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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
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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 % w! ~( {+ k9 }) r1 J
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    解释的不错
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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。
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    , J5 e0 n5 d5 @ 关键要素6 U4 {: L7 a; ?. C
    1. **基线条件(Base Case)**3 V2 C8 m: E# Y* l- X' D
       - 递归终止的条件,防止无限循环, ]6 e4 J6 w4 \( ?8 a
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    + B' L& Z4 `$ N4 C# q. L! E, q; M1 F
    2. **递归条件(Recursive Case)**
    % E& Y) T# z2 `% t, P4 `4 r& ~   - 将原问题分解为更小的子问题% t! v+ b! b# K! c
       - 例如:n! = n × (n-1)!
    0 {* t. \) u) W/ f, x( S( h- H0 e9 O. ~2 f0 r. e  b
    经典示例:计算阶乘
    . c! \8 d- p2 f( Ppython
    6 D$ e; J$ l4 C1 t' o! @def factorial(n):
    $ ?: s: x% x! Z1 Z    if n == 0:        # 基线条件0 E- ]" w  E3 L4 k$ g
            return 14 U6 [4 R3 G( _. G% C; ^/ P
        else:             # 递归条件1 x8 s+ C! A8 C! A" ~
            return n * factorial(n-1)
    ' z3 Y! P- i* d* U执行过程(以计算 3! 为例):4 u/ W" S6 S2 {* E1 z: [
    factorial(3)$ m0 E: {% b, k" I# z  [8 N2 A
    3 * factorial(2)
    % D  [0 N  f1 s( ~! s; m  c6 c6 U8 N3 * (2 * factorial(1))
    " A! }; {! M# E' v1 V( f3 * (2 * (1 * factorial(0)))" Q1 T" a* w; s! _: u4 d& q+ a2 @
    3 * (2 * (1 * 1)) = 6
    ( Z: k3 q( X) N  ~$ j6 g' |0 \2 A$ }: E# Z* O7 h" ?
    递归思维要点5 E6 z8 N0 D: [% V8 G8 Z7 w0 t
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! T8 @) i; j! y5 w2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    . u3 m4 ~5 V9 c! @3. **递推过程**:不断向下分解问题(递)5 L6 T* l# i) ]7 X7 p* e1 j
    4. **回溯过程**:组合子问题结果返回(归)# h' r( D# ~7 ~9 G

    : `% [2 f% i8 N$ e 注意事项3 Q: x/ \3 e- Z+ ?1 B  A+ ?
    必须要有终止条件
    ) \" a" @* f! p- t递归深度过大可能导致栈溢出(Python默认递归深度约1000层)$ K5 R8 N' w' D
    某些问题用递归更直观(如树遍历),但效率可能不如迭代4 j8 c! m, K- b! p  ^: m
    尾递归优化可以提升效率(但Python不支持)& m  m$ K6 p; k% Z3 j  m  J4 _; U
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    递归 vs 迭代; s+ b: Y' q( ?* O2 ]* X9 l3 z* I
    |          | 递归                          | 迭代               |- q9 z" E9 N! p3 }# D' u
    |----------|-----------------------------|------------------|) }7 t0 _7 v9 h4 a0 T7 \
    | 实现方式    | 函数自调用                        | 循环结构            |. }2 D; e6 q1 p" E" i
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    8 g- S6 f( t/ E  s! k6 [| 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         |0 m) r: G& D0 n7 u/ H
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |, D: r8 w' @" h, o( E2 O
    + J/ t- I/ F' S  U8 z3 _
    经典递归应用场景* j$ ^/ {" T7 `* p9 o& H- c
    1. 文件系统遍历(目录树结构)
    5 v  ~  H1 Q/ U7 Q* }2. 快速排序/归并排序算法% Z* m- x3 D* @# s
    3. 汉诺塔问题
    ( |( p  y. b$ o5 O! R( B4. 二叉树遍历(前序/中序/后序)
    4 j7 i5 s2 l& `, t& I! w5. 生成所有可能的组合(回溯算法)
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    9 z. o% o. J4 d% r6 K2 W( O. W; {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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    14 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,  f4 L' r  n0 j
    我推理机的核心算法应该是二叉树遍历的变种。- u9 g3 H4 Q/ n) J4 `
    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 t3 l! C3 e1 I; v
    Key Idea of Recursion
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    & b8 S- w9 w( Y6 {A recursive function solves a problem by:
    : Z  R" {9 w5 q# E' u
    / a8 J7 `. E% |& H$ R0 \8 M- k    Breaking the problem into smaller instances of the same problem.
    $ F) {" J! G7 J# b* v" t  X+ r6 ~+ W- y- |
        Solving the smallest instance directly (base case).# @* c- m& R, }( s0 k. T
    ) N4 n3 M) ?  [" E8 z
        Combining the results of smaller instances to solve the larger problem.' }3 }9 t7 J& a7 x: a# k9 g1 w1 K( W

