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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
     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
    / V7 ?) k, c( x3 M; n  y+ d% t7 p5 ~2 u9 d8 w, ?  W& J0 Z
    解释的不错
    ) \( m3 z% a+ O1 t9 H3 \, ~+ K; {6 i* n) W$ f7 K
    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。
    9 m0 j" c9 }' U' O2 [8 p; x/ N
    3 A1 d# i" Y4 I& a7 B4 V( |. N3 K 关键要素
    3 W7 K, T, @- i5 S- e6 p1. **基线条件(Base Case)**
    % ?4 {& c: k, `" C( l+ N   - 递归终止的条件,防止无限循环
    & I; `. i% S0 F. h: b, r2 u7 z+ m+ u   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    % Y/ L" }6 ?- \0 ~# A  q
    ' j$ D" j) |( D0 P2. **递归条件(Recursive Case)**4 r8 F( }% b7 a& f* N$ @  g
       - 将原问题分解为更小的子问题+ Z3 S6 s- k# |, r) N9 q
       - 例如:n! = n × (n-1)!. K" f2 j  j1 ^" u6 `; P

    ; T6 @1 g+ g& @: X 经典示例:计算阶乘
    . o) |8 Y7 ]' o% m, npython; e9 I$ ^3 c0 A+ H( {# r1 s8 k
    def factorial(n):9 w7 ~$ v/ v( b' r
        if n == 0:        # 基线条件. v. p4 v# Y4 ]7 f- A6 z& F
            return 1
    8 G& Z: [' p) Y; I% J6 d- A6 C    else:             # 递归条件
    % ~. @* I+ q3 C        return n * factorial(n-1); G) M5 ~/ c7 ~2 e) x8 q3 b
    执行过程(以计算 3! 为例):' E* V' }4 F* O, ?1 l
    factorial(3)4 A  O' ?/ s  y* o+ D' T
    3 * factorial(2)
    4 h$ b, P1 N1 G3 * (2 * factorial(1))2 j8 s% w8 f( Y( I# Y' z
    3 * (2 * (1 * factorial(0))). x* R2 u; G4 t9 [
    3 * (2 * (1 * 1)) = 62 p" x& r  r; x! }! u# |
    & x3 x; U8 v# g) N
    递归思维要点
    6 Q9 X, r4 b. ?  w& y1. **信任递归**:假设子问题已经解决,专注当前层逻辑- Z& {; b5 L4 ~* M
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    9 k; F. j* ^. P) v' M: }- N4 P3. **递推过程**:不断向下分解问题(递)
    7 u* v/ K! @8 S4 T6 D9 `* p7 H2 z4 u4. **回溯过程**:组合子问题结果返回(归)
    , E& u, P7 L3 c: q9 Y
    6 [4 f6 U. W; I* d! R* ~$ i: k 注意事项
    - i, g' {  u, ]9 k必须要有终止条件
    , S( Y8 H2 @$ T% J6 u递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    1 j8 M0 O: O# `/ j某些问题用递归更直观(如树遍历),但效率可能不如迭代& t" V; l" a6 ?) W
    尾递归优化可以提升效率(但Python不支持)
    / v( V' q% V9 |5 P" m) F, B
    5 Y" s: z9 E/ R5 | 递归 vs 迭代: u- I9 j( d8 Z( E
    |          | 递归                          | 迭代               |
    % ?$ C; ]4 H  X, p7 K, ~|----------|-----------------------------|------------------|7 ~  ?* R& j3 u& w
    | 实现方式    | 函数自调用                        | 循环结构            |
    2 J( U1 w2 T+ ~/ b2 d, b| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 k* [$ q- @2 C# p5 ]" V1 v
    | 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         |2 R: b, Z# y/ n" b% `& t* U$ v+ T
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    6 Z$ E  B; d/ C% r5 M+ R+ S% ?! Y! H: o& `" A& A
    经典递归应用场景
    & N2 l$ @3 V* h. H" Y( H5 i1. 文件系统遍历(目录树结构)0 r/ d- \+ ~7 d
    2. 快速排序/归并排序算法
    * ]. H& d3 F0 j4 R3. 汉诺塔问题
    9 h9 p" i  y3 N1 s4. 二叉树遍历(前序/中序/后序)
    - C7 h5 j% C# i+ [. T1 f8 [5. 生成所有可能的组合(回溯算法)* K9 M8 x7 \  ]

