设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2205|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 + I7 ^* |  \+ X3 q2 o# v- H& D

    6 u; l0 K6 Q  ]4 S: j, X解释的不错
    0 F/ y2 Q# g7 ~$ F) g
    2 ]$ s+ d' E7 Z2 \& I  q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。
    7 {; L2 h0 T0 n, S8 u
    , v( n1 K( {8 K# _5 R 关键要素- t* |& p4 s6 z5 q3 M
    1. **基线条件(Base Case)**% j2 l7 w' p& i
       - 递归终止的条件,防止无限循环* K& R+ R( f  v+ Z+ y) t$ J
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 13 ^4 [4 f$ J# |6 ]: G$ Q
    . {( s& C/ F+ ]( q/ |1 z+ S
    2. **递归条件(Recursive Case)**. }+ E6 |; F* ?& ^& ^
       - 将原问题分解为更小的子问题
    6 N, b+ z0 Y( u: l7 {0 ?+ r: j   - 例如:n! = n × (n-1)!
    5 q, }  c: ?# p. Y  K& T1 A5 c9 h" U
    经典示例:计算阶乘& G. Q+ k5 v( t9 x! u. y
    python
    - t" e$ Z3 q: n' ^def factorial(n):
    3 i- o7 W) W+ b# z; m    if n == 0:        # 基线条件
    6 T9 E9 C1 P2 `; Z7 }        return 1
    6 r$ H  O# Q7 f5 ^, N- h    else:             # 递归条件: d( e6 P# M5 N
            return n * factorial(n-1)
    ; @: K. e% u% q6 k0 n# Y; f- j执行过程(以计算 3! 为例):4 p3 H: K  W+ O( m
    factorial(3)
    1 S) _" a- t& A% W2 `3 * factorial(2)
      L+ W5 q2 Y, S, R0 c3 * (2 * factorial(1))
    % V# i- c4 N# f0 {9 y3 * (2 * (1 * factorial(0)))
    6 Y+ F7 G3 F. t* y. s3 * (2 * (1 * 1)) = 66 y8 y, ^4 q6 r& }

    0 @6 n# F: Q4 r! R- c 递归思维要点
    ( ?2 n3 U4 W0 H1. **信任递归**:假设子问题已经解决,专注当前层逻辑$ p' c. X* T# r
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ' F8 k, `0 v4 o8 ^1 g3. **递推过程**:不断向下分解问题(递)! T# O% T6 P% C7 A3 g# M* s/ {
    4. **回溯过程**:组合子问题结果返回(归)+ g+ @/ W; j+ X. i% N7 `# C) d
    " W! |' F( u5 w2 j* r+ C
    注意事项5 H9 o/ |1 H) X- t. O5 v7 M( m
    必须要有终止条件3 W. p! j/ X- `: y! T7 T5 J8 s
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    4 H7 W" K4 Q# ?7 \( P9 w( D  p某些问题用递归更直观(如树遍历),但效率可能不如迭代: H" @- G3 C/ H6 p/ A8 I
    尾递归优化可以提升效率(但Python不支持)1 d! Y; f& V6 p/ w* N& T& U

    . t; D  L" D8 F 递归 vs 迭代
    ! P0 Q* x; m$ [6 h2 v|          | 递归                          | 迭代               |8 o# B3 X8 z1 G$ C4 c
    |----------|-----------------------------|------------------|
    ) Y. E. L% c* q0 d| 实现方式    | 函数自调用                        | 循环结构            |3 I  I# f/ v# m+ Z" A
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |- ]* ]9 O1 L5 N: L5 \' b  k
    | 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         |  E6 _( E& d6 H( y
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    + i1 i( T/ V) |) U; U, @: I4 o- d
    9 H: e6 H# r9 [# U" J 经典递归应用场景; R. M# E/ k8 S  Z
    1. 文件系统遍历(目录树结构), {- n7 K$ C! g% x* B+ [" n" z" g
    2. 快速排序/归并排序算法
    " g! Z8 }* n# }7 J# I0 `& t3. 汉诺塔问题7 U! @' L0 o5 O$ R* d* H  s
    4. 二叉树遍历(前序/中序/后序)
    9 C7 x. I7 Z' Q8 n6 V3 @5. 生成所有可能的组合(回溯算法)
    % F0 G! |5 M, W/ m1 f, ~0 X8 k: U( a' f! ^0 y( E
    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