    3 {6 j4 `# N+ u: u( }/ K" uComponents of a Recursive Function2 n! x. [7 ]3 ]+ i* g2 `

    1 g, q# E. g( g; U9 I5 R. }    Base Case:4 c. a( l1 }9 F( F, B( \( B
    1 ?$ J; }) h- T! Z
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    0 q; z8 A4 j- a8 p1 S' h7 ^; ^  r5 q: O4 e
            It acts as the stopping condition to prevent infinite recursion.* V8 L" E0 _2 n6 V) t% g" X) u& A
    7 i1 r4 B8 R4 C: S3 Y8 v% k
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.# m# G/ }# O6 I
    + @6 G; B# m5 M
        Recursive Case:
    2 \# u. Z6 e* i# D" V% N4 q% E; ]5 M
            This is where the function calls itself with a smaller or simpler version of the problem.+ S& c- ^( r0 M" Q1 g1 l- h$ |

    ; I+ P7 i. O$ |# r& t        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 I3 E- V7 `0 R+ \
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    Example: Factorial Calculation0 X) t, W; Y4 V; P

    9 L/ |2 _5 q( `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:
    9 s; {( m- f' i! S. M6 t
    # Z" x3 w" N5 R: ^    Base case: 0! = 1* }- u# z6 f0 I% ]0 S1 y

    ) N. [% H3 A0 J: @7 k; l) g) Z$ ?$ E    Recursive case: n! = n * (n-1)!7 g8 m- N- D4 B
    $ ^4 L+ E$ u' e; G: K
    Here’s how it looks in code (Python):
    0 B8 Z+ ~3 O6 A% R; H0 a! }6 ppython2 w0 v% u- {+ v  |( Y
    4 a" L$ Z% \2 S, B. D6 i

    3 m" Q5 |- J, y$ bdef factorial(n):9 }0 s9 \1 c2 o4 n- ^; s  Q
        # Base case
    5 ~5 P3 N- Y5 J" I$ R    if n == 0:
    2 Q9 ^- n# C% U3 a( X- t        return 1# |) d+ p- ^  Q' i" b- T" M2 ^
        # Recursive case
    8 ?! @5 T+ v! x  J# Q/ V! r' Y    else:; X+ T% D; {: @/ g! l* X$ _9 I
            return n * factorial(n - 1)
    3 H, m) v% M( U1 _( ]" b" h
    $ P& B! [. ^0 v: B# Example usage
    # J4 ]6 p5 l4 C& mprint(factorial(5))  # Output: 120
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    + J& k/ r% _$ w  d4 L3 O. ^How Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.
    6 U8 c6 V* P1 z8 d  O# `2 w  }/ o& C2 r7 @% j
        Once the base case is reached, the function starts returning values back up the call stack.% }7 a2 y3 K. E
    / e/ _, {6 f6 s* o+ @  k
        These returned values are combined to produce the final result.. d& L: q0 o$ c3 b

    ; T# d5 a* p) D) {% y3 |, PFor factorial(5):4 d% ^  U; u* D: T
    ( x) m9 Z8 a% I$ g6 X9 K8 i

    # }& z# b! u% f5 c6 K7 w% {factorial(5) = 5 * factorial(4)+ s& M4 K! Y- a) q  W
    factorial(4) = 4 * factorial(3)" n# \8 e5 k# f1 p' N5 u
    factorial(3) = 3 * factorial(2): b" U0 J; M  ~: B4 W2 G
    factorial(2) = 2 * factorial(1)
    / E3 f9 X: S" X- cfactorial(1) = 1 * factorial(0)
    0 v, W  Z- p; o! y2 ~, Xfactorial(0) = 1  # Base case8 F5 s+ E+ l. U5 n- B

