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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
    2 D' V. C/ Z) I, V+ }; |( e) u; i, [& d- |' j! D
    解释的不错
    3 Y& |) n9 m3 v1 _. J/ u) f& g( K& L
    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。. G- _) y9 n5 N+ G/ U

    : x0 b& d6 U. e  o+ S/ P5 s 关键要素
      U: s5 q6 I) \% l1. **基线条件(Base Case)**4 k5 M& Q  W% R
       - 递归终止的条件,防止无限循环
    : a( l/ @$ X9 |/ }   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    - }% E5 f. U+ y* {4 P/ T$ \( ^0 U. Q0 ?$ t" k8 y
    2. **递归条件(Recursive Case)**
    3 E; @) t  [" i! R: X3 O   - 将原问题分解为更小的子问题( l: s, p) {. `
       - 例如:n! = n × (n-1)!7 L1 @9 u2 F) U+ z5 `" s& o

    . ]9 S/ o4 I. g. f2 B2 I 经典示例:计算阶乘
    # d% ?) i# _+ o( P+ U- Tpython- L/ ^. b8 M2 k% w5 N/ y7 c& Y
    def factorial(n):/ r- c+ z5 |* [. O% V) W1 i& G
        if n == 0:        # 基线条件
    & O/ S. _- r- b. R        return 1
    ! k: x9 ^5 w% h" \, z* g    else:             # 递归条件
    6 q. p+ r/ q- |. ?& j2 v& p        return n * factorial(n-1)
    * x4 ^0 A6 E* Q/ {8 t: i2 ]/ y执行过程(以计算 3! 为例):4 i( _+ M  |4 R  ^9 r$ _  k
    factorial(3)
    2 Q# t  Y$ u# v, a3 * factorial(2)
    3 x3 B+ f2 t4 u( P6 _3 * (2 * factorial(1))
    4 h$ U! p" P5 E9 n# u$ \1 O3 * (2 * (1 * factorial(0)))) C: G: H. o5 G7 i$ m2 n
    3 * (2 * (1 * 1)) = 6
    , J6 k: H) `9 c) O/ j$ Z' r3 ]
    $ g+ }& y0 @6 q: b. s, C( x* ? 递归思维要点
    9 D9 [6 ]5 o7 Y! J8 G" O1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    5 h7 s" I7 g, [6 y* y" d* a9 N" m2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    0 M* e" L; i6 J1 [, B7 r3. **递推过程**:不断向下分解问题(递)( v5 p7 J9 x. w$ |& O3 G
    4. **回溯过程**:组合子问题结果返回(归): v9 T7 x2 I) Z: O1 B. E! k1 u7 D/ E

    . `7 s7 e/ B2 u6 U5 J9 D 注意事项$ B2 E7 ]0 H, h! [0 z8 W8 z
    必须要有终止条件1 \* E' H! S4 Q5 X: ?
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ a2 }( q2 o/ Z0 T
    某些问题用递归更直观(如树遍历),但效率可能不如迭代8 U# ~% U, D# {7 B
    尾递归优化可以提升效率(但Python不支持)
    # s0 [5 p, n% ?* c2 l4 U% M8 Y- ^" Q1 ]4 b: n
    递归 vs 迭代/ ]3 ?/ E  T% ]" ^% }$ V
    |          | 递归                          | 迭代               |$ |1 }9 o* j1 ~& R, T9 m% _/ |
    |----------|-----------------------------|------------------|. }5 o3 S- l6 D8 _* }7 U+ |  G& s
    | 实现方式    | 函数自调用                        | 循环结构            |+ r' N# N, m$ R
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ( t1 _4 c. W  c# z# Y| 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         |
    : B( h+ v/ ^8 V0 y* a( u| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
      @9 O5 f& K0 M# _& s
    . \# @8 Z: p; R 经典递归应用场景
    . w1 Q! G0 G  |  w& E" i1. 文件系统遍历(目录树结构)
    5 _' `+ h! ]4 v' _" ]- d; ]. A2. 快速排序/归并排序算法1 v) h) K* P: g' z: e7 ~4 {
    3. 汉诺塔问题
    4 ^0 F- R; l4 W' e4. 二叉树遍历(前序/中序/后序)# @. c! o/ |0 W  w
    5. 生成所有可能的组合(回溯算法)
      {: s. ~$ v0 p" E: A6 Z- j: ?7 F+ `# k1 |
    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,
    + ?+ @3 i) E% H) k3 P; A+ J  ^7 I, E我推理机的核心算法应该是二叉树遍历的变种。: s$ }" M9 u7 l- q9 Z; a+ F
    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:
    2 {% J( Z* K* p$ N# _Key Idea of Recursion. i4 ^! N6 x" c: p. q
    2 ^" T7 q( H: O% m& D
    A recursive function solves a problem by:( j6 J$ l+ u  U' x

