设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1545|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 * V  N( h* Z, {3 V% \$ l! n
    ! j: `* o2 n# E2 O8 }) y
    解释的不错  X0 f# h! C7 u6 J0 v
    , S# }# `! V" h4 t
    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。
    4 C0 C6 R( ^4 X$ j+ w; V
    ! r6 _5 P0 Z5 R 关键要素$ |! L) d. f& V6 z. ~8 Q* e* F
    1. **基线条件(Base Case)**
    ( g& V$ m1 g1 `5 G. X4 g1 o& ?! g   - 递归终止的条件,防止无限循环
    5 y  b3 e, f1 m2 K! ]5 C3 [   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    6 U6 b6 r: p/ ^8 A6 _0 D' j1 V9 k5 M) x. S9 h
    2. **递归条件(Recursive Case)**
    ; [9 d) i8 g- E   - 将原问题分解为更小的子问题
    $ I) }) D0 R; t& k/ G+ G   - 例如:n! = n × (n-1)!
    6 N+ C/ j5 g/ T
    # y* i6 H9 M% c! p' N+ e6 C4 P 经典示例:计算阶乘
    2 \  Y( D+ H/ A0 h; V" mpython5 B9 t0 E8 ~, T7 ^  C3 R2 K9 G- v3 Y
    def factorial(n):
    * ?8 \9 d8 @. v    if n == 0:        # 基线条件
      R8 \/ e, L& h" Q  ~        return 1/ t  M) a! h2 T$ G: X
        else:             # 递归条件
    5 t+ L7 {0 {% }& B. K        return n * factorial(n-1)
    ! R& M& v( H2 J; ]$ j执行过程(以计算 3! 为例):, O3 s" Y1 r) J6 u
    factorial(3)" G" T! T" K. d( j
    3 * factorial(2)$ D3 _' K: I9 X' ^4 L
    3 * (2 * factorial(1))/ o8 K8 M: B& m9 |* z
    3 * (2 * (1 * factorial(0)))3 K' V! l. Z( V6 y
    3 * (2 * (1 * 1)) = 6
    5 X" {  v: |. X- }- x
    $ \+ @. X% L# H, Z8 L5 L* x 递归思维要点
    # M) l9 Z- R  O/ I1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    % }8 ~) j( ~8 C) }& R- r/ s2. **栈结构**:每次调用都会创建新的栈帧(内存空间)" @) e% ^( K0 X* c" N
    3. **递推过程**:不断向下分解问题(递)
    / n2 {/ ]- {6 Z- |4. **回溯过程**:组合子问题结果返回(归)5 x# R5 R6 a, {) p% R# l
    6 |4 h* @8 I0 n# }1 X' p
    注意事项4 Q  e$ G% }5 ]5 O( [/ M, V
    必须要有终止条件
    ( m+ g6 h5 b5 }) P6 I) b递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    : W! G8 i$ p2 _9 W某些问题用递归更直观(如树遍历),但效率可能不如迭代9 d# {% o- D0 D6 `
    尾递归优化可以提升效率(但Python不支持)& }% {1 S: a6 E( e2 b5 |
    ' h5 I8 j' t* P; t% F1 {
    递归 vs 迭代9 d  O) e4 c& _3 M& Q
    |          | 递归                          | 迭代               |
    + Q. y: ]  H/ K3 E: u' {! J|----------|-----------------------------|------------------|4 _/ C/ _3 M0 D# O$ @, y$ S/ A% g$ K
    | 实现方式    | 函数自调用                        | 循环结构            |% a1 ~, \- h3 ^  j3 Y% S$ ^6 k. y) R
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ( v  ]( t: ?5 z1 |0 e5 O| 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         |
    - G' y9 t' Q6 x3 l9 ]4 [/ I| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |0 f1 j  v8 n3 U% F; ]3 S+ @
    ; Z/ _0 F" p5 Y" d% N2 G% J
    经典递归应用场景5 `0 q/ l' O) }$ X+ @
    1. 文件系统遍历(目录树结构)
    4 u: Z* ?/ z3 ]5 F0 M2. 快速排序/归并排序算法
    + F% t- Q3 Z& ?5 M3. 汉诺塔问题
      S3 s' s) N1 |: E' E4. 二叉树遍历(前序/中序/后序)
    + @; y" k2 w1 O: s$ Q# E: g5. 生成所有可能的组合(回溯算法)1 I! g5 q) m1 I" _  P" e/ a: |

