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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 # V7 H- }, |; u& {/ F% t- ~! m
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    解释的不错" F* E$ ]5 g7 y  C" I

    ( j& ?2 `3 A+ w, |7 w8 h% T* M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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    关键要素1 ~. `! X( O- H
    1. **基线条件(Base Case)**  ]0 v; F) X/ e9 K5 h4 K9 \8 s
       - 递归终止的条件,防止无限循环
    1 q$ e- f9 `: r& ?% ^% K! w& @   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1. q% D) Z; ~& n3 K

    0 \0 t/ t: ^; \" J) |, {7 P2. **递归条件(Recursive Case)**
    , E0 P2 s* ?5 q4 z   - 将原问题分解为更小的子问题/ o, l/ h4 y1 p- ~  V. f: E
       - 例如:n! = n × (n-1)!
    ' A6 x4 m$ p# R( t! p  @; A) X. O. N( H8 |- c* P% D0 v3 `3 Q
    经典示例:计算阶乘- N7 P4 \; a$ R! A9 I, f9 M( |! D
    python; N5 A: {% p& `  u# t
    def factorial(n):8 s( g9 E, E9 ~3 k! G
        if n == 0:        # 基线条件
    * C9 M# o8 L8 f) w        return 13 J  }+ l% |5 ]% U8 x" v
        else:             # 递归条件
    2 A' I* g9 Q1 U, j        return n * factorial(n-1)' ^+ k* r6 }* ^# t$ ]! t3 u
    执行过程(以计算 3! 为例):9 ~3 D) P1 Q2 e6 E; _+ P3 a0 q
    factorial(3)
    . @% s9 M) s6 m5 W3 * factorial(2)% k$ W; C0 v; K! i- I# L
    3 * (2 * factorial(1))
    $ D0 c" N1 E7 h+ m1 j6 A3 * (2 * (1 * factorial(0)))5 F+ E- u- n, V2 ~
    3 * (2 * (1 * 1)) = 6
    # ~8 p* g( p1 i- j! B3 Q& g$ I1 ?( Y4 D
    ! S- x0 b3 t; U5 U 递归思维要点
    # c! F9 \# n2 `( Q) l1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    " r# _. |1 W% v2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 _" \0 e: Q) X/ Z; J- Z4 X8 ^2 h6 a
    3. **递推过程**:不断向下分解问题(递)5 i2 a% ^0 L/ u/ f
    4. **回溯过程**:组合子问题结果返回(归)$ C9 H9 E% U4 n$ r( [, k2 H
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    注意事项+ n! E" K8 g/ @. b+ S
    必须要有终止条件
    ! F. P" D: E6 A' L# N递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    + W8 S' ~! F/ j8 ]$ Q' x某些问题用递归更直观(如树遍历),但效率可能不如迭代1 T8 I0 f. _) A
    尾递归优化可以提升效率(但Python不支持). g' H+ k6 B% Z- ^$ L( ]# j% Y/ x

    1 z- n4 L! v  ~& q: @" ], Y 递归 vs 迭代2 A" ]9 x6 [9 M0 T
    |          | 递归                          | 迭代               |  Q" I- N% n8 A# W
    |----------|-----------------------------|------------------|
    ; j- _/ r. N" Z" c| 实现方式    | 函数自调用                        | 循环结构            |. f1 _3 X7 r, b; B) ^) X
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |- e& r$ g2 J" c& e) u) x
    | 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         |
    4 n, g6 C5 |6 J| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    2 f7 w. |% F( R" D/ h  _7 g  D& _! X9 M2 c/ E. V8 y. N9 f' c/ b, M
    经典递归应用场景# A9 V7 A7 O0 F" e: U; Q
    1. 文件系统遍历(目录树结构); _7 H) I; I3 V% N
    2. 快速排序/归并排序算法9 Y: S1 J$ F3 {5 E
    3. 汉诺塔问题
    7 ]* u7 W6 H! X7 k# V# i2 V; E4. 二叉树遍历(前序/中序/后序)$ [6 e. U, P: w9 S, _6 Y% r
    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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    2 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,& G, v. @' r6 K$ D* N
    我推理机的核心算法应该是二叉树遍历的变种。0 Y" D! Q% O6 B+ `( [4 B
    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:/ c0 h$ Y! H' p; z6 c) x* k$ n' W
    Key Idea of Recursion  M4 |! R) L/ Q# F* p

