设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2573|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 . F1 p: `$ h" ~4 R* n. z8 _

    $ u" V) T! A; D" a" K0 s解释的不错
    6 C% p* b  x) D9 v, s8 C: t9 ], L1 K$ r' n: [: y
    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。
    / B/ k- N5 A6 x4 K" H
    5 j8 ^0 R, V; U! X# v, [2 y  {) A7 V4 E 关键要素
    ' z- L  z% M2 R- g7 k3 N) s) R1. **基线条件(Base Case)**8 d/ t- x9 V* j  d" U+ t0 S
       - 递归终止的条件,防止无限循环
    ; L8 o0 D4 W( f7 g/ U2 ^8 Y   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ; I% U1 k. ^# ?, J" U5 K6 q2 q/ B% W+ ]- j& V
    2. **递归条件(Recursive Case)**
    ; ]& y& @, `1 g+ @, m   - 将原问题分解为更小的子问题, N3 |3 G) B) S4 X0 D1 |* w3 x
       - 例如:n! = n × (n-1)!
    % n! H- m2 \2 x+ z& m# @2 S) X+ [& Z0 X
    经典示例:计算阶乘
    ( X  v  C, b2 v6 }5 \2 H  d/ Hpython
    4 r% j8 `  C# V9 [5 ^5 @  z$ Pdef factorial(n):* i# t1 a5 h5 }7 Q. m3 o; O" T/ d5 n
        if n == 0:        # 基线条件
    ; v0 K* }" u: D. ^        return 1
    0 ~0 g) g7 G# c; d8 S    else:             # 递归条件
    7 [1 q( [) n, ^: l+ J6 M6 D/ T: J        return n * factorial(n-1)9 O7 s8 t) Z! k) w/ e
    执行过程(以计算 3! 为例):% E6 z9 N) G2 l  B6 G
    factorial(3)& K1 M* A- c8 T0 h) [1 O5 J
    3 * factorial(2)
    ; `; H# T  v/ g6 Z* d) s9 O! y. t! C3 * (2 * factorial(1))
    2 j- Q3 w0 ~* A5 v6 n% ?; ^3 * (2 * (1 * factorial(0)))
    0 F" s0 G/ K7 ^+ ^4 S3 @/ m3 * (2 * (1 * 1)) = 6
    % ^" }/ l  e8 M+ e( f
    / P3 |/ |$ z, ?& H: ]" y1 a 递归思维要点% C1 R# z  g$ Z  }
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ' ?" u/ F1 {3 q5 o$ ^: Q2. **栈结构**:每次调用都会创建新的栈帧(内存空间); ]/ a" v1 p( b- j0 b
    3. **递推过程**:不断向下分解问题(递)8 q3 E" q8 d% t" d
    4. **回溯过程**:组合子问题结果返回(归)
    ; I! W0 P" L& e1 I7 A3 J
    + i% B" b5 Y9 e7 j 注意事项
    * e" ^1 s8 v2 \3 z必须要有终止条件2 O: |9 M, s3 b" i1 c
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层); U3 M6 t( B3 Z8 o+ {$ ^- c( P
    某些问题用递归更直观(如树遍历),但效率可能不如迭代, T0 H+ L$ J  G. u$ ^# K+ [( M$ h1 t4 H
    尾递归优化可以提升效率(但Python不支持). a9 Q; r& e# @  s
    6 W, V# @6 e1 @8 S. r
    递归 vs 迭代9 ~: G6 Z5 P3 {/ m. W6 J" ?! c
    |          | 递归                          | 迭代               |+ e/ r  @' t6 Z/ B* R. C
    |----------|-----------------------------|------------------|/ j8 n9 F: c9 o0 ~+ m& z
    | 实现方式    | 函数自调用                        | 循环结构            |
    , P4 f; r  M+ t0 J) @| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ( j# p# z1 l6 v1 R& F* H| 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         |
    9 [7 v. X& `( L1 U$ l2 z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ; I& T/ }$ ?. x9 }( P: ?+ m
    4 a% l3 L( @4 V/ q% X 经典递归应用场景2 S) w) Y  P' K* Z/ }; ~+ i$ j
    1. 文件系统遍历(目录树结构)
    ' C) O9 j! ?* y6 C) m2. 快速排序/归并排序算法1 S, F4 i) Y3 i/ v. P6 [
    3. 汉诺塔问题1 o" z$ `; G1 Y4 O& @
    4. 二叉树遍历(前序/中序/后序)
    2 W) x. j0 @1 u& F4 e5. 生成所有可能的组合(回溯算法)# t" q! E1 e3 \/ [
    3 k% J' Z+ W" _& r# B" N
    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

