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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
  • 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
    2 K: p: @* s* Y( v; h; J' T4 {  T8 W7 `$ }7 i* q- ?
    解释的不错" ~7 A5 Z5 s' D

    5 F( ?+ Y$ s7 O7 ]$ d! U9 U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。
    + o; h* l$ X: S# H7 Y7 t
    , B' D# G; p5 J6 e+ J 关键要素
    " m$ s2 n  O5 t% B& M% l. A+ {1. **基线条件(Base Case)**
    # i) X- i1 [' Z9 q' p2 f   - 递归终止的条件,防止无限循环
    3 z# m4 `9 W8 W8 |1 `! t7 U2 I   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% Y' h2 g3 L; Y2 d9 s

    * T& b# ?% N& J, h( q4 J2. **递归条件(Recursive Case)**- \+ a% m& B9 v& q* J9 t3 Z1 t, l, H6 Z
       - 将原问题分解为更小的子问题1 }% L8 l; e$ o( e9 Q
       - 例如:n! = n × (n-1)!
    4 \$ i, l0 p, f+ |+ [
    0 ?1 C% G$ A8 z- Z 经典示例:计算阶乘! g! J/ O% v3 W( x
    python
    : |+ F: e1 P* H9 l/ n" q5 U$ hdef factorial(n):
    4 b  m( x  S) f, [    if n == 0:        # 基线条件
    ' ~: c" `. h$ p- u7 t2 r        return 1
    5 Y) w% N! e, h, g0 x    else:             # 递归条件) R' J; B0 h, k# S, `
            return n * factorial(n-1), h( n* ^3 v$ s' z5 c) @( Y
    执行过程(以计算 3! 为例):
    / Y  s1 n4 y3 B. @+ O7 [factorial(3)
    6 q4 W# v0 P. ^3 s( B" b3 * factorial(2)" B4 l% n6 l% `
    3 * (2 * factorial(1))5 n" ~* T9 b/ U$ P+ ^1 u
    3 * (2 * (1 * factorial(0)))
    7 ^# h! x, l& Z3 R5 M! M$ |* e/ z: }3 * (2 * (1 * 1)) = 6
    : X0 s/ C1 J# k% i' P% a0 f4 m
    9 P! s8 B, @- ] 递归思维要点
    , X: u' J; x1 u/ ?/ E: v; C1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    - |9 L1 p+ Q6 b+ i5 @2. **栈结构**:每次调用都会创建新的栈帧(内存空间)# E2 d" s' V5 q9 _- _3 s' [
    3. **递推过程**:不断向下分解问题(递)$ U* A6 C8 d+ I+ G
    4. **回溯过程**:组合子问题结果返回(归), A9 H; ^5 X( C5 d5 q

    9 Y8 [0 K$ O% L" A 注意事项
    % f0 `3 H( U8 ]/ ^$ G必须要有终止条件
    ) Q  ?# h0 d* }% r9 t! n3 f6 d# C1 O; E递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  r: |8 o- v' e# X
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    - a( P& [- T% r' h2 |尾递归优化可以提升效率(但Python不支持)1 p8 G8 m: B+ I: W

    7 ?2 \6 u3 |5 _ 递归 vs 迭代
    / R1 F+ d1 Y, x|          | 递归                          | 迭代               |7 Z5 o8 t1 D5 [8 b, V* T6 B
    |----------|-----------------------------|------------------|: u7 v$ _) f% D& B" N
    | 实现方式    | 函数自调用                        | 循环结构            |
    # m6 V# A; I8 {& ^| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |9 \& v/ B* W+ F6 ~" 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         |; c& X, S/ u# ~: n6 i
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |0 ~6 Z8 T: E) s7 R3 q! K) t8 c& G6 V- W
    " M% ]* Y+ V9 a* x2 n
    经典递归应用场景
    8 M2 {% \) }( ]1. 文件系统遍历(目录树结构)
    : \+ O1 \: v* [0 a+ e2. 快速排序/归并排序算法5 [( W, c* A; ^& d7 I4 P6 X7 U
    3. 汉诺塔问题
    0 Z- H7 r& I( e/ h$ v  a4. 二叉树遍历(前序/中序/后序), _% b) m3 G9 h3 ^+ I& a
    5. 生成所有可能的组合(回溯算法)9 B0 b& a& O: p
    & f+ ?9 g4 j, t/ K) l7 P
    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,; u5 r' P9 A/ |! V- M; @
    我推理机的核心算法应该是二叉树遍历的变种。  U  W% Y; R- ?3 N5 j
    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:
    * p4 H  n1 y$ A: dKey Idea of Recursion2 k2 a/ E1 q2 V' ~' u

