设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 2465|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 ]' c+ L2 h5 Q+ l6 j# a5 [0 U2 [8 u) n6 o7 T3 Z1 _
    解释的不错
    . b0 V* N2 `% Z, y: w! @1 w5 O- l
    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。5 T) D2 g) [! Y- q
    0 K9 D1 t4 h9 J4 v! H. N
    关键要素
    % N" x, y1 T- I' M  I6 q9 y1. **基线条件(Base Case)**
    . ^0 O, C& a  M3 [& \   - 递归终止的条件,防止无限循环1 o4 Q. Q* d, Y6 \/ y. P
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1/ c$ @. C( ^/ Z8 O
    1 j$ b# G. D5 K' R5 k  D
    2. **递归条件(Recursive Case)**, U. J' |6 `* c5 D; U' }# S  U
       - 将原问题分解为更小的子问题* v: k/ H$ I/ ~# d* R
       - 例如:n! = n × (n-1)!4 [! |2 ]( r' a, w
    ' t2 P8 u; m( w
    经典示例:计算阶乘8 _- a( q- ?; A% o
    python
    / ~* z& ?* J8 {0 V: Hdef factorial(n):/ L5 j- ]5 t: Q
        if n == 0:        # 基线条件
    4 U6 h, M' \6 {4 u5 x' V0 e        return 1
    9 Z: i% N4 y, H; C8 r) [8 B. ~    else:             # 递归条件
    ! u  ~& f# B6 V; p2 f# K6 Q9 r        return n * factorial(n-1)
    9 g. W: B$ B3 \2 \. i执行过程(以计算 3! 为例):
    4 A. L+ d2 I4 o( K+ sfactorial(3)
    - @( Q" _( _  [7 I: ^7 j+ N3 * factorial(2)1 T+ o  I6 J! r5 E1 v# ?
    3 * (2 * factorial(1))
    / t7 n, R# K$ \. c: Y1 H3 * (2 * (1 * factorial(0)))' m, @, ^7 r- o- s7 v( D0 e5 D
    3 * (2 * (1 * 1)) = 6; h/ O" D3 m  p+ n5 N) k4 O% p

    6 K" Z7 q2 _4 p$ T/ R 递归思维要点+ I7 T0 ]: N2 p! A; F; g
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! o" M2 k) B" D+ z2. **栈结构**:每次调用都会创建新的栈帧(内存空间)# @; g$ }) ]2 x9 V3 u
    3. **递推过程**:不断向下分解问题(递)# W4 i4 J2 e% c4 m
    4. **回溯过程**:组合子问题结果返回(归)" `2 Z# [$ V; W1 E& D

    - l2 {! U& o. w9 ~- N4 g 注意事项! u/ H# T# p2 u/ Q9 ~
    必须要有终止条件& _- \# J5 J+ s
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    0 @* _4 V1 [" v! P9 W( y1 i- `某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ) b& ?8 i' R+ m$ O; h尾递归优化可以提升效率(但Python不支持)
    : e" z: C) X9 I: b3 t
    7 J1 r$ K4 I. v6 c+ l 递归 vs 迭代
      E! M: C( L' n' [|          | 递归                          | 迭代               |
    4 a1 m) z6 Q2 v- X) X. z. |0 B5 I|----------|-----------------------------|------------------|
    / [* P; U& F  ^| 实现方式    | 函数自调用                        | 循环结构            |
    & J6 i1 r4 Q. H7 Y. ]| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |% S" I4 _7 M- s4 d- X# R4 n2 A
    | 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         |& H; l$ P0 C3 X! P+ W
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    + U+ A) f: r9 ]" T- l- V  y+ o) q0 Z0 p7 l. c2 _- c# N0 ~! _
    经典递归应用场景
    ' q& w" ?5 D: e6 n! t! X/ R1. 文件系统遍历(目录树结构)
      k8 @2 @5 }; V$ R5 o2. 快速排序/归并排序算法% ~0 g. Y0 X9 Q& P' X% r7 u) T
    3. 汉诺塔问题4 K5 R: t  o, {6 B  J
    4. 二叉树遍历(前序/中序/后序)7 z8 W' K; D% j% |
    5. 生成所有可能的组合(回溯算法)
    * r. ]" I) C+ @) i' p/ z/ l/ @, Z5 j0 u# |4 `
    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