    ; l2 V3 P3 C- T2 V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
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    昨天 06:31
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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,
    2 ^0 g( ^" n& X4 l. \" z; E我推理机的核心算法应该是二叉树遍历的变种。
    ! {* l$ w7 B; F9 _' ]/ R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:( Q3 A+ f1 O5 D- @4 R$ `7 d; k
    Key Idea of Recursion
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    A recursive function solves a problem by:
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    * V+ }0 t: u" U4 c+ I7 b    Breaking the problem into smaller instances of the same problem.( c3 j, C/ m* S8 e; L! W1 v
    , f2 e! j: h5 t, y9 ?6 v$ L0 a8 [
        Solving the smallest instance directly (base case).
    " }8 d( d$ @8 i0 _
    " {' v; C8 o7 ^    Combining the results of smaller instances to solve the larger problem.
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    % H7 l* {! C6 f, G5 J! OComponents of a Recursive Function" |$ h8 z5 e, K# R9 ]2 k/ c; _$ u
    ! r) b. \# M4 F" `! Y, p9 _
        Base Case:% ?5 F4 M9 d( S% k
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. R! L+ x4 o* S* u# ~
    $ }5 ~; H* m' C$ m: P. ^
            It acts as the stopping condition to prevent infinite recursion.
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    - m+ K; c! N- H3 ~3 X- S9 K        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ( r# s$ P0 w' b/ Q- y- ?( B, L! ~  b5 x* ~) S9 y
        Recursive Case:
    8 G  T# i/ _* `9 U+ y% W0 y! j9 L1 G) L# Y) I
            This is where the function calls itself with a smaller or simpler version of the problem.. |- [) U+ l1 b8 A
    , o0 K0 S( y4 C4 n: ~4 v7 d+ X" [
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).& k( n. S7 _4 ?( [
    / Z* F0 Y" ?# s; X; x8 s: y
    Example: Factorial Calculation. w0 n, u" G9 z% e# t7 W9 X

    " A9 j# a# T+ q3 |9 f: aThe 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 {5 u! F5 }( c5 R" T6 y
    6 `2 |2 W+ r/ }; V5 [8 V  s
        Base case: 0! = 1
    + f% b. ?, r6 b) j1 [4 F
    2 B: V9 A; C# B* H    Recursive case: n! = n * (n-1)!$ z) P+ D$ ]! I/ @7 Q% m" h
    4 f$ V# V9 K1 s( Y
    Here’s how it looks in code (Python):
    * g. D5 L6 y% Upython
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    : |- s" g6 \3 U' j- x# _% Bdef factorial(n):
    9 m' ~  c/ D& R    # Base case+ a% v: r3 X8 a. N2 `, F3 b& y
        if n == 0:
    / r- D+ B5 d# ?        return 1" {! V& C0 K3 b8 ?
        # Recursive case
    ; B2 }" ~5 S2 L( ^% C+ d- h% J: Z    else:
    # o- C+ _- \  o. |& J+ g        return n * factorial(n - 1)# W. S# e, W  R5 N
    1 c; R- K: X' J, h% R0 m
    # Example usage
    1 P, ~, V5 F! T/ W% C) r. A# }print(factorial(5))  # Output: 120
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    How Recursion Works
    7 n- G: q: k! P: a$ u2 e) S
    - R8 R# r$ e6 @! \/ s    The function keeps calling itself with smaller inputs until it reaches the base case.
    & }8 x# ^, U4 e: l$ N( n, Z
    ; y- o' I1 G  W" |. T4 U6 K' N    Once the base case is reached, the function starts returning values back up the call stack.
    ) ?* b3 P- d* y; W$ ~7 n, M: i+ }2 X' O' l; z
        These returned values are combined to produce the final result.: }+ v/ S1 }6 C6 D

