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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 ) F4 R8 z; I& K, _! S
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    解释的不错
    # d$ p/ G  A1 l: t  O7 o1 \" t% j, T% j6 j/ t. t( q
    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 D% u/ n+ i" \) y0 \
    # K8 z8 z7 ^0 I& i/ |9 g
    关键要素
      ]1 g0 k' P& I- k( D1. **基线条件(Base Case)**3 c. x. j4 x  V2 Z7 v; l4 \3 z
       - 递归终止的条件,防止无限循环+ _* G, i$ q$ I( _. h3 w1 S3 x& R$ T
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ; W- j' w3 G+ d; k9 F- S* G' I3 R3 ?, \3 |& y6 r8 l
    2. **递归条件(Recursive Case)**
    " ]3 T2 X3 R8 T2 O- c" k   - 将原问题分解为更小的子问题% w  w4 ^  t. U8 L
       - 例如:n! = n × (n-1)!& @  ]% a% y: @  F2 ?

    # `4 I! o. y6 p' L1 f5 K 经典示例:计算阶乘
    / b/ ?5 R& g: a  f1 r; Gpython
    8 D1 H# [% t' edef factorial(n):! r3 s. I1 V) T$ F% Z6 ^# o& ^3 w
        if n == 0:        # 基线条件
    : b8 O# e# H! O" d  v! d        return 1
    ) o1 P, _% V. N' h* _    else:             # 递归条件
    7 V) u' y$ {0 j  W, u" h        return n * factorial(n-1)$ ^. o& Z& h9 J8 U: W
    执行过程(以计算 3! 为例):
    * I) x1 z; n4 xfactorial(3)
    ; I, n7 x+ u+ u) W, P! l3 * factorial(2): D9 B; n* W$ r( X! O' ~& d  b
    3 * (2 * factorial(1))+ E, [3 }( G, [( s/ q9 X
    3 * (2 * (1 * factorial(0)))* v7 C+ G+ {, y8 g; p
    3 * (2 * (1 * 1)) = 6
    + ?. X( L4 M! x! z3 b
    2 M1 w# ^+ r9 l6 q: f 递归思维要点
    ; ~  \( j: h. l. h2 @  T1. **信任递归**:假设子问题已经解决,专注当前层逻辑2 c# a, A- j1 K9 ]) \, m
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)  D* Z: ~1 U3 N
    3. **递推过程**:不断向下分解问题(递)
    ( W: s  g' t$ J/ p/ [% l4. **回溯过程**:组合子问题结果返回(归)
    0 i4 K5 Y; N: M- M: z) ^3 m+ x7 ^2 N- L2 f4 q6 m# f* q
    注意事项
    0 Y3 c; V; k( X8 @% a必须要有终止条件
    * U$ `( L5 X' T: T& N; ~递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    8 D2 c# P; K  a. B. o" x% i  q某些问题用递归更直观(如树遍历),但效率可能不如迭代& w7 Y9 i* ~) V% z0 B
    尾递归优化可以提升效率(但Python不支持)
    ' |3 Y5 M# F( O3 }( v9 V! S- c3 D, s. u  G# g# w! T; W7 D
    递归 vs 迭代
    7 T/ B# @; k8 y. d2 N6 E. N, A0 @|          | 递归                          | 迭代               |
    9 w  j/ ^) w# G+ x' \|----------|-----------------------------|------------------|& J# K2 ]# m9 l7 d: U$ B. m2 o" X
    | 实现方式    | 函数自调用                        | 循环结构            |' z* T/ i% |0 _0 B
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
      w+ G) v* e/ k6 e| 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         |
    + O1 A0 B/ y' r. V; `( @| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |0 c- S  {6 X2 u+ _  j3 e! b
    ( R& r9 f  P6 c, F( y
    经典递归应用场景+ |) l) P3 J  t( \& K4 R
    1. 文件系统遍历(目录树结构)0 E1 q) J2 ~2 [. T2 F6 t7 U
    2. 快速排序/归并排序算法
    3 r1 O; R; i% R$ h3. 汉诺塔问题2 r$ ^" M* o2 g
    4. 二叉树遍历(前序/中序/后序)6 ^6 b8 m/ j5 b) J2 n9 O! x6 B9 `
    5. 生成所有可能的组合(回溯算法)
    2 X% x# f3 Q2 u: s8 @3 r4 i
      K' I. x7 s% D+ M, T" {, \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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    昨天 04:31
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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,
    " q6 r! d: f" j* \$ Y. e8 {我推理机的核心算法应该是二叉树遍历的变种。
    . ?( q; u8 w1 w1 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:
    / t9 R  q) @9 |$ Q  z3 y; ^Key Idea of Recursion
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    ( ~7 D7 }+ [& t/ m9 r2 BA recursive function solves a problem by:
    : G5 G, `+ K/ E/ Q: w' C
    ) D# [" `6 f. m' v    Breaking the problem into smaller instances of the same problem.8 S6 L, V& R% o7 O7 f
    % t% ~( k1 K& x! U+ Y9 P, P  j
        Solving the smallest instance directly (base case).
    * [+ |5 [) m7 s5 f5 T1 L0 Y4 K( h# t7 ^) ?7 B  B! a
        Combining the results of smaller instances to solve the larger problem.
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    " i! c3 D4 m+ y  m+ c- ~. vComponents of a Recursive Function6 [- o2 C. e2 P( R: J
    % h5 S6 }7 t3 y* ?- X
        Base Case:- K$ d' ~! W7 t$ L( w

