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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
    $ l$ ~+ D6 m; S' {, h
    " e- v9 f8 o7 N! E: y5 t. x6 w) X解释的不错
    : @5 @! N5 _& m8 d$ V+ \/ ^
    7 `/ ~6 N  u, O+ G9 c" t1 F8 c$ i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。( L8 b, c' e  w, Q1 h

    # l8 s! W4 E4 z: C- i9 T: u 关键要素; j7 v& `; ^1 Z* g$ b4 ]. I. J6 n3 z
    1. **基线条件(Base Case)**
    7 O0 F8 _  i0 U6 w+ f; c- s   - 递归终止的条件,防止无限循环2 u2 H2 _9 i. A# i* u; t
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1$ A, Q) H+ R2 m4 f; R. P! F2 m

    ; f8 O* l4 D! [8 T* r6 K& W( ]6 `2. **递归条件(Recursive Case)**
    4 j: X1 l( X! g  o  O3 `  b/ B. @   - 将原问题分解为更小的子问题7 s4 J" P% d+ T) O' p9 Z
       - 例如:n! = n × (n-1)!
    7 s, k+ u1 O+ |! u2 Z0 H2 m; |; w  C6 U
    经典示例:计算阶乘
    : {; O& p  v1 h9 b# o7 T. `- X" Epython9 [& _+ n1 K# {7 l' P. Q6 E
    def factorial(n):
    ) s! B  Z6 w; C* B2 [: Y# O% i) X7 ]    if n == 0:        # 基线条件
    7 f6 D- [0 T5 C- h: y        return 1
    , T5 `6 {& `2 V& r- d  S    else:             # 递归条件
    8 M) O# w9 s1 W" f% v        return n * factorial(n-1)
    & z. x! J; t4 V执行过程(以计算 3! 为例):
    $ o0 i) ~# ?4 h7 k2 ^factorial(3)
    * Z* B* w. z. M  R- |; x3 * factorial(2)
    * l' I6 ^; O) P$ s$ u3 v0 _3 * (2 * factorial(1)); _/ R! U0 O7 ^
    3 * (2 * (1 * factorial(0)))
    2 m3 C0 J" \% J: R' C- G5 r( f3 * (2 * (1 * 1)) = 6
    6 S% K% o) p$ E; U. L; e; A6 y7 }5 ^
    5 {+ A6 y/ f! j& x9 i  B, v 递归思维要点
    , z0 h0 u! d) ^1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ( E0 j% N. p1 A( W4 D! E2. **栈结构**:每次调用都会创建新的栈帧(内存空间)! k- f0 f, B  V4 h
    3. **递推过程**:不断向下分解问题(递)
    ( z" x  x7 {. o6 H4. **回溯过程**:组合子问题结果返回(归)8 a' f4 t: V' _, V! J0 N
    2 H1 k4 |3 v) u2 Y2 B
    注意事项( ]. }3 `$ k  Q2 a% K
    必须要有终止条件8 A  @- I. S0 G) _" p
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)4 g8 e+ Y7 R( E! _6 w
    某些问题用递归更直观(如树遍历),但效率可能不如迭代7 O! N! e% S2 _5 b
    尾递归优化可以提升效率(但Python不支持)
    , b: {) S1 C% N" O. x/ a/ e( f: z9 e2 N5 {0 r  s! M
    递归 vs 迭代
    6 I# Z9 [: [! x4 c; j  m* ]) w|          | 递归                          | 迭代               |
    . h! i- d0 R0 {1 i. c|----------|-----------------------------|------------------|
    # }8 a0 `8 {3 O% O( J& t" N| 实现方式    | 函数自调用                        | 循环结构            |
    , ?/ T' R. y" S3 s& P| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    " G  u  e, Z9 _| 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         |1 _1 N- N/ b" e# z, X8 U
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& ~, n7 v& c- [+ ^1 w
    4 u7 V: d  {$ Y. d* H
    经典递归应用场景
    2 e$ C6 S8 b( S# R1. 文件系统遍历(目录树结构)& a1 g5 T2 O3 k4 O
    2. 快速排序/归并排序算法
    4 A. \0 ?- a# `/ d3. 汉诺塔问题
    1 ?5 E5 B; s+ {7 S4 a, U4. 二叉树遍历(前序/中序/后序): N; O) o. f& u1 d5 M2 U/ _
    5. 生成所有可能的组合(回溯算法)9 E7 h5 G& d3 D( N% Q0 ~! s3 _
    2 v7 n' @. n5 @% s& A8 r1 W5 k
    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,2 f+ H3 \. I  K+ n9 g' ^0 m: w
    我推理机的核心算法应该是二叉树遍历的变种。* u* @! s/ v( t1 h5 I6 ]
    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:4 b; f4 D1 x  M* E% |
    Key Idea of Recursion
    * z* |$ @) z% D! }# A( [) j# ^/ [- \$ Y  }' p/ R" d
    A recursive function solves a problem by:( ?  I2 ?1 U6 P# j$ x3 {% t3 L, d

