设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1647|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 : ?( \. f9 T1 ~/ G
    # p, c; \$ {# |. p" n' h4 |
    解释的不错8 O* j7 `+ o5 d" C
    # P; S1 e. S/ b3 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。
    - T" N1 |  T  m# s9 L& Y" X
      a: p, \+ t- a2 }( Z1 m 关键要素3 _% r3 T" l" W
    1. **基线条件(Base Case)**
    * w& k, y" e; X8 t1 B   - 递归终止的条件,防止无限循环
    % A* f( G% ?* p9 Q/ b   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1- L% Z( a* z5 H" v2 Q
    - y! h9 y* w0 u
    2. **递归条件(Recursive Case)**
    ' R! f9 s3 s9 {2 Q   - 将原问题分解为更小的子问题
    0 z! U% I0 v7 ]+ `. O   - 例如:n! = n × (n-1)!5 W* ]- s+ b5 P, }% x% n) Q& }! Y) z
    / P% h) L: [6 F8 i; p
    经典示例:计算阶乘
    & t# t5 Y/ R% e3 U! C' x5 `python0 [8 z% W& T$ g8 X3 ^
    def factorial(n):
    ( J! _* |$ x5 l# D4 ]4 _" ?& L8 Z    if n == 0:        # 基线条件1 Z1 `* h/ l+ G! w- h( ?+ i
            return 1
    5 _9 Y) [* S% f6 H& C0 h% y2 p    else:             # 递归条件% s/ W. k( G! p1 V
            return n * factorial(n-1)( E4 ?) O8 G+ A( d; _
    执行过程(以计算 3! 为例):
    * y6 n1 ^. V: w5 X) R  j) k8 ?factorial(3)* n6 i& U* {  c8 t( c" e- q# I4 B
    3 * factorial(2)+ n4 w: b3 n/ l1 c
    3 * (2 * factorial(1))
    . u$ K4 O4 G( a  |0 \( g$ Y9 ~  _3 * (2 * (1 * factorial(0)))$ g  l' p. ~* \' \( I5 t
    3 * (2 * (1 * 1)) = 6
    8 n) X. p- f" c3 g: [+ V* e
    5 w/ x5 C3 h1 |8 X$ _+ A 递归思维要点
    ) o5 e5 \" R. @* B, G1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    - h6 K7 c9 i! Q  P9 N0 A2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 J, A- ]; H1 v- Q+ d. G( U+ ^
    3. **递推过程**:不断向下分解问题(递)4 [4 e8 \+ j. o  ^& X
    4. **回溯过程**:组合子问题结果返回(归), g1 M; c! W- b; P+ L8 t* j
    ; P/ _7 N2 i- ^) A7 A2 B! ^
    注意事项6 B0 k2 Q5 Q0 W( g3 E3 [
    必须要有终止条件
    1 @) b: e* P9 ?& E递归深度过大可能导致栈溢出(Python默认递归深度约1000层)8 \$ H% M8 x) `& Q- p) I
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    * Q' E( x  s* I: `. r5 C2 Q4 k尾递归优化可以提升效率(但Python不支持)
    3 y& P8 F8 J1 u- v  S+ a
    - }, N, p1 V* V9 L7 T- G5 {9 D 递归 vs 迭代& O0 {3 O0 W0 `; h
    |          | 递归                          | 迭代               |0 e# ~$ U6 H2 e  m2 J
    |----------|-----------------------------|------------------|
    9 O2 e) R, d* L3 o| 实现方式    | 函数自调用                        | 循环结构            |
    ! b& g7 o' v+ b1 D| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    5 k  d: M' Y5 s- F; K5 P, m1 m4 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         |  t3 @: F3 b7 l# l- E
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |, k4 m& ^. K( i5 a( [
    . d2 O4 d$ f& A' @
    经典递归应用场景7 d* [/ R, @) O0 I: R' I  J
    1. 文件系统遍历(目录树结构)( _! e& n7 U2 J  P8 B
    2. 快速排序/归并排序算法8 C  v  ?: {# P- N
    3. 汉诺塔问题
    5 {9 q. G+ r& C7 a7 ?/ g4. 二叉树遍历(前序/中序/后序)
      L. S5 ^; x+ e& @+ |2 h, e/ H5. 生成所有可能的组合(回溯算法)/ @3 z3 x' g1 |9 m) W$ ~1 _/ g9 }& W
    7 Y& Y6 f# E7 z6 m5 U- Q( ~
    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
    擦汗
    7 小时前
  • 签到天数: 3128 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,
    ( Z, ~  u: f) H8 P# s6 g4 U' S我推理机的核心算法应该是二叉树遍历的变种。
    1 \$ ]$ m5 n  h- n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:% S5 H5 \8 v5 u* I
    Key Idea of Recursion3 E; P6 I  C" G6 G0 U3 F% k

