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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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  • TA的每日心情
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    2025-9-8 05:08
  • 签到天数: 3 天

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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
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    解释的不错
    9 o& s' B9 m" S$ }# C2 ]
    . T) U0 B+ g0 v# e2 S5 r/ E( [4 W) 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。* I) B6 s  a5 @& i4 h' R5 q: o- I$ u! E! r

    % Q# ?& ~# ~4 n! W6 Y) l1 \ 关键要素3 X( q: g2 I) G5 a3 ]
    1. **基线条件(Base Case)**
    ( E) D; D2 k' R, B; F% d   - 递归终止的条件,防止无限循环
    ! |+ p5 T" W. {- D# G   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: e% F4 A/ y, k* a
    - m1 s. S: D" _, C6 g: S
    2. **递归条件(Recursive Case)**1 C: I1 w) s) E( v
       - 将原问题分解为更小的子问题
    3 K9 Z7 ~  _. b" B   - 例如:n! = n × (n-1)!, b6 _2 {' R; L
    4 F# U0 ^8 X3 }+ ^# J$ b
    经典示例:计算阶乘: Q, [3 F: k0 X* X
    python& ]7 i8 O& p. O, o
    def factorial(n):
    9 _! V- ]: T# d$ f+ F/ j/ M; Q( ~    if n == 0:        # 基线条件
    . p/ `8 R" P8 @2 J        return 1
    5 U1 {6 W8 G8 N. C: u- g    else:             # 递归条件
    4 T: k. i1 _0 m- T/ u        return n * factorial(n-1)/ J4 O  x8 ?+ B
    执行过程(以计算 3! 为例):
    2 {4 k+ r- w" cfactorial(3)
    3 b. B, u4 S+ K9 e8 i+ u4 q3 * factorial(2), c- D9 \4 @) Q. G8 a
    3 * (2 * factorial(1))9 O" ~* d  C* @% u! h' ]- L
    3 * (2 * (1 * factorial(0)))
    - J  K" Y+ {2 o8 n9 c3 * (2 * (1 * 1)) = 6' O) T6 E  ~, ?% v( o

    ( K8 }  O% y6 |/ M; D, P! w% L 递归思维要点
    % ~  m8 t! i' K3 r5 r1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    $ ~" J, O1 C: |. D2. **栈结构**:每次调用都会创建新的栈帧(内存空间)6 y; ]* m# R- a8 \: V3 q7 Z
    3. **递推过程**:不断向下分解问题(递)1 t, r/ _* m8 a6 }3 ~# z
    4. **回溯过程**:组合子问题结果返回(归)$ J* V7 _/ q6 N) u
    7 B1 j# ?# s, D: G# z
    注意事项8 J/ i7 ]! T8 D0 v" [) ]; h
    必须要有终止条件
    : p4 Y* O5 a+ U2 w! u递归深度过大可能导致栈溢出(Python默认递归深度约1000层)" I% n, a) T2 C/ o1 W6 s# \* ~" K
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    8 d4 z* K$ i8 o' @# }尾递归优化可以提升效率(但Python不支持)
    ! P: e* h6 O1 h3 x2 q/ R
    1 i; T& O$ e+ U# r1 N 递归 vs 迭代
    0 \6 Z( |6 h: A: R  w3 F! c|          | 递归                          | 迭代               |; u2 l. ^) H$ {
    |----------|-----------------------------|------------------|
    : V$ |# D  q; v( Y: ^: @  P| 实现方式    | 函数自调用                        | 循环结构            |8 d, _; @" q: u9 a
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |/ \( e/ k! H. _4 W
    | 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         |7 I4 e4 g2 ]+ B' j/ o6 {: `
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |/ r& J  S; ]. R

    7 a* O2 R' L& }) L6 V 经典递归应用场景
    / s& g+ R2 \( ^4 z1. 文件系统遍历(目录树结构)
    . D  P& m; z! w4 }! p& q5 y2 x2. 快速排序/归并排序算法
    1 Y' K2 L/ K1 k# T3. 汉诺塔问题
    * K* ^( u8 v( J% v4. 二叉树遍历(前序/中序/后序)9 u* z2 Q0 `/ `0 q# G, E. U
    5. 生成所有可能的组合(回溯算法)5 u& N  N' M$ G! K+ F
      I( R5 [$ c! C
    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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  • TA的每日心情
    难过
    昨天 07:51
  • 签到天数: 3161 天

