设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2733|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 E, o( K; `# N! I, \* n% O
    4 w3 o1 I- u5 {
    解释的不错3 y6 n& n% W0 q4 L
    # ]9 }  k  o- c. A$ P# g4 d" S; M* 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。# u3 e1 ]- u8 |" T1 m7 G$ i, n+ k
    % g! I' z$ l6 [$ K8 n* W4 ^$ f' v
    关键要素
      Y! J) a5 |+ v  z1. **基线条件(Base Case)**! [% s9 [( v) ~6 u3 }3 m
       - 递归终止的条件,防止无限循环9 x+ f7 Z! w' g$ V
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    3 r7 a0 M- @: k; Y  E  s9 ]" d
    " C, c0 x+ |5 R, L' Q2. **递归条件(Recursive Case)**+ v0 h. Z  k. h& h  y2 E
       - 将原问题分解为更小的子问题
    * o, A3 i) H4 V. d5 G  A   - 例如:n! = n × (n-1)!2 S' r; W8 V2 G' l
    - a3 w; O- n# i. U
    经典示例:计算阶乘& Y/ x  }5 l7 P* n3 ^8 t- |8 Z: F  G
    python9 k2 E( q% O( Y
    def factorial(n):
    ) w1 @. G, ?, S! ]; ?2 B1 z. q" m! {    if n == 0:        # 基线条件
    5 M5 U; L6 O; T5 |' E        return 1' I; |& P1 A! W+ ^1 v) \5 H
        else:             # 递归条件
    2 Q( V5 J- a' ?) L- v' Q, X6 b6 K4 E5 T        return n * factorial(n-1)
    . D  E# ?& O  _: R! f执行过程(以计算 3! 为例):  y& t; M8 w/ {6 o1 \( y, r) u, D
    factorial(3)" u; H* x) j' K' F
    3 * factorial(2)
    5 _9 x, Z# T& h' a8 t# f2 c# b- X3 * (2 * factorial(1))
    ) T7 e  v# G4 O' ]3 * (2 * (1 * factorial(0)))
    + c( E' S8 |  V: z% q$ P  o* `3 * (2 * (1 * 1)) = 6
    4 q; J. }) u+ T4 ?' [) N- n& ]4 ]& y/ ~- s) o
    递归思维要点3 i, S5 z: O3 U; p
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    0 b6 [( p6 }9 _1 ~8 r; F2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    8 t# o, `/ L& W3. **递推过程**:不断向下分解问题(递)9 Y% ~( }3 Z/ b
    4. **回溯过程**:组合子问题结果返回(归)1 P- S" P: P0 x: k4 [- n

    7 t% l- _: Y" _! g 注意事项4 t" ]3 m3 U, R  b
    必须要有终止条件
    1 o* }, n' v, x/ A5 t递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    & _. y% u1 @8 z2 Y4 W' A# T某些问题用递归更直观(如树遍历),但效率可能不如迭代/ n( \' a& s* q8 y3 E& S  G# \
    尾递归优化可以提升效率(但Python不支持)
    5 X! Y& c/ R, K: _1 G( {8 P
    $ m9 z% ^  P4 s( H+ @" X 递归 vs 迭代
    9 p. _% [5 u. X: s2 \|          | 递归                          | 迭代               |
    # O4 S$ L( a' ]6 Z0 n|----------|-----------------------------|------------------|
    ' j; H# {$ \( @8 O% z* c| 实现方式    | 函数自调用                        | 循环结构            |% D$ p8 {) o1 {( j3 _9 R* n
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    - J9 ^9 P0 \5 J2 F& k3 q6 Y| 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         |5 Y: P: e& u0 b$ G
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |6 R, f" ?8 s3 l

    8 E- I8 V# ~  G0 _ 经典递归应用场景6 P: B8 G9 a0 D. D2 V1 R7 B, Z" O
    1. 文件系统遍历(目录树结构)2 W- M" F2 M+ N4 D
    2. 快速排序/归并排序算法
    5 ]$ `! ^; q, [- \, {3. 汉诺塔问题& d! P& S% C# E& j- a+ G  w$ A
    4. 二叉树遍历(前序/中序/后序), e/ _% F8 ^% a7 i' s
    5. 生成所有可能的组合(回溯算法)
    + d; m" ]+ R* ~; c; l, Q# I$ D' E' u( g; n* l
    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