    % N1 g9 A( R, `. v/ j6 \0 SA recursive function solves a problem by:+ Y+ p3 o8 c) N% n) I
    9 n2 ~+ O* z5 Z
        Breaking the problem into smaller instances of the same problem.8 h5 i2 I9 ~% o8 B9 e$ J( w( W; Q
      ]8 B+ r# _0 n& X* }, O! B9 v
        Solving the smallest instance directly (base case).
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        Combining the results of smaller instances to solve the larger problem.
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    . ]1 b, j/ V" ]5 i4 U. o3 iComponents of a Recursive Function, N. A7 V2 r( L1 Q

    # m) g+ A! J5 i    Base Case:& b  N& \2 M( ?6 J: j  [

    : C: _& A- j, D! B        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    3 O$ q; N+ b6 I5 O* ]& P. R  L$ W% @$ w" [; a# x3 T
            It acts as the stopping condition to prevent infinite recursion.
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    2 K* d. Q; ^9 L- G) ~: e# R        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:$ \+ ~/ M- A2 t+ w/ A
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            This is where the function calls itself with a smaller or simpler version of the problem.
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    ( H8 R9 y7 u1 y5 C8 b2 U9 k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).; o/ P- C; B# }6 ~( C6 V; G

    2 D& R  g1 S# \- X+ }3 z- ?& R7 L# ?Example: Factorial Calculation
    & [& Y. p8 ~+ |: b
    ; y* J) X  G" U4 VThe 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:
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    " D% G4 c: ]5 Z    Base case: 0! = 1
    $ _7 W- L' X" J4 \. V( m- n& V5 |3 u3 Z% F1 Y  H
        Recursive case: n! = n * (n-1)!/ L2 d+ p, i9 p% f
    2 K- j2 \- o( R/ e7 D4 y8 u4 x1 J+ {
    Here’s how it looks in code (Python):3 y  ?6 S/ D9 W7 C) f) w
    python6 B% i7 R9 g9 S4 T  f2 U
    " a! [: `- d. \8 U% w
    0 G, R" L$ n" T" E" p" F3 O4 P6 M0 T9 q
    def factorial(n):
    ) T' a* m. @" t# N: s) r6 c    # Base case/ [$ V% a: O& C$ t
        if n == 0:/ `' P( n0 C$ {( B- K5 J1 f  H
            return 1  k9 Y5 j( W. l- d2 `9 _9 \7 \
        # Recursive case- H. U' T: u( {* ^& J1 j
        else:
      V/ U. l- J0 ^/ u/ H4 @        return n * factorial(n - 1)
    7 b, Z- p# ?! C# F
    1 f/ S; h6 X1 g2 W1 v# Example usage; u8 @; g9 s" N9 f/ _4 ?6 J
    print(factorial(5))  # Output: 1205 P0 i4 {& l( V9 z* {  U

    ) k* P) W3 Z! [" D3 Y; A# |& JHow Recursion Works
    3 x. w: X& I  T; n+ F4 V7 Z7 k
    ( W8 Y' {5 o5 X* W% N: j    The function keeps calling itself with smaller inputs until it reaches the base case.
    - d4 ?, g& z% {2 m
    1 E6 X  j* e1 a) ~; x    Once the base case is reached, the function starts returning values back up the call stack.
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        These returned values are combined to produce the final result.7 L( X* s* t* t, M9 K
    5 e( ^3 C8 m$ @/ _4 T8 e2 d
    For factorial(5):; n: b) H6 p2 i4 f, K. n
    , U& A  H& i5 z! ]# v

