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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 6 ?2 m$ v) Q; d! T# i4 M6 g6 ^+ k
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    解释的不错4 G' k$ x# U- N! e( R
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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。0 G5 e, s( X  M

    ( V" l8 S# p, r4 [ 关键要素( L' O) D8 k& R1 l
    1. **基线条件(Base Case)**
    ! D& E6 D% ?) {8 ?! u   - 递归终止的条件,防止无限循环
    # }9 \2 @. [0 d, d* E9 S   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1+ i, k+ I. `8 d, [0 t
    * q: i$ E, T0 P* H1 \0 N* Z
    2. **递归条件(Recursive Case)**8 B2 \9 V" y9 W( p* \) r9 y# v6 s9 u
       - 将原问题分解为更小的子问题3 d7 ?5 @- K' P+ ^; \8 @
       - 例如:n! = n × (n-1)!1 r  r( B$ L4 X
    . [/ j/ C$ q. ]9 c, L" U5 T& Q
    经典示例:计算阶乘) d6 y3 f. ~2 `* f, o
    python
    ' D+ x9 a9 i+ idef factorial(n):- e# R# R8 O2 U( L, U
        if n == 0:        # 基线条件
    3 P  E3 ^) N5 c3 z1 \7 ?0 Q        return 1
    9 A- d  G  k1 Q8 H( p/ _, O3 B    else:             # 递归条件3 C. r/ P5 G+ p7 ]: h- J' p8 ]
            return n * factorial(n-1)
    " G0 S! X) z; I执行过程(以计算 3! 为例):
    ! S# H3 Z" D. s# k# z: S) Wfactorial(3)
    9 T& i  r0 q# b4 Y  l# X2 o3 * factorial(2)
    & `5 n* m, @1 P# A3 * (2 * factorial(1))
    % X  X% q$ d6 |7 y, z5 b3 * (2 * (1 * factorial(0))): F2 i0 L/ P7 c6 [$ ?% s9 ^% o
    3 * (2 * (1 * 1)) = 6. ~4 ~2 ]+ y% p& T8 J2 ?) O# P% d
    0 k' I: H: J% m. X, w
    递归思维要点
    : D, M! k0 A6 p7 {& G1. **信任递归**:假设子问题已经解决,专注当前层逻辑1 H. [6 A6 W1 A/ J8 e) }. |
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    * }9 v$ o) q  o, l) S- O$ Q3. **递推过程**:不断向下分解问题(递)1 W  E5 g* @7 j/ k  N) r0 u
    4. **回溯过程**:组合子问题结果返回(归)% c- m2 R& e8 W3 u
    * }% P' L# p1 d; W5 ~& Q2 Z' P
    注意事项6 l5 u  s/ U: ]1 U
    必须要有终止条件, b- y6 p9 I8 B6 R! d$ V( Z
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    2 b7 X" m# X, ~0 F/ f; y某些问题用递归更直观(如树遍历),但效率可能不如迭代
    8 k2 T9 `  Z; S- M7 j/ w尾递归优化可以提升效率(但Python不支持)5 ^5 C3 R2 s- c  G- C
    8 T5 q3 [; Q' i" `! i4 y$ w6 `
    递归 vs 迭代; {/ X' X$ Y' H
    |          | 递归                          | 迭代               |1 z. n4 ]- f/ d( o' k, |# ~
    |----------|-----------------------------|------------------|7 [, B" F9 C+ X: U
    | 实现方式    | 函数自调用                        | 循环结构            |
    1 f" o/ L% R, ^! _6 b| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    * n  ]+ D& [9 P& ^| 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         |. N5 \& V" f: D( ~
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ; T9 l3 J9 {  Z% _1 D/ T/ P  e/ I0 t& }3 ]" M2 ?
    经典递归应用场景
    ; ?# \* q: {4 f- C- j/ F1. 文件系统遍历(目录树结构)8 s0 x7 j  w' a6 o& m& O- {# Q- D' y6 {
    2. 快速排序/归并排序算法
    % x- U2 t6 h7 r. `1 s* p3. 汉诺塔问题9 I4 j0 @3 V9 y1 J+ O; Z
    4. 二叉树遍历(前序/中序/后序)
    4 _, n( [8 X3 X1 t5. 生成所有可能的组合(回溯算法)* P' O- G) k$ p' \3 c% T: O" x3 ^

