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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
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    解释的不错9 b9 a$ x; m3 B' F8 m
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    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。
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    关键要素9 p$ g) c  S" O0 R* p
    1. **基线条件(Base Case)**5 X9 E- }5 f3 z3 j
       - 递归终止的条件,防止无限循环# O) e: j5 r, M! u
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1$ N/ g1 M9 a( h
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    2. **递归条件(Recursive Case)**
    8 a3 d$ w' B/ R9 n   - 将原问题分解为更小的子问题1 w6 @$ _9 V2 ^% n1 p/ M
       - 例如:n! = n × (n-1)!# `; ]6 h% ~4 M: D7 Y7 @

    2 J! O. ?# B. g8 Z' | 经典示例:计算阶乘8 K1 F* e( O2 ^" w( ]4 S# M. ~5 F& Y
    python" b* K& A* u' H5 \- x
    def factorial(n):/ I) O  L) u* }7 l
        if n == 0:        # 基线条件
    $ g% l" i/ p% \5 Z) T        return 1
    $ _' U( R# L/ q( k& u8 |    else:             # 递归条件
    # J/ T2 ~/ a- _        return n * factorial(n-1)+ J4 _5 d& X. s* _! \- l- j8 A
    执行过程(以计算 3! 为例):4 f& M" V+ Q% ~8 F" Y! I8 Z! l
    factorial(3)
    8 P  A* y7 k) W( h# \4 a$ S& X3 * factorial(2)) B- C$ ^3 ~$ Z  @. M. u; N
    3 * (2 * factorial(1))! o8 F! w6 t, T7 e* {' u
    3 * (2 * (1 * factorial(0)))6 R9 w$ F: l$ \8 G+ V- K+ M
    3 * (2 * (1 * 1)) = 6" T4 }. u; Z0 x7 z3 k3 X1 t, y
    ! X  r; {% v6 Z5 P; q) m, H' @+ T$ V
    递归思维要点
    $ h; T* l, x/ V! L+ D+ n1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    8 W& x1 g8 h2 [9 r2 [2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    6 X/ d, i; Q: r' X( A4 v) U4 n3. **递推过程**:不断向下分解问题(递)5 U; M; o& {) o3 l. ?* S9 s
    4. **回溯过程**:组合子问题结果返回(归)
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    3 i% |7 S% t+ ~' @% N/ l' T% I 注意事项
      }' Z3 M, Y0 a& Q: u必须要有终止条件
    + A0 Z4 p4 `6 c- |( |8 {0 D/ z递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ) h, M5 t# V# T8 y某些问题用递归更直观(如树遍历),但效率可能不如迭代
    - d& q' r) e/ [' ?" E尾递归优化可以提升效率(但Python不支持)
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    - T2 `: r  U- i 递归 vs 迭代. Z" {3 i9 o$ C% Y  [
    |          | 递归                          | 迭代               |' ]9 F) E5 s5 b# O4 H* B2 Q
    |----------|-----------------------------|------------------|
    / I4 D: y/ Q/ T9 B- m+ ?1 Q| 实现方式    | 函数自调用                        | 循环结构            |
    0 i1 C& w) T& j6 ]' f! ?  || 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |- Y( I# r) Q$ b1 ?' n
    | 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         |8 f8 y5 r" A. D" O1 }& E' A# K" _
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |8 }6 t/ y1 @7 k7 i# G1 W+ H) X' P

    / n5 J- X0 {! f3 i& }7 ^ 经典递归应用场景
    6 Y* g' y4 D9 A" k1 q% h; W1. 文件系统遍历(目录树结构)( P; K, T- V' {5 j$ A
    2. 快速排序/归并排序算法
    ; k' c4 G3 V) J4 U3. 汉诺塔问题7 @: v( g. D" u5 C& Z. o% Z. l; ]
    4. 二叉树遍历(前序/中序/后序)+ R" i3 }8 E# j0 y; V) `
    5. 生成所有可能的组合(回溯算法)) J2 U$ \4 C- \1 m5 o2 L( T; o