    9 l/ C3 q; @, G+ Cfactorial(5) = 5 * factorial(4)
    " Z7 K# C# Z0 n* W' N6 Ffactorial(4) = 4 * factorial(3)
    & G( t4 F. ^" m: T0 R: D" l* qfactorial(3) = 3 * factorial(2)
    6 E+ v1 f0 k4 p; e% yfactorial(2) = 2 * factorial(1)* g8 p  h* G" |. L' h7 n; m
    factorial(1) = 1 * factorial(0). ?9 d( U9 r% a: X& r
    factorial(0) = 1  # Base case
    : W# V1 U# f7 b. K6 E" C  D3 W0 t0 B, a8 \! X( `
    Then, the results are combined:
    . z, A: `  h  L5 H0 Q3 \+ h) Y7 y: y) `
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    factorial(1) = 1 * 1 = 1
    ( l; S/ \* T3 r3 m. _4 afactorial(2) = 2 * 1 = 2: a  N% E  Z( s. X1 }
    factorial(3) = 3 * 2 = 6
    ( C7 K8 h/ P/ F+ cfactorial(4) = 4 * 6 = 24
    . K; {1 h$ k7 U  M1 ]2 Mfactorial(5) = 5 * 24 = 120
    4 b7 V* ^8 }' K- ^4 P6 y& m$ L' ~0 \! ~" J8 ^0 p: R4 k
    Advantages of Recursion
    ) p; o4 k0 |$ N
    $ I: J" V1 O! _; w    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).( J( i& q" [& f, m# c6 a# l" s/ V
    7 s9 k! r7 O1 Z5 `% I/ ?
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion) _. j5 P; d1 r5 R5 H! D7 y9 A

    : C2 T( g4 r' R5 _6 R    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 j8 c; T0 z/ ]7 n: f
    5 `9 _2 N; M/ A5 W# s) h. U, v' a    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    + O' Q2 z% S; T! o0 @- {4 [, J  Q3 I4 U( P4 V9 c% @8 ]  h8 C' `
    When to Use Recursion
    * k/ f" j0 n8 e/ G) Q0 {
    ' K$ G% ]) S" O7 k4 c5 J9 ^' x+ T    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    * k3 r- \! X; s0 @* |' ?# A, i  h3 l# {% F3 z9 e2 _
        Problems with a clear base case and recursive case.
    5 C9 f% L7 g" H. I: X3 ~  m8 \) F% U0 K
    7 |- D. K" k. [  s; u3 y, Y8 BExample: Fibonacci Sequence8 `, ]3 e) \1 S  J% M3 Q5 m
    / F/ j  p- ~  `$ ?7 n5 N
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    5 [  m8 j5 J4 `  `& ^
    1 H1 \! }8 }1 s- Y6 V    Base case: fib(0) = 0, fib(1) = 1
    & {: b7 B- C! P! b% [, j% n9 m2 t/ l8 [) b1 h+ n9 E0 g* w
        Recursive case: fib(n) = fib(n-1) + fib(n-2)! x/ [1 c5 S/ v- q5 K; O( ]

    % ^; s: K; `% Y4 M& K1 p# M& ~python
    1 K' q7 O( \3 B3 r2 K6 L5 v& R% R% c& w1 S, \. ]" Z1 X$ L
    7 ~+ @# ^, D9 @) A
    def fibonacci(n):
    2 J6 s' |) ?$ U1 d8 x" j$ N7 B    # Base cases2 b5 _3 N" R: T/ S& a/ d
        if n == 0:( S3 I$ F+ ~* B2 z4 Q+ J1 T6 A: `7 l# `
            return 0- F7 e, g  V/ N: `) P/ S
        elif n == 1:- i1 d/ W& u8 E% [- L8 c: E5 Y' G
            return 1% Z, g5 V& w. D
        # Recursive case
    % K% P  \" a" `, l: b    else:
    & \) a$ P7 @* c+ F8 Z; r        return fibonacci(n - 1) + fibonacci(n - 2)
    0 t. |4 k  t& v1 F; Q: V
      m- g" g# \5 z3 d/ L7 {, _# s5 t# Example usage
    * ]9 G7 W- a, ]print(fibonacci(6))  # Output: 8" T. a1 ^- c7 ~" z3 o8 ?0 I) _

    1 @2 Z! N4 S: |8 h; ^Tail Recursion
    & m" G& K  X- m1 a! ^: L
    : A9 a" D( |9 j4 dTail 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).
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    0 ~+ H( \# o: h$ }5 N( AIn 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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