    / x) U6 J  h$ o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,- j9 E+ B! a) d- j! U
    我推理机的核心算法应该是二叉树遍历的变种。
    3 D  E( T. o& P( J' Q, j+ I1 X' 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:
    , `+ o  L9 O7 E- iKey Idea of Recursion
    ) _* |0 d+ ~; R% \, d
    / _" o) s% v  d& `; F" d, [A recursive function solves a problem by:3 l+ V; O5 n  a) Z0 E6 c; I8 ^
    5 Z  l1 H' i' ~8 o( B4 D+ U$ s
        Breaking the problem into smaller instances of the same problem.. `4 I9 A- B% C% P
    # k4 h* p6 t5 j0 w
        Solving the smallest instance directly (base case).0 @' I5 F8 x+ e

    - j1 J( t9 w6 v( f) ~    Combining the results of smaller instances to solve the larger problem.
    , C7 n& T# F: i/ n. u' r: R. `& `: Z. X* c) ], J( U0 P/ L( C3 q
    Components of a Recursive Function3 v. z- C+ Y# G% _4 G
    2 G; l0 B$ J# j
        Base Case:
    / ?4 {! Z1 A7 U
    4 E( Z. M% J6 v- h: I9 K        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    % N3 U+ T3 o3 f0 H: _/ U2 k: I8 N; o  q5 f/ _
            It acts as the stopping condition to prevent infinite recursion.
    # ]" f0 H4 B4 W
    - }- l: e$ Z8 R: g( E        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    / `; H# d% ~* P/ C
    & o+ ^% Q; F- H/ }6 ?1 T( \    Recursive Case:
    2 N3 f. L6 p" I+ I$ p: b
    + i0 m& c7 z/ v! C$ y        This is where the function calls itself with a smaller or simpler version of the problem.8 w5 \; _# {$ U8 X9 r- }3 d: o

    7 ~* S) b, r9 J* `* _/ B$ L! E        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 _; o/ o) J- k* w- B9 N

    ( G. L" F2 V' x6 s+ E: I( UExample: Factorial Calculation
    & M' j6 I) m: y4 N! I" R
    . ?% Q8 _' g$ J6 |$ J, k# d! C# HThe 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:
    4 o) p  G, F+ Q, ~5 [, C- o4 b& [# e7 H7 ^6 g: l
        Base case: 0! = 1
    ; u5 P' y& n% T6 U2 w5 s4 d; Z4 ^' p
        Recursive case: n! = n * (n-1)!+ Z2 U6 e( g2 [9 z( l, j% i

    * S; q! k0 E6 o2 D( r2 ~Here’s how it looks in code (Python):
    . \7 Y8 m" V6 o  B& vpython) T1 ]6 q0 p6 E3 p- I0 c

    % s3 T2 [6 U2 X' W  `- J, V5 e! C! c3 Q5 W5 M
    def factorial(n):
    / t' j4 A+ V. N    # Base case- K) x* A+ G( ^: t+ ]5 o! B5 k- K$ l. X* N
        if n == 0:
    6 ^% q4 m8 T' E8 C+ Z4 H        return 1$ ~! ^, E8 C4 U7 g! z
        # Recursive case
    4 ~0 X5 f& |9 B4 b/ o0 o) b) Y    else:- ?% @& |3 W+ H. l' K0 o( T
            return n * factorial(n - 1)
    : l. X( K$ Y7 Y. D; R, M, I2 o& Z* D2 u1 b/ N& F! M( R! J+ N2 T
    # Example usage
    ( e0 G+ O$ p6 b$ xprint(factorial(5))  # Output: 120
    ) W+ d$ [% i1 C
    : y8 f/ j& ^: k" |How Recursion Works9 `0 C. v$ U$ Q$ S
    % Y: y$ Z+ D( e
        The function keeps calling itself with smaller inputs until it reaches the base case.% u1 p! @: ]( h" @  y( \( }8 ~% Z
    ' R3 p) R1 `) Q# t8 P5 u9 r0 Q
        Once the base case is reached, the function starts returning values back up the call stack.
    & C4 R+ |$ c3 ?$ `9 [  B+ A% _. j" B7 ]3 s9 N
        These returned values are combined to produce the final result.  n/ A# I+ j5 r, \2 O6 h8 ]: y7 I4 h