    $ t: N( J) L! X! p# T1 c6 t9 C+ M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
    奋斗
    1 小时前
  • 签到天数: 3120 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,
    ( c% P. Y- b& _* B6 b6 \1 j- g我推理机的核心算法应该是二叉树遍历的变种。
    * I, F5 m# n" p' E- l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:, C. t) w2 m8 G: J& X; U7 I
    Key Idea of Recursion
    1 N! }: V2 w# X) C
    2 a# k9 ?5 d3 }5 B/ e  lA recursive function solves a problem by:7 B3 |& ]0 e; l7 X- b% ]+ |( J& V

    + f/ |8 E5 Y% m2 y* }: s7 q: R    Breaking the problem into smaller instances of the same problem.4 M9 U" |5 G4 v9 c2 t; `

    % E$ J9 X; \! T. a# N    Solving the smallest instance directly (base case).
    ! `& \& e; R. |7 O7 v0 ^$ Z' a: }  I# W7 w$ G
        Combining the results of smaller instances to solve the larger problem., \' _3 E4 G0 N0 a4 x1 P( \6 w- k" |' l
    / b% j5 l4 W9 J8 f, x
    Components of a Recursive Function, d0 j( F) O; s; D' h) \3 R
    5 C9 W) b$ L- }6 K7 _6 \
        Base Case:4 v: @" f( |  k; E# |8 v: ?; z0 f
    9 |* _' \$ `5 |: {  p3 i5 Q6 _; M
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.! u; J: P3 T$ Y8 Q; t3 u

    ' ?$ Z1 P2 q, @* R7 [0 d        It acts as the stopping condition to prevent infinite recursion.$ k. k/ }) D1 O) e* d0 U. |

    3 d1 J( k/ U1 ]# T/ x5 h        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ E/ b% F9 o8 _. V' B5 W' m- Z  f
    ( N  _5 }6 \9 H( {7 n
        Recursive Case:! `4 `1 X' g+ F/ Q

    1 J1 L- D6 _% p3 E; x        This is where the function calls itself with a smaller or simpler version of the problem.
    " ]2 T& Y8 |1 ?  k! H5 Y% l' r0 c7 \" ]
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    % G, d# o" O! h+ s* j7 ]; C0 z" C, r: v- A/ B" V4 ]2 S
    Example: Factorial Calculation9 b% p' `) P- a: A2 ^/ L

    3 @' ^8 g1 g6 i) a: \' @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:" Z7 u" B5 ]7 E' ]' G
    0 X7 @3 O1 `3 a! k0 w* ?
        Base case: 0! = 1
    - |; F/ h0 e1 z' T2 X& {
    ( [9 Q: A. R. S- X  A& B9 T9 z    Recursive case: n! = n * (n-1)!
    2 l1 x# C: d' W8 @( p& X' l+ L8 R; L7 V( S. t
    Here’s how it looks in code (Python):
    7 h( j0 q, R! y. S. ^python
    " p" |( i( U1 g3 }  A
    8 [/ I; K8 Q  l7 `) x" H. ^
    ( S, _& {8 s7 U# [! Gdef factorial(n):6 p( R. n4 D1 X5 o1 f- d& j0 ?4 ?+ ^
        # Base case
    $ P- p9 V, S" f& N  D% A$ P/ \    if n == 0:5 v+ e* i+ D# r$ ?( @
            return 1( ~2 y! j6 w. l' z7 @
        # Recursive case
    ' ]! S- E4 [  g. d    else:0 |7 t( a  }( B9 ?6 y
            return n * factorial(n - 1)
    / C, w: M! X* d. I$ G; ~' C" f
    ; i( L2 k4 z9 ^5 w: I6 P# Example usage( d! @( O5 t: n5 I
    print(factorial(5))  # Output: 120
    % e( \0 ^/ v7 K6 ]' n. O: F7 k, `: |/ R! S6 n
    How Recursion Works- e4 B  S. _  o8 f- o
    5 j' I' N# Y- Y- O. G& |6 d( i
        The function keeps calling itself with smaller inputs until it reaches the base case.
    9 e% e; D3 o: I6 p5 {. u8 B' W/ P) h: \* ]
        Once the base case is reached, the function starts returning values back up the call stack.2 W  K4 F6 l& N, k, N

    & l* V6 Z  K; k. R2 K    These returned values are combined to produce the final result.
    # g; Y6 a3 W" Z+ V1 z6 c9 N+ m8 m8 }1 k! M7 D
    For factorial(5):
    $ `8 \) Z" G( K* i5 X2 Y0 t3 e0 I6 ]/ u! i" l! y0 W

