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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| 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 r, R7 h, }! O  U* t0 K) k
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    解释的不错" F( G# a- [4 O; N
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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。3 {; \2 T5 _+ o$ }" D- e* E! D

    ) o  R( u/ a4 W0 {( T4 m! _ 关键要素2 i0 y( Q  C  \1 r- U' ^! C' D
    1. **基线条件(Base Case)**# I4 ^& k0 W4 D! C( U" e9 ^+ b
       - 递归终止的条件,防止无限循环
    . B/ M# d0 H+ [4 ^* t   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 11 u! ?7 H8 V- }4 x& a  V
    $ m2 t# K) ~, N% P# t) [7 Z" P1 g! S
    2. **递归条件(Recursive Case)**$ q0 t$ J8 v) I) F
       - 将原问题分解为更小的子问题7 H1 ^3 ~+ M; j6 a
       - 例如:n! = n × (n-1)!* Z; y& O" p" ?* G$ ?# u/ o
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    经典示例:计算阶乘
    ! \- b# m: Z, t" w% |python0 M( |& q5 h! W& @" j7 b5 F
    def factorial(n):
    1 h" r& P5 e" t6 a0 [& _8 ?    if n == 0:        # 基线条件
    8 M# D* A" @. U; ]9 G        return 1
    4 R% F5 l/ w; ^9 m3 t    else:             # 递归条件
    / ]' n9 [) U) G! \9 f% x        return n * factorial(n-1)
    4 t/ \" x2 i1 x执行过程(以计算 3! 为例):) W2 s; x$ b7 T% a# B
    factorial(3)
    ! A7 j$ R4 Z# H/ s7 Z6 a3 * factorial(2)
    3 O4 e' t  |4 D, [; I3 * (2 * factorial(1))7 n( w6 D& x) r: Y) |
    3 * (2 * (1 * factorial(0)))
    / Z9 T6 F/ W: b3 * (2 * (1 * 1)) = 6' Q  u& N# }8 ?- A# j: V" P  k
    5 u; i* a1 W3 d, t0 Q$ A% L
    递归思维要点+ U9 l3 m2 J# X" l9 W2 k7 j
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑9 T; p4 y4 h/ }! |
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- K, L* U% T, I/ G" z! I; r
    3. **递推过程**:不断向下分解问题(递), i7 V; @# ]3 N0 g6 [& g# _, i
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项
    - G$ S$ I7 U7 v/ d: [必须要有终止条件
    " f9 n2 J: N2 N" i3 p( ?* V递归深度过大可能导致栈溢出(Python默认递归深度约1000层)4 E! D$ U  p4 f! Z  n( S' M" \
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    " |2 R1 d* V& f7 ?7 q6 A$ v' F: w尾递归优化可以提升效率(但Python不支持)4 d1 h) G* o$ V7 ^7 X3 O& a# z

    1 T; R* S" l6 e: ?+ T 递归 vs 迭代) ]9 ^' f! p, G0 V$ G6 G
    |          | 递归                          | 迭代               |
    # h$ Z7 N% f( Y! }4 @. ]|----------|-----------------------------|------------------|
    ' R  x+ H' f! i3 B( C) X+ d, I| 实现方式    | 函数自调用                        | 循环结构            |, R& w0 N2 n6 \. ]: x
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    % g2 z& R  Z6 d" Z  }6 Z| 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         |3 X: U) Y  t# e3 P
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ! C( |6 ~( W. J$ c& D
    8 }. F: d/ R8 L$ B% b* E; U 经典递归应用场景9 C! m1 D( k# _5 N7 H' T" l
    1. 文件系统遍历(目录树结构)
    - h+ n! V/ h; |, D* I4 p2. 快速排序/归并排序算法; ^( X0 p! X7 I, ~: [  C2 v
    3. 汉诺塔问题
    ( P; i  r& c2 m9 L7 H+ a- ^4. 二叉树遍历(前序/中序/后序)7 `( U& k, m$ P
    5. 生成所有可能的组合(回溯算法)
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    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 07:05
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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,5 D( K% A# e8 p3 I
    我推理机的核心算法应该是二叉树遍历的变种。9 G  X3 z0 C& C+ A2 m
    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' b6 o1 ^" D) p" q3 _- g! {
    Key Idea of Recursion
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    A recursive function solves a problem by:
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    $ y$ F0 z& x: n. E7 T    Breaking the problem into smaller instances of the same problem.
    , K, a# @6 K6 l7 K! ]4 \# r  y5 M
    % O6 S" v! H' q/ S5 G5 z    Solving the smallest instance directly (base case).& F9 B# I) z7 ?0 x8 _+ _
    ) L  @. D; V/ k) J0 g, x$ u, S" M
        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function
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        Base Case:" y! S" b' `0 q" r- A' ^
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.3 [4 a1 f8 `7 r& h) y4 f
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            It acts as the stopping condition to prevent infinite recursion.
    % D' }/ K$ w9 ^1 Y' ?- R2 `$ c3 J. M  }* L
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 A( @7 `8 D/ r

