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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 / w5 X# P2 @% g8 ^+ u: t

    , I6 Z3 Y+ \$ V! g5 r3 c! g# A解释的不错
    ) ~1 F/ s5 Y2 R- q: x; H
    & c9 u' K. [, g2 y2 g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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    9 g, Q+ p" |  L/ o' ?4 k. U  e 关键要素- t( }3 ^* R/ C8 i3 D
    1. **基线条件(Base Case)**
    " i+ z# O% L* n, z" Q0 k- a   - 递归终止的条件,防止无限循环
    / Y) w  T2 V- n8 a7 K5 y/ G   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 10 f5 g9 L; W) `7 a& U$ b0 y* G& o, F

    8 Z. J: U1 q$ C6 T' u2. **递归条件(Recursive Case)**
    - \, r6 ~5 h7 N' k; M- }* n. n   - 将原问题分解为更小的子问题
    # M0 ?2 h0 |9 I6 t& O5 c; `( Z' m2 Y   - 例如:n! = n × (n-1)!0 P/ |7 s0 Y( {! X! h! |
    # S4 y% \0 u$ |% b$ C7 B
    经典示例:计算阶乘
    * }1 d, `0 ]9 U2 ~( E/ ]) o! n7 }python# L! ^' c2 c) w4 N4 j. K+ L% w
    def factorial(n):
    2 `. z7 Y. x' ~# \1 T2 b! k7 P    if n == 0:        # 基线条件2 `( t; _0 g$ q+ ~/ T9 D
            return 10 k8 m, }6 M; h* X- A( {
        else:             # 递归条件
      B' L- T" U  _9 c        return n * factorial(n-1)
    ! \. t$ [7 P: e执行过程(以计算 3! 为例):
    - W( k& d% u8 rfactorial(3)
    + ^' u) y5 o/ w2 J: K* q. z. S0 m3 * factorial(2)
    + j2 Q; O" Q% k: m3 * (2 * factorial(1))7 r% [! a; ~3 u: W9 R: C3 |
    3 * (2 * (1 * factorial(0)))
    % V  ?: k7 S0 I4 g3 * (2 * (1 * 1)) = 6% s0 f" E+ B; a3 n7 x0 n  y( e
    1 a' C- m# l+ v  e- C
    递归思维要点, m$ |8 a; B- P; f& C* i
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 \0 i5 y* ^8 K
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( P2 I4 a; T2 J) T: q
    3. **递推过程**:不断向下分解问题(递)
    7 _5 W# g5 s- U4 L1 R2 k# G4. **回溯过程**:组合子问题结果返回(归)
    , P5 m8 q( V8 a6 r( d
    4 d7 \' x9 H" }1 Z6 r$ U. w 注意事项' q# P/ ^2 u! x
    必须要有终止条件
    3 c8 _- W% Z3 Z8 E6 e3 _3 k递归深度过大可能导致栈溢出(Python默认递归深度约1000层)# L) z& T0 Y0 w1 x
    某些问题用递归更直观(如树遍历),但效率可能不如迭代' e, L% B: A2 N2 I: A6 d$ @( s7 b
    尾递归优化可以提升效率(但Python不支持)& ?  e) z6 Z$ ^: b8 c

    ' b) d) N7 F) { 递归 vs 迭代
    8 ^8 p( Z# @$ S- I' k, U* T|          | 递归                          | 迭代               |
    9 f) ?# P  z( T2 Q" u) j|----------|-----------------------------|------------------|
    0 g  }3 F, e- h$ R/ _| 实现方式    | 函数自调用                        | 循环结构            |9 j, a- B7 {1 ^' s' b
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |8 S$ Y, Z4 p( D/ Q" y- e9 f
    | 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 v: M, W+ W3 f1 _/ @
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |5 z4 M$ F' T3 c4 V: F
    * r) {  a, _8 c" d, _8 k; {* ?
    经典递归应用场景
    0 a+ m2 c6 G9 d1. 文件系统遍历(目录树结构)# j2 ~  }6 s4 G& U0 R$ P, x
    2. 快速排序/归并排序算法& d2 t& i! T+ y) M, r- y
    3. 汉诺塔问题
    ! |: s4 c) F% i: k4. 二叉树遍历(前序/中序/后序)6 }. c2 C- t9 F7 y" m8 C4 }6 q
    5. 生成所有可能的组合(回溯算法)
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    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

