设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2381|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
    1 |7 f. D9 p+ g' ?, o- Q
    . K' n# e/ T* f, z9 v9 y+ T解释的不错
    4 ?' p& j( j  W- }6 ?, n' X! z! W& {. ?
    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。
    0 d; p4 `  }8 z1 X8 L' }; q( a# ?" J4 c9 W# h
    关键要素  ]* o+ T+ A( G; ]' ~0 B0 `
    1. **基线条件(Base Case)**
    # ]0 ?& w) D# A8 ~1 F& _   - 递归终止的条件,防止无限循环
    - O8 W- H# K2 ~   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    " a4 w! d/ t% Z% k/ _, c0 F0 A# S3 \: [+ D+ N% W
    2. **递归条件(Recursive Case)**: o( f  S; q% U9 a
       - 将原问题分解为更小的子问题; f0 _3 _. M% K! [. `" M; ~4 A
       - 例如:n! = n × (n-1)!0 ~8 `6 ~, |& T1 ]- A

    : U# t. e, o* h' Y6 A# k' ? 经典示例:计算阶乘' \2 j1 v7 P& }5 V2 b
    python
      f( i& v/ j4 D5 K+ ?, Odef factorial(n):! D8 u/ s4 _5 m% L: D$ n# D9 S8 M7 v
        if n == 0:        # 基线条件
    9 H9 ]- s% m3 u+ [. C        return 1
    " J( ]: I3 A" A8 q1 \) x( s    else:             # 递归条件$ A+ [8 p6 `* O& ]$ c$ a0 Q/ D
            return n * factorial(n-1), X6 O. ]. X/ s* `
    执行过程(以计算 3! 为例):8 ?4 c* {4 E7 n5 @* a
    factorial(3)
      Y1 ]7 }% Q2 W3 * factorial(2)
    % \; [) A# A2 w1 F* p. F3 * (2 * factorial(1))7 E( h2 [& R3 ^7 c; Q5 X
    3 * (2 * (1 * factorial(0)))
    . P* r! k" B: i. ]  ^* e3 * (2 * (1 * 1)) = 6
    9 |! d4 P$ ~' |! g/ d  r! V* F7 Q: K
    7 B( v' L# B6 ^: r* e 递归思维要点/ z6 H2 }4 x; Y& e8 L
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    7 a9 y5 U1 _" z7 Z2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    1 X% b4 n/ Q# f6 ~! O3 l- J8 \3. **递推过程**:不断向下分解问题(递)
    : o( n+ I; i; l+ d4. **回溯过程**:组合子问题结果返回(归)0 L' C+ S. U* X2 S
    , L; P) O* T5 Q3 k
    注意事项) u: K% ^) r2 R2 B9 l
    必须要有终止条件8 c- t/ I( k: z6 [+ B* W1 H
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    7 S1 R* U& N% @! {9 A* e某些问题用递归更直观(如树遍历),但效率可能不如迭代
    + C/ M* g3 k1 P7 {尾递归优化可以提升效率(但Python不支持)  Y; C$ ?4 i1 x

    , z/ H! x( U3 U1 g 递归 vs 迭代& z$ v: L, R7 i) Z4 D$ w& p
    |          | 递归                          | 迭代               |
    4 Z2 @1 X; }; h. m9 Y: C* E8 ?|----------|-----------------------------|------------------|
    5 B2 @/ g9 t( g| 实现方式    | 函数自调用                        | 循环结构            |
    5 m3 z0 ~* M/ ^+ Z" I9 B& l| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    1 y/ P7 U% r" Q* P- t1 V3 t2 E| 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         |
    / f9 `( I" c6 J! y5 H* n& d| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |5 ~7 R3 V! X  `& p6 K& m9 R+ F3 s

