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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 7 y0 d" Q3 Y7 P! c2 o4 j

    ' G( A* w: S- y$ }4 y2 p解释的不错
    ; ~- F6 u/ I/ w6 D  o+ x
    . ^6 x& N& L3 E  f2 _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。
    8 ^# p  T2 E% W8 i' Y
    # g# d$ Q& z. g1 u 关键要素- F% U( P# L! v5 S2 h
    1. **基线条件(Base Case)**
    ; h* j! ?3 K) q: R4 V5 v7 u   - 递归终止的条件,防止无限循环5 {, k- q; O  O' Z2 X
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      f7 x; @7 G& l. \$ M. K. g( c5 R6 }2 [- b* f7 M
    2. **递归条件(Recursive Case)**
      i& S; e( Y5 G' w5 h9 i9 c   - 将原问题分解为更小的子问题% T: G' I+ u# A" C* C
       - 例如:n! = n × (n-1)!
    # G& [! a4 G$ d9 u- x0 D$ z5 h7 {! ^, W* M  ~2 `+ d5 B1 J8 r& U
    经典示例:计算阶乘
    0 p% f* B2 ~# P8 Cpython1 p$ p. u) B, H1 C
    def factorial(n):" ?/ B* ^5 `' x
        if n == 0:        # 基线条件
    , K2 M; Z, N: H$ Y6 b& n7 L        return 1' {& T6 o4 I- S' @
        else:             # 递归条件
    ; g0 G" A* X( Z; S4 }8 e# O1 n        return n * factorial(n-1)
    6 E6 s6 x3 K) ]7 [/ \6 D5 I2 o7 ?" V3 I执行过程(以计算 3! 为例):
    3 b. v1 i' A2 w+ j1 ^factorial(3)
    + u  |; B, O$ G! c, W1 |3 * factorial(2)3 x, E6 C  q' x8 i: w/ {+ K0 O
    3 * (2 * factorial(1))! J* `  Q0 ^  H9 _9 J
    3 * (2 * (1 * factorial(0)))5 @+ w  g# K! ^0 f0 N. Y) d0 a5 F  p
    3 * (2 * (1 * 1)) = 6
    2 x5 W3 I/ o5 |3 y* A1 h) r0 L* K
    递归思维要点
    8 b4 l( G# z2 m3 e. e+ m- v4 ^" f1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 b2 w$ s6 S6 p+ ?/ _/ e- P
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    % N& [8 X. h' f; }, u3. **递推过程**:不断向下分解问题(递)
    ' U, Q, N3 y4 j' m/ _4. **回溯过程**:组合子问题结果返回(归)/ ]0 [3 @/ Q$ D; B% F

    2 a; \. c; @4 n+ q- w9 {' u 注意事项
    - \% E% E5 C  Q) }5 j6 V# R必须要有终止条件
    6 o4 A$ W' D5 R9 S6 k递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( L6 {, O0 h& ]! Y" F- A
    某些问题用递归更直观(如树遍历),但效率可能不如迭代% W/ C8 H% a9 x, L
    尾递归优化可以提升效率(但Python不支持)
    5 k  e6 Y/ F+ `1 ^' ]
    . z1 |  a3 S0 z% t# ~3 M! n% t 递归 vs 迭代- y% o* h! C; R$ v; D, h- D
    |          | 递归                          | 迭代               |$ r; _* P7 O+ E& |; x( e1 L5 N, }8 w
    |----------|-----------------------------|------------------|1 Y! u3 h6 z6 H+ t8 m% O7 ?
    | 实现方式    | 函数自调用                        | 循环结构            |; N* I: C1 v! w, i( V2 G+ b$ x
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    # W- M; C3 Q$ Q% t' v) O( F: q$ i% n- L| 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 O" W1 o6 v' t" E5 h| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ) z% I* [0 S. d% X. D4 i. W
    . g1 Q/ E! [9 I  N7 a( P 经典递归应用场景, C7 g; V+ j( v( b
    1. 文件系统遍历(目录树结构)
    ) |  a; \% |6 S" \5 V) E6 e! e2. 快速排序/归并排序算法5 N! |* `% T! p" U/ w
    3. 汉诺塔问题6 Q! H6 K+ U# ~" _1 Q
    4. 二叉树遍历(前序/中序/后序)
    9 ?4 _7 s0 e* U( _5. 生成所有可能的组合(回溯算法), ^) j0 m! s1 D9 z. R) X! _