    ; n- m- A* p( t5 J4 l7 l    Recursive Case:
    - P% B1 K8 k: `+ \! x- l4 q% x% i7 m9 q% ?1 Q3 H/ U( g
            This is where the function calls itself with a smaller or simpler version of the problem.9 S! C, z& b4 r9 y- ~# |
    8 {9 J. J: @: L& c
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).' W! ]/ H3 W+ e$ v( n$ K3 L" V

    + {: K' ?  s1 n; g! wExample: Factorial Calculation
    3 t# R& d+ u9 \. v0 L/ w. h# g" q* {8 o: d2 O- O$ e% 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:5 ]) V; l4 C7 S& D% l

    $ S. Z7 n5 }! X1 Y% ^: W+ d    Base case: 0! = 1  V( K" L$ z* H. t2 ]

    , S3 W6 C4 F0 d' A8 O    Recursive case: n! = n * (n-1)!2 n$ ~, |; s6 B9 U0 I: X9 p
    ( Q+ @. \& M% C
    Here’s how it looks in code (Python):4 x( c5 m. a, J3 d) G' S
    python0 C7 }  N, H" `" T

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    / f$ m' J# ~$ _- L: p1 A. udef factorial(n):
    ) K5 ~( [& d. k0 Q4 F8 a! m    # Base case
    0 M% P% |4 B! y$ u5 Z8 N    if n == 0:
    ! ]+ O, O6 e" u3 a( r        return 1  y- X2 J$ v& v' S- v
        # Recursive case( t' a4 `$ u9 L* E; C
        else:7 u$ y& f1 J% q- c( S, p9 }' j
            return n * factorial(n - 1)9 r; J7 y4 h! {' H

    3 q) k, S5 G, S4 i& |" s# Example usage
    6 r5 d. O. M" |print(factorial(5))  # Output: 120! B* }5 i! D) c) Z( C
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    How Recursion Works9 u' i8 m/ D- L  I

    # u+ i' g; j1 R/ X* k+ ]    The function keeps calling itself with smaller inputs until it reaches the base case.
    5 }/ Y; e* S% y/ `4 s
    ' m5 ?$ n4 J! q# m$ G4 h    Once the base case is reached, the function starts returning values back up the call stack.
    : q. V2 J# c0 A+ ]% t3 R
    3 V! x) ~# d) I5 F2 t$ {4 h( e- t    These returned values are combined to produce the final result.! j* Y3 b/ H2 P4 ], ~

    + A; \$ ~7 u3 t3 q* BFor factorial(5):
    7 b. Q0 x3 C* |8 F. `! v: Q9 s" q( R9 Q
    - }5 O' f' c" D$ M3 V3 l; V1 |9 v9 t, Q
    factorial(5) = 5 * factorial(4)
      |2 ?7 l3 w6 G  A. v: r( kfactorial(4) = 4 * factorial(3)) y! |: G4 w. I2 ]
    factorial(3) = 3 * factorial(2)
    3 A% ^# S" [  S9 O  b% U0 lfactorial(2) = 2 * factorial(1)8 x: G+ k8 b) f7 ?! P
    factorial(1) = 1 * factorial(0)* p' R3 s1 W! C6 U% @8 w+ B& ]# o
    factorial(0) = 1  # Base case8 B2 E7 H! J) B: w; ?

