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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 6 V, s+ _. x: I' e' U+ q
    $ w+ x1 X2 {+ z" i$ N
    解释的不错7 _7 S* f, u5 D( Y  {: d7 L3 x
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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。, d' w; R: r8 k4 k" O7 |
    + _, m- C5 m- n6 q
    关键要素# ~) l' K5 V9 T5 s( w/ E
    1. **基线条件(Base Case)**
    ! C3 ?# U8 `5 }/ P' @$ V" R6 V   - 递归终止的条件,防止无限循环/ }) r  {3 h0 ~8 T: ^
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    - F# b* }) s  D2 j3 C$ \
    1 l( Y4 G- A4 J/ o: w! x2. **递归条件(Recursive Case)**
    " W% Q" t9 \! s   - 将原问题分解为更小的子问题) K7 M, D$ N# J! Q, p
       - 例如:n! = n × (n-1)!' o) _) r2 ]; l1 _" U
    # m5 ?0 g& f5 G9 O$ ~! B7 \. {, E
    经典示例:计算阶乘3 O" q/ n  V5 `6 u: ^! e
    python' J% b' R6 t, J4 p( u2 `- Q) }3 p
    def factorial(n):
    . I; ]4 n) ?. B! a2 L    if n == 0:        # 基线条件8 Y4 H) t+ x4 @* L" R4 ^8 e
            return 1% F* N4 W2 L6 Z, _) u
        else:             # 递归条件) x/ P: \* D; K+ D7 i
            return n * factorial(n-1)
    8 w5 O  F0 B3 y* o执行过程(以计算 3! 为例):
    ! s, m$ T6 V' R6 }$ S3 bfactorial(3)
    9 D1 ~8 ^3 B4 Z: B' O4 b& ]) b3 * factorial(2)8 \+ R3 ]- Y' v4 ?: r
    3 * (2 * factorial(1))
    7 L6 I7 h, f1 J4 h+ q6 P: V3 * (2 * (1 * factorial(0)))4 ]3 [. c7 t3 I
    3 * (2 * (1 * 1)) = 6
    8 Y9 I6 r, |& }- e7 F! X5 P! x9 x  a, k7 h' y/ [( m! e
    递归思维要点2 `( G" t# p8 b6 C5 C. s. [1 f& q. I
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ) S# T5 V, h* r: C2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( a7 f3 x% c$ F& f2 a
    3. **递推过程**:不断向下分解问题(递)
    ) \6 `0 t# C5 A1 F5 Y4. **回溯过程**:组合子问题结果返回(归)
    7 ]0 f% _! C% W7 l) o$ L
    : e( x& B; Y' I( X 注意事项/ l$ y$ \' U0 T& k1 e
    必须要有终止条件* o0 d2 E+ Q: }% X$ l
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)8 B; {! |3 W5 _# K1 X" t
    某些问题用递归更直观(如树遍历),但效率可能不如迭代$ I' F9 M+ K) k# j
    尾递归优化可以提升效率(但Python不支持)
    : o6 y8 r) [/ _2 R) x& t+ i  J, \/ D" Q7 e' S4 u
    递归 vs 迭代
    / w% F8 g( `0 a6 M$ \; Q|          | 递归                          | 迭代               |) w- [" S; K6 q' n5 d
    |----------|-----------------------------|------------------|
    , u2 B7 C! u5 ?| 实现方式    | 函数自调用                        | 循环结构            |; ]$ I: z! k5 F3 T; Q+ U! ~  ^% P, c& A+ _
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |/ `- j2 l8 x$ P8 B) M  W
    | 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         |
    ) g1 w  b) G: _, s0 f, f| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    4 Q9 v) a& L) i" |! s& C: F7 j0 V8 }! _& n5 q
    经典递归应用场景
    7 I: p9 Y" Z* g3 t; m8 h1. 文件系统遍历(目录树结构)
      M& l" y% U( N- P2. 快速排序/归并排序算法0 J; q6 c, n8 U) N8 _. ^" Q
    3. 汉诺塔问题
    4 G. S& l! k7 y* V4. 二叉树遍历(前序/中序/后序): E' w1 Z: D9 t6 N& Z) s7 R: j
    5. 生成所有可能的组合(回溯算法)
    % U1 _: U; U! J1 H" ^: Y; U$ p( ?0 |" Q. }  Q& o: U% Q
    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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    郁闷
    15 小时前
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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,
    : g9 {4 `% u# U& U, C我推理机的核心算法应该是二叉树遍历的变种。
    , l7 c- s( [1 `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:  }- H& `4 b) B# `8 k! X8 ^/ R
    Key Idea of Recursion
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    A recursive function solves a problem by:
    % ~0 {+ T3 L& g9 l; @3 E9 q5 |! e& n  }' J& K3 W
        Breaking the problem into smaller instances of the same problem.
    ' M, I; T% v  B. ~3 A' a' J
    ( M+ Q# O% n5 o) {. J    Solving the smallest instance directly (base case)./ o$ l" B! W  m- U2 s/ ?/ T/ K
    . i  z9 g0 O9 v+ }# g% \; g
        Combining the results of smaller instances to solve the larger problem.7 D7 ?: Y5 k( ^% C8 G
    ! {+ w" j2 M% q/ o1 @+ j
    Components of a Recursive Function
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        Base Case:
    + a& t/ i, a- E* ^# r$ D" g& F/ ]% I7 w; \( F7 I* k
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    $ i  P: L7 g$ j$ i7 c* l  u! v/ a7 m. v5 `) y; T7 Q! k: r* p7 r
            It acts as the stopping condition to prevent infinite recursion.
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    - h: O9 j' j& k5 `' ]        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.$ ~; h! _( r* K9 q5 _& G6 x1 q& ]