    8 d+ j' h7 t- P7 P) {1 V2 [+ Y+ g2 e    Breaking the problem into smaller instances of the same problem.. H9 X" `9 [- ?0 Q3 W7 [
    0 [1 k* T2 _& ^$ j; o8 t& Z
        Solving the smallest instance directly (base case).
    # D& [4 G6 h5 K2 [. q3 d
    3 l7 f  ]/ \' v    Combining the results of smaller instances to solve the larger problem., \4 O; @+ Q: y7 m1 N" v$ n5 U

    1 D" |& m7 P9 J. r( dComponents of a Recursive Function
    8 i: }7 k: f* J1 V. T' C, G. ]5 X1 U4 T# @/ u
        Base Case:/ F5 ~2 R( n- P! X
    ( [. z& H2 |! l3 T
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.( K3 j' S5 f- q7 Y3 z6 x  b

    , G# E  t" d! m7 }% e        It acts as the stopping condition to prevent infinite recursion.
    ( W6 P6 E8 ^6 b' _" f# {" Z% t6 k! V
    ; d- h# M9 B& w: o; s        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.9 `% P' Z4 z: S3 v  E

    * H, [! A; ^1 V! v    Recursive Case:
    . u7 q* Y$ y2 [( m; C
    8 I; O0 _: n4 R9 Q( H3 m        This is where the function calls itself with a smaller or simpler version of the problem." E1 S: w  M5 y

    ; z6 {  L: R9 C) H  t7 e4 \- H! Z  h- F        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    & }: u  |. G: P% q$ k" o/ [3 N: M2 C5 P# b. b: N/ x
    Example: Factorial Calculation
    + D9 H# F9 J7 D5 c# P. Q+ J& r: z' C0 K$ X- x3 S3 V$ i5 ^. L
    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:
    " V+ r  Z- i( e& s! k8 A: x5 ?  p& h
        Base case: 0! = 1. i2 S' X% {, z
    + N' D  \# m% ~6 l: Y6 v' u; I
        Recursive case: n! = n * (n-1)!+ [/ s9 r$ P1 K* T/ i* o; I
    + q- m5 L7 l5 n- y8 J
    Here’s how it looks in code (Python):6 e- W  z0 {3 T( \0 f: `
    python# M: H1 D' D. ^$ A

    ) d* @4 l; H% _* y, D& O4 `8 x$ }: b' S: q6 b& z( D0 R
    def factorial(n):2 f9 ]2 k6 j/ J4 f. S  \1 R
        # Base case
    - k# b7 C7 Q5 w# ]# S, u& k5 n    if n == 0:
    & `6 D" x3 ^' |% }        return 1
    & D5 e0 U6 m9 X1 w& q2 D3 I    # Recursive case
    " h2 I0 a$ M8 \$ a$ y% \* l    else:, w8 T5 ?9 c* O" U: U+ N2 L" i2 G
            return n * factorial(n - 1)
    ( @& w& @( b; V3 ^$ z
    . N1 Q4 C( L) V* ?4 K7 E# Example usage
    7 z: @9 I; D" B: B# Tprint(factorial(5))  # Output: 1204 r- D, S9 E" A' k5 p3 K

    0 i9 N% T& p* {0 c' Y% l7 zHow Recursion Works
    3 M7 x/ M! |  T* `  {9 X' x
    % a" x7 H% o( P+ _0 ]. o    The function keeps calling itself with smaller inputs until it reaches the base case.4 f- x! `' V! b2 ~

    " a; L+ K1 Z& ?( }    Once the base case is reached, the function starts returning values back up the call stack.* o( C/ p; O- ]1 [2 y" h0 E6 z

    4 g  I0 F: O8 d3 J! x2 g  Q    These returned values are combined to produce the final result.
    / e2 C2 E( G( G; I- ?$ I- {
    * e5 V( N* ~' M2 R7 m- ^. lFor factorial(5):; e3 I) t. {4 B0 l; @

