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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
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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
    7 h; ]1 n. v- z
    1 V3 A( Q/ ~; y9 H( ?; L解释的不错
    $ u+ i. A$ o* s: v6 l% U6 _9 }7 x+ @/ B% j5 [' b1 r) u. E
    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。0 ~) n# O  x1 b6 d! i$ G' u

    3 q( \3 z4 o0 j 关键要素
    8 i) W; E# b! e1. **基线条件(Base Case)**
    2 @6 V4 H$ a9 J   - 递归终止的条件,防止无限循环8 y+ e3 i0 x9 a( Z
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      V) ~' e# c: [  f$ ?/ c7 O
    7 Y2 s, d" {: {6 Q, H/ ]2. **递归条件(Recursive Case)**7 e6 G0 k) H0 K) c
       - 将原问题分解为更小的子问题
    0 F: G3 n$ T+ L* p+ V+ p9 E   - 例如:n! = n × (n-1)!, g5 T  u2 B- a6 _9 y) W! Z; q

    3 S9 J) f! |$ ^9 R' x) I 经典示例:计算阶乘
    ( w6 z+ s# V9 C( [7 e1 [1 v5 ipython
    : J2 g: x6 J9 Zdef factorial(n):+ \- H4 r4 ^8 J+ y0 O
        if n == 0:        # 基线条件
    ! H8 I) O3 t0 }: J  q8 f/ j        return 1
    * \+ N! _, D5 s3 F! H; w! X0 W    else:             # 递归条件
    . `0 Z6 [! s  }9 l: n        return n * factorial(n-1)& P0 ?- R1 Y  u0 K# `* w
    执行过程(以计算 3! 为例):: Z" U2 @2 d1 k6 l
    factorial(3)) M/ y# P, L! m7 p! w# }. ~
    3 * factorial(2)# B, I* m5 l" A1 d, ?0 c/ A  I) w
    3 * (2 * factorial(1))7 u4 [6 ^, U1 u$ ^5 E; ~. v
    3 * (2 * (1 * factorial(0))), b4 f- w& R3 E: D
    3 * (2 * (1 * 1)) = 6
    5 y; y( |! }7 K- z/ r' l) W* ?+ r. Z1 j
    递归思维要点
    ) O9 `4 |9 {; q1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    9 S3 r9 V$ o- d3 T& F1 q2. **栈结构**:每次调用都会创建新的栈帧(内存空间)7 `# {7 T+ o$ m" p2 p9 F" T
    3. **递推过程**:不断向下分解问题(递)
    + e6 N5 D& S+ u9 d8 N* L* b" M3 ^4. **回溯过程**:组合子问题结果返回(归)
    . V0 f/ E* j, j7 A0 n# n, i' o9 A  I7 c; D' F) k
    注意事项
    , ^  h* E# A! t* ]必须要有终止条件0 B. t! p, ^0 |3 L9 i  B' m
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    8 o7 g* Y9 v, v/ X& d某些问题用递归更直观(如树遍历),但效率可能不如迭代1 ]# R* n3 y9 Q
    尾递归优化可以提升效率(但Python不支持)( v; ^% Y- f. m( k5 [6 k# b7 {4 K

    / F3 t. w# D2 c" i2 Q 递归 vs 迭代
    9 |' l( {0 S7 a- l# i/ ]' W2 {|          | 递归                          | 迭代               |- z4 m( E9 C! n) x
    |----------|-----------------------------|------------------|
    ! a5 W. \8 |8 v- J) x| 实现方式    | 函数自调用                        | 循环结构            |8 ~3 w& C3 H4 a5 T' t2 ]
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    4 X' a2 O" A* R1 p  r: z; L| 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         |
    9 H& G- |& o. H# ?( T8 Q| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    2 J% z( k* s! w7 }+ c7 }8 [& |2 V3 Q( f' T1 q0 i4 d0 j
    经典递归应用场景
    0 U- e4 @. x) j) d) |3 D1. 文件系统遍历(目录树结构)
    % y* b6 |; c: g' d2. 快速排序/归并排序算法1 C9 p6 P8 D- P, t
    3. 汉诺塔问题2 I" d" t  U% H+ a5 u+ k0 W
    4. 二叉树遍历(前序/中序/后序)
    5 W( P! N; s  q/ ?7 z: o4 S5. 生成所有可能的组合(回溯算法)8 t0 Z, X8 {1 x; y7 U6 z0 O2 Q( N
    : J7 ]; F" [' \( K) Z1 \% B) H( s
    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,
    % o8 M! r  E4 B  l; G, D& D: T) R我推理机的核心算法应该是二叉树遍历的变种。' a4 B$ d4 R/ I* @1 Q
    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:; |' c( N2 _! U
    Key Idea of Recursion& O# |9 j% c: ~. a, y% M
    * j6 X  @+ R$ U4 ]% |* D
    A recursive function solves a problem by:
    ! ^; V2 u/ V# S8 v8 ]: v' |! ^* n4 W; B* c; F0 a
        Breaking the problem into smaller instances of the same problem.
    $ C4 |/ o8 [! C' Y! o/ F
    & Y& N4 t* E- j8 [- y$ ^    Solving the smallest instance directly (base case).) W+ V/ A0 h8 p8 T, [4 R