    , p9 f% `3 i+ S: q  Z    Recursive Case:0 m" A/ E% N2 k. y

    $ L. g; d" I2 ?) |# U        This is where the function calls itself with a smaller or simpler version of the problem.) ^/ t% `( p5 A) E( v$ h

    ( j# O8 c2 H9 L# H        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    9 @$ k: A9 W: E0 w. g
    $ I" W% w* J2 P) E( qExample: Factorial Calculation9 @# B( i* q9 m
    : G$ ^+ P0 ^  h& `
    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:0 D0 N# X3 P8 Z7 ?. s7 R9 U

    / t5 L  o6 n( X. D: {    Base case: 0! = 1/ A4 A0 c; f$ |0 ~
    5 A+ n- w* X- Q  h, ?& s
        Recursive case: n! = n * (n-1)!
    2 w0 ?% U: z4 d( m8 n6 Y' j0 u/ V5 t* a
    Here’s how it looks in code (Python):
      H" q0 m% M% L8 mpython, b2 v# r7 s. f3 p1 N+ `1 \

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    def factorial(n):+ }$ G% [8 G* P2 P+ P* R( N* Z
        # Base case2 Z  g; t, a6 [; U  _: t
        if n == 0:
    3 w& {2 `) H0 Z# r/ g6 B6 b        return 1
    / \* [( h" {( u! `4 A$ x) p2 w    # Recursive case. H! Y* c+ F. W$ K# u
        else:
      y" z4 h4 m7 T0 V% m        return n * factorial(n - 1)
    1 p4 |7 F8 Y  |4 e+ o2 K; e- T, U' u4 O* C$ N4 t: D* T
    # Example usage
    ( I6 ]$ O2 t1 f$ o1 U3 N: eprint(factorial(5))  # Output: 1202 T3 D6 H# M5 V# S# S: m2 p

    $ z# l. ?: z3 A8 a3 e7 V5 ?How Recursion Works# _7 Z; [, {# d1 V' i8 H5 d+ ]

    1 I- Z% y/ y# [5 ?3 D/ D4 f    The function keeps calling itself with smaller inputs until it reaches the base case.4 r% u0 o- D* r' w6 ^
    2 S4 y0 q9 V* ?+ }
        Once the base case is reached, the function starts returning values back up the call stack.
    : C2 t; A, W1 N; c0 u
    ' R0 r: t% S% y3 [0 I    These returned values are combined to produce the final result.
    ! g% A8 ?& @( p9 j4 P4 F7 q
    6 x: g. |6 G" _$ h5 E" x! {; d6 @For factorial(5):/ Q3 n9 K1 T9 T" [: T1 d. R