    3 天前
  • 签到天数: 3217 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,7 B  t/ `+ S! b( Q$ ~
    我推理机的核心算法应该是二叉树遍历的变种。
    8 Z/ t+ d( ], _3 ]% \1 R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:" a" V- y7 x, t0 @* D4 v4 c3 {, H
    Key Idea of Recursion
    5 `$ X. B: f' _' p
    1 g# A* c. r0 ?4 y' C3 _' u/ RA recursive function solves a problem by:
    8 |1 `# [7 u* A% Z7 [/ H
    * P) h& e% l& c8 V1 F/ x* F& E) e    Breaking the problem into smaller instances of the same problem.
    9 x$ N  X% g) o! @  B: B( b) E) g  A, c( A
        Solving the smallest instance directly (base case).
    : L8 ~5 o3 m0 S& w! l
    3 }  d- d8 i  K- @- j    Combining the results of smaller instances to solve the larger problem.# }0 |$ t! Z4 Y( Q3 a; N+ z  K, c

      b7 u( p, g; j. K) I& u- `( ZComponents of a Recursive Function
    ! k- D+ n: h+ h! r$ O7 A
    ; r. D# o2 ^" T* D1 C0 S/ B    Base Case:% s1 z7 t' L$ m1 Y+ m4 v
    : v+ l7 f' B/ J
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
      Z0 W( L( `, H
    % L4 b) S5 {: D1 R0 R7 Q' T5 n        It acts as the stopping condition to prevent infinite recursion.
    5 \4 q, h" y) O1 ^( }" T! J
    / r: @/ `5 a5 Z5 |3 k: _        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.' a( n8 d- h7 D
      {; V) m/ `9 Z1 E5 x  D, [  M
        Recursive Case:- h) h9 U" {3 a7 y& l* L

    ! p# w! P& i- ~        This is where the function calls itself with a smaller or simpler version of the problem.
    9 T# |5 t8 q# K5 P' X# A# I/ q9 J
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    & e" W# e5 O- J% \6 y2 D6 ^, R/ d% g3 O. p! ]1 j
    Example: Factorial Calculation# h  k: t( g: |

      d0 T3 E) D( N, v. |( j0 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:: Y( n" w% e& P; J

    ! V' z9 f' R. }* `+ d! d% }) z! l    Base case: 0! = 1
    4 T- N5 A7 J  z" e/ U
    0 X  ]5 @4 x9 d    Recursive case: n! = n * (n-1)!
    8 S+ O0 [) g+ p( B( {
    ! U; s: i2 c1 r5 o/ W$ f1 tHere’s how it looks in code (Python):0 R$ D! j+ t2 _, v2 j5 ~6 e, B
    python* m/ a0 j7 g+ B9 k# G) j
    4 j" H- ]& S- I# h
    2 f9 W1 F' F: l; B
    def factorial(n):
    ! A. Q* y5 s0 H. m    # Base case
    4 s9 t6 y6 d* X  c    if n == 0:- q7 r+ g: U2 L- d
            return 1
    . j3 i# v" c& k    # Recursive case% U: O: U: _  U" I2 _
        else:2 e* v1 l# T% x9 k4 ]5 L" Q
            return n * factorial(n - 1)* X% n2 F& [8 c0 Y

    - H2 e' A( R5 E) i# Example usage0 e) R- f( C9 v0 Y/ }
    print(factorial(5))  # Output: 120
    5 `0 G2 \8 m6 X8 \/ R: z0 t+ m' I& G* Y6 s
    How Recursion Works* \* w1 s" S; I4 j+ f: e

    & v* \2 [8 V3 O0 t- @    The function keeps calling itself with smaller inputs until it reaches the base case.6 R7 R" a2 ~# x% c. Z% e! O
    . Z/ O$ h- P: H% C3 B! }1 {
        Once the base case is reached, the function starts returning values back up the call stack.
    ! Z( P2 H' Q' R" v( R5 ^, S$ d
    , t# x% h5 e# A# v4 ?    These returned values are combined to produce the final result.
    $ ^" }: F* b# D4 y2 _4 T* X' f, \* e6 T
    For factorial(5):3 Q8 b$ z& P( p# J/ x, z
    + U. N' T. I8 P' f0 _7 |( B