    前天 07:21
  • 签到天数: 3224 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,3 E/ ?1 j+ `* T, R1 K1 u; W
    我推理机的核心算法应该是二叉树遍历的变种。
    ' V; K: {4 l6 }& ]$ i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:% O+ B0 g% _3 }7 R
    Key Idea of Recursion
    & t1 F" e: ]2 U4 D, b! s6 |) ~) w+ `
    A recursive function solves a problem by:' R6 B3 F. u' v+ i6 u9 M
    + s. l2 P" E( e: ~. j9 |6 U, o
        Breaking the problem into smaller instances of the same problem.
    0 g) H7 w4 r* e- k* w9 K5 X, R8 T- l8 F6 b8 O, }
        Solving the smallest instance directly (base case)." E; g, n) A8 b) e( `

    4 O7 H$ f2 A* \4 z/ d    Combining the results of smaller instances to solve the larger problem.
    ' A4 i( a0 X- P" ^; ~' B1 U
    * j2 R# }2 @; ~5 o9 q3 J' cComponents of a Recursive Function
    * i8 u3 d: m+ G+ q, @% i+ C) F* \, {4 f$ I7 B+ i8 `0 Y
        Base Case:( ?  D- b6 S) [2 g/ O6 t0 m
    ; K4 H) k3 _& f  X0 z9 a% ~2 X
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.7 c! a: H- a$ s; Q& b( {, |+ I

    9 A9 B- Q$ T, P2 n( F        It acts as the stopping condition to prevent infinite recursion.# O- V, W5 ~8 X$ b) H+ T( M$ {
    " J" k) M7 O. T8 ?& b
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    . B, K& i( _; p4 L
    & T7 d+ `" ~9 E' B) e    Recursive Case:5 d) {, ]' @' k5 n/ C. T! |. `( p
    # i  e0 F2 d/ Y
            This is where the function calls itself with a smaller or simpler version of the problem.. U. ?1 ^0 p' ?) w, L& n+ u4 s
    7 r0 B( q, K$ l) {: {. l: L2 ~
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).9 E1 B9 p$ `# J; P3 c

    . b3 A4 S5 V+ eExample: Factorial Calculation
    7 h, b" H- h0 K, ]2 O9 F4 j
    . I6 T4 E* E3 f% V# c5 @) m; ]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:
    $ L; a3 P$ B& w8 l0 k" B" K
    : x$ R( z& W' O7 n# c' L5 R    Base case: 0! = 1
    7 l; m* @) E: }: M
    7 Y& U; D: B2 l5 O0 S  P5 V    Recursive case: n! = n * (n-1)!0 A+ g) L9 q1 U0 l1 `& ^
    , J$ K' O  A, Q* f! m' ^+ \
    Here’s how it looks in code (Python):) l5 `5 J0 X  m1 f
    python9 E6 z, k6 g; w% a- R# v

    7 v: W! s6 o$ |& d# L4 q. a
    + p; v" H5 W+ n! e* h. ], s: t; sdef factorial(n):
    % p8 _3 Y7 v  j+ x1 B6 f    # Base case
    # w/ z% V* H; m" Y    if n == 0:  H1 Z0 N- k4 X1 B9 y4 p) R* D
            return 1
    ; y. m8 e, w% K7 ]0 m$ ?    # Recursive case5 e1 V' \; w6 r( L1 K) J, ?
        else:
    / |# |: }" J# E/ K* I. I* n$ q  E7 M        return n * factorial(n - 1)
    3 K: _- x7 B% B; \" j% s/ H
    5 i$ T- Z: d# j$ T# Example usage/ n& `" ^. H8 ]
    print(factorial(5))  # Output: 120
    ; q9 F( u( A0 d
    3 g# {6 ~9 r) _# k7 yHow Recursion Works. z* M# n. Z5 @7 Z: q

    ( g' l6 q& C" Y; p1 H& P    The function keeps calling itself with smaller inputs until it reaches the base case.2 t, X4 [8 [* Y
    8 v2 l- c! k0 q, m  |% `! D- @
        Once the base case is reached, the function starts returning values back up the call stack.
    - U3 V2 i  j% t- n. c; ^0 G- @: s7 n6 K2 q( P2 `
        These returned values are combined to produce the final result.
    8 M) a! h8 N  S5 h! T, y7 G; B- o+ N# d: W+ j! r
    For factorial(5):
    / w' L7 E: k6 c+ r6 o8 p/ U0 o$ k% Y" t5 t1 _: a

