial(n - 1)* P4 \' M5 Z- J; p& s) h
    # J5 W; S( Q  u4 A0 T- z
    # Example usage( B* [- }$ F" _5 `  j$ ~% m
    print(factorial(5))  # Output: 120) p: i( e. w; A+ d5 K, l

    " d% S6 g. ^- U2 |How Recursion Works
    ! Q* q8 a1 B8 @+ I% k, K: }( V+ p2 ?; p, c$ Y) _
        The function keeps calling itself with smaller inputs until it reaches the base case.
    $ q( @* s! R2 V0 O+ T
    ) B3 j) Q5 ^7 c6 W, h    Once the base case is reached, the function starts returning values back up the call stack.5 }' S$ \% h# x* a( a
    8 a0 v+ l2 s! G( D) P1 D
        These returned values are combined to produce the final result.( ?: w0 A, V2 z' r+ e. a) m# k$ c+ a

    1 a: K; N8 ^1 |* Z% `0 NFor factorial(5):) i2 k4 T1 I0 @

    7 x( L5 `8 ]; W# I
    - ~: Q! Y# D& I% K6 S+ y; R$ Ifactorial(5) = 5 * factorial(4)
    8 D  E$ a  u& ~factorial(4) = 4 * factorial(3)0 l6 z# n* ]2 J3 M4 z( O) A% G& a
    factorial(3) = 3 * factorial(2)
    ( o% K/ ^; @5 Q! v7 E0 \factorial(2) = 2 * factorial(1)
    + w+ g( y; [5 mfactorial(1) = 1 * factorial(0)
    ! r0 k3 e. @9 [. W3 m1 t9 Rfactorial(0) = 1  # Base case5 Q% j0 W. l9 C( F' S% N
    , S1 [8 H  k/ X
    Then, the results are combined:
    # b, S% f5 p  a" `% G* U7 f
    " }% k$ E- h( Z9 c7 ^( O- V# }4 z* F1 K3 m0 d; b
    factorial(1) = 1 * 1 = 1
    7 w, r. V/ S1 zfactorial(2) = 2 * 1 = 2
    / V# ^; s6 }, g4 H: V4 T6 L1 V3 r% Vfactorial(3) = 3 * 2 = 6. o/ A$ \8 p6 ~. @
    factorial(4) = 4 * 6 = 245 z* P+ Q3 s/ M0 w/ J( h
    factorial(5) = 5 * 24 = 120% m: ]; @) ?' ?# ^6 ^
    . p+ E' K1 p6 `" S* ~7 F- }
    Advantages of Recursion
    . z) c0 c( @$ H
    - N2 S, _6 \: C2 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).
    9 Q1 B; F6 I2 E4 n9 b5 y; m
    ; Y7 n7 b2 \- Y    Readability: Recursive code can be more readable and concise compared to iterative solutions.8 x# l5 R3 @3 H0 t* }

    1 I$ V( ?: x" c; A' G( @. L  T6 p, w2 IDisadvantages of Recursion0 n7 E5 ]% v, t: d+ Q$ H9 s  q

    8 q* |8 @6 N1 g8 H3 {* L$ [- D. g# R    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.6 b% s3 ^7 w' H# m8 S) N
    ) o" c; z- l: @: N& q4 o# a
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ! e. O% q$ F. f1 P* q- e9 l4 h4 P# h6 B$ v
    When to Use Recursion
    , W4 J# [2 R/ G# Y( K  `8 V$ D
    ) r  i' |3 v3 U    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' K/ c$ O7 V: E
    4 E0 ]5 m" x7 ]+ A& ?) J
        Problems with a clear base case and recursive case.
    * M7 A/ U& H: ^! h0 h- W2 R4 x* _- g! A9 B$ c
    Example: Fibonacci Sequence
    7 e" s; v! |, L6 @! a9 T9 Y
    ! [) T+ H9 _# [2 kThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    7 B$ u& p7 {! _' H: C. w% O; ^1 r1 g
        Base case: fib(0) = 0, fib(1) = 1: g0 ?% I: I2 l' p  @, F, p& o7 x4 p

    9 c+ X. ^( X7 @9 m    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    . O1 B9 Q* I, _- E( p9 F9 Y
    * S& J' [2 o& a% q5 xpython% m* o; o4 Z- V) p: W
    : J# ?1 r" S- J, [& r! N( R

    ! p4 ~& q8 n) E0 n* Vdef fibonacci(n):
      j# P% g6 y" `2 L1 b4 t    # Base cases
    , y+ X' L. Z. F, a7 ?2 o    if n == 0:
    - [' ^- B! Q) l! y; l6 U        return 0# z& u. X8 V+ a5 E3 \
        elif n == 1:
    ! I& d$ P( @) @8 ]* e        return 1
    6 A7 F. I9 R" T' b; {6 P    # Recursive case9 |5 U$ j0 O* Q( F
        else:: j- |! _5 j- S  F2 K, B7 G
            return fibonacci(n - 1) + fibonacci(n - 2)- ^3 D2 F+ J1 ?4 L* X
    * P+ m  z& D( [' q( o* a6 ^: U
    # Example usage
    5 c2 `( O2 N/ @7 A3 u" kprint(fibonacci(6))  # Output: 8
    % S; D' M8 e# n) @" \+ y% l6 m5 F% J/ O3 B* ^7 l3 K4 o5 S
    Tail Recursion: H* h2 |1 w$ z/ H9 r
    3 d5 S5 G4 H$ B! ?5 w  N/ L7 O+ t# k1 h
    Tail 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).
    6 O# K; M) F9 w; a" Y3 a: O9 b
    4 D$ w7 e& G5 X* aIn 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, 2026-2-3 21:32 , Processed in 0.055407 second(s), 18 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表