    昨天 08:50
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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," Q" c& N  B, M& u
    我推理机的核心算法应该是二叉树遍历的变种。
    5 O4 o9 h5 I) f+ E, t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; `( o/ l! q: H' i7 }% S: C8 e$ \
    Key Idea of Recursion- V+ i/ S; e! D9 ?8 o( @
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    A recursive function solves a problem by:5 y" A! y& a% f0 |

    . x6 W+ [5 k. T: c* q    Breaking the problem into smaller instances of the same problem.
    , `$ ?% O7 `1 H" g; x- [6 S) b5 B2 {$ K+ G( w' ?' d4 X! I$ q
        Solving the smallest instance directly (base case).
    5 \" c% U8 u) m6 u+ }  b% q
    ! ]" a' z) o2 c6 I0 Z    Combining the results of smaller instances to solve the larger problem.
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    - u" |  r  X7 [# j! wComponents of a Recursive Function# F& a9 B1 l# V% u

    2 }+ _. m# ~9 w5 m0 E7 R' M    Base Case:$ y" d: w2 i- i( o' L# S
    ( h$ E+ t2 J1 @) R: W3 M! O& Y
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    % V& u2 B7 I% I
    " r$ D$ e# p6 e( D7 |, n        It acts as the stopping condition to prevent infinite recursion.# T$ ?. Y, ~' \8 c9 K
    " l, [- X: n+ r! v7 G
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 @$ ^7 s# {6 w: q; n: g

    : S; ~! K- C1 a1 c. Z" X    Recursive Case:8 ~+ n6 M9 ^2 |4 z' j8 Z
    0 i9 P( F/ B# h
            This is where the function calls itself with a smaller or simpler version of the problem.
    ( V( E: V5 K2 [: y& I+ L' C! s, x7 H0 v: T
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).# |1 f8 N, [: Q: y
    2 P4 r# Z. L5 z
    Example: Factorial Calculation: Z- P0 @4 |% i3 o1 P4 `% r9 y) T

    2 T8 L6 L" j4 ], C" k/ r( ^+ ?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:1 M% F+ w4 U, x: I( B
    : P# h# }( {% D! j9 J
        Base case: 0! = 1
    . ~# {+ e) ?& p* C8 B, ]0 m* T- O0 |% S- ^
        Recursive case: n! = n * (n-1)!# K4 ?4 d) t1 r/ ?3 V. U8 M

    ; B$ m) K/ Z4 L1 Q* |4 N- k  |5 iHere’s how it looks in code (Python):! H/ U+ R6 f2 f$ ?0 r% ]7 ], M& k
    python' A! w5 l: y6 a) n( F: g# W

    ; y) \* e( P( c, `
    % ~# `* X7 A6 s' Udef factorial(n):
    4 e3 o7 j6 E7 y; R2 h$ D" w4 K9 h$ h    # Base case5 A* v3 }0 {8 `. X( {; W
        if n == 0:9 o# Y+ p' i2 @9 V2 d3 b
            return 12 ?9 B& z& S4 {& s( A; W# c
        # Recursive case
    . @0 Z: U6 j# J* h, x$ _/ c    else:7 h! }6 @2 u5 \
            return n * factorial(n - 1)( L5 Q9 o# A8 P+ \

    : M, A+ O: I# U7 Y4 O# Example usage
    / n+ K+ t1 F* Q$ qprint(factorial(5))  # Output: 120
    * @) N7 a0 x. \) p
    & j: Y$ P5 ?7 V. W2 Q) E! z* vHow Recursion Works
    % ^' j0 p  i7 D& Y; J, m7 Q
    ' _' Y/ \: R- l) y    The function keeps calling itself with smaller inputs until it reaches the base case., A% D+ V3 y7 G$ i3 o
    & G, N2 Z! L3 }" m* d
        Once the base case is reached, the function starts returning values back up the call stack.) T3 t1 ~+ G4 X0 {
    : {, K0 B# J& n, }+ ^
        These returned values are combined to produce the final result.
    $ s0 I% o0 O* J" f% ]  P# L! q. \
    4 b: n; A; b- |. F% p" UFor factorial(5):
    " M' U! y/ d) `1 T9 B5 x4 P+ `9 X7 ~# B

