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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
      n# i( r" c) Z7 b3 Z+ P! x1 a7 s3 B9 N+ ?0 K8 j3 n% D* Y' Q9 p
    解释的不错! Z4 W* {) }% K" p$ w
    3 {3 T/ s( Q1 Y; ]- j
    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。& K% B' F: w' a  j

    $ c0 r- Z( [- e& k. J8 D- c6 D 关键要素
    ! n& L8 M# ?9 ^& ?1. **基线条件(Base Case)**2 }' L, z- A+ ]5 b# [
       - 递归终止的条件,防止无限循环
    9 p7 q- r8 H' y; Y% ?% Y   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    , c) w+ w6 F& I1 f" d8 l# ]" f/ z* I- P
    2. **递归条件(Recursive Case)**: l8 f! v9 X! E0 o: ?
       - 将原问题分解为更小的子问题( R: H3 I! i& `
       - 例如:n! = n × (n-1)!& H6 L, O2 S! c6 a4 v1 D0 {% p: s

    ( U% @% `/ P' x% n5 x/ R/ p, n 经典示例:计算阶乘
    7 c4 ]( n% q  _9 n& j9 I, ^python
    9 l* y% J9 m0 Y5 mdef factorial(n):
    / l" b5 E! {5 @$ `: }    if n == 0:        # 基线条件
    # v( {; Z6 \% N( y; q        return 1
    ' E. T! M+ M& N. q7 m    else:             # 递归条件! ?9 x$ H+ `# N4 `9 y: O: A; n  k
            return n * factorial(n-1)9 I5 `2 Q; |% W3 P# u# U
    执行过程(以计算 3! 为例):3 U' L& Z& A% F' ?4 N4 O
    factorial(3)
    3 Y( H; C+ X' g" @( i2 T6 R4 b( B$ T3 * factorial(2)
    * @$ |% s3 A9 s3 * (2 * factorial(1))
    2 P% H) f9 m3 J% ?! S3 * (2 * (1 * factorial(0))); `" q4 P/ z7 p% G" L
    3 * (2 * (1 * 1)) = 6: @/ f7 E* t7 X$ E
    + N+ c) w6 U: ^- c2 I( \+ m
    递归思维要点% A" P  V) G3 @& g3 j
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ; F% _& {, r/ P. v* e" N; u4 T2. **栈结构**:每次调用都会创建新的栈帧(内存空间)1 p/ Q' ^( i2 L' d# P- \
    3. **递推过程**:不断向下分解问题(递)4 R6 i7 A; H+ [/ p/ [5 B, g  ]2 C$ _
    4. **回溯过程**:组合子问题结果返回(归). [! {# E  U  y9 ^& ~; W9 A
      \( o$ z; V$ ~' r. I' t; W
    注意事项- R2 a" Z) U* H# K$ G( ~
    必须要有终止条件
      B0 e2 [% ]7 {  B7 t' Z递归深度过大可能导致栈溢出(Python默认递归深度约1000层)8 V0 p. h  }& {; o, U( x
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    2 C# n, U+ Z8 d: T4 q* a: C7 t2 ~尾递归优化可以提升效率(但Python不支持)
    % v# X( j+ O3 w; G  L/ i8 m+ G; d
    递归 vs 迭代$ o* Q1 Q& l. l% N4 W" Q5 }
    |          | 递归                          | 迭代               |
    . i7 x. s9 x+ z) N4 y|----------|-----------------------------|------------------|
    # P: b2 o, U% R- x& O| 实现方式    | 函数自调用                        | 循环结构            |
    - q; T" S, f8 Q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    * Y- w7 v+ z1 i| 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         |
    6 r9 U! H+ K% T2 K9 ^) t9 X8 q| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |- K9 w$ W4 V  b# }