    ( S; t8 {/ x2 u5 B  l- jThen, the results are combined:. v! Z% ?2 ]% P2 `0 P
    . l; _/ f. S, q9 I6 J8 e

    , y: k4 t" d: o1 ?: f' x2 G7 sfactorial(1) = 1 * 1 = 1
    6 h$ b7 y* M/ b- e9 \: _factorial(2) = 2 * 1 = 2" T, i) E$ U# E- C- W5 z
    factorial(3) = 3 * 2 = 6
    ! ?6 ]& u2 v& `factorial(4) = 4 * 6 = 248 e7 i+ k; O% N: }
    factorial(5) = 5 * 24 = 120
    . O5 L- D6 E) U. j9 ?9 c7 k. H1 d& |( z# X5 ~
    Advantages of Recursion: s2 e' h4 e; I2 K5 T8 m0 `. W- I

      ~! B# ~0 j, Q6 `1 z4 {0 ^    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).& R9 {, u7 [- w% t

    ) g5 o8 b* _6 R2 A0 x7 i# P    Readability: Recursive code can be more readable and concise compared to iterative solutions.5 R% `5 G6 O; D! [9 q* R( {* l% W
    ! B7 {) f3 O$ H8 H$ e' [" E& b
    Disadvantages of Recursion
    2 o. ]4 v/ W0 E4 I" l! n( o7 X- ?4 L2 ^/ b( c1 ~
        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 R* q. u0 I- |/ x6 c4 [

    2 L9 a9 D/ I+ N( b* }    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 ]: Y8 o6 h, R
    9 X. f$ F, k* f* w7 h, r8 n" u' o
    When to Use Recursion
    & @) i2 G' l& S$ g
    , T& @# w' f8 X6 C    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    9 Z. @4 l% I) |& c
    3 v: H2 d; T# R1 e0 K" A3 O    Problems with a clear base case and recursive case.
    4 s; N- n% r% z8 e/ ~4 A& ]$ r' d+ ^
    Example: Fibonacci Sequence3 H/ X! _  w5 H3 Z& z

    1 ^9 G' H: n! P. V8 b% f9 ]The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:6 R5 f1 q5 Y3 C8 t& ^7 S. w

    : a- D2 J( R& _: d+ ~2 M5 [    Base case: fib(0) = 0, fib(1) = 1' I. f, \6 m/ z6 e0 y' Z( K9 b, @& X

    * {& l  x2 v2 U4 ?1 ?1 r    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! z1 T/ H$ }2 M2 w7 H# _, W/ H" B( F9 A1 a% p2 W; U2 b
    python
    # D+ U" ^, c3 D, F6 M8 r* @# \, z# o% ^# {% q+ f8 b5 I% j2 g2 Q
    % j9 \6 K2 T" o/ }, r
    def fibonacci(n):4 d% I/ j, \5 @6 V/ r
        # Base cases
    - }! Q+ y6 {; }/ O" X$ P    if n == 0:' l4 P5 L: [3 a5 v
            return 02 e9 T, o! B5 x+ P5 l0 g
        elif n == 1:4 p/ F* G$ W' i. b3 S
            return 1
    * v& |6 _, V: V    # Recursive case
    * D7 C* Y7 q8 m/ u: ?. z/ n    else:
    6 i0 n; ~, K& D4 ]% ]- i        return fibonacci(n - 1) + fibonacci(n - 2)$ i# u; r1 m' ]) F' i

    + y8 H  b/ Y4 ^6 ?: Z; O# Example usage
    4 T( x/ z2 X8 ~/ o0 r6 {print(fibonacci(6))  # Output: 8+ f: N3 J8 M! v. F) D4 ]

    2 P4 N1 E* \7 gTail Recursion
    5 V/ A+ l% W4 J/ W+ C* ^0 |4 D
    4 [+ v9 ?' S6 W' ^2 }2 |6 p( HTail 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).
    2 \/ V) s0 ]( x1 k) t* ~
    6 I+ f# g, g7 L) |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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