    + T2 v$ k! I+ i8 K& b/ Y9 [A recursive function solves a problem by:9 ^% k0 G' P! i2 E( ?! s* ~
    3 G" ^6 a- ?0 X; [! X
        Breaking the problem into smaller instances of the same problem.
    - |- a/ y4 b; |4 Z5 |  ]! I5 F9 e) G  C, a" l  M0 ^; e
        Solving the smallest instance directly (base case).
    / }5 U8 _1 w$ S) V9 d8 x7 ^) j1 @+ T4 {# B+ i
        Combining the results of smaller instances to solve the larger problem.1 r& d, i5 m$ d
    & h+ `$ k6 O, S: i, V1 R
    Components of a Recursive Function
    : s8 m; D( A. Z7 T- L8 J: ~8 @, m4 n" Z3 x8 Q- j$ B* }8 o: ]
        Base Case:+ B3 K" p3 x: e% i  ^  I

    7 @1 k8 o' y- Q. F$ i; ^        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; z0 V0 E. s0 n5 ?* E
    # Y, p4 H6 k+ N3 l( t        It acts as the stopping condition to prevent infinite recursion.  m: J. \8 n) @
    ( G' Z" K6 G! [; y
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.* H* U8 [$ x5 O9 P

    0 @9 w3 z3 @. \$ `' }" P4 |    Recursive Case:( G" N- ^' \+ ?4 W

    " p7 o1 j+ o. o+ F9 S        This is where the function calls itself with a smaller or simpler version of the problem.0 R1 U/ q0 S2 B, y3 K2 r1 {

    " |3 m5 F# X# n. m1 M+ v; _6 W( u        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).3 O, ^) q( f' {/ d4 R1 R- w# }
    9 [! V5 G% S. K
    Example: Factorial Calculation
    & ^- T1 f) ]1 M
    6 l6 y: ]3 P) m' N/ {- w1 v7 @/ cThe 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:
    9 C  [/ M+ @- Z8 d9 m, [3 Q. h% F9 ~) v: b
        Base case: 0! = 1
    0 t0 V5 S) j, ^) n5 r! ^
    2 P3 a) W5 G" j7 ^6 p    Recursive case: n! = n * (n-1)!6 z- y& T9 ~9 W/ q, S8 [) i
    & n$ J' |" a8 i# u4 b1 H/ x4 z) F
    Here’s how it looks in code (Python):
    9 l' H+ P* K. Y$ |python
      F. j) C" i7 E# q5 L" c- `' v7 S' O# A( F: {1 k% O& U

    4 Q( L/ q5 w, l" J8 _0 mdef factorial(n):
      E; e( F0 q+ {) n4 }' x. z8 c    # Base case
    8 F/ m& \' g2 X% H    if n == 0:% k; q  R4 n7 l5 T# K, j  f
            return 1
    # H- }, s) @0 b% q  v4 a! ?3 Q    # Recursive case! q1 P) n6 ~% ~3 |; c) h
        else:5 v" q; Y$ H& g, j) s
            return n * factorial(n - 1)+ x6 w* k( ^; [. a# E

    9 m- P' G6 ^& `7 v; o# Example usage' V, }* o9 {- T) D( J& c# y
    print(factorial(5))  # Output: 120
    . J) K. c) R7 a8 ]* F. \6 B: j2 s
    How Recursion Works* t+ u) o6 o' q! @5 i* O4 W" \
    2 s5 C4 f, U8 x1 i  X# S+ x! e3 @1 f
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ! v& g) _3 y0 @  M5 Z
    + ]6 E& M, \. c& l1 \" P    Once the base case is reached, the function starts returning values back up the call stack." G: L( |+ Q1 ]# Y  R; M6 e7 k
    + o& ?' j8 C& A3 t  f1 c
        These returned values are combined to produce the final result.) n9 Q" k9 s( \7 @/ b
    0 Z5 p# B! ~  N/ h, l6 ?/ A$ u" k- Q
    For factorial(5):
    $ A0 C9 S! O( T2 m8 e4 N+ S$ |8 `: P, _5 B9 v7 i  @* f