    " ^( r# n- V2 l: l" c) tThen, the results are combined:
    - n: |' f0 r, e* q% ?' }: t  d. ?) u9 @2 L( U1 k

    , ?- b, d# S3 ?2 P5 \$ |factorial(1) = 1 * 1 = 15 z" `: Z& N% `+ k
    factorial(2) = 2 * 1 = 2
    7 X6 v) `; b1 d: |factorial(3) = 3 * 2 = 6
    / k4 N3 R8 x# F' h& _9 A' Z. _factorial(4) = 4 * 6 = 24
    ' e% L5 P+ i8 ^, V/ \factorial(5) = 5 * 24 = 120
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    Advantages of Recursion. }% u/ [: t) K7 O
    2 s8 ]) `+ c0 C+ Y
        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).
    * [& |$ X7 y3 t3 Y- l4 `. D) x
    : \% z! ]  D4 B' O$ e7 l    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion9 M9 L+ f6 T/ {; w: K
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        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.6 M7 ^/ F- D3 o. p7 c2 y; l7 g
    : o; E8 @& M  m* u% y7 v* j, j0 D! |, Y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 s. H& Q# O* ?1 x9 n

    9 r& W& l' B8 W8 OWhen to Use Recursion; n, @/ G; @8 y0 Y% `1 A. i
    2 j; ^& c4 K/ F
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).- E( ]* [6 J4 a8 t: j

    0 U  o! ^% l. u+ D    Problems with a clear base case and recursive case./ B' f7 M# e0 _% s* v) g5 A
    # E! s4 t3 L/ c
    Example: Fibonacci Sequence1 D& X6 o: D7 M& i4 _1 \0 ~! `
    , L* p. O3 Q+ S5 d3 \
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:; ~* ]6 J( D5 F& D/ q' q

    6 I+ m; W& M! C) d9 B$ D* j    Base case: fib(0) = 0, fib(1) = 1
    6 D+ |% T0 J' k3 x( J6 b
    5 E# s  y& k$ _% Z# l" W    Recursive case: fib(n) = fib(n-1) + fib(n-2)5 V* K" b# l' y  B; @4 ^# n: A
    5 b  i# u4 x+ n8 e
    python
    6 r2 g* |5 H) P! g  w" r" j- Q2 a$ q& v/ m; j" G2 R( v
    / G( n7 L; M' f. k
    def fibonacci(n):
    # S9 u- Y& q7 F# w" U    # Base cases
    # ^5 w& \4 ~% W1 p    if n == 0:
    / H, X- n' ^  k- D2 f        return 0
    / h  e+ Q& [. T& d. Y  ~$ H6 b    elif n == 1:
    / W; f& @  S4 X. d) z8 \9 z6 k! V        return 1$ D9 p% M. y0 w! h8 p
        # Recursive case9 U6 o" D. A" A/ \( d
        else:
    8 E+ @, F- ?; P. u" `- h/ Y        return fibonacci(n - 1) + fibonacci(n - 2); v: a+ F# }$ z

      y6 r: w: j; E8 U" Q- R3 z8 {6 b# Example usage
    6 u; O0 j) {, yprint(fibonacci(6))  # Output: 8# ]) }+ O. A% e$ \# o1 D4 `
    , N( ]( M, |1 R- H0 l
    Tail Recursion
    % B, l% N- M  b5 w) `- o7 |4 A5 ]6 v" i5 t  B% g, ]3 x9 _
    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).9 o6 U) m% a$ O; Y; Y% S9 E) T3 c
      m( P$ G6 {$ w& }; O
    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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