    * |. N3 c9 M2 r( k( yA recursive function solves a problem by:1 _' q& x/ y1 r1 t2 K

    * F$ w, E1 s+ [    Breaking the problem into smaller instances of the same problem.
    9 l) {& _; E( i) ^. N# J3 G0 w4 O& |5 z) a: @: Q
        Solving the smallest instance directly (base case).- z) {2 }4 \& f# y& K( B+ M( B" `

    3 ]9 ^5 a; U' I# r0 i    Combining the results of smaller instances to solve the larger problem.
    - d# |$ }5 C% Q1 W' H( `# M6 X
    ! w( L: `( y4 N; y. QComponents of a Recursive Function
    $ `% u1 f0 L" E0 ~) m$ y/ O1 _! N/ u/ a# t
        Base Case:
      p$ `( k0 T" o  S" D# C7 d% H- O9 f, E0 A9 X
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* C- J4 d1 n7 o. ?" S! v- {

    & J* |" Z8 p" n4 g6 v        It acts as the stopping condition to prevent infinite recursion.
    4 [' _6 w! a$ S0 B. }9 v# t& R
    9 L/ L5 N7 ~1 d) _. F/ j4 b4 n  \        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.- z" E% A3 ^5 r

    - C$ A5 x/ s1 c, m    Recursive Case:  p$ y4 Z+ _  D5 X

    - i# m; ~+ R4 O2 q% y. ^7 E        This is where the function calls itself with a smaller or simpler version of the problem.5 r" r: |% w; y# M

    ( L3 b# V2 G7 _. B5 b/ h        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    9 I( E* K2 z/ Q, ~/ Q9 @6 d; b) D3 `- R4 j' H4 j
    Example: Factorial Calculation
    2 o2 c9 l* N/ v8 k% d1 f- O. B0 t: L& ]# I. K& \  k  p. L6 Y0 _
    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:6 `& i0 l2 x% \& |+ O

    # a' |3 z9 J2 T" \    Base case: 0! = 1: ?0 k$ c! A3 M) t0 K

    , C; }& y9 w$ y0 h- B% j: Y    Recursive case: n! = n * (n-1)!
    + ^! C1 a4 ~; x" D  V# |& @" l; k9 a
    2 e/ X  x$ s4 @3 g- z" @Here’s how it looks in code (Python):
    * I+ k# i1 z' R5 w* K: Y0 Ypython+ e: P/ n5 p  G% K7 w9 L

    , y$ a  Y- b% Q# D+ M0 {* Y8 d$ H  x) H
    def factorial(n):
    ) }- e/ _, Y2 N7 P$ N1 |; d6 @    # Base case
    7 v& y7 r8 r$ [# Z3 ^/ f& q' V) t8 Y3 k    if n == 0:& o# j8 C7 @2 a! }4 N& v3 {& i; y) f
            return 1$ W' F$ L) ^& `: d5 {8 y
        # Recursive case1 c; {1 u3 L6 H) l
        else:
    ) [# ~' x& y- ~9 ~        return n * factorial(n - 1)
    5 M; u$ X( P; [. @; X1 O. [6 `- [* J' M/ A" B1 W
    # Example usage! z* ^6 r/ I9 S" k+ g/ l
    print(factorial(5))  # Output: 120
    : Z; X) {# P1 }- |
    . v# U* }) [! U+ T$ i9 ?How Recursion Works+ f1 j8 }- D2 K" v

    1 n$ f# a! q2 O( ^6 }; F8 f% m    The function keeps calling itself with smaller inputs until it reaches the base case.- d8 u- n# h3 o$ W! c+ h- E

    ; ^0 H4 G' Z9 d& g% ~' \    Once the base case is reached, the function starts returning values back up the call stack.
    $ u9 w. H$ I! I( K) `, D6 t# n8 ]
    8 |, A# M" W- v* Y. e* r    These returned values are combined to produce the final result.
    ! H+ e5 {! s7 y. U! S: e) c5 L! N! L
    For factorial(5):& b+ b! P6 S) h( [2 _9 K

