设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1497|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 * S( c- _1 U  V3 D9 t$ \6 h6 [
    # C8 d( P  L7 t$ |5 Q
    解释的不错- H, m/ @3 m* @; i+ J: n

    + w1 _# @/ Q! n7 L" m7 h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。% w9 M1 v# S6 [) z8 a

    / V  z& s& X# S* p 关键要素; x) N, c- g3 O+ L, v
    1. **基线条件(Base Case)**  u* s+ }4 Z8 A2 L
       - 递归终止的条件,防止无限循环
    % _" I( A9 y/ L) U. B7 M" k   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% R$ n  M8 m" f" |9 L
    $ ]. E4 B1 `+ C' R( J; C
    2. **递归条件(Recursive Case)**
    - }) Y/ V& S3 i; h* i$ A   - 将原问题分解为更小的子问题, I2 ^0 o4 R5 j8 N
       - 例如:n! = n × (n-1)!& [7 J# b) {$ C1 Z7 w% b: S$ U7 B- `
    , b: T% D5 b  E( C5 D) D
    经典示例:计算阶乘6 y" h* D0 u5 K5 o1 N. e
    python, ?! c6 K: ?5 ~2 T8 a
    def factorial(n):
    3 _, E( ?1 ^7 [3 _% u5 j    if n == 0:        # 基线条件
    $ L' {% `  {5 r1 j' w        return 1
    3 n) b5 Y- w* o5 w& F    else:             # 递归条件
    # C/ i9 l) g4 O        return n * factorial(n-1)
    + x7 @; F7 E5 x# }& U2 C+ \* T9 w执行过程(以计算 3! 为例):6 u) V3 E6 a# u: n$ R+ R
    factorial(3)
    ! ]2 `- U0 q6 K3 * factorial(2)
    - \3 x( n# C# A% A5 J' M( b' K9 b# ?3 * (2 * factorial(1))4 c3 \& e8 f) F' G, o
    3 * (2 * (1 * factorial(0)))
    6 t2 \3 L8 L3 ]( Y% g3 * (2 * (1 * 1)) = 6" j4 B" a$ p3 j+ {8 v1 d
    $ R0 b8 h, a. ~2 |4 }
    递归思维要点
    " H1 }% C; [, q8 {0 {1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ) a4 _# e' h. Q2 v" }2 k0 s( k2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 d6 m; w$ {: c( ]
    3. **递推过程**:不断向下分解问题(递)
    + F+ b+ C% Q) W3 u4. **回溯过程**:组合子问题结果返回(归)
    ' }. r9 s- @% W% l  q2 ?: n4 x
    ' D2 R$ E2 ?2 g& ~ 注意事项* _) j# x  J$ b: s0 L5 Z
    必须要有终止条件- |% D# l! J: L$ c
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)) F0 X1 \1 h, }' l" T* @
    某些问题用递归更直观(如树遍历),但效率可能不如迭代, e1 }. g: Z7 p( m" [
    尾递归优化可以提升效率(但Python不支持)
    - Y1 `1 P# ]0 V/ e9 U7 a3 w
    + f' R% z) e  O2 v4 ` 递归 vs 迭代
    - x1 s$ m: O1 _7 P! s6 J|          | 递归                          | 迭代               |+ D4 A! f1 K  a' W
    |----------|-----------------------------|------------------|
    ; a" S( h. i( |: v* S8 d. R# W6 s: I( S| 实现方式    | 函数自调用                        | 循环结构            |
    * g5 F) S, D% G| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    $ n/ L) z, h9 t3 ]. N, ]| 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         |
    - R) I( W7 b" [6 I| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ' N$ x" b3 O0 P  j' G& o. s% X
    " z: W  t% I1 j  B5 f: _ 经典递归应用场景7 v: B4 X  d- T
    1. 文件系统遍历(目录树结构)
    1 _, ~/ n) W  S2. 快速排序/归并排序算法
    - v  g6 v3 ]! L, }6 N! U3. 汉诺塔问题& L, |$ G  r, ?* b2 x
    4. 二叉树遍历(前序/中序/后序)# B+ K; Z) q4 L
    5. 生成所有可能的组合(回溯算法)
    * M5 Q  \, D4 d' Q+ f; k3 O7 J$ Q
    # k1 q4 C0 x. F; `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
    7 小时前
  • 签到天数: 3116 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,4 Y7 u0 W! @* p% p* q
    我推理机的核心算法应该是二叉树遍历的变种。2 l( |3 C) `/ c
    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:4 Q4 l( X+ @! P  N
    Key Idea of Recursion
    # _$ `5 e/ h0 f+ Y6 q
    + V/ ^, u* A2 FA recursive function solves a problem by:# }- ~' m  c! b/ i! \
    , Q- W( p% ]1 Y0 S" t# `
        Breaking the problem into smaller instances of the same problem.
    ; e8 i) X" j8 c- j4 ^0 N* D+ p7 h( j+ \+ x1 b* g
        Solving the smallest instance directly (base case).4 C/ r9 }* Y+ [5 _+ Z- x! E
    3 S2 G+ X" R5 }' e( H) g0 T: L
        Combining the results of smaller instances to solve the larger problem., u2 A6 k9 R$ e. e9 b