    11 小时前
  • 签到天数: 3243 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,/ p6 _' e8 ?9 p, ?
    我推理机的核心算法应该是二叉树遍历的变种。: ~1 j: ?2 {& q
    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' V3 f* g4 U5 U0 [4 c* b
    Key Idea of Recursion
    - F4 P7 Q6 V% h9 @
    % I5 `3 h% A' fA recursive function solves a problem by:
    ) K* U8 F. a2 ]6 L% g& x
    + \! _) l3 A1 J% N    Breaking the problem into smaller instances of the same problem., D9 O; {; T3 J) b6 b

    & f0 L; X; v- Y: y5 R    Solving the smallest instance directly (base case).
    4 [1 S9 Y* k; h2 f' w" s6 n+ u9 K& W/ B! e
        Combining the results of smaller instances to solve the larger problem.4 z. A+ h/ a. c5 h* |9 P( n2 Z

    9 z+ j0 Y7 v* j8 z  N1 vComponents of a Recursive Function* E6 s' o# k0 n' e# G. b9 V! r
    # M; }& t# E% w6 z3 N1 t
        Base Case:
    " G. g+ D% s/ D+ E7 q
    ' i* x/ R/ \$ @: [$ S/ U        This is the simplest, smallest instance of the problem that can be solved directly without further recursion./ ]4 W$ W$ j/ n; \
    5 g  f+ L$ q1 |: y3 T, x# I2 G, B
            It acts as the stopping condition to prevent infinite recursion.
    3 U3 O7 w: @+ t* G
    , h* d  G  P8 F# S9 k, a5 {' g4 C        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 D8 t/ E3 K9 ?7 i7 D  {+ m4 z

    6 ?# F8 Z3 {9 `, P4 E$ D    Recursive Case:) `  r) \1 u! ]+ e/ M% x6 S. P. n
    - x* `* a1 {/ o$ J
            This is where the function calls itself with a smaller or simpler version of the problem.
    7 R% G5 M2 S  M- D+ K, W* a
    $ E; \& A% b! n( |7 n        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    8 I. _6 V) h* L6 O$ s. i1 @# c7 H
    ' w. s; u! W# u( n: B5 f6 hExample: Factorial Calculation
    0 c. Z+ B  b6 W- z& @" E, Y
    $ h$ W7 M# _* z$ ^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:
    & w# Z' A. H, j! J1 i4 E
    ( h( J0 M8 f/ \0 h5 K    Base case: 0! = 1- F' U( R3 d0 f, k" y. R

    ! i3 o; |6 K( t" \. D    Recursive case: n! = n * (n-1)!
    ' Z- G4 T$ F( J# `' c) u  y7 T* P( k0 R0 ~) [4 H
    Here’s how it looks in code (Python):
    8 [1 L- z- W/ I1 V; Jpython
    9 Z3 J, m  m" V# ]6 j8 a' O. C% }0 w' k( Y1 C) p/ I

    ( h7 c! d  O) R2 ]def factorial(n):
    4 V# b- Z1 R# E1 L* j' F    # Base case
    5 D" q/ i3 m! D2 m; W6 _    if n == 0:
    1 r2 \" R$ t( Z& G1 S6 k0 B/ d        return 1
    1 u9 D9 q! U0 g4 E    # Recursive case
    5 E' b1 C- B( z! f2 |5 I3 n    else:
    2 s  a5 O: Z9 ]6 O% Y$ c        return n * factorial(n - 1)& j4 f! Z0 a8 |

    + `5 r* f4 F, b. q/ K# Example usage3 i' A* }: U0 j! s. H) \
    print(factorial(5))  # Output: 120
    * R, a( H( L! s+ x5 e; n  N7 l9 w: ], x, O9 E9 t/ p* y4 S! K
    How Recursion Works
    : P/ q" E7 ~: X& ^* }' l
    3 P& J3 t& i- d  W+ b' G, d+ @0 m: V    The function keeps calling itself with smaller inputs until it reaches the base case.$ Z3 \4 x% z+ E4 [% }

    # g! O! F- v6 j8 q! l    Once the base case is reached, the function starts returning values back up the call stack.; l; F8 e6 m$ j, F- ?& g( c
    2 R/ V1 M$ b) q0 @; _6 U
        These returned values are combined to produce the final result.
    / k  z# Y  B/ s; X% H! r9 H
    $ c* G# T2 s1 O6 p& G0 vFor factorial(5):) }% I4 @$ [; V& A2 u2 p' c