    - j% d  Z$ \, Y0 Q5 e$ }6 h 经典递归应用场景, f1 m( y7 u) @8 Q' ]
    1. 文件系统遍历(目录树结构)6 B/ V3 V5 E" t7 `! c0 [
    2. 快速排序/归并排序算法5 b* F5 p7 x5 Z. \# b& q
    3. 汉诺塔问题8 ^4 E3 e/ k7 g* b* U
    4. 二叉树遍历(前序/中序/后序)! \- w' P# n5 S  B
    5. 生成所有可能的组合(回溯算法)8 D' N, ]2 @" P
    0 n2 d9 Y4 T: |: J# 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
    开心
    3 小时前
  • 签到天数: 3210 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,
    0 ?/ p) F8 V' W我推理机的核心算法应该是二叉树遍历的变种。2 ^' P# Q9 F& ?& E  F/ e0 a
    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 g$ B, q, f; m1 W3 wKey Idea of Recursion6 t7 E- H5 ]) ^
    ( N: [, f& z( y9 g
    A recursive function solves a problem by:4 V# _, }4 y- j+ T; ], L8 _
    + T$ w" x5 \+ l! F* H/ c% ]. ~
        Breaking the problem into smaller instances of the same problem.
    . b' F/ i3 T1 O/ x' u
    : A0 c) K: p% m" Q0 P    Solving the smallest instance directly (base case).2 F. `5 ?. P! q! f2 u5 W' P9 a

    ( @! ^8 z8 t' Y% D8 x    Combining the results of smaller instances to solve the larger problem.! f9 Q0 d- B8 ~3 @1 [

    5 m) z/ D4 u+ g! d" s! {Components of a Recursive Function
    - @6 j- Y+ A3 Q$ M* s, O
    " R8 X5 b( U; ?3 E0 \) f  m    Base Case:
    $ i: n- V# R: {+ F) ~# M, e
    8 ^1 `; c8 Z, Z0 A. C8 y. t        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    - ~7 O  g3 |& l) |
    7 W6 N: G& ]; u        It acts as the stopping condition to prevent infinite recursion.
    4 k: k9 z* p" Z1 J. L- l; ~, J8 u5 R( ]* E( O
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    6 ]6 h9 P8 G- L$ e+ X
    # V$ F9 r; y: ?) w    Recursive Case:
    # D1 _4 C, Q) n  B6 G8 H  j& \0 }
    % D* Q7 B$ b) o$ d4 u% O* m        This is where the function calls itself with a smaller or simpler version of the problem.
    $ a$ ]8 r0 M( x" U  a7 \8 ^/ J4 |, r4 S
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    & h0 d8 S3 L- D' S% W6 {
    # b9 Y7 C. F* T1 ^. T  |- ?Example: Factorial Calculation% N" o1 t1 ?3 l7 r" V2 L3 r3 N
      j' a, [$ B1 E( D# ~  E+ I. }
    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% i0 K3 x) d

    + J1 ^" V+ T8 j' {! R/ R  A    Base case: 0! = 12 Q5 p- A. v3 E
    8 p/ D2 F& H+ `& @' m
        Recursive case: n! = n * (n-1)!9 t7 a" R; ^; Q2 F% u2 l
    ! y- {5 S1 U& T5 m
    Here’s how it looks in code (Python):
    5 X2 G; F5 N% Rpython5 S, s# [8 c' P3 p( i
    - E3 m- I9 }! e9 K

    * \. R6 d5 |, C( I7 m4 u! |def factorial(n):
    $ u2 c% P9 J; i  a2 ^" j    # Base case% U# y- Z% Z* C0 P4 a, J
        if n == 0:- @$ |7 L3 [3 Z" R$ T: N
            return 1
    6 s; {8 @' q7 l2 J    # Recursive case! m2 O3 ]: k6 I& Z+ B
        else:/ K& S# T( G/ T6 L
            return n * factorial(n - 1)
    & r) b" B% H5 e% h7 w7 J: a6 d- }' |9 a( T
    # Example usage
    1 t( [+ s! r1 _print(factorial(5))  # Output: 120! u# b) b" f; y  G% b# _8 Y
    2 V2 s  m: ?; d7 x( v3 V+ V
    How Recursion Works7 v( @6 i& y: q! S" T

    ) X' E/ i9 |# L0 r) C: x    The function keeps calling itself with smaller inputs until it reaches the base case.
    1 J! k8 l- S7 p. E% L: {* B+ i; N1 n  T! _
        Once the base case is reached, the function starts returning values back up the call stack.
    $ N/ w* l9 y( Q
    7 }% L2 W# |5 U% Q- m0 W( o    These returned values are combined to produce the final result.1 T# }) V0 R) c9 @3 Q4 l