    8 小时前
  • 签到天数: 3192 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,
    9 Y# `! C1 I% Y/ S我推理机的核心算法应该是二叉树遍历的变种。$ X" s8 c9 {7 u5 I% O$ S: H
    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 m2 Y, L! E- R' C" [( A# I
    Key Idea of Recursion
      L6 n0 X$ }4 [/ k
    . X9 b9 o  |" M, O5 a' X5 pA recursive function solves a problem by:  t6 ^2 V! x. z4 ^- O4 [, \
    9 p$ V& {+ [7 i3 ?) Y
        Breaking the problem into smaller instances of the same problem.
    % P: m" ]3 U: K: F- \# W" H0 v  [( E; z; A
        Solving the smallest instance directly (base case).& j, R% e! z" P6 _3 i0 h- X0 a

    5 r# U% |- P  M9 c: ~* O0 Q0 p& _    Combining the results of smaller instances to solve the larger problem.
    . {6 L, g0 F# Z1 P/ J3 u5 p0 t  z& A9 p
    Components of a Recursive Function! Q  K. h. D/ y. d4 t
    - _) u/ `. B1 t6 i% h8 d9 h+ d
        Base Case:3 {' ]9 a5 _; D

      R- ?* ]$ C; u/ C        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    / P. E8 C7 y- n  s. }; ^
    ; l! o/ d8 j; J/ y- g+ l9 S        It acts as the stopping condition to prevent infinite recursion.8 r; P, t% n$ d- L/ j' P  ?

    9 }; t# Z5 `, x# X+ |, Z8 ], y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    - l" N- T& \0 y2 n# c$ T
      C, k6 W- U% ^2 P, e: L0 N! I    Recursive Case:2 ~/ `/ Z. t2 g6 M* q- `6 F4 |

    - Q( u1 v0 e0 q0 O! O. Z. T        This is where the function calls itself with a smaller or simpler version of the problem." n' K: Y. K4 ?  P# T
    1 C9 t5 x: }3 p; q9 }% Y8 B+ T2 R
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    $ H: c. F& M9 B6 C/ [: t( X1 N8 C% g& @! {, b
    Example: Factorial Calculation
    ) a/ l! O3 }  m, }1 m. o1 Y: `( b% U0 x5 L- p
    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:
    , }  x9 v2 ]% R" @( L" p( [/ `3 k5 u) S+ [+ V% l: C
        Base case: 0! = 1( `3 z2 }& @6 q/ U& r

    ) B; @. [* D+ b7 y+ v! l9 }0 t    Recursive case: n! = n * (n-1)!7 W( ^8 g6 Y1 j& J

    # W/ `! ]. e0 x* ?Here’s how it looks in code (Python):: C- A) S0 d8 m3 l! Z
    python
    0 E* F% ^& k0 T$ x" j( q% D; O+ k2 B( D
    5 t" h9 F: i3 W0 E- x* |- e: |
    def factorial(n):
    : q6 W% v! I9 c; T! l, _6 L- F    # Base case
    : j7 R0 h' A7 b& Y- _) @3 E0 p    if n == 0:
    7 O, Q/ L" D* Y9 l+ k( P& |# G- x        return 1, ^: ~# e1 u5 t
        # Recursive case+ o5 l2 N2 p, z/ ^" w8 ]
        else:
    . S; E: o  L9 B! @! C* c" R% u        return n * factorial(n - 1)! U: {7 x% \( e6 W& z

    & U  H) s4 p& ]1 r1 a5 w7 T# Example usage
    / i" j1 ?* d; O3 J6 Kprint(factorial(5))  # Output: 120
      j- ~6 _" {0 {! d* y
    & q2 g# R8 Z6 sHow Recursion Works
    : T. R! G8 P1 a8 \" s1 W
    $ \' _' Y) h* `3 j' m; R    The function keeps calling itself with smaller inputs until it reaches the base case.3 n3 W; X# G. T0 u) r
    4 }* H* |1 s2 h0 Q, b  ?* q
        Once the base case is reached, the function starts returning values back up the call stack.
    . k" e7 d; }, u8 E: u: z/ e$ z+ x1 e' U* Y3 l( v4 ]
        These returned values are combined to produce the final result.
    0 P8 Y+ \( i. |3 y
    5 a! C; \" J1 bFor factorial(5):
    + c* J0 t3 |$ T* Q% g$ }+ r# e8 G. u& ]

