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[科技前沿] 突然想到让deepseek来解释一下递归

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    2025-9-8 05:08
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     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
    5 v9 e% y8 ^. T2 z* F; q! ]8 e+ j  \- b: Z
    解释的不错
    ' L( H0 k$ @! V! b2 {. D% Q
    2 L* l& ^- G" _  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。
    8 m" t9 b1 N7 J) c" T" j
    4 W2 a+ k7 R+ R 关键要素
    : l3 ~- B# T) R6 b0 U1. **基线条件(Base Case)**
    5 f+ s2 x. k- }4 L0 u   - 递归终止的条件,防止无限循环8 l' ~& C. ?# B) i
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    , c3 |+ t7 U5 w( N7 v' Z
    2 Z5 O6 v. }7 n  ]5 e2. **递归条件(Recursive Case)**, N( Z5 k% q! G% C& t( ^7 f
       - 将原问题分解为更小的子问题) N9 K- N& e! |# e+ d" w
       - 例如:n! = n × (n-1)!* J  o4 @3 E4 \$ b. ?. y2 U

      J+ \8 V0 z$ t. C2 y* z 经典示例:计算阶乘
    & ?! }+ G1 [1 ]% I& mpython, t9 d. ?9 o! o+ o) W& w
    def factorial(n):  e9 _: J0 t) K3 {. {
        if n == 0:        # 基线条件; t% _5 k  p7 q9 @2 F0 u
            return 12 N, m; q1 R5 S( _% w9 I9 V  w
        else:             # 递归条件3 c2 U. J6 e$ y$ r5 l
            return n * factorial(n-1)% N# `4 F4 @3 k' V
    执行过程(以计算 3! 为例):
    3 S4 g- V7 K6 s5 [# R% ~$ x" h$ xfactorial(3)1 o& Y4 ?8 S# a% A2 T
    3 * factorial(2)
    ( P0 H  ]$ D- |+ j; m) g) {3 * (2 * factorial(1))
    7 S$ R/ o* d" p6 _5 @3 * (2 * (1 * factorial(0)))
    5 T5 c1 ~9 s6 `+ E. Z4 w3 * (2 * (1 * 1)) = 6
      @( G' D' `  ?6 @9 |' T" E; Y# S
    递归思维要点
    ) h9 w  y  J* t2 Q0 r1 a1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    / t% b# t0 i9 p) m2 x' G2. **栈结构**:每次调用都会创建新的栈帧(内存空间)7 ~1 A- `' X, Q0 I4 d9 S- w
    3. **递推过程**:不断向下分解问题(递)
    5 k& u% w3 X3 ]! S8 f# a9 ^- ~4. **回溯过程**:组合子问题结果返回(归)' y6 ^. n: c$ [$ z8 b
    5 [, v1 r/ ?. y0 N
    注意事项
    ( p6 Y0 [& {* v& b, f0 ?( m. @必须要有终止条件
    # i. i4 O3 e9 _* g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" p6 w+ S/ D0 x某些问题用递归更直观(如树遍历),但效率可能不如迭代/ ~" I* B  Y0 V
    尾递归优化可以提升效率(但Python不支持)9 w% s1 L4 f$ y4 A, S; K; {

    ! f7 R4 }* Y; U9 `/ ~ 递归 vs 迭代
    7 o9 H) H* p# f3 Y|          | 递归                          | 迭代               |
    4 R$ G$ a; r. W0 ^4 a& s|----------|-----------------------------|------------------|( p" Z! w9 f3 o& D$ H6 T! B
    | 实现方式    | 函数自调用                        | 循环结构            |
    9 S' ^* f2 R* u. }- l! o| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    5 ~2 Z' V* L' l6 d) 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         |. R6 k' k' A4 C+ X5 c/ T% k
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    9 v2 e# r, c2 g, y% m  ?- m" S4 L  `' u- X
    经典递归应用场景
    # O0 f4 ]* Q+ ^+ }1. 文件系统遍历(目录树结构)2 ^" k# {7 ?( R! n1 Z
    2. 快速排序/归并排序算法; r7 v0 L% p" P
    3. 汉诺塔问题
    % K! Q3 A- B* k& \/ i$ i+ i# o" U4. 二叉树遍历(前序/中序/后序); g" }+ ~; Q$ T( u# G
    5. 生成所有可能的组合(回溯算法)$ k  T6 d. O# U8 s4 a
    6 A. d# _7 g% Z3 J  r5 ?
    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。