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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,
    # V: \7 L/ k5 ?8 K我推理机的核心算法应该是二叉树遍历的变种。
    # b; U* l1 q6 a  l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:
    + O/ v' x5 U1 B0 e2 uKey Idea of Recursion6 L" A8 N0 I/ s& n% g5 n# n, g

    % s% ]2 V+ C9 c% e8 T$ S  xA recursive function solves a problem by:, x5 p7 d1 {/ V5 a3 C2 b
    8 o/ d1 k. b) Z. }6 y4 [# k
        Breaking the problem into smaller instances of the same problem.
    3 \2 M2 n* t( N/ w  l8 R# B' M
    * ?; i0 B0 h1 K9 g9 i, U    Solving the smallest instance directly (base case).
    - i" u8 G  h  D! Y: ?# @- ~$ R
    , u( A& z- U5 \. R9 `    Combining the results of smaller instances to solve the larger problem.$ N& Q2 A/ y- n8 B8 r. L
    1 j" j% I3 i; ]
    Components of a Recursive Function
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        Base Case:0 C, W. E8 F/ l5 _4 y4 W
    $ y( O4 {3 m5 Q+ h
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.7 e1 l' V' Y5 o! a$ v/ e: A

    5 Y% X& E0 f5 E        It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    8 }6 E9 F1 w" L' @2 L, }    Recursive Case:8 Y1 f( l; y* L4 A) e
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            This is where the function calls itself with a smaller or simpler version of the problem.% U, H0 }# K  k/ [3 j' R! e

    ; E% D. o& q+ f        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% Q# G% x1 |$ Y& ~! {0 t

    ' A/ Y# M, d; z# {# u( SExample: Factorial Calculation0 x2 b, e2 x* W- m

    , W0 m) {6 t. E4 H* T, p. ?  sThe 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:
    8 J7 [- @1 p+ O# Y1 K% @0 J" e7 V& F' n& k4 A
        Base case: 0! = 14 H& y' v; O4 ]7 s1 R# k
    8 t: U' l3 |( M4 K& s  V. i: n
        Recursive case: n! = n * (n-1)!
    . a7 f. M, W7 C: q  d- ]2 L/ f. r- o  q* _; W/ h
    Here’s how it looks in code (Python):7 Y% u0 C6 p; Q. T+ ^* j$ }
    python2 W+ b& B0 s, C( b& d
    6 ~# h# l7 B, `

    ' x9 s  @5 z, o( z- r' o  _# J+ T4 Y4 Udef factorial(n):
    ' a6 Y& H$ u: k    # Base case' W' X/ L. S& p, Z
        if n == 0:  [& @- R" p% D# _
            return 1
    1 h2 [9 s3 l: [- w2 O6 g/ `    # Recursive case
    6 ~3 d" ~$ j, z7 w* H$ V( \    else:8 Q" y7 m3 P+ {, ?& ]# ?7 e! ~
            return n * factorial(n - 1)
    2 ]: q* O5 ?% P
    3 `  l" T6 {7 A+ ^# Example usage- D/ W2 g1 G- B# [& q1 B4 t2 U5 ]* x
    print(factorial(5))  # Output: 1207 u0 ?8 L, l2 b' H8 L$ P+ D- M
    8 |0 X7 q% u( [' O6 P
    How Recursion Works% [* ^1 }8 ?6 k. x4 x+ E# I: h
    $ k/ c( I) j1 R( w! @
        The function keeps calling itself with smaller inputs until it reaches the base case.
    : Q( y- d# X+ X# Z' G" G& b8 P& w. y! r# g5 u$ b; _
        Once the base case is reached, the function starts returning values back up the call stack.
    & `  g' I; a+ z4 \, ~: A+ v* L2 r" Y6 U) X* ]+ @
        These returned values are combined to produce the final result.+ W7 S" B5 l+ e
    6 f3 k7 m/ f/ I/ X/ R8 X8 {
    For factorial(5):
    . h( k/ ^) W1 v7 U& R3 w0 ?& Q5 k4 U( H* }3 T
    ; [. V+ r- V" b4 w+ a+ k+ C4 \
    factorial(5) = 5 * factorial(4)8 U/ F6 o- |7 N$ g
    factorial(4) = 4 * factorial(3)' x' Y5 M7 u6 K4 d( j
    factorial(3) = 3 * factorial(2)# s/ d4 ?+ n( B, A- i, W
    factorial(2) = 2 * factorial(1)
    0 q0 r. L/ A+ f- v3 {$ Efactorial(1) = 1 * factorial(0)
    * ^: ]5 e6 Z9 Z: _0 kfactorial(0) = 1  # Base case2 {  o" j2 Q' h$ s

