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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
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    4 K: R2 ]1 e$ F解释的不错& [( I1 Z6 f* H" V* ~, N, |& x6 p4 \

    2 ~+ ?5 o2 J4 e& ?; S+ l  Q( y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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    ) c( l( s* B; E) E6 j 关键要素7 ]3 i$ `7 i8 b% ?, E3 W' e
    1. **基线条件(Base Case)**/ C* T* Q! l; x9 k$ R7 p
       - 递归终止的条件,防止无限循环# C# S6 D3 s9 Q( ^5 y6 b- i7 v
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 14 v( J# I; W/ J
    & D4 H' T: u% M( W- c2 ]/ l
    2. **递归条件(Recursive Case)**" c- D. {! C5 a- Q/ k  o0 k- ^& ?* z
       - 将原问题分解为更小的子问题, R7 t# {' z7 E7 `& p" H
       - 例如:n! = n × (n-1)!. }. F: @7 ]$ B. k$ f* m2 B
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    经典示例:计算阶乘5 B6 E/ m+ x9 k. \
    python$ b. M2 j, q8 M1 W$ Z% t$ g
    def factorial(n):. R# w; n; |# x- _3 z3 Q/ A
        if n == 0:        # 基线条件0 a; {& c/ g- J$ i2 [
            return 1. B% p, n- s$ m. N7 ~
        else:             # 递归条件- o, `9 W, _. C  p% ?! q  I
            return n * factorial(n-1)0 [3 r* @: k3 T7 ^- E' m8 O  J
    执行过程(以计算 3! 为例):
    8 L" a5 I: }* b! \: ifactorial(3)6 z4 X# I3 h8 e/ F
    3 * factorial(2)9 ?# K7 x( g5 Y4 r" {6 n
    3 * (2 * factorial(1))3 K7 \# g2 b7 j5 z1 W+ C
    3 * (2 * (1 * factorial(0)))5 l# s9 c; \  C, D/ d  z7 S3 N
    3 * (2 * (1 * 1)) = 6
    ( e& v- ]4 V* F6 u" J; q- K
    / G3 J+ S7 K" J3 t 递归思维要点
    5 z6 M3 w0 k; L5 I& \) }! o1. **信任递归**:假设子问题已经解决,专注当前层逻辑
      L% P( c& ]% T9 Z  o  I2. **栈结构**:每次调用都会创建新的栈帧(内存空间). K6 h2 K& B8 R/ P6 S$ {
    3. **递推过程**:不断向下分解问题(递)8 h3 g$ Y: b: K
    4. **回溯过程**:组合子问题结果返回(归)7 l! ]7 k- U& G- f, ~

    0 Q/ [: ]7 {: o+ o 注意事项! m( y- I4 x: v0 h4 Z3 B3 v
    必须要有终止条件! E# ]4 D/ s2 a5 F8 V
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ! h% n0 L6 b+ O! C" M9 Z' |% w某些问题用递归更直观(如树遍历),但效率可能不如迭代* j' k* v9 T1 N2 ]4 T5 |% ?# k
    尾递归优化可以提升效率(但Python不支持)
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    5 g- g+ q6 K1 U4 w. c9 Z* } 递归 vs 迭代: p' X. C3 N8 \4 p, t3 ~
    |          | 递归                          | 迭代               |
    : [/ x3 {( E* ~  J7 @3 a$ Y  x|----------|-----------------------------|------------------|5 o( Z! D. ~! _$ ^
    | 实现方式    | 函数自调用                        | 循环结构            |
    - ~; V, a* [4 H9 T2 ~- a# H| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# {7 s9 w( e8 f# f9 z  N
    | 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         |0 S1 T. H7 L+ b* Y! m% c+ N+ [1 j
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |6 h+ u$ p8 a7 J+ I  T

    . l4 }8 g0 d) Z) c8 k  E. W 经典递归应用场景1 I# O( C3 o8 d2 q+ ?( Q* I) p
    1. 文件系统遍历(目录树结构)
    $ ~; b# L' y) c3 X2 C$ m; w2. 快速排序/归并排序算法% e" B$ c& {! f0 [+ N4 v6 o
    3. 汉诺塔问题# }1 l2 |- N$ `( Q  B  a* ]8 P
    4. 二叉树遍历(前序/中序/后序)
    ' s* D0 U* q+ a+ I  |: X5. 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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    慵懒
    昨天 09:43
  • 签到天数: 3167 天