    ' A; \5 G0 [4 e3 s9 ?For factorial(5):0 Q+ t6 c2 R( A; A
    " O# s* m6 W3 y+ [" ~* X
    ; N9 e+ j( u9 Z2 I2 j) `" c& e" P9 b: Q
    factorial(5) = 5 * factorial(4)
      H  [% I! R/ m5 ofactorial(4) = 4 * factorial(3), z6 z. W/ m0 [6 l
    factorial(3) = 3 * factorial(2)( y2 o  V/ p/ ^1 v! ]& M% I/ M; ]" g
    factorial(2) = 2 * factorial(1)
    + F( \% p4 _, U! @7 t6 J) ~factorial(1) = 1 * factorial(0)
    8 y# n' @9 [) I& D: ?factorial(0) = 1  # Base case
    8 W* c: z& N( G6 _2 I4 o
    ( K% k/ l2 W0 a5 d4 qThen, the results are combined:
    . @1 W) s3 r$ s; b
    / @8 ?/ c9 D: q( v+ M4 E/ [
    ) F& z3 K9 B3 C6 G& d5 ]4 W% {factorial(1) = 1 * 1 = 1  F: y' Z$ {; e$ v
    factorial(2) = 2 * 1 = 2
    - x8 d+ Z: q% H; o! yfactorial(3) = 3 * 2 = 6
    ; [: v3 u2 u1 [5 e. vfactorial(4) = 4 * 6 = 24, I; y2 l/ R. h
    factorial(5) = 5 * 24 = 120
    . r9 v: R7 s' v! x' I$ G8 F+ |
    ) |( X9 J* P4 r" @% DAdvantages of Recursion
    ) O0 C; M  I' t" H. j
    0 _1 F/ Z8 P$ e( A0 e" Z6 Y+ x    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).4 A/ b2 u6 F* _1 k

    0 a! g. V: i0 C& p, f1 `5 n    Readability: Recursive code can be more readable and concise compared to iterative solutions.: S0 t% ^% M7 X8 M
    8 M& H# ], u0 L$ s% U9 K
    Disadvantages of Recursion9 c, h. K9 v/ \; o4 o6 }+ l
    : r$ p9 l1 ~8 k/ u
        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.
    ! C8 ]6 {  y' i; ~! k; K  W0 {% ?4 I6 J
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)./ M2 I* V9 r* u

    4 T6 V# @3 K9 h% O8 s. M8 v  d% zWhen to Use Recursion
    ; S( K) K# _' K8 ^8 Z$ m8 B& i7 p! H  |+ W
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ; X$ ~, y2 ^1 x+ |
    9 S: V) a' M9 K( t! I, z  [$ z. z    Problems with a clear base case and recursive case.2 |  q$ [! i. O) {

    : b$ r: X6 i- P3 l: t# W" C3 tExample: Fibonacci Sequence
    5 ]# k+ p# |# p
    1 g5 Q+ ?( _' o$ MThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    8 Z/ ], A/ _0 |+ a4 ^% S- e" N# O. e
        Base case: fib(0) = 0, fib(1) = 1
    9 Z  u4 d% f" W! A. y: R: D8 m9 O0 T7 i
        Recursive case: fib(n) = fib(n-1) + fib(n-2)& Z4 O9 U$ P! L$ ~7 ?

    / r' y! e1 k3 t0 T/ rpython4 i  L5 ]* H6 e, S4 |8 d5 v0 N

    % r+ }* w; \, W% v. e: q( C( {, \7 C! t! N. I& z
    def fibonacci(n):: G, |9 j7 {7 ]9 s/ i& Y: l# E
        # Base cases) h& ]" K& O* w, B' n5 Y& P" U
        if n == 0:
    0 v4 @) i# b9 z- H- |+ t        return 00 W! _: f$ f6 y# F! k0 `# v
        elif n == 1:
    8 I. ]. b/ i3 y4 E' b( v        return 1( m! \* y; v% T- Q
        # Recursive case
    . n1 @- Y, O2 |  }    else:
    9 U$ o  @: C# v; I/ W6 u        return fibonacci(n - 1) + fibonacci(n - 2)
    9 P& H# e/ U0 F* C( H+ B1 ^4 R% P* q0 q8 K  C
    # Example usage2 c' t6 M7 H" ?
    print(fibonacci(6))  # Output: 8& f6 `8 M9 _' T2 c

    : h, I- h* {. A! o3 JTail Recursion
    ! E( l4 B, C9 q  ^/ M7 O7 x4 A7 R2 A, ^- ^% o( f) X# z% g; A2 D8 [
    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).. h4 O/ `2 p' `
    & [1 l' s9 i. }$ W! ?
    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, 2026-3-31 10:44 , Processed in 0.077910 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