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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
    * F  p! z: Z# B: P0 M, i
    $ J- t1 r. p  Q9 I8 t解释的不错* c2 `6 u/ l8 p# a7 h# A5 G6 N8 N
      `+ J  r8 c8 }" K. _/ h4 C- U; A
    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。
    2 o! T  v( s) |. ~& u/ W) E: y+ L( Q4 U5 b9 ~9 Z3 f
    关键要素# I8 P9 Q$ D% D
    1. **基线条件(Base Case)**! M. ?9 L% \& F, v5 e3 z
       - 递归终止的条件,防止无限循环/ v/ J# B  R2 l) V. `' v; L
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1" h& y2 F% o4 m  m# f$ s

    , u+ R# c4 C. s. a. L+ R$ Y& L2. **递归条件(Recursive Case)**
    3 k8 m2 A+ [( d# a6 x7 L9 F7 Q   - 将原问题分解为更小的子问题' [; b! f4 ]- h/ S: W6 c
       - 例如:n! = n × (n-1)!
    ) |/ L: `( W$ t+ d: i8 q3 u8 W  {2 D( n" q) Z  R+ L
    经典示例:计算阶乘6 v, ?" M* v$ r; |+ G
    python
    ) d& |# A# ~  z' K/ odef factorial(n):
    - l! ]( ?1 ^8 }) B2 g8 ]  @    if n == 0:        # 基线条件* K/ B7 s. z" {4 W
            return 1, h! {: c' J( s& L
        else:             # 递归条件
    ' N/ y# m$ c; Q4 }6 W) }# T4 {8 ~        return n * factorial(n-1)
    # H- W7 h. d) b/ b3 s; P- J4 x执行过程(以计算 3! 为例):, a9 I* h$ W5 t, i9 }
    factorial(3)- Y" `& ^9 X6 T9 v! |
    3 * factorial(2)  j6 B  f0 U# h$ S- c+ K
    3 * (2 * factorial(1))
    / r0 z4 }, g+ j6 q3 * (2 * (1 * factorial(0)))
    5 L( i# ?0 }% }  G3 * (2 * (1 * 1)) = 6
    $ q8 }1 w' J% |* t8 o1 F4 ^% `4 B+ H
    递归思维要点
    6 |& |; k0 b( T# ?" o1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    1 I- S' F  y* N* ^* h/ g$ k2. **栈结构**:每次调用都会创建新的栈帧(内存空间)& j! F2 E% n# v
    3. **递推过程**:不断向下分解问题(递)$ X" ~+ Z# e3 t8 ~
    4. **回溯过程**:组合子问题结果返回(归); C1 A, c; G% O3 A3 c, ?

    6 v8 p3 J& m* p1 @8 E 注意事项" O* k, [; z/ J) P
    必须要有终止条件$ s* l  O& t% s! j! L1 j- F
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ( h; U8 }) x# o3 c6 u某些问题用递归更直观(如树遍历),但效率可能不如迭代
    " G6 j" R& K% I+ U/ l尾递归优化可以提升效率(但Python不支持)# W  }4 I& w% A, r7 s. r& d
      N0 S( K( K; X; P, e! m% ~# Y
    递归 vs 迭代
    7 |/ @- l; ~( {: X$ k1 n  K' h$ @* V|          | 递归                          | 迭代               |
    , U7 d- Z5 B- }( n$ q1 {, z0 j|----------|-----------------------------|------------------|
    0 z' o$ E- N7 ~9 r" e7 D3 z6 L1 g| 实现方式    | 函数自调用                        | 循环结构            |) g8 N& d$ c, @% @: S! M: O, Z( E
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    * w3 f! e, w7 @- x7 p/ 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         |
    # {5 B0 ~* {# [0 k8 y6 |. z2 u+ t| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
      v2 @7 R4 ?; G# g7 o& l. c, L2 c) n, ]- A) `) {( R
    经典递归应用场景4 g7 ~: a0 T0 R
    1. 文件系统遍历(目录树结构)( r% X4 E  W$ [
    2. 快速排序/归并排序算法
    2 L6 }5 _% s+ U# }$ _0 @3. 汉诺塔问题
    1 F& ~  q2 g+ L  \- I4. 二叉树遍历(前序/中序/后序)
      [! ~+ W, ?# U1 f0 r5 c5. 生成所有可能的组合(回溯算法)
    / _6 s7 k& N: m3 M7 Y4 w+ e0 E
    / A# y( k, T2 I; \* _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