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    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,
    1 `) m8 C, [% b4 ^' I我推理机的核心算法应该是二叉树遍历的变种。
    3 i1 k4 r$ D1 }- C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。
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    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:/ r9 F9 a0 Z" P' i0 ~$ y: M' [
    Key Idea of Recursion
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    A recursive function solves a problem by:' q2 _- F; A2 M+ v0 \1 F

    9 b; V. \( o) _- W$ H    Breaking the problem into smaller instances of the same problem.
    0 M$ x# o) q' f- v! P9 e2 f* T
    / P2 H' A* J! K- D4 i    Solving the smallest instance directly (base case).
    ) \- \' o1 F) x' \) m0 L1 d
      m4 G$ P! c4 z- j  c" O    Combining the results of smaller instances to solve the larger problem.* ~" E7 [. t8 ?6 }# C- R
    8 K! K; S1 M9 N
    Components of a Recursive Function8 H1 z; K  z3 o8 @
    - ?) K% r; u! a# N; i! r4 N
        Base Case:
      t* [" R2 G+ c' T) v0 l
    / q; F4 ^& Z3 |4 P        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* L) B9 E# D' e" q0 M+ }

    5 N( V) C4 l# ^+ ]% H        It acts as the stopping condition to prevent infinite recursion.
    5 G# I- c# Y, t& l1 b+ o. Q+ R8 q* P. t! J& X+ y* U. m# E
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    7 R0 w7 ^% ~1 ?: R6 o" _* x( t% E
    % Y  P0 s8 z# b' N  O# E4 E    Recursive Case:
    0 F/ E& i. x6 f( k$ T- B) _3 Z0 ?2 I  K
            This is where the function calls itself with a smaller or simpler version of the problem.
    0 D3 Y8 I4 x9 {: D. ]# \9 Y/ M' B; q6 {1 _; l
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 y0 X: B1 B9 e- S5 C2 P" F
    " K3 H5 Z( @. v
    Example: Factorial Calculation3 U& F. t6 n- [2 {( H& L5 q

    6 ?. U+ Y, u+ ~( {+ S. }) JThe 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:
    6 {# n1 M/ }, u# \1 y$ G. j. }2 ^! \
        Base case: 0! = 1* ~: E4 S* q2 w
    ! |' y1 i8 j0 E) h! M# U
        Recursive case: n! = n * (n-1)!
    " w( L1 U+ J9 z' s
    / K2 m6 T. o6 ]4 ^. k& BHere’s how it looks in code (Python):/ @; c+ r3 \* R1 H; y
    python: L0 [' S  I, ^: s- g3 ^$ q
    5 x: H4 V4 q; K$ L

    # b1 G. y( S& r. S( \' w+ V: t, vdef factorial(n):/ l! X/ r  l6 i, k7 `( g1 w$ B
        # Base case! H+ k  ^$ d7 Z) c( H0 n
        if n == 0:  U0 r+ S/ w9 p7 ^7 M! o7 @8 o: V
            return 16 f0 M! q* @* s+ d- t7 `4 l
        # Recursive case
    $ w+ K: P: ?$ M, u" A9 W    else:
    ! L4 _/ I% f% X3 T        return n * factorial(n - 1)
    5 K/ u8 ]6 p: ^$ r  s9 x3 s% r2 H8 D0 A2 l
    # Example usage+ k9 ?' Z7 A/ z% {
    print(factorial(5))  # Output: 1208 i+ X% p# c6 `( S" z- m3 f) I

    + ]6 q+ \5 a2 u6 l: GHow Recursion Works
    % Q/ X. }/ |/ T( p1 S5 ^/ h  ]$ ^( J  p' d
        The function keeps calling itself with smaller inputs until it reaches the base case.
    / [4 A5 ?" M0 w, ~# y! p5 _) E9 O8 L( l$ Z
        Once the base case is reached, the function starts returning values back up the call stack.
    ) i( m% L; M) i8 w  [' O/ D# Y" h/ j7 L* b
        These returned values are combined to produce the final result.6 E3 x& [1 r: b

    4 g8 U% Z( y3 _0 }For factorial(5):3 f( ^0 k1 c) J1 C
    1 ]; r5 U9 w1 P. g3 d! H