    ; q0 m/ l& l9 u& X$ n0 B- L6 M1 C4 {7 I& j2 Q0 N
    factorial(5) = 5 * factorial(4)
    - O! Q5 o) i2 c0 K) H. G- dfactorial(4) = 4 * factorial(3)" w0 g5 c9 W* n2 k
    factorial(3) = 3 * factorial(2); J: r6 N1 E9 h: g5 _  T6 E, N
    factorial(2) = 2 * factorial(1)
    / A; X0 A+ K! O5 ?# X3 S! A% w3 xfactorial(1) = 1 * factorial(0)% b( K  F+ l  ]. u; j3 B  U  X5 w
    factorial(0) = 1  # Base case
    $ W9 N6 ?, K2 X+ |
    1 E9 b& v( w# Y9 LThen, the results are combined:, ]) _3 C) o" U8 c

    0 ^0 O0 q8 {5 v1 l7 C' N: m3 R6 i. ~; A0 D% v  _& I& b0 h" }
    factorial(1) = 1 * 1 = 1
    : j+ y# `1 g" |8 F; B1 S! Lfactorial(2) = 2 * 1 = 2' R2 a, L5 E: g( E' y4 k
    factorial(3) = 3 * 2 = 65 {, O% b) o) b6 e* t
    factorial(4) = 4 * 6 = 24  L! I3 G2 O, A* O  J4 }
    factorial(5) = 5 * 24 = 120
    1 Y0 l9 b* W( r9 {0 M$ i
      k7 Q& o0 k; q; H6 Z% ]. _Advantages of Recursion, N0 x4 z# F" j& [, u( ^6 F
    ; P, @" A5 }8 z/ 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).( @' L6 u1 m  w  x! d( F! ]
    + X8 b0 ]% l: k
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    1 T. W. H+ L! i6 q6 V' {, p  V5 S. @$ u) p: Q& [( Z
    Disadvantages of Recursion! \* j8 m* M  }2 \

    ) |7 c6 N9 {6 C3 ]( ]0 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.- D1 K) T, z6 F: @, r
    " L# u( {1 U* y) V. M
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ; l) k4 B1 n4 B& z4 q
      l: c4 T* Q- |+ \$ oWhen to Use Recursion5 l8 ^3 I( w) D
    8 I' _8 E+ l+ [0 t
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).6 t5 t; x6 \9 W4 i
    ' H* W1 o6 ^, r% j, {' k
        Problems with a clear base case and recursive case." }9 e- |! K) l& U4 ?& v
    - S* U; I8 K6 z$ ~% G7 Y
    Example: Fibonacci Sequence: p9 T$ K% E% h

    ( K0 H4 j0 d' I7 dThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:( x( r- p$ c/ P+ b" L% ~: K6 v

    , k1 \' B  e0 @7 ]( e, R$ ^( F! {8 n    Base case: fib(0) = 0, fib(1) = 1
    9 ], E8 |& d8 y/ k
    5 w6 _. E! b  p/ _8 X    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    4 A& I9 `. R* ~1 e3 G
    ; f( }* O2 q8 O' T5 O& qpython
    * `. n( i+ h7 C# o
    5 k: J1 |' P. j( s8 t5 z  B; F. Y. p; q
    7 @% \$ r' D3 r% s0 z& A7 e- Udef fibonacci(n):4 ?' j1 p. o- a3 y  V
        # Base cases
    6 B$ K! @) ^  n4 x    if n == 0:( E% [" ]6 i: m5 e- h! X: G9 }' U
            return 0
    8 g8 u% p( a8 F* J: i- ]9 _    elif n == 1:
    : q/ S! l5 z3 B1 |7 s        return 1, |0 M& W7 F/ Z3 [6 Q+ H. b
        # Recursive case% h. g, l9 V* M0 i9 G( i2 k# `
        else:
    , v* D- }+ g% a9 |! K4 C& c        return fibonacci(n - 1) + fibonacci(n - 2)5 h, p2 _) d- h/ Z- u+ Y

    . l7 b( f. p+ J3 J" \- j# Example usage8 X/ }, B. W$ \5 J8 b! I( A# N
    print(fibonacci(6))  # Output: 8
    " Z) ~6 [+ D" G0 f; H  b9 N; A- [3 Y+ x4 U$ i
    Tail Recursion7 V9 u+ g) `( a) l: x

    * x8 {9 z6 O% ?' z4 J! kTail 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 N' l2 X3 j! x! b" B1 x
    . W, N# `5 S8 C: s& f
    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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