    4 S- X3 d+ j  A: y; jComponents of a Recursive Function
    3 w" n7 L6 k# k/ \( @" `
    5 N/ M% ~; i* k    Base Case:$ ?$ c6 l, Y/ C
    ' D' Q5 b- `: @& g
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.' m2 I' [: z8 u* u5 V  j+ R4 p' Z/ j

    1 |( f, d- d6 D# |        It acts as the stopping condition to prevent infinite recursion.
    ( y  ^) \+ e( j3 U
    0 N: y; k8 P7 B) w        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    0 J3 Q7 v% z6 G! e7 M# \! X
    : z$ _- e% u- Z1 o2 n+ E    Recursive Case:, s: O+ y) N! S4 N0 P
    5 V% k6 e9 R' G, v: {$ o
            This is where the function calls itself with a smaller or simpler version of the problem.: k( \0 y2 o; ]& Q& i1 c
    5 L2 ~3 G: r8 D: z
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' K  E6 Q. y5 h# g# B4 p

    - N9 F+ ]/ [& {5 p1 L- z& ?% kExample: Factorial Calculation2 [4 Q8 e- Y+ [) O$ w9 |1 }% f: ~/ c
    6 m' E8 w, Y; l7 y4 p, X8 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:3 W% N2 d# t+ ]; u; L

    4 R, Z! B' a% X    Base case: 0! = 1# a' U" Z2 M; ^% q. n
    , o# R+ D. m( v1 h6 k( }; Z6 ?
        Recursive case: n! = n * (n-1)!- C- r8 j" _' O! q

      Q1 S) m& ^: y* f: CHere’s how it looks in code (Python):* d9 U* n& F/ Q$ l! }3 R% ^
    python
    " t$ e, {; |6 H5 u0 ?6 ?2 D- L+ c, k

    - i6 T3 S4 j8 z, w% s" }def factorial(n):/ ?0 k3 h. Z8 x7 I9 B2 p. |
        # Base case
    + G. A% s+ U  V5 l9 ~    if n == 0:4 G. W3 g0 Q% Y
            return 1
    # M( `& o" o$ j* s9 H. m! W    # Recursive case  J" x; I; L! l5 ]. j
        else:
    3 l( L5 {2 y. {, C8 l( h& ~        return n * factorial(n - 1)
    ' q& O8 \  a9 @, Y. V1 n
    ( h# ^. g, {8 Q- k( H$ F/ I1 O% s; y# Example usage
    ; L1 _- ]( B6 ?6 o5 v; r: {# }/ C5 fprint(factorial(5))  # Output: 120% l3 B4 _. W( F+ D' I
    4 I+ C% r" V6 X7 g( n
    How Recursion Works; V! c, \! m* i0 u

    % Q1 F' a! N/ B0 X0 R- H$ f    The function keeps calling itself with smaller inputs until it reaches the base case.$ D+ h7 ~* E, `$ [5 Q+ T) ?

    % b5 h9 j2 G2 v& r( J$ W" b    Once the base case is reached, the function starts returning values back up the call stack.$ Y+ N, [2 i+ n; K
    4 d) `  J/ M) u; R& F- c) t2 L. A
        These returned values are combined to produce the final result.
    - a" N1 E7 u; z  ~& [& u% r- W7 S% l4 c* v; Y: t" y6 m5 ^
    For factorial(5):
    & t# |6 A$ u7 u2 J5 W1 Q) W5 f2 P3 i