    # I# {. j2 g6 C4 }# `* @        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.7 O. K/ C9 h7 F# ?1 \; e- K

    : a( w& g8 C( F        It acts as the stopping condition to prevent infinite recursion.
    2 T& N* e0 V" K* M7 W
      R4 p/ U- ~% h        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    5 E( J3 z, G" S* r: I  Q0 a, A8 B3 R  r# y8 K
        Recursive Case:
    9 h/ j8 |" C2 C( j0 q. Z4 L9 m$ {1 e+ ~. J0 c) I
            This is where the function calls itself with a smaller or simpler version of the problem.7 W- z$ |  q& {4 j$ }4 Q
    + w7 G! O8 y3 b* ^. S' ?
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 B! q% o% `# D& S: h' j  B1 R

    ) H: r. A- @( @* p3 X" {Example: Factorial Calculation
    / O) I( E4 |  Q+ Q. `- k. ]1 a- q7 M) y! g7 U6 f  i
    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:4 U3 h; b8 e( O: p# K/ N+ e3 ^
    . [- O& b8 \! i* B4 y
        Base case: 0! = 17 d8 Q/ l- Q' H, n& E" y

    * I( B8 e: |$ r! ?8 y6 j    Recursive case: n! = n * (n-1)!+ W2 u: @2 q4 _) a8 @! l4 s

    ; C* U8 ^7 p! F* O- X4 jHere’s how it looks in code (Python):4 U' L4 p5 }; p" f
    python
    2 f5 l7 ?6 j+ z/ U
    4 F0 T2 e3 `0 P9 q' f6 @
    + r+ q9 D5 z9 _6 A7 Tdef factorial(n):2 L6 l9 @# D5 U6 c
        # Base case$ k0 Y% e5 v) W$ \8 S& N. _
        if n == 0:. K% ^& ]- o( S4 J1 c2 g& `
            return 14 }0 ~6 @2 h3 i" m
        # Recursive case( q+ e: X1 x7 Q$ Q+ j
        else:
    . c: W1 z9 k; P3 C        return n * factorial(n - 1)
    * X$ q; `" W+ M, x" i+ w# l; \7 q; T
    # Example usage) i2 ~% @& S) C& H% k. t) I
    print(factorial(5))  # Output: 1204 i1 A2 Q0 R/ k% w/ \+ p9 m* F  M
    & [+ K) U% B" S' n+ Q# e( c
    How Recursion Works
    7 j& W$ ?- i& ]  z: ^2 A4 J
    9 R" V4 E* V! }' H    The function keeps calling itself with smaller inputs until it reaches the base case.# b1 R- V. o. b. a/ f* S! F! O

    5 m4 a3 ]" i( O  k    Once the base case is reached, the function starts returning values back up the call stack.
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    7 G6 @2 h6 Z/ w1 }    These returned values are combined to produce the final result.+ S  [2 [/ b, ?7 f9 e" m6 F1 f