    : k( ?# C$ K" S( |" @+ a; KFor factorial(5):
      V; D7 K, I; H; |9 K
    * V! t) s6 t1 z( _" ~+ W3 }
    % y; y+ a/ E/ hfactorial(5) = 5 * factorial(4)
    9 k  F, j4 v! D4 Ofactorial(4) = 4 * factorial(3)$ w( _5 I1 e1 Q8 q0 c% n! @7 a
    factorial(3) = 3 * factorial(2)2 `% D; ~( d' D; A" T. y0 b1 P
    factorial(2) = 2 * factorial(1)
    / l3 y( A* Y/ M- j% {. w' A+ }% |' w3 Ufactorial(1) = 1 * factorial(0)
    - U2 U3 k/ n: @& i4 V; }factorial(0) = 1  # Base case
    9 L6 l# O, \/ K; D5 v# d( [" g9 V. [  L9 l6 N
    Then, the results are combined:" {. Z2 J" a- O9 n+ ]% y

    / ~7 U8 v; I) n; J( C6 B" [6 m4 U. Z# W+ `5 a
    factorial(1) = 1 * 1 = 1- l6 X8 `! d( t0 J" A$ G# S
    factorial(2) = 2 * 1 = 2+ n, i" `. L3 H9 \: j5 ?( S
    factorial(3) = 3 * 2 = 6  m4 l- c. a0 ^8 o3 c6 S+ w/ j
    factorial(4) = 4 * 6 = 24+ s: n- S3 r, c9 q' Z  d
    factorial(5) = 5 * 24 = 120* Q" G/ d7 _# p$ [- @9 t
    9 }0 s' |0 @3 z
    Advantages of Recursion
    $ |4 ]' ~! y9 k# b/ R+ \' {. [! p8 e0 q* m" Z& n
        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).
      \! w+ i) Q5 Z* T% q
      t9 \- m( E% p1 Q& C) O    Readability: Recursive code can be more readable and concise compared to iterative solutions.3 P+ f. G# w6 N4 a
    ; F* p3 h# i: c. Z
    Disadvantages of Recursion
    ; o- R( A) m- j8 k% Y5 a7 d# U! m0 ^2 y& @- S2 O$ v5 A# ^2 k! 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.
    2 ^3 q5 O- B* |" p+ w% p& O" }) s% T% _* T! m' Y4 K3 I
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    " o% o) v+ k7 B
    & @; t2 I8 R& B5 D5 M# hWhen to Use Recursion, c) T% t& y; R* z

    2 o9 u, h3 I6 C& R    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).- w2 B8 `2 E8 Y0 e- i; c

    . J, Y! D- R! ~$ k    Problems with a clear base case and recursive case.
      H  r2 W* D! [3 }  v3 K5 j3 n/ H& j6 n7 x; I5 J
    Example: Fibonacci Sequence3 i0 H. D6 d9 N2 {' c* D7 p

    8 Y1 _( F, M: V' W% d8 SThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:; a; Q& Z! E8 N: [- H3 C

    . E- k; ]. g3 m* ^9 ~    Base case: fib(0) = 0, fib(1) = 1
    ( _. i8 W2 H/ q# j7 N* m( K8 {5 b9 a# s( F+ w7 @! E4 ~- l
        Recursive case: fib(n) = fib(n-1) + fib(n-2)/ r7 x4 N# [* q& j$ i

    $ v! I: j% \) Dpython/ T' e9 I" ?2 q+ U
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    : T, L$ o" @0 Z' y  j
    def fibonacci(n):' F, O% S0 U# i. m
        # Base cases* ^  o6 R3 F# t! V* p+ Y
        if n == 0:
    3 R# o9 n- j' `1 o" t; k        return 05 y$ k& Y8 j! Q! _$ k
        elif n == 1:1 ], P! e1 s$ f- }* F1 w  G# r
            return 1
    + S1 `- g- a* M2 w- ~, A: d    # Recursive case
    - W5 t+ o/ |6 _    else:
    5 q- `: h2 k/ d; D8 a& u4 e( }        return fibonacci(n - 1) + fibonacci(n - 2)  v' h' t. n" ~' W6 `% p6 f: E3 l
    ( J; u5 ~8 ~( P
    # Example usage
    + L" T, y/ [3 {) W/ Z9 Z0 Pprint(fibonacci(6))  # Output: 8
    " `6 ]: v* Y* `1 a% B# Z% F8 P. P1 M& B0 r. a5 O
    Tail Recursion6 I/ s2 Y- J' i+ U
    3 S  \. v9 W4 }: N. R/ Z. D; Y
    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).
    * m% a4 S0 |3 G! f! b" n4 z# ?$ Z  t' I& q$ G' h9 Y) h
    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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