    - o% I1 t) `, i7 N# w+ q: u- U9 n    Breaking the problem into smaller instances of the same problem.
    4 u4 B" B" a+ R4 j1 Q$ B" Z3 V& O0 p8 W9 j6 T- K% t6 q
        Solving the smallest instance directly (base case).
    ; f7 i. B6 J& B) p4 i4 G$ N, G% ~7 k/ A$ K) O# {
        Combining the results of smaller instances to solve the larger problem.6 p: l; Z5 Z3 o  ^: I

    + d5 o0 s; [% F% e) VComponents of a Recursive Function9 J" d7 O; C% I6 O: G% r8 i

    8 }6 e3 O7 d$ c( `" c    Base Case:
    5 N6 `" A; s. x+ Y4 W: G1 S# a% ]* r% T$ B
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    # Q" B8 O( w4 B# l+ @9 H* t. Z7 i+ ^3 h2 J: V" s; m* q, O
            It acts as the stopping condition to prevent infinite recursion.
    * y) r6 n3 o6 i/ d; \( C' S
    ( k1 w- A# B) M& f  e; y/ x6 s, [        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    8 j+ R% J# G$ y9 t0 H
      ]# [  c; C- d6 U+ h8 s; ?0 O' }    Recursive Case:4 O5 V# d- c7 g! c$ X

      z% I, m3 U; }% Q' H# B) ^8 ]        This is where the function calls itself with a smaller or simpler version of the problem.
    & X8 F* t+ P' v* K- v
    0 Q  E* [" M/ \: e! Q7 h0 H        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ' |9 |0 w. B4 J) R9 ]1 }0 V2 a. W; m( Z$ f
    Example: Factorial Calculation) j  \+ g2 Z+ G& X  c
    ) @  ~8 I6 [) N' |4 c
    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:
    ; ~, }5 R* A. l) h5 h4 U# m! a1 E4 a& ]) J
        Base case: 0! = 1# s& n% [1 c2 o

    # Y' s# h$ o! w$ b- H8 w5 L$ X    Recursive case: n! = n * (n-1)!
    / ~; v, m/ |0 J5 {. L( i0 r$ K3 s& D# v4 A$ u" \
    Here’s how it looks in code (Python):) [/ g6 K  r% o2 \4 P
    python
    1 K& p& q. H4 m7 ^$ ?! f5 k& ?( Q
    : b8 }9 B% S! m2 d' B  z
      o# a  e5 ]* gdef factorial(n):
    % {1 i9 V# _. P# c    # Base case) c& N5 n% R! P. V# g. r9 `
        if n == 0:5 f; l6 C! Q/ C( d9 `& T/ ~
            return 1
    . m- v3 l; ?2 V    # Recursive case
    ' H0 S/ B# j8 O8 i  w    else:: D3 s  P/ ]' m3 @0 F
            return n * factorial(n - 1)
    0 N- q1 q7 y, @9 o- t4 c# R" h6 ?  Y; v1 O$ u! ]  R: w9 d
    # Example usage9 m- V% [- M6 i* s3 Z) }
    print(factorial(5))  # Output: 120* `& s9 E" h' `5 D1 m4 y. D- L0 W7 e
    # `( @. ?- a/ \- ~0 ]" b. w
    How Recursion Works2 Q" {- C7 H' v3 x
    $ V& V4 N7 K: }
        The function keeps calling itself with smaller inputs until it reaches the base case.4 `, |0 X" X, M, A- G
    " b( [. ~9 l9 R/ K$ G9 F" w
        Once the base case is reached, the function starts returning values back up the call stack.
    % Y0 M# R& K8 G6 Z0 S& ^3 R1 W6 H4 c4 v/ P' u  I* F+ x
        These returned values are combined to produce the final result.# _0 A# z0 B8 Q8 [, p