    昨天 07:10
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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," {  f( J2 N  }" o( q" K
    我推理机的核心算法应该是二叉树遍历的变种。- }$ b) |* M8 l# A% [
    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:
    8 B5 h) J8 |6 T, I# s9 e- g9 |: V! dKey Idea of Recursion' x$ [- s* w/ R4 a
    / ^# l& ]; P& D, q; U
    A recursive function solves a problem by:
    6 ]2 i- e5 x- D
    ' {# ?) b; V  s" J    Breaking the problem into smaller instances of the same problem.2 S3 \; [+ W9 V' Y

    . Q' e0 @4 C+ ]: j3 N( K% I! ?. i    Solving the smallest instance directly (base case).
    " e; j, {- N% n- c
    - J/ A( m& p+ U    Combining the results of smaller instances to solve the larger problem.
    $ i; z  }/ |% }& ^9 L$ T+ {9 Q# s7 p9 T  e: N5 J+ r: m! Y( v
    Components of a Recursive Function  V) M7 {+ L) m7 k7 F
    8 Q7 F- z- T0 n3 |! H+ z) X
        Base Case:5 x/ S' \) T; i4 b' Z
    9 J# U- q0 w4 |  I  J
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.+ w& P8 }$ P/ S( ?- G! v6 Z2 k. S

    2 P" @; U) z. }1 f% N2 G        It acts as the stopping condition to prevent infinite recursion.9 m: {- E5 e/ C: M, T' q
    . i0 ^1 U) f8 z" T+ ]; k# N
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 ^4 |' U; P2 J2 l4 W7 ?2 u; G7 l' v" z
    , m( ?. M1 i  v
        Recursive Case:  \( N, ~) ~! L4 W

    : {0 T$ N% i% a$ _  }. m        This is where the function calls itself with a smaller or simpler version of the problem.1 t% X: U6 Q' l5 l/ A

    / A7 e# I- T/ c' a& K; V        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    2 e9 ^; K/ R+ t7 L& M3 v& t9 k* m
    Example: Factorial Calculation9 _9 U& D! U3 f9 I4 B3 Q& N! E

    & ]" {0 S7 n6 R  ^: l4 H( ~( cThe 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:
    " f4 ~( A' b/ m+ c/ v! Z
    4 s5 B) d" O( c" k    Base case: 0! = 1
    5 I/ `9 G0 U% {* d" r
    , f) N* x/ I4 I+ c% h    Recursive case: n! = n * (n-1)!
      O1 t) b) J9 o9 ]" d2 v! X9 ~% a6 ?
    Here’s how it looks in code (Python):
    % [5 Z7 I9 ]& r. Epython
    * r; \# |$ R; P+ o5 n
    8 G7 l; _3 T" K5 h6 W) _
    $ e- F% n( ]# j  t+ @7 ]5 @def factorial(n):; g$ E) F# \6 s7 i; w
        # Base case
    ; d% y- T0 I$ D    if n == 0:
    / F, P& I" b8 r# F        return 1) M- D4 B9 C7 e1 B. h! _
        # Recursive case+ N+ B  o+ g* U/ |
        else:
    & d5 ]+ P* b+ D7 R- P" S        return n * factorial(n - 1)" B7 o; B; m: x; b& x

    . g0 I6 g, e9 t+ T! N# Example usage5 p# x$ S( V2 G
    print(factorial(5))  # Output: 120+ O( l' G, y! N/ Q; W