    - {# h( E0 L8 b( A, qfactorial(5) = 5 * factorial(4)
    4 A: A3 h- h7 g; u' z, @( bfactorial(4) = 4 * factorial(3)
    7 f% p+ Y. _& G% x) B1 g2 xfactorial(3) = 3 * factorial(2)
    . a5 z$ `: h" z: ~2 Pfactorial(2) = 2 * factorial(1)
    - N+ Q  M+ }1 ?% w6 n1 q5 Efactorial(1) = 1 * factorial(0)
    . X3 {' S, D5 k; cfactorial(0) = 1  # Base case( u1 n" |3 [9 }* |" _2 ?$ Y

    ) ^' H. A7 F2 Q, l' a5 ~: RThen, the results are combined:' Y( X1 U7 P  A' H! Q
    1 I7 [! v; _( v7 [! P  ^& y
    % k+ Q9 E$ C5 i* O; U! S
    factorial(1) = 1 * 1 = 1
    - ~3 i; z- M: Cfactorial(2) = 2 * 1 = 23 s, j/ B* w1 d. ]% L0 D  y: c
    factorial(3) = 3 * 2 = 6
    ; ]7 m+ w# A* |factorial(4) = 4 * 6 = 24: z7 H) |* }; E, ~* b4 o: r( g2 y* ?
    factorial(5) = 5 * 24 = 120, D- d3 e; q  c9 [  R# E
    * k  G4 N- f6 z' D
    Advantages of Recursion
    0 e8 ~9 S) M4 |& U0 s( K: u2 n" z4 `& g2 N5 {/ [) R: U
        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).+ U$ W5 H6 F1 U! ]( \

    . {9 z3 o. H8 e7 Q. U    Readability: Recursive code can be more readable and concise compared to iterative solutions.$ _, {3 V. G/ y+ m# [

    ) L  ~' C" k" d& y! d; sDisadvantages of Recursion( U0 @) F" r+ \

    1 d& O, F9 |2 y    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.- B' _3 v5 _2 W; W* I6 \- J1 o' a3 O

    9 X1 |6 _$ V) N9 ^    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ) G9 |% q& X. r$ X& d. V- @3 Y  Y; ~! ]. z+ h: C3 m2 P  i; u. n
    When to Use Recursion
    ; S6 j. K. l3 B2 V  l6 Z
    # \+ W8 A$ x- t7 z2 _    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    + _) I! O2 _- V& [) P
    % i* M. z0 H3 j# ^& x    Problems with a clear base case and recursive case.4 U- S* h0 s% e" Z7 N

    , u7 d1 Y2 P( w3 K% k) Z4 qExample: Fibonacci Sequence( s2 \2 @7 n# k  ~
    2 Y% J6 A' n& O6 `
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    % h( U% T9 l' ?# d, x7 T& Q+ G6 Q, ^4 n9 g# w
        Base case: fib(0) = 0, fib(1) = 1
      e# t+ x, {, J3 ]8 L% m3 b$ @% B5 j% _9 G3 w! M
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    " ~0 _8 N  \5 u- y. r9 B$ a7 ]  @# A4 e9 S
    python) g4 D4 s; ]1 ^" p* F7 `8 g3 J

    ! G! A5 l1 d( |  E; E9 k& G
      C: M% P2 y. O7 l1 g% [  kdef fibonacci(n):+ s4 ?+ W) ^, K) ]
        # Base cases( N1 `. r/ H3 ^$ W. N
        if n == 0:
    # p" c# ]1 T- f* U8 z: K        return 02 ^4 R6 E' M: C" A  ?' H$ I/ }
        elif n == 1:
    " b& `" Z6 }8 i9 k        return 1( I4 A" x1 s& X& [# o
        # Recursive case
    ) r/ P& O! f- p# m    else:
    ; p. u/ \7 `  f/ U4 j        return fibonacci(n - 1) + fibonacci(n - 2)
    ( K% r( H$ x7 \) r) u( y; Z  Q8 d. m
    # Example usage
    * {5 M1 x# W4 {% `. }4 a$ [! Nprint(fibonacci(6))  # Output: 80 u: n9 }" {# b" m* }" S

    2 e% `3 L2 x" a2 G' P/ d1 _Tail Recursion
    ! k9 e2 I/ \3 X5 U$ m0 p9 I" O6 S0 n
    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).8 q! G' x  U/ n- O
    - e7 ?* Z; ~: }; q) R7 T0 o6 l, C
    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, 2026-4-27 06:47 , Processed in 0.060497 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