    - ]" w; v- T$ A. ffactorial(5) = 5 * factorial(4)( o# D9 Z7 F* M; v7 s! D6 Z. |
    factorial(4) = 4 * factorial(3)
    5 y1 ~6 i( L# p9 ?5 J: tfactorial(3) = 3 * factorial(2)2 G. `9 o9 _* ~: D' f
    factorial(2) = 2 * factorial(1)
    5 d9 L' V8 \! X! Q7 tfactorial(1) = 1 * factorial(0)
    ' d9 k/ x# E3 t, H( f# r0 z3 ^factorial(0) = 1  # Base case6 H. F( ~1 G$ x- S
    . B7 k& T7 h0 U: B9 c" F
    Then, the results are combined:
    8 O0 R" j" w) U) m9 a  u4 ^/ V0 _" L8 h- H% l
    9 ~; }9 N. ]% g
    factorial(1) = 1 * 1 = 16 l% S. b" s$ V4 Y2 Y6 o  }
    factorial(2) = 2 * 1 = 2
    ' a' T5 T6 y* u1 j' g; yfactorial(3) = 3 * 2 = 6
    : v1 A# V0 e+ T7 [4 D% p+ ?% yfactorial(4) = 4 * 6 = 241 e3 u& W! s: p5 G# B8 I
    factorial(5) = 5 * 24 = 120
    + \- _, m! {  B; c; g3 N+ {: C$ Y3 ]8 H
    Advantages of Recursion4 d2 T# V9 q7 H. m) `9 S* s0 Z

      ]0 B+ f4 f7 X8 c; ]    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).
    " S& q# A) [; s5 L' i; a, n6 a0 ^2 E2 M( R. \
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ; y& k. N8 u4 g: k3 w* U! V; n1 Z5 d7 B- ?( Y. ?
    Disadvantages of Recursion8 y0 A) O: G% q$ w# c

    * @0 _) h  W4 i" v3 j    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.
    # n# g. W2 T, Q5 R. E
    1 I$ ~2 f$ V! ^! I4 E# }' M    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    3 n2 o/ w: q6 N$ u( h2 B% T7 q9 [+ v& U, ]9 V+ p; @$ s# U" ?! z
    When to Use Recursion
    * y5 g  x% g5 C& P( r4 H
    8 c& ^$ ?( m/ {+ |9 E# ?    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    - W5 P4 T' R! U  d8 ~
    1 x2 u8 o$ o& I    Problems with a clear base case and recursive case.& H; n8 Z" f# I+ b
    0 p+ H& @. ^9 l, ~& y2 v
    Example: Fibonacci Sequence
    * H8 p% B( e: J- i- Y
    ; k; {/ q- H+ Y3 K& uThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 r) l& i5 `' C2 ]. G: W
    6 K% o( K5 K6 X. B: S7 p) i
        Base case: fib(0) = 0, fib(1) = 1* }8 Z7 _- ~% j1 \
    % T$ e* n$ s5 s2 [5 T# g
        Recursive case: fib(n) = fib(n-1) + fib(n-2)$ ~6 @* b% L+ ^" g0 U

    * J7 g  Y9 z. x, z: {python) t  z, F" B/ @% b9 J5 v

    5 H$ ]$ S9 Y* H* l' i8 n) c( y: _6 Q! t9 g: j. M& e
    def fibonacci(n):
    2 \* l& A7 K. q3 X5 L% {    # Base cases) I# _$ F) j7 B: E; N7 U0 d1 A
        if n == 0:
    9 R3 A/ c, J, R8 p/ c- _* L2 t        return 0
    ' ~- q7 v1 D) Z& }4 g! E# B    elif n == 1:- H5 D4 y1 A( a4 d
            return 1
    3 Z/ I+ i$ s; n    # Recursive case: G/ Z9 r4 N: K$ U' q
        else:% s+ U8 B4 h# }3 A, n" Q  u5 t! q
            return fibonacci(n - 1) + fibonacci(n - 2). S# {: G' w) s& z

    2 F( K6 |. m4 n# ~# Example usage3 a& q% p) `, ^- |. j
    print(fibonacci(6))  # Output: 8, d6 g6 `3 |3 ?# b! d% ]
    6 r% o5 T% ?, H! J. G  T" a
    Tail Recursion
    " e' ~5 X) o/ @, w! v- [3 v! I  C. g+ k3 D( I& s5 \) A
    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).
    ) r. R$ A4 O& ?
    8 g2 i1 `; U2 X6 s9 oIn 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, 2025-12-26 14:10 , Processed in 0.031447 second(s), 20 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表