    8 V  |0 d6 D0 G* v7 `$ @7 Vfactorial(5) = 5 * factorial(4)/ s3 f3 o) g( U/ K: E
    factorial(4) = 4 * factorial(3)4 s% t' T' A# f' Q0 Q8 N
    factorial(3) = 3 * factorial(2)9 _! h1 y" K0 N6 n  A
    factorial(2) = 2 * factorial(1)( c' j' m* W+ Q4 u; `& {
    factorial(1) = 1 * factorial(0)
    ; b+ k/ ~7 g' X0 V$ N" A" Afactorial(0) = 1  # Base case$ X* ?2 Z( I7 l
    * H  t. }7 r8 \# S
    Then, the results are combined:
    ; E2 K& u, V3 m# ~$ W2 A5 I0 q+ {( X( v6 {5 |2 b; q; T1 D

    ) H9 p; e' w: F4 tfactorial(1) = 1 * 1 = 1/ R4 P: |1 o, i) A
    factorial(2) = 2 * 1 = 2) _; W& Z7 e% j4 B0 V' ]
    factorial(3) = 3 * 2 = 6
    " c0 R- y- c5 K  Rfactorial(4) = 4 * 6 = 24
    0 w6 a+ B: x8 jfactorial(5) = 5 * 24 = 1202 h5 C8 N  ~9 t  N

    3 l- W' W% i7 W% XAdvantages of Recursion7 B" Q' ?4 c7 A
    . C4 Q" [8 _% o+ O
        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).* a/ R4 \/ H/ E, a4 C2 @- U1 {/ U3 h

    2 H8 A0 W% f2 Z$ N! S    Readability: Recursive code can be more readable and concise compared to iterative solutions.7 P' G5 w5 x& G% v1 S8 C

    " @; U9 J0 W6 P" CDisadvantages of Recursion
    % t4 ]4 G% B- c- B  Q& n9 A! t$ R
        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.
    ' T8 F2 [; B5 o/ K3 e2 @
    9 p6 U) T/ ~4 ~( y& o, W6 _    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    1 t9 X( R. ?" z) M, J7 s. G! d2 A1 _
    3 G# t$ Y* ^; D1 z6 F  H( kWhen to Use Recursion
    3 i/ ]: R  `5 T: s9 P: ?9 O: `- d* P6 A/ x+ w% [
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    + U5 ^: H& q6 B/ M5 J- u
    : Q6 r* S- u3 f. y3 a: N    Problems with a clear base case and recursive case.2 L5 P! E. Y$ l! |2 V
    : {4 A0 S2 G6 O3 s- Z1 x
    Example: Fibonacci Sequence
    : u" J" G! m7 l& s0 L& c
    , @, _; x5 Y( c# w7 }, f# Q8 U: I' QThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . M4 b) m4 d. R( |* ~* [
    ; w5 ~, j! g6 d4 v$ B! `+ f1 f# r    Base case: fib(0) = 0, fib(1) = 14 F& `' K7 i- \2 _6 M# X$ R

    9 z2 V; S" r9 G$ C5 Y    Recursive case: fib(n) = fib(n-1) + fib(n-2)# }2 F" `" b- V3 ~6 n, A  @8 X

    $ t# H6 w1 e, X  |7 m, y8 fpython9 e/ |% e" W/ b8 h0 f

    * {7 S1 z2 H' G) m
    - y9 G; p9 z! }) l! d$ Pdef fibonacci(n):
    3 Z4 U0 ]2 ~3 b  `" U  G; {    # Base cases+ i: u+ z  t8 H8 m7 K. A+ `9 g8 x
        if n == 0:
    ; @7 p6 ]  b. T( T$ {' ?        return 0
    / [3 W/ G5 c+ L" W' w. r3 `    elif n == 1:
    : ~6 ]1 Y4 _/ r, c! \. }+ Y: }/ O9 {        return 18 a) ]# W# A# C9 |
        # Recursive case( L7 o" j% a+ q" n4 ]
        else:
    - z2 w" S% ?/ X) V1 P/ S  \' o        return fibonacci(n - 1) + fibonacci(n - 2)
      P9 q8 ]5 q# r0 A7 _
    4 z0 T6 w/ ^3 L' D: u# Example usage
    4 F8 t* B2 p5 x$ A! nprint(fibonacci(6))  # Output: 8
    ) s: B  Z/ ~5 ?6 L% ^3 \, ?+ L& R2 `+ F  J& g6 \5 X
    Tail Recursion& g# i& D" a% q4 O& C2 m  n
    & M. M& B/ V! P+ l1 D* h! q8 [9 C
    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).
    ! w- [4 m, h' `( t; u' N
    1 B6 `) P7 c% g0 d- ^! r1 J& t( l. RIn 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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