    ) P* e3 _, p( o4 Ufactorial(5) = 5 * factorial(4)0 ^& Q. D. U7 {0 V% ?- v# n
    factorial(4) = 4 * factorial(3)
    - L2 Z# @0 j; B3 H7 lfactorial(3) = 3 * factorial(2)
    ) B+ \/ M9 K8 e7 A8 M: |( S# f0 v6 Wfactorial(2) = 2 * factorial(1). v8 ]  ^) O' G- U$ A
    factorial(1) = 1 * factorial(0)
    * B7 h! y5 b( W! b7 k! {factorial(0) = 1  # Base case" a0 i6 n: R2 ?/ ]2 L
    ( B/ [* ^& b0 A- r0 u
    Then, the results are combined:! D8 W. ]. a2 y1 Y9 |) f* P
    $ p/ H8 I7 _1 j' C% E9 V
    & O$ D" g) C5 ]) K( t- T
    factorial(1) = 1 * 1 = 1
    7 w, E7 s4 h' Ufactorial(2) = 2 * 1 = 2
    , Y' c8 U. q, U  Qfactorial(3) = 3 * 2 = 6
    7 G9 n  x; g1 z" ?! \! V9 O& ifactorial(4) = 4 * 6 = 24
    ' q( f, J+ R0 Q' ?% Yfactorial(5) = 5 * 24 = 120
    # |3 I0 a: f& h" J
      j' N8 c3 c- N2 C1 kAdvantages of Recursion2 D- g7 |2 q% O& k; b0 }- o

    5 C" r# |1 p3 v0 L! Y    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).% k4 z5 o+ h: K" N4 n
    % D- |7 N! s0 k7 q9 Y" s
        Readability: Recursive code can be more readable and concise compared to iterative solutions.6 {0 n; V( W; t

    ( Y% [' E0 C! D" X2 t. uDisadvantages of Recursion' A; s: U0 G3 X9 h, J; {0 A

    , E6 N! m/ j$ ~2 @* R  a7 b    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.! n5 N& \+ o2 A# ^3 a/ t& F

    # P& k( G6 C' f  h" }3 n    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).! [" d& u" r& P- `+ W$ v

    # ?- R: I2 B( V1 v/ aWhen to Use Recursion
    8 ]% k* Q  A; h4 T2 j2 Q1 \  J' E
    1 x# z  K! U+ ?    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ) t9 ?' f! \8 t/ i; N5 @) w2 D, c
    + d) _6 t0 {" J. k: I9 W; Z    Problems with a clear base case and recursive case.
    ) b# R- X3 @1 Y' x+ G: T: u5 w) N3 g3 b: F5 J! m- @( [4 v
    Example: Fibonacci Sequence
    ( U+ R/ u# [8 U. s$ u3 b: r- |+ h9 A9 D
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:) K, o0 g2 A- a/ u* L
    $ B7 x) Y; k5 S' ^0 u+ q" ~& G7 N
        Base case: fib(0) = 0, fib(1) = 19 G6 s8 q0 y7 w* `5 k1 |

    ' ]7 O: g8 ?+ V' S9 x    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    $ Z, b! _8 N" D+ m# X) m) }
    % |2 a$ }1 c9 z$ d0 mpython$ \- l8 j3 z! F$ G

    4 ?4 E" K1 _/ L8 I! y% e. C5 \$ ?  g$ x( F  U9 j
    def fibonacci(n):, r( O/ {8 Z5 \! |6 w' V+ S
        # Base cases
    & }5 E0 t3 u. z( X0 @" h    if n == 0:- Z( a2 f3 I# j; l
            return 0
    ( p& u2 k- n' T) F2 v; t8 s  D; G0 x' i    elif n == 1:, @; i, o- _9 e3 a
            return 1( r9 Z( R9 r8 G4 ^5 u4 F7 q4 y
        # Recursive case+ u* I* n( X4 d* W9 [( S; Q/ u
        else:
    3 F, K: w+ a9 J# N( W1 j        return fibonacci(n - 1) + fibonacci(n - 2)* {$ m4 [. ^) J, O: f3 n

    : }* _+ c+ K3 e% o' v# Example usage
    . \; t8 S% u- p# G* Q5 ~$ g' Jprint(fibonacci(6))  # Output: 8* n5 I; D# ~- d% ~
    # Y% `7 M5 d2 r# y0 K, x1 S
    Tail Recursion, W7 Q& f) R; j4 r

    0 l9 Q. s: g& u# O( \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).
    1 l  k6 K' e1 V- q
    1 @$ z4 X0 @4 n* N4 y5 qIn 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, 2025-12-17 08:59 , Processed in 0.031398 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