    2 U# U+ @  w8 C% [ 经典递归应用场景
    6 {0 |7 u% f  L- \/ E1. 文件系统遍历(目录树结构): g+ e% {/ }7 G# c" ^7 V1 w( P
    2. 快速排序/归并排序算法
    7 F* F! x$ N; k5 m3. 汉诺塔问题
    ! I6 Y# O$ Q: [- k/ K& _; B4. 二叉树遍历(前序/中序/后序)
    # B! E+ d- o" _' {  ?7 n2 A+ u5. 生成所有可能的组合(回溯算法)
      K; S9 b# s9 o: v" S. y5 n7 n% w$ ^) Z7 C
    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,( G+ r# o9 Q2 \; A* R+ K, s& {
    我推理机的核心算法应该是二叉树遍历的变种。
    ; {  G9 G$ k: r& a" P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:# W* }0 q- _( i. h2 H4 W- C
    Key Idea of Recursion5 t* l2 A: G2 D! s
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    A recursive function solves a problem by:
    ) S2 t$ S/ j6 S, F( @' X( g8 t; R3 }6 T" j  z, A8 F* O
        Breaking the problem into smaller instances of the same problem.
    * v+ H9 _9 p! D* \% ?* l9 b' j/ J/ j7 J5 @. y
        Solving the smallest instance directly (base case).  K' H1 d3 u7 }7 y8 q2 E  x4 [

    - j" @/ w* O: q- T    Combining the results of smaller instances to solve the larger problem.
    . _: a; _$ B, H/ [8 P! `$ X; u  k; m: }. f
    Components of a Recursive Function! y  o. Q3 {# L' g7 K# a5 H  M

    ' t9 f* R7 P$ T0 M    Base Case:$ g1 Z5 u4 e! G. K6 j  ]: J
    ; j1 f6 J9 l: C. M! a4 t7 W! k
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; [* I( }* C) c2 Y3 J& n( p' D% g7 y+ b1 C& w4 Z
            It acts as the stopping condition to prevent infinite recursion.
    % c, O& n$ N0 U/ |1 m
    ( m( v+ Y# E- a+ z. n( e. O% ]        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.) e4 M" T" K) m9 A; ^
    8 l9 k- D8 i2 x1 F
        Recursive Case:3 a( z5 t0 y3 f& l: R7 g, Z2 i

    * A- {5 m2 f2 E        This is where the function calls itself with a smaller or simpler version of the problem.
    " }( k9 c% q2 M3 |/ O8 A  @! n) p! {% _/ i) Q. [% D' X% k0 X0 l9 A
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).5 ~+ j- Q5 l  H6 }8 p* G; d( L' J6 F

    * r- i3 P0 x- s2 p7 oExample: Factorial Calculation" c/ [+ z8 j$ R& z" ~# n
    : g8 m# y8 b% 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:9 Y6 m( H! f, \

    . |! ^& i. G2 e% s8 A    Base case: 0! = 1
    , {% v. e( C2 x) Z# L( w6 `2 u+ b0 z$ l! G2 b) m3 ^3 G* e6 `
        Recursive case: n! = n * (n-1)!
    - I# k! d1 d8 V% Y( p0 ]* k
    ! A/ k$ u3 c; P  S( K( s  bHere’s how it looks in code (Python):( Q1 Z/ T7 I& o% m  S0 F+ \
    python2 ?, b% E3 y# T! j3 `1 u
    ! I2 {7 _7 d( n: G3 z. M

    & s: V4 U2 ~. N4 |def factorial(n):& W8 V4 r$ A+ j0 q# L
        # Base case0 ^/ Y3 b: a7 `0 I# w; Z+ K. Y( h% N
        if n == 0:
    " N% ~6 E! P  O% ^# n, X( u% X7 w, k% `        return 1
    ( X' o* s! b" B( Z- n    # Recursive case, g; _) a. }1 U1 O9 G$ B+ Z
        else:& j# L& |- K& a: S& ]9 d
            return n * factorial(n - 1)! a3 M1 e" e% l; G6 Z. ?
    . }) e$ B% Q) u6 Y( s. @# i3 B0 F0 K
    # Example usage/ O- M8 z; ~0 y/ P$ y
    print(factorial(5))  # Output: 120
    ; z4 x1 l/ G9 _8 f* o+ M+ e3 d! b- E; y( o- p8 H& A4 {
    How Recursion Works" s# g1 _$ ~/ t: G3 J/ o$ N

    : M" o7 }9 l' z% `% |' F    The function keeps calling itself with smaller inputs until it reaches the base case./ K' \0 j$ [' E5 n# n$ c: d$ q
    * ^! N. ^* D+ R" r+ K" U  \5 W8 r# R
        Once the base case is reached, the function starts returning values back up the call stack.
    . }/ V8 F$ F" }0 [% k3 `4 f3 y. [3 f2 }7 n
        These returned values are combined to produce the final result.
    + r8 K0 n' h) Q6 X( o% p
    " i) v! y6 D0 @8 v& {7 L5 m# Q1 u; SFor factorial(5):2 u5 B2 @6 I7 a; F3 e
    " {7 O" h" k# I# B- q: K1 }