    + o$ |! l. c) }% z4 {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,  ]* d2 V* R4 f$ M
    我推理机的核心算法应该是二叉树遍历的变种。
    + q, B! f/ N$ O& A1 N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:' M- V+ L3 Q) }1 b8 E' ^
    Key Idea of Recursion# T: F+ A& H* X# ?1 U6 `% o
    , `) B1 U! t) d9 {# K, }
    A recursive function solves a problem by:
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        Breaking the problem into smaller instances of the same problem.8 X; |# g; X) y2 _* \' }$ ?
    : ^1 h/ j* C' c5 _. U1 S( I
        Solving the smallest instance directly (base case).
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    5 N; Z- W; p# ^# ?6 q3 r: A, {    Combining the results of smaller instances to solve the larger problem.
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    - W; x! X5 [# f: Q& a) k$ c( Q: xComponents of a Recursive Function4 {% h: u3 A1 _, @7 C8 D/ L7 z

    2 E; F" o. f. ?: m% {  c4 s" x3 C    Base Case:5 m" r2 u# [* i# f' H

    & B9 D2 M# O7 t  `' O        This is the simplest, smallest instance of the problem that can be solved directly without further recursion./ J, p* o# W+ B: r) k6 E

    & ]) b( [3 v& T9 E6 n& `* u9 k        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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        Recursive Case:
    # R6 }4 e7 D; a& v6 u( J& A% x3 o9 S" v/ C
            This is where the function calls itself with a smaller or simpler version of the problem.
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    3 W8 O3 e5 {; J: ^. T: T2 |        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation: [& x8 [# n, f. U# j! j5 o
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    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:8 Q! I0 J5 _9 J$ m# B! I
    5 \3 ]4 P+ S  A# c/ q8 N: }2 F
        Base case: 0! = 1
    $ y5 h+ \0 a" v% I. `1 n# l2 v  e# c; e
        Recursive case: n! = n * (n-1)!
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    7 z6 i" |- p. v" ?! j2 c: LHere’s how it looks in code (Python):
    # u1 z; w/ _! a& T+ b  S+ l4 W: d, Upython
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    7 Q' Z! K' E3 \$ }" q( h. {
    : Y! }% @$ W4 _2 @5 {def factorial(n):
    9 m( k4 e- p9 B4 B$ X. K    # Base case& ~" J, J! @+ X, ?  g% \9 z, G
        if n == 0:
    # J( D8 ~5 W1 c7 [! {        return 1
    : W& Z. q- V0 j* z  }6 h, \) a9 ^    # Recursive case
    % O' j* R# r' `    else:
    2 S+ p+ M6 M0 k9 ?( ?5 R        return n * factorial(n - 1)5 S) ?1 e7 Z# m; k3 j0 f" Y0 Z
    ' f8 Z. i; B, K& o
    # Example usage- `; C% e1 b5 Z( K( z
    print(factorial(5))  # Output: 120
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    How Recursion Works6 j0 V4 M7 x+ [& Y5 S9 S  w  Y

    & X6 I; M7 t( s8 N% B" `    The function keeps calling itself with smaller inputs until it reaches the base case.3 d  h# S5 B8 H

    % A" s0 Y" j. j8 V0 z% }    Once the base case is reached, the function starts returning values back up the call stack.
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    # R% u5 b4 ]6 G* o$ V    These returned values are combined to produce the final result.
      X0 \9 {, Z9 c& R
    # u0 k3 [  y5 }For factorial(5):
    # x' \, t# ?9 b* P/ }# t" I
    / [* b" w& D: s- M) n# v% |) G4 O8 `$ h" g7 u
    factorial(5) = 5 * factorial(4)0 ?9 g* C8 z9 c; |& m8 O* Y5 ?3 e2 p& X
    factorial(4) = 4 * factorial(3)
    & R' p# ~5 k: S) Rfactorial(3) = 3 * factorial(2)
    # x' W: i: A/ ~" ufactorial(2) = 2 * factorial(1)
    7 t7 ?0 [" j( e. r9 S( Kfactorial(1) = 1 * factorial(0)
    1 }8 i5 M& z+ X7 }- ofactorial(0) = 1  # Base case
    % ^4 n& h- u% x" b. [
    ; |7 _6 [8 Z& k5 y- XThen, the results are combined:+ L4 k& w1 y9 d8 Z( F