    / n( w  C$ F4 ~. W6 oThen, the results are combined:  b! c, K8 K- K8 V+ l

    ' L1 k/ \9 q/ A0 I  a- ?" H0 h! I9 Q2 D1 r7 n; F' @
    factorial(1) = 1 * 1 = 1/ s2 P- o0 F: W6 g. f* {/ \
    factorial(2) = 2 * 1 = 2
    . W# g; i9 h7 a' g9 f% q+ ofactorial(3) = 3 * 2 = 6
    9 S+ `1 _# V$ d9 F  Yfactorial(4) = 4 * 6 = 24
    0 E1 b) W4 M% a, T2 o+ M& Ofactorial(5) = 5 * 24 = 120
    ; [9 |2 u& D; S* l- i, U5 p
    / ~8 W1 i+ |" L( R4 IAdvantages of Recursion: X# }+ L5 K1 Z* i4 ?3 w* z

    + p" {% t- G5 h, \% g    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).0 h6 R, @. j) _4 ^. y) \* [
    / p( M; r+ ~0 S
        Readability: Recursive code can be more readable and concise compared to iterative solutions.6 B( ^) C- F8 R, w8 V- n. B+ B

    4 s+ A8 @( Y3 ?% O9 Q. mDisadvantages of Recursion
    6 Y, G& j7 Q0 ~( {
    . l6 j+ Q. r9 N$ t    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.- Y" @6 x( }) X: |) i8 U- K2 f

    ' S0 {5 g/ i: S$ V; i6 D6 T    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)./ L% T% @% ^4 U$ }

    + s0 \  U; h' Z2 jWhen to Use Recursion) R. `( c# c: X1 l0 d2 A5 @
    $ e' Q. w3 O3 a3 j; \9 X
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).( E( G) Q# A: A2 s

    8 o* P1 ?5 w3 {) ?& h$ `    Problems with a clear base case and recursive case.
    : w7 |! s6 t3 d! ^7 ^
    + g' E+ S9 H6 y7 H2 r- _0 _* `, xExample: Fibonacci Sequence
    ( [3 p. Q' v! i2 E7 O- |" L% ^
    ' y0 {, s0 p" w2 ~The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:' B( `1 n' @# o; n, `

    5 R8 u; N3 V0 z6 U1 j. c. r- c    Base case: fib(0) = 0, fib(1) = 1
    & K6 h# ~7 U5 R, [; q
    ' n3 e7 T( h% T, h    Recursive case: fib(n) = fib(n-1) + fib(n-2): ?' F' r# G5 Y+ z0 q" ~

    ; B9 v" b9 Q! A: Q5 x' ypython0 X. }  _. u  F1 _

    1 n9 `( X' \9 k  W& ^/ T3 E
    4 o: O* b. \6 y+ Ddef fibonacci(n):- h. j6 c' V- ]: V5 L8 T4 U
        # Base cases
    " H" R  [  b. q0 H% c    if n == 0:
    5 C4 u3 e- X5 h0 w$ s& M; M        return 0, w/ z0 g9 O+ t( _8 E8 H
        elif n == 1:% \5 X' p" a' V9 N" f! e
            return 1
    7 [4 a, F% P1 }" p) q* g  g# {8 b    # Recursive case
    . H- F6 p% ~+ n1 p. F* w    else:
      w0 V" g  k: |8 p        return fibonacci(n - 1) + fibonacci(n - 2)) I/ A. O5 ~% ?' b! B9 k

    2 x$ X: r  E: {+ a7 `$ x# Example usage
    # \  n6 G* ~8 b+ |( t, aprint(fibonacci(6))  # Output: 83 l4 q- i, q7 R; `

    6 f1 a5 y/ L- W8 ~0 _Tail Recursion
    2 C7 k0 L- z+ s8 D- Z. ?; ~! g+ M0 G% {
    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).2 p/ w0 g" @2 N

    & z. u# @5 h+ K' P, IIn 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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