    : Y; I; i& N- Gfactorial(5) = 5 * factorial(4)8 M, l/ G5 }! Z7 A" y! ?$ U) R
    factorial(4) = 4 * factorial(3)
    ' _9 o) r% E* h7 Vfactorial(3) = 3 * factorial(2)
    1 a, s3 g1 G, o7 _0 lfactorial(2) = 2 * factorial(1)6 `5 z+ F. ^2 v9 C0 ^6 [% w
    factorial(1) = 1 * factorial(0)
    9 X; N/ V4 H! r. ^8 d$ y8 qfactorial(0) = 1  # Base case0 _+ T% w9 k7 Z

    $ C( U8 D2 @( Q* T( wThen, the results are combined:9 T- G: E7 c% u  ^' k% ~% j

    $ f. z1 h: H. Z- E- A% v& s& W4 w( ^! k& v/ L" Z
    factorial(1) = 1 * 1 = 1
    , J% |& k  }3 C& V; p9 i/ Y) V2 ?3 yfactorial(2) = 2 * 1 = 26 V2 ^; G& G+ M4 k0 u
    factorial(3) = 3 * 2 = 6. W% Y# q5 `* Z8 T
    factorial(4) = 4 * 6 = 24
    ! Y+ ?! @0 ^% A  U, K6 Jfactorial(5) = 5 * 24 = 120
    6 v7 P$ O6 O8 m% V8 c( e3 V9 d. ?+ L1 y/ J( I
    Advantages of Recursion( ?/ C, D' V) L) a
    6 w/ u; R/ W: X7 r9 Q- j; x
        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).& p: A2 X0 k  \4 @  [/ k/ _# S: t

    7 X5 C; ?! }& l  \$ x    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    5 w  q# [% ]; i$ W4 E/ h3 @" a4 T, `+ f3 v' F. S3 I
    Disadvantages of Recursion! d0 N4 H" m& W/ N& d4 T7 U) Z0 _
    6 a4 c3 @! v6 c0 O
        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.
    $ ?' M+ `) A0 o1 X# s
    # I1 N) ~! ]; K. ^7 C' P    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).$ e+ H/ G+ V% l2 ]

    6 v/ g* B* Q9 E- L5 A8 WWhen to Use Recursion
    4 H1 N' }& ^. f
    ! L; d6 A2 W( _* m2 U. {% f    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    9 b, l4 U( g6 [! U$ M3 d; [2 h! j$ |! n$ O5 J2 Y
        Problems with a clear base case and recursive case.
    " V; B" w  B, d0 g( q3 b; a; e4 a% S
    Example: Fibonacci Sequence0 n8 D7 I" M$ k

    ' z. R5 c$ d# n; uThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 t' V) X* ?, i! ?% R' e

    ( q- U& `- q9 F    Base case: fib(0) = 0, fib(1) = 1: H" x8 K2 I/ ?' p- i

    , z. w3 K  C8 c! P6 _    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    % E; I5 J; {1 o% F& |6 ]0 F' G+ s1 G9 l9 C  u4 X' Z
    python: {  d# p( j3 [* ~7 ^, u7 N
    ; q# z* p  V& [) h/ d

    - \# R, T6 F# h2 Cdef fibonacci(n):
    ( x% u+ p6 N% U$ z4 O/ ~1 g    # Base cases
    7 [/ q9 z- O" a& B* O    if n == 0:
    9 j8 `- L! t+ b6 k+ w        return 03 L6 F; v' o' W& S
        elif n == 1:
    # `; |- E0 z. w        return 1
    : i1 A( m7 O. ]- d$ R    # Recursive case9 A' [) U4 B* ?2 x# f# V+ Q
        else:
    " f  k. v, z" K( F1 N! M        return fibonacci(n - 1) + fibonacci(n - 2)
    + R! O9 |* n9 k# l! u; d' Q  n) i: I
    # Example usage4 y) I: y4 z; i3 Z( G3 i0 m1 j
    print(fibonacci(6))  # Output: 8! z2 L. ?: u6 ^, ~' H2 O3 j

    1 {1 k  t  l& m! U8 F- lTail Recursion
    + o! d, }- i* b. c
    . O5 j$ R7 ]2 v7 S! b% bTail 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).
    . F% z1 t# P: m: @- l% e
    : u. E& U6 c! @! ~$ w3 HIn 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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