    ( V* [) A& C! x6 N5 N. ofactorial(5) = 5 * factorial(4)
    ! q9 c8 u2 L! L8 `0 L; Lfactorial(4) = 4 * factorial(3)# `6 p/ p2 t4 `7 C) v0 X) N1 R  i
    factorial(3) = 3 * factorial(2). I2 r7 L% ]: r6 Z* t- H
    factorial(2) = 2 * factorial(1)4 e2 Z, c% F7 W2 ~! l" m, m6 S
    factorial(1) = 1 * factorial(0)2 ]0 m+ F( H- P: \' E
    factorial(0) = 1  # Base case
    0 j" L( L) n8 C" z! R6 L' U/ [& }1 J3 P* ~4 g( {
    Then, the results are combined:
    & g$ P% A9 u7 h9 Y$ Z5 x5 b3 f1 e5 v1 |. d8 f

    ( u& r: \5 B, S! h0 A3 m* _factorial(1) = 1 * 1 = 1
    4 A! U$ n; \2 [. U( Zfactorial(2) = 2 * 1 = 2
      C  H+ U  w" p( K9 y9 i# ]# zfactorial(3) = 3 * 2 = 6
    + o+ q. t5 u! \" R& ?1 E* R' Efactorial(4) = 4 * 6 = 245 P3 ?% w* w( @- J$ _
    factorial(5) = 5 * 24 = 120
    0 {1 H) R+ b4 t* R; z+ e' e  A# q' h
    * z/ u& h/ p3 BAdvantages of Recursion
    $ d  g) }( x7 Z4 x
    8 u/ z4 i: B) [# R/ t* ~    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).' ^: j& X$ w+ Z6 B
    9 {  d0 Z; i5 Y. v6 ]; X% c
        Readability: Recursive code can be more readable and concise compared to iterative solutions.) W+ M4 x' K6 U9 Q& S

    : ]5 h2 C( e- _4 U7 s% E/ hDisadvantages of Recursion! W/ h  Q5 k$ a, y3 ~

    $ o' ~& c+ o( @! X* ?  ?    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.7 O4 v/ i3 K1 ?  \# w

    3 h3 z2 Y6 N- `/ H7 _+ H    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).: ^* _7 v( \. M

    ' m; ~$ Q9 ~' s' d: hWhen to Use Recursion- m! A3 j3 ~. e) e9 a- d. v; I4 ^
    0 m" ?( L, g! g; x1 |( {" D
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . O7 P0 x, {9 D2 N* ^% T% m7 R3 K! W& F! \% X8 v
        Problems with a clear base case and recursive case.- h+ p* r# _" F$ |; n& X

    / i6 g! x* N& H* u- O8 @Example: Fibonacci Sequence& n, p/ l/ l( }6 z$ d

    ( Q, h) C' Y5 X+ k5 YThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    1 y5 T! d) O. t) i& J8 e5 ^9 J; z0 {, E3 y
    3 F4 n' a$ s3 U. L, q' B* `4 y6 n+ J    Base case: fib(0) = 0, fib(1) = 1' Y6 e5 ^2 D* b2 H. c( Q

    3 X: |5 L& p4 y+ o9 O, a- \: N    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    : V! |5 |+ R- n9 @8 a3 n* f
    5 }! r- Z& f( ]python
    6 O. _9 K; a! C7 M' s
    8 i+ J( S& X" ?, Q7 p
    , X! d7 z; a+ x' a, P  e' ?( Tdef fibonacci(n):& a8 I  X* h# K( W
        # Base cases$ W% `' d) i8 w6 J/ ?; a9 _
        if n == 0:
    , E! r9 V# d- c3 ^& `        return 0! F- N" e% \: J9 X9 M" T" H
        elif n == 1:$ l# r7 Z& Z8 g5 ]$ _9 k
            return 1
    8 ^8 Y$ P- i6 ~6 P1 h3 ^4 G) @! P8 H    # Recursive case
      N- r$ s4 ^  f5 S$ a+ A    else:
    5 h6 ]  }1 u3 x% E+ \1 L! X1 x        return fibonacci(n - 1) + fibonacci(n - 2)* y  d7 W) b% D8 S8 V7 B

    4 d! ]/ m+ x! Z9 v/ W1 }3 H; \/ z# Example usage0 H) @+ D' r1 m! K9 X
    print(fibonacci(6))  # Output: 8& c: A' O3 |) l* W* {( h

    8 c/ v" m+ d2 L% WTail Recursion7 e( s. C' v8 k3 R, c1 N

    1 Y- e8 {  @5 Q' ^$ nTail 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 ~) j# l' ~+ k: |/ a2 V: F9 j4 H
    7 e$ ]& `3 _- ^' R4 M8 s% yIn 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.
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    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。
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