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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,
    6 `  J! |! _$ B; v5 p% v( t我推理机的核心算法应该是二叉树遍历的变种。
      I! I' r7 _$ R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:
    % W7 g1 D- d  V1 R, J; u# |Key Idea of Recursion
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    ! t' y# [8 W* f+ r# V$ n4 c6 u$ e! BA recursive function solves a problem by:
    / r) K* Y% x) G
    2 i! f: w* w9 P6 t    Breaking the problem into smaller instances of the same problem.5 @5 o' J4 N5 e) {9 ~$ L
    5 k9 O3 [0 i1 a+ H* S& k
        Solving the smallest instance directly (base case).
    ( ]+ }7 I4 {$ m0 S6 i
    % }  J# O" }) M    Combining the results of smaller instances to solve the larger problem.: }# E, N. M) u- L1 P) Z) ~3 F

    / c" @% N) j; f1 iComponents of a Recursive Function
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    ) U. O' _4 r9 H    Base Case:" X! l- h$ O9 a) U8 R  M
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    * `, q9 P7 t9 f2 C0 n! T5 `0 q
    " U8 c. P+ r7 t' ], v6 P! n        It acts as the stopping condition to prevent infinite recursion.) Z1 Q3 c6 g- q$ k7 l0 C! p! }8 L

    8 H9 T# X! R7 l$ {        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.' r, d! R$ F4 }

    * r  l4 G2 t+ Z6 |: C) p. |    Recursive Case:; Y) U% d  ]0 y. s0 i# Y

    1 s- ^0 o- I; y& l        This is where the function calls itself with a smaller or simpler version of the problem.
    , l+ ]. j1 V3 a9 b1 ]0 S$ S. N
    6 T( m- U- ?* X' q) I! p5 D        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    8 I% K5 J- E0 s1 X; t4 IExample: Factorial Calculation; q! r) I% r1 `' |8 }
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    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:
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        Base case: 0! = 12 _7 F) M' H' V# J3 H
    1 }+ `) K! q9 n7 B. O6 \- X) Y- N- k* \/ i
        Recursive case: n! = n * (n-1)!
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    Here’s how it looks in code (Python):7 Y" g& i$ S& P; R0 ^+ O
    python
    , o8 t+ w5 E# f+ _5 y5 d$ i# l& `' o+ B- C2 t+ s% i! Q& G

    6 e$ @/ y# z5 H* Bdef factorial(n):
    6 ]4 e9 O" {: A, Q3 R( m    # Base case. v. Z3 R3 u1 ?: T& Z, G
        if n == 0:) P, @; n) S* l8 q2 E' C- T' f) P' Y
            return 1+ |4 A: j' M1 j; q7 p$ H
        # Recursive case: h0 E$ I+ W/ o  v' }4 N
        else:
    0 ]# m2 v5 w' N* {/ `/ E; R        return n * factorial(n - 1)& [" ?! `" ^; ?& c! e0 C! [; ^

    7 I+ I) l: ?# Y0 L5 v6 y: H: A& K# Example usage& B' O1 q% Q6 r& D) }/ Y. W
    print(factorial(5))  # Output: 120
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    2 S9 l7 x6 u6 U5 ?How Recursion Works
    ! _& D# Q% o  D& s) c4 W" ~" o8 ^. j* x/ o  a
        The function keeps calling itself with smaller inputs until it reaches the base case.
    + J+ Z2 u: T( h, n) p4 p7 R+ p& U. v
    + ?; g7 q" [$ V- y' @: G    Once the base case is reached, the function starts returning values back up the call stack.
    5 O! E0 H3 F$ `
      U) w. I5 k" {, S! X    These returned values are combined to produce the final result./ g9 U' {- H9 \) e' }/ {7 M- a
      d# p& c- p1 b& P- n2 g
    For factorial(5):- ~6 U6 c5 H$ t3 C