    ) z& o' q1 o  }7 N2 v% v$ U' L    Combining the results of smaller instances to solve the larger problem.
    ! H2 D1 M" n+ k" L' h" w
    9 n& r7 Q" L( v/ nComponents of a Recursive Function4 J2 z2 _4 O) w! F
    - E" d6 _5 T$ B  z* B% a) X
        Base Case:4 b7 F& {8 j1 q; B' l! n

    3 j6 U4 p) F+ U( C5 O5 e( Z7 Q        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    8 G3 E$ q- ?* z% |5 u: h. Y2 n4 G8 j: D$ \8 m  r) |; L
            It acts as the stopping condition to prevent infinite recursion.
    ( {4 V! e8 ^# c6 W$ o% z$ K9 D+ s
    % i7 ^) ~1 d# ?- L/ U        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    9 ^/ v$ a, ^1 P. |# N+ E0 k
    ( A: V# p- r* U    Recursive Case:/ C% N* y3 U3 F

    5 Z2 j+ S9 q) s( u. t# N: q* l        This is where the function calls itself with a smaller or simpler version of the problem.
    8 w; u" W* H8 Q7 u; R. x7 U. ?4 r7 d0 I
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    4 ~" x4 l* n: |, w! v- E; m: ^* I# Q. E
    Example: Factorial Calculation8 g/ L7 i2 T# \

    + Y, u! \6 q, M6 F- 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:
    5 Z6 w9 l& z0 r9 V4 [/ y1 a/ T7 c; g5 G8 m. N( ^
        Base case: 0! = 1( N% O- k0 f; ^

      {, |9 w7 b# J% M) v/ J  J$ \9 t    Recursive case: n! = n * (n-1)!
    0 {1 j! y- _" s$ f9 w
    4 {6 c, [/ H9 w6 |- ~- _7 JHere’s how it looks in code (Python):
    / n/ @$ n& c. B$ hpython
    6 A+ ~- c: y* I, v+ _
    0 C, Q# B; N( S
    9 F6 [# A- B) C8 pdef factorial(n):
    & f# B: M! D3 t- C0 g    # Base case
    ) ^. _9 f8 C6 u  {$ c4 m    if n == 0:5 U$ s' T7 Z* S. o. W' u' k: O' v
            return 14 i+ G: f" u3 P  a6 J+ s; R
        # Recursive case
    3 {; y7 Y* a+ u    else:
    0 e* f: c4 F( P        return n * factorial(n - 1)2 x0 ^: r; Q4 }% P) H

    # i  \; q% Y: _/ w2 y: e8 r. W# Example usage
    + w& ]4 b+ [. c9 H$ S3 eprint(factorial(5))  # Output: 120  f- W9 c9 a" s9 u/ P$ |