    3 I$ G/ `$ u5 c/ O, U# o' J% C7 ]# Z! |+ B7 |+ [  p/ F
    factorial(1) = 1 * 1 = 1$ B3 W0 u9 x5 @/ [9 C
    factorial(2) = 2 * 1 = 2
    7 S7 c  [9 q& w% B5 Z! pfactorial(3) = 3 * 2 = 6
    ( d: O; n) G( E( d& x3 Efactorial(4) = 4 * 6 = 24
    9 Y" p- U$ {" Sfactorial(5) = 5 * 24 = 120
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    3 o1 u, n9 g: n. v, t' Z/ E! rAdvantages of Recursion$ q* n5 M$ e9 Z$ l4 I: x

    7 q+ F; _" y8 F* F: w3 w5 `    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)./ ~# P/ T( D+ m; f( G  ^& u& b, V

    ! R8 E6 c8 q# E4 K- U. V. J' S    Readability: Recursive code can be more readable and concise compared to iterative solutions.7 j1 k; H5 \& ?! c

    / N% }6 H, ?9 b3 Y7 j6 _Disadvantages of Recursion: J, Y+ J3 A( h& M9 ~# T0 G) K
      S# r+ W& U/ _( }
        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.
    ) W# K+ V* D6 h0 i
    , y$ S- g) a5 K    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)., b- y3 H8 m4 x5 N

    5 X& P, k$ V8 y" E1 u$ PWhen to Use Recursion
    9 s' b5 s; c# D  s- s9 e- ]" i" V( Q, E, @# ^2 g  |
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& e& B' b: S: z" q) U5 Y5 l

    ( o- V0 S$ F! h. U/ I5 ~    Problems with a clear base case and recursive case.) C% e% i  O% g  h$ r0 V. ~# J. Q
    9 `3 C! C" m# k  z: w, O6 Z9 k
    Example: Fibonacci Sequence
    & k0 F, P8 T3 n* O; |
    ( P+ {( u1 `# _3 U+ G$ M+ e3 mThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    5 N" z4 y  M0 d6 K" w5 V
    ) S' W4 P9 u& H5 p  ~    Base case: fib(0) = 0, fib(1) = 1# t5 e2 G  Z6 z/ _% j$ W* O

    9 B0 \: x4 c6 C: e5 x* W    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    : ^/ f7 }* ]9 y8 [+ B0 k1 xpython
    ( k/ N4 Z2 U/ H5 Q) D" ~0 Z  g+ D' z/ i6 f% M

    $ V& ?+ d  @' idef fibonacci(n):
    1 q% c2 O* x' Z. Z" O: u# F/ s    # Base cases6 w) i) y2 D' W5 V9 E- N
        if n == 0:6 [/ @/ {- @/ n, h0 \* b( a0 k8 ?1 L1 g
            return 0: A; K. V2 G3 L0 O* ^4 ], @
        elif n == 1:+ z9 B" L+ b; w/ w6 n/ y  {* }
            return 1
    ) t5 S) z, D0 u8 Q    # Recursive case# V0 j& y: H- M0 l' }
        else:& B4 Z6 Z1 N1 ]8 n  |
            return fibonacci(n - 1) + fibonacci(n - 2)! z) n+ O5 H3 H1 F2 G

    ( ~6 K3 ?2 }6 K3 A% x2 q# Example usage3 v, g" B, P, C! V9 Z; m0 ?
    print(fibonacci(6))  # Output: 8
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    Tail Recursion
    4 n% I! p* A+ D9 F0 K" w% d. Y6 |2 ^
    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).
    * R9 |8 X7 W  L$ e7 u1 D
    , `: [9 Q, a! TIn 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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