    , L* r  J! u& O# S6 T4 h9 d  H4 `6 ]; P& E+ O/ F
    factorial(5) = 5 * factorial(4)
    ( F  O* {+ g0 }5 f9 S  m' }factorial(4) = 4 * factorial(3)( h8 k, e2 S) n3 D* b/ O/ R
    factorial(3) = 3 * factorial(2)/ |/ g$ d4 n& O9 L8 _
    factorial(2) = 2 * factorial(1)
    ! T3 W; {6 D6 Hfactorial(1) = 1 * factorial(0)
    " N$ w( O8 ]6 l/ ?( x: r8 r, ffactorial(0) = 1  # Base case
    ) Q9 x) v$ ^7 c& t$ Q
    / M+ J8 g+ n, j4 Y( CThen, the results are combined:9 K1 |1 V$ c. V* ^' x" g
    6 P8 E  b0 r, f5 A, q* h

    " @+ p: v9 f+ D7 R) v" c$ E! r8 Qfactorial(1) = 1 * 1 = 10 C6 f. G1 o) s! c0 ?9 c1 |
    factorial(2) = 2 * 1 = 2
      O& b9 o0 j; f1 @/ |4 ufactorial(3) = 3 * 2 = 63 A" X4 t; P1 k; n, ~" [2 F( S
    factorial(4) = 4 * 6 = 242 U2 V3 o1 B, G1 C7 Y$ M* y% o
    factorial(5) = 5 * 24 = 120
    : `* [- I* F: _
    1 B2 K  m- C' K1 cAdvantages of Recursion
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    ' m& w% h, e* P* L" d  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).' z" Q. ~% C+ ^' T
    9 ?2 P% B' {! o& y
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    - G" h( v: m6 a& _' O4 B: b+ Y) E. r8 M0 k# H0 v8 g
    Disadvantages of Recursion
    2 {- ~6 a' Z5 ?1 X
    ( W+ h1 K9 s7 s3 h  m8 x0 p    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.
    3 [8 Q& s, }- |5 Y0 H4 J7 A: r* q" r
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)./ `8 @8 F% |" j( l
    $ C" `# f4 U, Z) d' S4 j/ i
    When to Use Recursion3 @. H0 Q. l# x: u  S* [
    : V" I2 Z7 W- L5 @
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ) m, u7 B  X1 H" c2 Q! j
    7 j8 s* U" |5 H& y- e2 G    Problems with a clear base case and recursive case.8 u$ a: Y6 O- {6 i$ k; P, Y3 L: k
    $ n0 [0 i4 b9 c3 G: X0 i: h$ f* w
    Example: Fibonacci Sequence
    5 O" ~# g) j# j) B! R* m
    ) f6 B7 [, c4 `The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:  i, f4 q2 ?0 R2 T% ]1 d" X

    * ^6 A/ ~2 [, Q! C6 g; f, {# }    Base case: fib(0) = 0, fib(1) = 1
    5 t6 f# k, [9 O9 B+ v! O+ I, R
        Recursive case: fib(n) = fib(n-1) + fib(n-2)' v! O( D% t3 C+ R
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    python! m; X" `, Q6 X9 D# ^

    9 ~; u% p9 \: G" j1 {$ j; V+ P6 T% B
    def fibonacci(n):
    , S+ g; J3 c$ E    # Base cases
    ; p$ R' ?# `* y% U* G  X    if n == 0:0 @4 B, Z* _9 m' E" d
            return 0
    7 ^) j+ H2 e6 A& X    elif n == 1:
    ( [: V8 q- b4 L  c% Q        return 1
    . x/ C9 a2 R' [8 ^    # Recursive case/ A% J; @/ ?7 _/ p0 R
        else:6 e. z. n3 |! s, B
            return fibonacci(n - 1) + fibonacci(n - 2)( k/ x; z. N  C0 G( h/ |

    $ O, N& S+ H$ |# Example usage* Z+ p5 k5 ]& M. D$ x
    print(fibonacci(6))  # Output: 88 I9 X% L. o/ _* {
    , D, {; [- _# C$ a
    Tail Recursion2 D! F, O5 p3 ?" {; v
    ) z0 w* U& b8 T( j" w1 {. D9 M
    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).! @4 p. r9 O8 N) q  q
    + @( w& M  o% L/ w! W' f/ L# ^
    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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