    0 E( a6 c3 Z6 s3 L$ n+ FHow Recursion Works0 H' E/ s: i( o

    ! y% ^' n* F4 d/ ]% {4 g    The function keeps calling itself with smaller inputs until it reaches the base case.
    8 E1 q5 H' r' Y$ G# n- L  i4 m& x+ ^# s* n
        Once the base case is reached, the function starts returning values back up the call stack.
    4 A, m1 a! h8 [% S* y! v( X- e
    / V1 E+ A# T7 M    These returned values are combined to produce the final result.: O! v8 J! ]& x7 y7 L
    $ T: U2 b( Y; g0 j! u4 G. L# f) \
    For factorial(5):
    3 N" [9 u2 H; Y2 t  O
    ( O4 N+ e& i& T. @! g2 Y- E
    6 G  N$ v8 K0 P& c; v0 A# @) |factorial(5) = 5 * factorial(4)7 L+ z7 E" R- ?4 h' ]1 p: K# P
    factorial(4) = 4 * factorial(3)
    6 i$ s, \) h- yfactorial(3) = 3 * factorial(2)3 N6 c1 L3 l! R' s4 v! l" e
    factorial(2) = 2 * factorial(1)
      a6 A8 }1 u/ K" I  afactorial(1) = 1 * factorial(0)
    ( `1 a/ ~' z) J- A, z  K" bfactorial(0) = 1  # Base case# ?  W3 b: A7 I9 l; n" d
    - C! ]7 `% w/ b0 @3 m, F
    Then, the results are combined:
    9 @; s+ _0 E1 W, E0 ^% b
    & z' V2 p3 T' x  t) Z+ w! b4 \
    8 V- ^, J& n. p- O. N6 _' d4 G- Jfactorial(1) = 1 * 1 = 1
    4 n6 k. ~3 E) e' M0 @3 n' j' lfactorial(2) = 2 * 1 = 28 U2 x5 Q# w, K7 P  I: s
    factorial(3) = 3 * 2 = 6
    " y) w' u9 z' i, ^; T0 }factorial(4) = 4 * 6 = 240 Z( X" R6 _/ c0 F! ^' P$ _7 n
    factorial(5) = 5 * 24 = 120
    , |# Q8 \) C2 T' B* E& q4 \- T, U# b4 q5 V7 X# I
    Advantages of Recursion
    2 @; o" e; ^8 h6 |& [- Q1 d& A+ e, o
    . G, i' N6 c$ m4 D8 N+ C! M: _    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).; c6 J: Z$ s6 e: \
    : ]" f* T' Z1 n% N: Y4 p7 r
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    9 R  G* T- X- r* ?
    ) ]5 ?; N# J4 m  C/ Z) [, sDisadvantages of Recursion
    $ t) E/ M1 x/ L3 Z- r$ E8 J3 B( M& i' Y7 ?$ A
        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.$ f+ o( o' V7 r1 [' b! v: n4 f
    . V7 [& r) N1 {
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
      I" d" C& X) c. Z1 t3 l# z8 j0 @3 D. Q% x% ?* x
    When to Use Recursion
    8 v. f& ^4 j7 l  {/ S) N8 p0 m) F* i* f, i
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)., q! ]3 R- q0 P+ c5 k

    - Y) c2 y9 s' e4 C$ x, Y2 p    Problems with a clear base case and recursive case., h0 m, f# s! u
    , p2 w2 A. X* k
    Example: Fibonacci Sequence
    . }# b) E/ f5 `/ g% {
    7 Z& o0 k/ e- R* u; M7 TThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:% S$ v' p& c5 t  v/ J$ {: l

    # ]  \" {2 l! e1 p7 c    Base case: fib(0) = 0, fib(1) = 1
    5 w+ B* f5 I& {
    8 Y8 }  O% M) b* D' x5 ]    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    5 h" B. L' }, `) F& N/ R% q
    4 z0 v; `+ r5 q; \python1 P  Y8 b8 g( f1 t4 h8 I
    $ |7 u& Y6 m4 Q/ [* g, s5 Y* u

    , O2 I3 K  g& a3 Z# C8 y0 Hdef fibonacci(n):
    9 i& R5 t/ P2 d: s    # Base cases" e* k2 W: Q% V# f) U
        if n == 0:% p4 H/ B, Y7 T# z
            return 08 x" k' \* W  v% G. k
        elif n == 1:# Q% ~! o; Z4 t, R% v0 n7 N5 R
            return 13 y# F5 V# X: h8 n1 f) T
        # Recursive case
    / G: L9 f" ^; E0 t* d* V& }3 B    else:
    4 f0 o9 E. n% A; t; n        return fibonacci(n - 1) + fibonacci(n - 2)6 {. G9 o4 b0 \
    4 `' _$ n: ^9 R' {
    # Example usage
    $ a! w9 e2 c) q: \( v  Pprint(fibonacci(6))  # Output: 8
    * S, o% m9 k  i, P+ d
    1 T: B# a) O* D  n( qTail Recursion
    9 e1 l! W& r0 }8 p
    / l; y5 L0 j/ V. PTail 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).
    0 B, C  C2 x& Q# x% ?2 K
    $ e2 t- ?$ [7 ]) _, V: g/ SIn 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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