    1 }* w3 b5 `8 ^$ _) ~) x2 efactorial(5) = 5 * factorial(4)% z3 A; d  W1 L  I: a
    factorial(4) = 4 * factorial(3)9 B( K# j" T( M, i! C( P. @. s
    factorial(3) = 3 * factorial(2)9 K  l' n( I2 g/ O% p8 W% J" Q5 k- ]
    factorial(2) = 2 * factorial(1)
    1 s8 t7 K$ l. `factorial(1) = 1 * factorial(0)# p+ R. d2 l" n" S. C$ C
    factorial(0) = 1  # Base case, ~- l. n3 u9 J' p# ]

    5 k1 _( \  Z( h% oThen, the results are combined:5 i) B  J' d3 d4 _, E2 q1 \
    5 }) |: x8 r2 J* S
    5 c. X- f7 |( b- H
    factorial(1) = 1 * 1 = 1) k6 F% O* U7 y$ G+ F' g  h( x
    factorial(2) = 2 * 1 = 25 h1 _8 k: e5 Y4 _( O3 p5 s
    factorial(3) = 3 * 2 = 6- q4 q( j4 Z( I6 E8 n& C: t, E+ e, ~
    factorial(4) = 4 * 6 = 24; Q. b$ V6 y1 n( C$ p
    factorial(5) = 5 * 24 = 120# D' K4 \1 Y3 n, `  d% U5 R8 I( a

    6 E/ ?2 o, g  t' YAdvantages of Recursion
    1 E/ ~- S4 F) E, t/ u4 d. Y( m5 Y7 t8 v8 W& h5 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).9 i: ]! F/ z0 {% e5 m9 O3 r' j

    5 ]2 c' ?& u. `* r- A0 l! I! x& z    Readability: Recursive code can be more readable and concise compared to iterative solutions.0 c" T/ |9 x; m

    $ Y5 H9 N6 w4 s% h' MDisadvantages of Recursion0 O- N! y. z( Q0 F5 F% u, ]
    7 z$ W' M$ g) X; d( P! B
        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.! I8 t. v! p) R5 i

    * w7 \& @* n% E8 C0 |    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 k6 Y  A0 `  G. c

    $ m( x, A! W- k8 M# D" rWhen to Use Recursion- L2 r. j/ k) Q) z9 x7 S# g/ ^; N

    2 y0 f  t5 t# F    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).: ^7 |1 R# q% {- F% m
    ! E8 D0 C; c& {. H
        Problems with a clear base case and recursive case.5 b% A- v' o5 |( A

    + ]: |$ t) B/ H; e% w* v& CExample: Fibonacci Sequence% ~# F* A7 E& s" u4 [
    + D, [3 L. ^; Q" l
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:8 b# b1 Y4 C+ T4 x) ]" o' L# t" r

    6 S  d, v) ^8 X, O" F2 x- W    Base case: fib(0) = 0, fib(1) = 1
    , R/ W, J+ X- _
    + T! U6 {- D* K6 p4 i    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ) k6 k: I  b3 S1 y) o
      T8 _% U: {6 }2 r3 T6 n  upython
    " ]9 B4 u! q, z- d; Y
    + M3 l" M* ?! t' {3 t8 a7 Z" U4 n* u" l' h* `2 V
    def fibonacci(n):
    3 S3 D, _) Z0 B4 S  M6 U; T    # Base cases
    : M& k0 F; \- m    if n == 0:) B1 {$ A. o) w% Y1 Q
            return 03 A# Y5 `' N* n1 O% O6 U- U
        elif n == 1:
    5 X& u$ }  X. P        return 1
    # p, b) y  w( `) V7 ~4 k    # Recursive case
      x! B, r$ ?; u4 X' A# A    else:
    4 ]& v; n* ^9 l' ]1 c/ o        return fibonacci(n - 1) + fibonacci(n - 2); H* K% L- D) Z

    ; b+ [1 O" K5 r. e; |0 U4 ]# Example usage
    1 R4 ~( F$ b* _! vprint(fibonacci(6))  # Output: 81 J8 O. U  u( O# W2 q
    " [3 D4 d& `- a
    Tail Recursion0 `+ |/ Y% N; e- L9 s
    9 P/ N. H: a$ b
    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).
    + d# e, v1 e6 Q4 z: l7 `- I
    8 n9 l5 g- p- {3 v+ uIn 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, 2026-4-12 01:37 , Processed in 0.056283 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