    ) S# D8 k. Z: S; J! xfactorial(5) = 5 * factorial(4)
    9 u% U4 c4 C/ j& |& zfactorial(4) = 4 * factorial(3)
    2 T; D/ q0 ^- h- l  Rfactorial(3) = 3 * factorial(2)2 K4 L: X* d3 s) _  m
    factorial(2) = 2 * factorial(1); M; o8 ^3 z* n: ]1 ]$ B
    factorial(1) = 1 * factorial(0); f2 A- F7 b, c% ^0 a% m* z6 b
    factorial(0) = 1  # Base case& [: C/ v0 |- J! b" Q; G1 q) P/ q$ q

    ( z" c: z- g+ i2 {Then, the results are combined:. B1 x6 \2 R3 b% _6 {5 [
    ! S% q* O+ T- y1 S2 H; g) _/ H

    ' l' n3 `+ X& u9 ?factorial(1) = 1 * 1 = 19 E1 ]4 n+ K& `7 n/ F1 P& s" u
    factorial(2) = 2 * 1 = 2
    4 \, e$ S. m( S6 V7 `2 R. wfactorial(3) = 3 * 2 = 6
    6 N" l$ z* u  J9 ^9 \  f$ V8 s( Tfactorial(4) = 4 * 6 = 24
    4 F, K" _2 z4 j& v7 w, Ofactorial(5) = 5 * 24 = 120$ q  r8 \1 |; E
    , j# d$ w- U' E& g/ A
    Advantages of Recursion7 ~/ c$ h! x4 r; A0 C5 ?
    7 U8 _. A2 V) Y, a& 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).. l+ l" r1 u7 G/ |. N0 V( ]4 O0 L
    ' ?2 @+ Z4 h) Q& ^- j  e
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    / D. ?& [- h! x' r
    8 G+ _: e' ~1 M, Z1 `. _: gDisadvantages of Recursion2 j" U' K- y7 S9 p* {
    $ _+ g3 p8 Z; x# k% @1 I
        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.% O6 q& }6 W' d0 o6 I( \* Z& X

    8 k6 [" O: I" P' \8 u/ `+ t* e    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    3 X' @$ k9 R# v7 }  w( Y) B
    / ~( \+ k/ W0 {  |% \8 MWhen to Use Recursion0 {" K8 j! Y* t3 ~- p- m

    & p' _+ [7 y8 x" p9 `8 c9 `% H    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    3 K) i; q$ {% i7 P/ C3 Q+ e
    + B: l# h( p) X. w. ]$ P    Problems with a clear base case and recursive case., q! h8 r' @! \; s+ |# @# l

    & H. R& X( g# Q3 X& ?- I) }Example: Fibonacci Sequence+ N  p8 o1 l' @& b; B

    9 J5 G# `% y. H, y2 c$ p, _. yThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 o: T' O0 e4 f; _# |
    / l/ z0 q6 W5 q8 |! E6 ?
        Base case: fib(0) = 0, fib(1) = 1
    * \5 Z, G: d# }% ^! O: _, `. m' H  V5 S
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    1 @& v9 p# E, o( V7 ^
      W# n2 ]- j8 [/ Xpython
    . i& N" b7 ^; }7 t; c
    1 d6 [9 J/ d! T) w2 ?& b3 a" r2 C; C2 m
    def fibonacci(n):
    # v: Q$ a+ ^  W. |3 ^# y  z% R3 E    # Base cases
    % i. p9 \4 Y' n% k    if n == 0:
    ) b  m7 O7 S) ?1 k        return 0
    $ F& ^% d6 T8 Y" r) ~- D: t" M    elif n == 1:/ A6 ]3 r; H' {% t: @3 P, p
            return 1' u7 l4 J. P4 H. O8 S6 T4 H. Q+ f! h
        # Recursive case* N' q$ {9 c- V1 Z# K
        else:
    8 X, T) u  ^2 g0 z        return fibonacci(n - 1) + fibonacci(n - 2)
    # L7 g# o% z1 A" g5 b1 X9 j9 p, \8 C4 {6 ?$ }2 M# j9 S
    # Example usage
    , @% ^: v$ S4 g# J6 kprint(fibonacci(6))  # Output: 8
    $ j  A9 r8 B, Z0 L% T7 h2 t: K' ?2 [" d. |+ h- B
    Tail Recursion
      [1 I+ V6 T' K& h5 {6 y7 r" h4 h1 s) d* @+ j/ p
    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).3 u5 \! P% |+ u3 I0 m! H* M
    ; z0 n% ^+ j2 k
    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, 2026-3-5 08:41 , Processed in 0.059223 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