    ! L4 O0 |6 h% p8 e. |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,
    # C) U! s: t4 |+ E% I( l我推理机的核心算法应该是二叉树遍历的变种。
    2 K) W% t  N5 |, |( u* k8 h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:2 d% B  W/ G8 W; x+ h' \& ]/ L
    Key Idea of Recursion6 i$ v3 \$ r3 W) u
    9 Z, h; Q2 n2 v) d  A$ I: `
    A recursive function solves a problem by:- r, S: D# s. i% w) j" S
    ; T8 S: H- p0 \# p* d; M
        Breaking the problem into smaller instances of the same problem.2 k$ f2 X% x0 u8 N' _, z
    , f6 V- k& k- I( U/ `. {
        Solving the smallest instance directly (base case).
    ; ~$ P2 ~1 X1 `4 a: I( [8 ?. C, x' y  D# n* A/ W8 H$ P
        Combining the results of smaller instances to solve the larger problem.0 u. E" i) P. e7 x" t# t

    8 A8 T4 T, x3 H1 I( ~4 t8 ^( a7 z" X* oComponents of a Recursive Function
    6 ^6 D* q$ `0 n5 d. N( \
      N+ k( x: N! E5 W; ^    Base Case:0 `" {% L$ l& L+ A) v% h
    ' M, G7 \% N3 J8 ^5 P
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ( q3 w2 f- H& M( H6 r2 H  g+ j+ R% ^/ ?
            It acts as the stopping condition to prevent infinite recursion.1 R3 R- n8 i" X. s& ]/ v, N
    0 P8 \) q9 I0 y# R4 H+ h3 O( d5 w& R; X
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    9 V0 p  X% {7 [% v7 x0 _
    % r1 j' v/ }" C* I    Recursive Case:2 l$ w2 Q+ E  z2 m

    & e, q! H; C+ Z) D        This is where the function calls itself with a smaller or simpler version of the problem.. j2 X$ R" {9 T$ L! P) [. `2 r

    ! H6 Z, J) {, p; g% {8 j        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    9 V) B8 M* S9 a, G3 y1 |* J% I4 e% \# P; `% i: o" `
    Example: Factorial Calculation
    / a+ @, {- Y- [; G9 j- ^0 Z, y* ^/ o' C7 P0 F+ f; A
    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:
    " K0 a' l' Q5 V: Q
    # Y$ L/ v3 v/ l$ }+ i  R    Base case: 0! = 1
    + m% `' h+ `8 s1 d  k4 P7 w" C6 f% O7 \3 {5 G: n  ]+ _
        Recursive case: n! = n * (n-1)!
    # a: l0 z# g6 o
    % G/ c, g2 y! C, @3 ZHere’s how it looks in code (Python):3 Y( U" W! d2 k0 i& L
    python
    1 W6 W6 ~4 @7 x& j+ x+ _* Z1 {1 t, {, R8 d  Q4 L0 A' s

    ( t( N, G* W1 t* Edef factorial(n):
    2 a7 @) M- w0 G0 t$ q    # Base case
    0 Q6 [" e1 g# [2 O    if n == 0:  x: M  |: `* y* ?
            return 1% Y- ?/ Q6 e# }2 K# s3 w: |0 j% \0 A# `
        # Recursive case
    ' h! K" I+ E# W8 b. Z+ H; W    else:
    ( N8 ?7 e3 M! _* N        return n * factorial(n - 1)" e/ L1 B' \7 W/ y5 E4 d! P3 \

    1 k+ W0 g% p9 F" a4 V8 B" v7 \# Example usage  R  `* ~' B) \1 B; }$ `; J
    print(factorial(5))  # Output: 120
    3 Z4 u/ O" }/ \, A" o* L, ?5 F& `/ {9 _/ y
    How Recursion Works
    3 b. x3 z& K- k5 @% C4 \1 p3 `0 B( x( X8 \9 z, O' i5 X3 J( |# [/ j
        The function keeps calling itself with smaller inputs until it reaches the base case.4 @% Q. v& E& c, D9 N) }

    ( F# R* h: V3 i% ]    Once the base case is reached, the function starts returning values back up the call stack.9 q: ^9 F5 Z2 s$ g2 w8 |