    $ L0 `0 ?7 w# f, h* UHow Recursion Works1 H/ @5 f6 h8 i' S$ N  h

    ) W0 x1 n8 h# p# @0 x# U    The function keeps calling itself with smaller inputs until it reaches the base case.
    6 w0 ~& y* G# l6 \% s; _. N. @/ o* K& V$ v3 H9 V4 O" D  a
        Once the base case is reached, the function starts returning values back up the call stack.
    6 L* P* u3 n) D. T6 b8 R' q1 @* M* `: D* o, K
        These returned values are combined to produce the final result.: h; X* Y/ o/ v
    / `, @6 P1 R5 Y; l
    For factorial(5):. ]% K$ K  c& ^: D
    ! v4 J. r0 X  G& O, X
    9 ^  y- M  u8 u0 G$ o
    factorial(5) = 5 * factorial(4)6 R2 t7 L- W  w: m% W  o9 u
    factorial(4) = 4 * factorial(3)$ `4 h* W1 u" z7 }
    factorial(3) = 3 * factorial(2)
    ' ?- g* \$ e4 t# Sfactorial(2) = 2 * factorial(1)1 D, `& C# j5 g( c0 |, k
    factorial(1) = 1 * factorial(0)
    ' G' @' n) ?$ w# r( bfactorial(0) = 1  # Base case, O! |  O  o3 t9 w) q2 t+ t4 ~
    . a' M) F6 p$ ?$ e
    Then, the results are combined:
    ; n! H# j2 ^2 G0 U  ]
    5 b4 K8 s% Q6 z6 W2 K. O; v- _9 b: h& f# }7 @
    factorial(1) = 1 * 1 = 1
    . {; b# q; P, y# d0 B+ v2 Lfactorial(2) = 2 * 1 = 2
    % ~# ?1 l" }& |( P: q* Qfactorial(3) = 3 * 2 = 6
    : z; b' B* Q, R7 Nfactorial(4) = 4 * 6 = 247 O6 ^% r4 \; a& b$ t" M4 b
    factorial(5) = 5 * 24 = 120) D7 k4 H/ a0 v  V% r) R( E: q
    & N9 ^8 x: \  O0 z
    Advantages of Recursion
    ) c; M1 |8 V( `3 N* c2 V) s' u3 G
    ' G! ^' [4 {( I% e6 e- 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).
    3 E7 B$ b+ \" l7 U4 s
    # G) R4 ~- o) c6 K9 h; M    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ; i% a. k9 n5 W6 A4 u+ I4 G
    ; }( t- i4 g3 {0 SDisadvantages of Recursion# D4 O! x9 f2 Y$ W
    * v! E, z8 J3 s- r9 u; J! [2 _
        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.
    % b# A" N, R5 }9 S2 w/ X. g
      W) l' y/ R+ O    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).* ]) F6 F' M4 \, j- Y" L$ K+ l* I
    / V0 E: m0 m0 f
    When to Use Recursion
    6 V7 @, Y- k+ I# {# c
    & t1 Y( O- q( U! S# a& S    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).8 v' ?: o3 G# w3 U# n' B$ \7 W

    7 a1 V' Q6 o+ H7 i# O    Problems with a clear base case and recursive case.3 X/ c$ W! a, C! S$ C

    , c! a$ T) Z0 E' J' }/ j+ IExample: Fibonacci Sequence% A/ ~+ N! H* z, l8 t
    % J- z8 y6 d: G
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 K2 |! R0 _; a* D+ h

    # O% c1 {9 w- ?* v  U9 `    Base case: fib(0) = 0, fib(1) = 1, M/ T! r8 I1 e/ i; U
    4 _+ @" H" a2 w+ ^. ~, N
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    2 k/ J* t7 w! I+ M$ G& C) X& V: }7 l4 f9 i) j' i" h2 s
    python! C3 }9 `1 n: E: |
    % }' j! R$ I. E8 T  U

    8 W$ M, P; v; t: B- u! Z( Pdef fibonacci(n):
    ; ?8 R; v4 R4 @0 |. s0 S% d    # Base cases
    + g. x  N- s; A5 v' Z0 l1 W& n    if n == 0:
    / A" `% L. T7 o5 z4 O2 f        return 0- u. n' O9 z0 e8 h
        elif n == 1:
    4 c9 j3 X6 w2 N        return 1
    - `7 _: i4 G; ^0 p  z- ^    # Recursive case8 \9 G$ Q2 I3 D( K4 ~, w
        else:
    + `* B4 }( R% f: J! m$ w, }, T- F        return fibonacci(n - 1) + fibonacci(n - 2)
    - N/ P/ N1 `; i$ \9 E1 J/ J# z2 v- ^  s% ~* z* U6 K' B
    # Example usage' B& M  l4 J, }8 `7 X- t
    print(fibonacci(6))  # Output: 8! H) T$ t0 C) ]$ k0 A/ ]. ~
    3 T/ N+ V& y! w
    Tail Recursion( v$ T" f1 b& v/ k/ ~) ~
    # p) D! K: ]$ R$ Y% a3 g
    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)., i# R8 b! n8 G* a
    , M) A+ o! K; ~+ A; F1 x9 G; Z
    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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