    * d$ w7 X9 a" z3 w% d1 D: g; S3 ~/ ?* v' @
    factorial(5) = 5 * factorial(4)- P" X' D$ p# K+ [$ x
    factorial(4) = 4 * factorial(3)# Q& a! ]4 b; w/ g! X* y" Q
    factorial(3) = 3 * factorial(2)
    ! n) C0 i& K' E0 Wfactorial(2) = 2 * factorial(1)
    3 q, X) S( {! Q- O$ A/ }7 H5 hfactorial(1) = 1 * factorial(0)
    6 [* c# C3 c) ]* y7 xfactorial(0) = 1  # Base case
    / A* u  V) g  ?1 U1 Y( v) v$ \. P8 ~5 ]) A
    Then, the results are combined:
    % E3 y: G) k$ i! [3 z9 c/ M
    4 v$ q8 p. P0 \' d0 y
    0 X3 y* m+ K3 {- Ifactorial(1) = 1 * 1 = 1
    , t* J' ~4 [4 B" N' K" r  f( K- _factorial(2) = 2 * 1 = 2
    # _% z5 Y- X* @0 E6 K5 efactorial(3) = 3 * 2 = 6
    7 ]9 e! f, R* J$ T2 |' pfactorial(4) = 4 * 6 = 245 `4 [% |( R  C" F: K; e& M2 b
    factorial(5) = 5 * 24 = 120
    7 C/ z* N0 f2 F9 k7 o1 _3 F* j( `) m' j' _9 F' o
    Advantages of Recursion
    6 O$ T2 k2 m" |9 {. e) s* H2 t% b- v$ c  g5 f, e
        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).
    + f8 [8 y: r3 Z9 W' O! U9 t; d1 A. U5 u  S9 n$ g
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    " d/ Z+ F& {  a4 i& {) I  ?* b1 ^( k5 o( h- t6 n& h9 E6 m4 T6 ?
    Disadvantages of Recursion
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    # o, b, O& Y2 O" C    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.$ f- ?* w9 |! [( q9 Z9 T5 |

    , A9 k" T2 U8 c# |, |. h    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    8 S: s0 K& v6 g% J( F7 U  z+ z: a: J, t. H5 |" w& S
    When to Use Recursion9 I! K* c# b, q2 L9 I& N, J8 j1 D

    & z/ {% X/ z& f6 {- o& `# n    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ' t5 x9 z6 G" A9 k. f" `" Q+ e
    " Y0 j# a$ i! k" Z5 g    Problems with a clear base case and recursive case.
    3 t! B2 x6 a" Z+ B( q& z: D& }
    ) M" d( o5 b4 P" U3 L7 }Example: Fibonacci Sequence
    1 z  T4 y) O" K) ]7 W5 Q) ], s$ ~% ^8 W; ~! i$ r. Q  ^
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:# B% Q, d: p6 t$ H
    ' n1 X6 L. w) n; r- i8 }
        Base case: fib(0) = 0, fib(1) = 1
    3 A  _, H+ \6 x) `2 B- U$ o, |- J) W+ M$ Z* C/ s
        Recursive case: fib(n) = fib(n-1) + fib(n-2)3 s& x4 J9 @3 U( N/ O9 j
    $ N0 u/ r2 k2 j8 L2 D, o7 ^
    python8 e: L9 j( U) s
    ; H" D% D' r9 S
    / x( a$ U* w/ g( T6 v
    def fibonacci(n):
    0 H/ R' Z  a. J5 c    # Base cases+ A/ y3 r5 l0 v
        if n == 0:
    9 Z2 d1 ^+ K3 T8 Q' M        return 0: |" P: Z, d9 {/ w+ [# `
        elif n == 1:2 N$ B/ t) o# a2 f  x
            return 1
    7 {8 V2 X' S' x+ r' ?    # Recursive case
    ( S' D# L4 f8 d0 P: }    else:% v# ~, Y) ]2 X4 B
            return fibonacci(n - 1) + fibonacci(n - 2)# p& g9 o! o4 n$ S* U* I+ R

      x6 F- I; v1 |: R1 j" A, v# Example usage
    4 d5 }1 w, }3 F' ?! P' J$ I2 g; Lprint(fibonacci(6))  # Output: 8
    % L+ u: y6 H0 @# `! |
    - ?/ J( o2 a, y1 {# d# M- mTail Recursion
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    & z, V* _6 N  s* OTail 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; q% ~( T% w% y% T: }! m& ]! g8 K
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    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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