    - u- Y: f* y8 |
    # a2 b+ g5 L) \! j9 ffactorial(5) = 5 * factorial(4)" K3 V: V) k) \$ i, ~3 _4 c/ E
    factorial(4) = 4 * factorial(3)0 A5 {) R9 F0 d" x. l' K$ u
    factorial(3) = 3 * factorial(2)
    ) R% b1 m  M) ~4 o$ g$ ~factorial(2) = 2 * factorial(1)
    ! w& X3 a: t. @% S- m( O3 f/ f3 efactorial(1) = 1 * factorial(0)6 j$ o; n" W  ~3 M
    factorial(0) = 1  # Base case
    $ x7 q0 Y1 [7 M" T! x9 H& C8 w
    Then, the results are combined:/ t- }) R. k' v2 Q' w) h5 F

    0 V& h7 e: ~. r/ B
    + T, V9 a* y5 G8 J3 k! }. ?factorial(1) = 1 * 1 = 1* {8 l' X9 s1 k4 v
    factorial(2) = 2 * 1 = 2+ c4 }0 ~& |/ s' u2 u* I
    factorial(3) = 3 * 2 = 6, V+ ~, O' `0 I# ~( g
    factorial(4) = 4 * 6 = 24
    , P* D- p5 M8 `: f' |6 }5 rfactorial(5) = 5 * 24 = 120
    5 |& P  e1 Q* g! C, L" U# `9 v% J* z! B% |
    Advantages of Recursion
    ' |( y# A* z0 p* ^. J9 i+ l0 x, g& D. T* b5 ]7 S. S5 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)./ S8 U8 d* Q- M( P& G: K

    8 l! w5 C) C4 n* i, s    Readability: Recursive code can be more readable and concise compared to iterative solutions.: k2 R5 t7 ]3 v9 ^
    % t3 \! I! n' d. K( T
    Disadvantages of Recursion, Y; R) u! ^0 Q0 l, N* ]) \; i5 {

    2 f- \2 w3 |8 H1 T) d    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.; J: E8 P9 J5 b) f' @  M

    7 t; D- p( Q2 J" ]5 \, `    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    # Q5 z& B3 o1 ?3 n8 ]! L9 D0 G5 o3 P9 ~5 F
    When to Use Recursion- f) \  n; T9 p- N9 L$ u) P$ G1 u; P
    & g( E) Y5 s9 J9 m: u0 c( h) R& e
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).! v9 j3 a9 S/ `8 u; H; b8 ]
    9 r. q' j. W3 w' P. y
        Problems with a clear base case and recursive case.
    8 P# Z4 V- m* X' I& U, L) I3 X
    4 V: q4 h5 d# j7 J  IExample: Fibonacci Sequence
    ; J: \# s) `0 r0 D* C1 B  m% I
    * T. e+ X$ g6 x9 {& }The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 ?) F: ^( ?- J% K7 g5 ^) L/ N8 ~
    ( x8 I/ R/ Q4 y0 ]( y
        Base case: fib(0) = 0, fib(1) = 1: k& g8 S/ Z$ [% b! P9 ^5 I8 j4 e

    ( j9 h) c) B# j/ v* Y6 Q: K    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    : |. X2 e  p& X% q8 D( Y3 y5 e( _. ?/ v
    python6 o. x8 S  `4 t+ p; p6 Z& O

    ( a+ |0 T6 ?7 ?! a/ u7 Y$ D6 \/ P1 Z$ L) X
    def fibonacci(n):. Q- y7 b$ P* e" f9 `
        # Base cases2 B$ N: O* o  D3 t: j( y
        if n == 0:
    % [* G& p+ b# p$ d5 [% b( \6 F        return 09 @4 K+ f9 l6 g; y" q$ W: y1 c
        elif n == 1:: ?$ {' d5 H3 ^7 m
            return 1: x" B, z5 l3 d! N) P3 }4 t+ k* u
        # Recursive case( ~8 h( b5 ?: J. r) r2 j1 M
        else:* ]( m. P$ D& C) d
            return fibonacci(n - 1) + fibonacci(n - 2)
    5 p% [. ]1 V. v3 g
    % H5 q5 a; h% a5 g. E# Example usage
    ! m" {/ C: P; Bprint(fibonacci(6))  # Output: 8/ H$ C6 c3 r' M1 s% d
    # A+ v7 A& z. f  |: R- [
    Tail Recursion
    : l  F8 {9 G7 M% u0 k/ e9 q
    ) J3 X& }  a  m) V( r# hTail 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).- x" L( F5 L" v7 a  h

    . V7 Q' e( n: I' d& R' s- K, n/ [, U8 jIn 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, 2026-5-19 18:14 , Processed in 0.060428 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