    ( ~; I/ }- _' \4 _( |3 ~% g5 ?For factorial(5):
    5 _+ U) ^  \, @, q& I# F# D) j' y2 {( a: w3 E2 M+ q
    $ P1 ^. Y1 M9 S. \: o2 ?
    factorial(5) = 5 * factorial(4)% T5 ?1 C, \+ z/ L! g4 [
    factorial(4) = 4 * factorial(3)/ F4 j4 |5 s0 G8 B
    factorial(3) = 3 * factorial(2)6 s$ r$ c6 p; f4 ]! E
    factorial(2) = 2 * factorial(1); H7 ~3 K* b. T7 s6 r9 y
    factorial(1) = 1 * factorial(0)
    4 }# e" y" S2 F! _factorial(0) = 1  # Base case
    ; E+ W: p1 _8 P
    # b& ^$ ^0 W9 I- H& V$ IThen, the results are combined:- [, y% j! H- G
    / Y9 H8 W$ j- }$ f2 G. U

    8 k% W0 F7 {5 [8 Lfactorial(1) = 1 * 1 = 1
    % B6 L5 o; i3 c* m8 ]factorial(2) = 2 * 1 = 20 ~  F) Q5 N2 o( R) z
    factorial(3) = 3 * 2 = 6
    2 p- E$ m( E. q8 S/ n, {factorial(4) = 4 * 6 = 24
    2 |- n9 G  I  ]0 x' T; Xfactorial(5) = 5 * 24 = 120
    ) g* y: _* w( a% ~
    0 N  y. z% c8 ?0 eAdvantages of Recursion# E+ k( T6 ~: D0 y; e
    ' {: R2 W$ w  `* C- Z+ Q
        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).. F% i, V1 `; f8 z/ I5 W

    # }+ r2 ?" O: j6 `6 X    Readability: Recursive code can be more readable and concise compared to iterative solutions.5 c) L3 h$ |5 H2 S2 h7 G! b2 I

    - F9 ]; f& h) O8 z# b; TDisadvantages of Recursion1 ?  a4 E* \7 R3 m, r3 r

    4 z# u( M; j# y6 K& J! ~    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.  G  l0 ], t4 f* f. `! G

    ' ]4 \- K! ]1 x$ y5 @/ e$ @' e    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    % i& q! E6 t  p
    3 P/ S: G' Z5 zWhen to Use Recursion* B7 ?, U7 z7 f+ X& K7 _

    # U& O$ Z" Q* f1 ~  }3 o    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    5 x4 J  m6 R  x) n, q
    : W: n2 H5 M5 w6 J4 f! V" Z    Problems with a clear base case and recursive case.2 P3 u- l5 R7 F/ M0 C  |* j0 G
    " A. L" l  P1 f' N% D4 `, c
    Example: Fibonacci Sequence7 T% I, D( p* |/ [
    + X% X5 i0 F( n* i
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 P# b1 Y& M, A$ z/ Y  ]4 G
    5 v; q+ z! n/ C! P4 P+ Y* m( z
        Base case: fib(0) = 0, fib(1) = 15 y7 {$ A$ [1 m/ b
    2 h: Z6 |: ~" x' T; {7 B  l  x
        Recursive case: fib(n) = fib(n-1) + fib(n-2)" n. @; p2 j) n0 U$ N+ ~: V
    2 u3 ?8 Y$ E6 j6 n7 ~
    python
    . h" F$ z$ N4 B3 n# c" G/ Q1 q  P8 V7 r0 H- [+ i6 A5 N! ^5 v

    1 o+ [6 _. P0 F* X! Edef fibonacci(n):  V( S# C- E" f5 X$ z% Q2 |' T
        # Base cases
    5 P0 C3 i9 K3 l  V; g    if n == 0:( L& e2 U6 S. ^( o: p' `
            return 0
    9 [  {* y6 J! [7 G+ E6 U! k  u2 Z8 y    elif n == 1:. u) ]  b" i0 e* g1 e
            return 1
    , V) S* ?5 j2 o8 w8 ?$ f+ I4 j    # Recursive case
    - J- N) _' Y4 U; t; a    else:
    3 ~4 X( C+ }2 u2 y- F        return fibonacci(n - 1) + fibonacci(n - 2)
    ) r( R, D5 i1 n. r; I1 k7 ~6 g& ~* `. @& A, f1 n
    # Example usage! q1 h: ]0 d. i- a
    print(fibonacci(6))  # Output: 8
    9 u! J0 Y) }  J. Q7 E* W" ]! K/ O. \6 ^/ N1 B; H
    Tail Recursion
    1 a1 A" s" ~. u+ K% U1 N* v9 v2 l4 c1 d) @8 q2 m
    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).8 u  c7 X. _3 Z8 I( k+ e

    , H8 ^+ y+ s$ l5 Y, ]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.
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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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