    ) R7 Q; Q9 w9 S; |9 efactorial(5) = 5 * factorial(4)  S9 t1 [) Q  F% \( X9 }1 O: B
    factorial(4) = 4 * factorial(3)
    + `. }" K- P( Y, }& L7 [factorial(3) = 3 * factorial(2)$ J+ R7 L, Y  V: i
    factorial(2) = 2 * factorial(1)  ^0 @, D$ T) M4 f5 l0 Y
    factorial(1) = 1 * factorial(0)2 ?) W! _$ E& \4 V9 w
    factorial(0) = 1  # Base case8 u4 J" I; u6 U( ^( h
    7 r% m% ]# Y1 O8 ], r- N
    Then, the results are combined:
    2 b6 M8 J. f* l# [7 \6 Z, \
    # Q7 k3 d9 c* r% O
    . a! r2 }" T: w; f3 x! X" `factorial(1) = 1 * 1 = 19 R0 w, F2 V" |2 f/ r' t3 v
    factorial(2) = 2 * 1 = 2- D7 t2 _7 [% r# `$ O: d3 X
    factorial(3) = 3 * 2 = 6
    & H2 l, _9 ?3 w' Z+ U/ `factorial(4) = 4 * 6 = 24
    3 q/ U6 ]9 T8 [, ^; r2 gfactorial(5) = 5 * 24 = 1203 L3 `: ?$ L. b9 p6 ?' c

    ) \$ @1 N6 p9 E; l: [$ QAdvantages of Recursion7 C8 m$ O9 J8 a+ s$ |" C
    ( x6 f# ~% v* O7 x$ C) R. ?
        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 ^. f% h1 F# Y: F
    , V0 A3 h& f, W1 t% t! V    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ( C, N5 Y5 y3 E( P' R: e: I% c( v4 q6 A  {- {+ U3 k; @
    Disadvantages of Recursion4 l% M; q* ~9 b/ B8 m/ ^

    9 X; V; w! k# j7 f$ z6 l9 q    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.0 o; P* P2 M* O& z1 T8 d' k
    ( u0 N3 T/ `" m1 j8 f  b5 X3 e" Q
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    $ S/ }, i4 i$ e) n) Z9 e9 p) g4 R& r5 }# R  t) E; C0 _
    When to Use Recursion9 v) t# [# [( P% P
    . A, {9 X- H0 G; H1 g
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ) M" _% m- G4 M2 D
    $ R( {' q0 e; i" Q    Problems with a clear base case and recursive case.
    3 ~/ \! X3 u& C$ z, m" V4 v0 Q4 R" w; `4 c9 I+ W; ?  m8 s  ^4 P& f
    Example: Fibonacci Sequence
    / d6 K0 h, e8 n8 ]0 m: l/ o
    / q2 s  t8 [$ {# i6 G; aThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:8 ~: M- t: A: O8 }8 L. _$ e

    * X0 A% X' u- m0 U# M    Base case: fib(0) = 0, fib(1) = 17 M. x1 m. \5 f, d2 q) _, b
    " H, B) H3 Q' ]& ]" I
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ) p2 `# z, ]; l9 C: i' n
    5 Z" k% K( c7 F+ c2 X/ _- Hpython
    ! `; I) f: `. z% V5 E( l# Z: X. {/ K9 ?
    - J8 V8 v) b* M; l
    def fibonacci(n):0 F9 _/ J, u( ~, x% k
        # Base cases# L: t+ a/ z0 X9 ~! _- r9 ~' d
        if n == 0:
    1 A/ q7 z2 j/ ~6 l3 W! Y: _        return 0- G2 ?0 ?) j" \3 Z& o
        elif n == 1:
    * X' ?& v0 W0 B( a" _$ f6 Z        return 1$ X( @1 C- z/ V3 U( _- q
        # Recursive case$ A) L& C! F/ ~! |0 t4 B
        else:8 |; }7 K3 Q9 _2 u1 q) ]5 r7 s6 E3 N
            return fibonacci(n - 1) + fibonacci(n - 2)
    ' `( X, s1 e7 h/ m' }8 E% v$ G4 [
    # Example usage8 M) d* ?. }+ M! X9 l
    print(fibonacci(6))  # Output: 82 d2 E0 z" C# U/ I0 F0 S

    ' O3 [2 p" o) H0 D: Y$ _Tail Recursion0 T: S* b  u3 b( n: U7 `: p
    ' M0 U: E% q+ c# j; O# x, l. E
    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).& |) G" ~$ f# M# t

    7 A: N0 h" C5 ~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, 2025-12-13 20:27 , Processed in 0.035451 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