    / ^0 f0 r( P; @* o% K7 z0 w    These returned values are combined to produce the final result.
    4 r+ h1 Y; k, u: U) A
    ! Z- S( w: l' g2 x/ JFor factorial(5):$ k( [1 l5 y) x, b3 a* C$ f
    1 }, H8 I. \8 \  f
    # `: I8 R& V: b5 z  G$ ?+ i
    factorial(5) = 5 * factorial(4)0 X# _; S! M9 t* u9 @8 N
    factorial(4) = 4 * factorial(3)& g1 L( v0 L9 f( X( T! V
    factorial(3) = 3 * factorial(2)
    ) c; a; w. o& W; x- U4 Bfactorial(2) = 2 * factorial(1)
    $ h/ g& s3 n: H0 d3 N6 C8 O; F0 K# Ffactorial(1) = 1 * factorial(0)4 W0 \* s8 a9 C' w/ y
    factorial(0) = 1  # Base case" u! [2 ~- Z& i1 E  k! r

    * `% H9 _) w$ h6 D& tThen, the results are combined:
    5 y( D0 \- u+ D( O
    ! F( ?: H5 a, v
    ( T. L+ B4 S8 ?9 V! x3 p% k) Lfactorial(1) = 1 * 1 = 1
    9 l! W' c2 ]  J' s: qfactorial(2) = 2 * 1 = 2
    + ^' l) a5 y# }) Lfactorial(3) = 3 * 2 = 6" l5 x8 W1 z* x8 u; o# k
    factorial(4) = 4 * 6 = 24
    2 U' A& o% n0 @6 `* x3 l' {factorial(5) = 5 * 24 = 120
    ( W/ l* d) L3 v. \2 q# B
    : `1 U. l$ e) R3 i! jAdvantages of Recursion
      ]- t) D# y' Y0 K
    . {- q1 D( `# }1 ?3 S; \    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).
    % y9 ^5 Z+ u' \/ P) [' Y' s  {  P) ^6 S- X+ X  s# A
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    " D' n3 q1 `2 @  C7 f7 X6 m
    & a" L3 ^% D' IDisadvantages of Recursion6 y5 X8 u& Q+ r' [
    1 i% H2 ^$ a  s
        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.6 V2 H1 D* D2 f0 B: _0 e
    * \0 ^# t$ f+ I! y# V
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).7 J6 t: \7 a  k- v7 e
    ; }% D6 g( Q) ?+ i/ |' G: i- }
    When to Use Recursion
    3 c+ B) s9 f/ r/ Y: U6 K, Q& n5 }$ z8 o1 O
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    / h/ f7 l0 i0 ?2 N0 U. }2 @1 _- M# `* Z( Q
        Problems with a clear base case and recursive case.
    1 q8 S! ]% H+ p% q' ]. s
    $ J4 m$ z8 C) I9 o: o7 pExample: Fibonacci Sequence
    0 L3 b! X% t* y, ~: U, r  q; T- J+ Y4 q7 y' h# C# X
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ! ^% F7 o5 }- x9 }8 q6 G  F
    7 ^6 `: ^" R& v/ r    Base case: fib(0) = 0, fib(1) = 1! l/ E/ B: o7 J

    * ^6 t0 J" N' S+ @# }    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ; f+ ]* v- }6 L3 o% S4 [8 S- H0 q$ _; M5 v. C& i8 k
    python8 s! b* k# P0 p; o! m) o- X
    / {+ V. I% `4 k* R8 Q6 c% k

      i/ Q- |! E+ I) Hdef fibonacci(n):
    ' D6 @' X; U+ _! t$ `$ x: g    # Base cases
    & m4 c, @( k( o. Z    if n == 0:
    2 f7 X3 @" E( B8 }. k        return 0
    . k9 A' O. @( O& K7 @2 v    elif n == 1:9 {6 `! N) P1 v0 o  O0 J" S7 k
            return 11 X+ I$ ?1 z0 l# z" ^7 G' j
        # Recursive case6 {& `0 S* C( u. `! A5 |6 {
        else:+ n) t" c4 e' M0 U/ b7 g. k! |' L
            return fibonacci(n - 1) + fibonacci(n - 2)
    , h. i6 _+ q: Q
    3 j# U4 S1 U* ]# Example usage( z- t. h6 O/ }3 O
    print(fibonacci(6))  # Output: 88 H5 S3 a" i. R8 d# g0 a
    ) a. v5 [+ o" [4 m% j# {
    Tail Recursion- Z7 t# |: G; n0 U/ ]6 ~! f, S+ t- ~
    ' R4 R7 q4 T; A0 U5 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).9 O9 }1 W7 Y( W
    4 C" T6 _& c  y; 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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