    6 C1 }5 }& L+ w& V4 V* I' l# JFor factorial(5):! F7 X% E' {/ P- a* ?
    ! @' v$ h7 @8 F, R2 j2 E; |0 {) T
    2 f! P8 N9 t: ]9 H4 P
    factorial(5) = 5 * factorial(4)# x7 W2 E( R  n  u) O) O
    factorial(4) = 4 * factorial(3)1 k2 u3 X# q- t' j8 ?' U3 L
    factorial(3) = 3 * factorial(2)1 C8 [- x+ M# a0 U' c( m
    factorial(2) = 2 * factorial(1)! y; }/ O- c$ b: v+ T* n9 `
    factorial(1) = 1 * factorial(0)  K" w- _! r; x! R  u3 c& b8 v" B
    factorial(0) = 1  # Base case
    2 q1 x$ }6 e- C( {+ S# `
    & A3 r& S5 h1 I( V7 R+ D" `Then, the results are combined:
    / R" Z$ l% W. j# m/ b4 V- V* y5 w( N/ [0 Q

    6 n8 H" Q3 n* b$ k3 qfactorial(1) = 1 * 1 = 1- C% p3 U; }: z% U
    factorial(2) = 2 * 1 = 2
    , m' y/ L; e0 @$ J2 X, cfactorial(3) = 3 * 2 = 6
    / ?3 H# f7 J, p0 Ifactorial(4) = 4 * 6 = 24
    ! i1 D  v# x7 j) x1 y0 n( Ifactorial(5) = 5 * 24 = 120
    : I2 h- i( k/ b; X+ u2 C9 {5 [' z; _6 u4 g/ X
    Advantages of Recursion$ C/ d% Y1 i1 k+ c9 R
    + `! T0 X" i8 I; ^+ c
        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).0 V$ ~" ]: d: h2 x$ q: ?

    , @8 O, |# c9 M9 q  N% \    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    * T, y2 S+ W3 V" `& r/ i$ j
    $ I; k, \% L, Z' |1 Y; {Disadvantages of Recursion
    3 h) b4 |  S4 m5 V" v3 j! Z$ S1 y
        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., a# H; N6 a0 e( {% p' R
    2 ~) x" E! c# ^9 z+ h
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    . a6 b" S) m% t" ?) d/ ^5 {( O3 T9 e4 G( a" a$ I1 G& S* L
    When to Use Recursion
    ' V1 d% e1 d( Q
    7 r0 \! k1 Y* ]; s7 J    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# q0 O7 A2 x7 H8 I+ D6 u$ ?

    6 k$ H2 ~5 F9 q( Y$ E8 [    Problems with a clear base case and recursive case.
    ) o- J% \8 V8 o; Y
    " R+ I& J) y; V9 U2 u2 @Example: Fibonacci Sequence0 _: e* u7 r  V! C# `( W( o

    / M/ k) `, w: |2 @1 [' yThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    0 g9 S: o; M" E  a+ Q. A5 X
    6 y% g; l) s( l0 x    Base case: fib(0) = 0, fib(1) = 1
    / ^) A+ R* \- ~3 Z5 x" w, l
    4 k& o) A/ K) x4 G" h1 S7 W    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    / m4 V( t' p. X! Q4 s, `6 f, J8 L( J1 N. W4 M
    python
    ; W. M. T  a/ _) k2 R8 U2 b% P. ?3 @# N

    7 j3 o) F) _# K$ t2 `& I5 s4 ldef fibonacci(n):
    " z9 k9 f2 D' K2 i/ |1 d    # Base cases6 w8 {' ], E1 d+ t7 E# [" V, J
        if n == 0:
    3 P+ U1 L1 E1 K        return 0& Z% S. x2 S! T1 h* W; V( R' S+ W( s
        elif n == 1:4 U" k  ^$ x* }9 H- i3 h
            return 1# X5 Y! E# {, o
        # Recursive case
    ' ^4 O. a: B7 H; W' w' d8 R4 k    else:0 l, p) T% ~- k: ^+ r3 j
            return fibonacci(n - 1) + fibonacci(n - 2)
    ' E* U: x& d! V/ s1 Y: r8 [5 C( W3 `; o$ O
    # Example usage
    # \+ V/ Q, U9 D. c5 X9 C5 @; Pprint(fibonacci(6))  # Output: 8
    7 h! v  E7 R5 d) C9 ~& _/ d; [2 `- X$ R! j9 d% h
    Tail Recursion
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    8 S7 e* w; E% _- u/ VTail 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).
    % t7 {. [8 U8 T% z9 F9 i/ K9 s5 s  A" k. ^- d4 n" q
    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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