    5 ]# v" N; R: Z8 d
    8 _6 V" _0 b9 F& d- rfactorial(5) = 5 * factorial(4)* e+ o* n5 H- N9 D
    factorial(4) = 4 * factorial(3)
    " p& Q% h. W6 \: A1 p8 m& lfactorial(3) = 3 * factorial(2)
    # e5 `; V. o2 w1 c* O0 L0 V+ Afactorial(2) = 2 * factorial(1): x7 g6 r3 }" V$ d$ r
    factorial(1) = 1 * factorial(0)& a2 u2 X/ m  V. O/ u( b
    factorial(0) = 1  # Base case
    - m4 F9 b% r: T2 C) g( [; M" X, S1 m. E4 d4 ^8 I/ d6 Y& \
    Then, the results are combined:, K1 ]" l* M  w1 o

    6 f1 w( Z7 f" M* n" d2 w3 U3 \/ X: _; s
    8 ^0 Q; a" G& L3 tfactorial(1) = 1 * 1 = 13 t" X! W2 ?1 h2 B7 F' K- R
    factorial(2) = 2 * 1 = 2" ~) @. E$ S! f$ m
    factorial(3) = 3 * 2 = 61 m0 z% N/ Y! [% v, C" S* J
    factorial(4) = 4 * 6 = 249 L  z; B( x9 |7 q9 X  Q
    factorial(5) = 5 * 24 = 120
    7 s, r! I( q1 p8 s/ I( T5 s7 ]' b
    Advantages of Recursion1 X7 r( O8 R! y0 l; N0 k
    ) t& C1 D0 f8 r, b& P* Q4 {
        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).5 Q! j( Y" K) t% Y6 T) v

    , ]9 m- Z" ?* r  Y1 @2 s    Readability: Recursive code can be more readable and concise compared to iterative solutions.+ `4 O( x- \  s6 f" J* ^

    5 {: ^; D$ l$ W. ^- M! uDisadvantages of Recursion! ^8 z5 P2 h; B' o

    2 g; I$ {& `# }* P1 l4 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.
    ! R% Y$ L  W; I- U
    " d; |0 X  a0 B9 C    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    / z5 T6 P- b4 y: M, W4 f6 {6 E. i1 b' y
    When to Use Recursion
    % ^# X3 E- u" d: W% I/ B; M2 k8 t8 N
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# s9 P* `; ?7 H
    0 J4 L" c, _( x& W. K$ M# w& e2 c
        Problems with a clear base case and recursive case.
    8 T! ~1 p4 g) U! ?; K9 x# a
    6 d. U, v/ T1 k+ q+ {Example: Fibonacci Sequence  [9 u# S( L3 r' A

    7 u2 o' D# N3 xThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 {4 r  x  r, E7 Z

    9 H: t' N/ N; Y( ~. t6 e    Base case: fib(0) = 0, fib(1) = 1+ ^3 I7 x0 {: b' z' [/ T
    8 v5 N% w6 F* t* I4 }( ?
        Recursive case: fib(n) = fib(n-1) + fib(n-2)7 y, d, U; V( b9 b& m- @$ t- X

    ) \6 l8 b% y' ^python
    3 ^; x5 P4 A# P! }  R1 b1 J. C7 X; m' l" P
    3 X1 ~1 s% |1 A! u
    def fibonacci(n):  h# u1 x3 k. v$ ^5 V3 V, r3 d
        # Base cases  E5 ^8 K# U0 y. d4 L& \
        if n == 0:
    0 f; I, z8 P* ?& T        return 0
    " O6 s( X6 R, y0 [! F, Q    elif n == 1:1 ]/ v4 C) |% Z# c+ y; j# k
            return 1
    0 q7 |$ ^( a1 d    # Recursive case5 N; m3 K* H0 Y
        else:/ A# s7 Q( s2 d: B# `) Q9 v( Q
            return fibonacci(n - 1) + fibonacci(n - 2)0 q/ Y# L! c" N2 I" P) G. Z
    ; X6 p: a* S" t# T: v' E
    # Example usage
    8 E/ B* F2 g9 @$ v" `; b8 ^print(fibonacci(6))  # Output: 8
    4 v$ K9 S. x) B- v6 y; d
    ( `4 c, S$ e7 S7 }6 |Tail Recursion. F: _6 v1 U1 h. G
    1 E; C4 v+ O0 Y$ s( s& X* Z% i
    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).5 M0 o4 B& g' H* Y" P9 O5 h

    , _! d) B: a: H/ T7 k% P. g  oIn 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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