    ; n& d$ F- K0 y; L2 QFor factorial(5):
    : ]. U- _2 |) @8 O) i* H# Q* U) D8 R5 {- T' _% `& p" E/ ?! e
    ! H4 D  u4 D- j1 ~8 ?2 i
    factorial(5) = 5 * factorial(4)
    # O( `7 j# d' N1 k: ~4 xfactorial(4) = 4 * factorial(3)
    : D' M- {6 ^: T3 w1 tfactorial(3) = 3 * factorial(2)
    0 M. Y* v4 p) |9 N4 efactorial(2) = 2 * factorial(1)/ P: o, M4 s7 |6 X$ L/ D& }
    factorial(1) = 1 * factorial(0)# u* [7 s, {' b  O/ }% Y; |% D
    factorial(0) = 1  # Base case
    ; Y0 X7 }" S, o7 B( T& k3 q) ~' ?3 Y+ e" {  O* f: u9 B8 ~5 }
    Then, the results are combined:4 r! P$ g/ j( @5 m# x

    0 w/ e7 g$ O6 ]/ l3 `
    . W+ @% [) v' U, I; ofactorial(1) = 1 * 1 = 19 f  Z0 G: s( f! N! W" o; {
    factorial(2) = 2 * 1 = 2* ]7 d" M) `/ C3 B) p
    factorial(3) = 3 * 2 = 6# F; e2 ~! s" W" ]$ A* I/ I
    factorial(4) = 4 * 6 = 24
    + ]5 U4 M# k% E+ z0 }4 ]% E6 Kfactorial(5) = 5 * 24 = 1206 P* ~* d4 f% V6 @9 |" S
    8 T8 ~! p0 v8 G' P1 C
    Advantages of Recursion
    . p& ]2 g, u+ o! D; f6 r+ B: }% u6 C6 K* `
        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).
    1 ~8 x7 [8 B. ?# ~  r3 k
    ; l7 S9 a% d) F' U9 H    Readability: Recursive code can be more readable and concise compared to iterative solutions., P/ h- _5 b% a& i3 x
    5 Y' K) W4 m$ z" \
    Disadvantages of Recursion
    . j# m% P9 w! Z' J) r7 J. y/ ?+ L
    * x3 o/ m4 c8 Y) G2 L1 p6 K    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.
    9 o, |+ }5 i- s# h! w) W, |" p/ q
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ; E/ X3 {" }/ |7 }, c
    - x4 s6 H% Q& E1 E2 `# L9 J: P$ SWhen to Use Recursion
    $ X" p/ F# b; ?# u- w6 b
    2 T+ I5 j% g% |3 g    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    # T: M0 W- Y" [+ e; S: b2 h, R9 J
    * L6 z8 H& I* _: C    Problems with a clear base case and recursive case.5 _0 K# n' H7 e$ u/ t
    ! y, u: R) ~# P/ L
    Example: Fibonacci Sequence
    0 W! A9 j3 I& e7 a2 y$ P  C# I8 L* Y) R
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:* O! @- I: S$ O/ P4 I* m
    $ p. d/ `9 }. y# b) k
        Base case: fib(0) = 0, fib(1) = 15 q1 y1 K* [8 t% S4 [0 Q$ b6 R$ E( I
      ~; c  {6 n: t+ W) Y  |/ ^
        Recursive case: fib(n) = fib(n-1) + fib(n-2)/ c+ o& c8 L- ^  x- a

    + k* w! a3 J9 cpython  A0 S* f  m4 T3 U3 c# q) x( x4 I5 B

    1 M/ U8 w0 _' j5 }
    & P3 _, h) p0 i8 Wdef fibonacci(n):
    # d  {0 I/ T7 x, U5 @6 z) x8 r) Q    # Base cases/ p9 M& @- Q* c) K; S) x
        if n == 0:
    5 L1 V( {$ x+ S  B3 q0 ^        return 0
    + W' M( p. t8 K    elif n == 1:; y2 E* \% o1 B7 M7 D
            return 1
    0 W: O2 |' f2 \3 {    # Recursive case
    3 Q, k3 }2 b% z+ D3 a5 g- _    else:
    4 Y. g4 r7 Q7 _        return fibonacci(n - 1) + fibonacci(n - 2)$ `& }5 a% |- ~9 b& o( N2 A

    / S) {8 r8 e2 L# Example usage
    * v; Y: d7 u6 M5 G/ G  `8 c& z# yprint(fibonacci(6))  # Output: 8
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    Tail Recursion
    2 v- g: e& h; [$ O( a! q
    ! W2 {) Z: S: Q4 W$ FTail 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).
    - M+ U2 n# V4 R% l# s$ }
      x* u6 ~9 i$ }( G) d0 Q0 iIn 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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