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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
     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 , D- U2 H: o3 s" f8 Q
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    解释的不错
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    5 n, O! ~) `& S) a4 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。* ~0 f! {% P+ s) l' V; q* \
    ( ~3 ?. N$ v& s$ R2 k2 V& d
    关键要素
    ) t5 s8 C* [' s- w1. **基线条件(Base Case)**  G4 c$ A4 \+ K7 K! f4 B
       - 递归终止的条件,防止无限循环) J  b1 D2 {# @9 V5 V4 k4 U
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    : q% t  @! T" t2 v! G8 h* X# f* t/ x3 O9 z/ W: p/ X: p% K" @7 m
    2. **递归条件(Recursive Case)**; z( J! Z* F, O: A
       - 将原问题分解为更小的子问题5 B0 m0 z% t8 b
       - 例如:n! = n × (n-1)!
    # N! X3 f" t. [% r- `' @$ ^$ m, J+ j& S  f$ {( E  Y
    经典示例:计算阶乘3 I+ |9 r' B3 \: _# v
    python
    3 x" m+ S, q6 Rdef factorial(n):8 ]8 d& Q# q5 {) O$ u$ H
        if n == 0:        # 基线条件3 y" X( P! F: n9 n' l) a  K
            return 1
    * b; d2 A; A! z0 X6 O    else:             # 递归条件
    9 F$ s6 F6 r5 E! N5 ^  V! H        return n * factorial(n-1)% @7 x  ~: v! |6 m7 r. y, Y6 Y
    执行过程(以计算 3! 为例):
    & q$ k. w% I# ^) I( Qfactorial(3). @" x& u& p! ]% r6 T8 M/ d
    3 * factorial(2)
    ! L( B, \& i# z& Z4 n3 * (2 * factorial(1))
    5 _- W. P8 W" F$ c( c6 ^3 * (2 * (1 * factorial(0)))
    8 [) _. X( U, U+ C! b' \7 f# M3 * (2 * (1 * 1)) = 6
    : s, ?% u+ E* U$ Z0 x' X2 o. x
    ' k7 n! S; r7 C. |. s* C 递归思维要点
      _8 U9 o% x( _  L. G( _1. **信任递归**:假设子问题已经解决,专注当前层逻辑5 u0 p, ~1 h* N% d
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    4 d  o1 e: M2 [' l( M3. **递推过程**:不断向下分解问题(递)
    " `/ a8 K2 r2 S' T4. **回溯过程**:组合子问题结果返回(归)2 N4 B$ L1 H2 h) \

    * W) E! J6 t5 _$ W 注意事项1 p5 g) m1 T4 W5 w/ x
    必须要有终止条件0 o( i' F* ?! @7 ~, K
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)# q! k! `) h& G2 |
    某些问题用递归更直观(如树遍历),但效率可能不如迭代- t6 }. W; u( A4 s8 j
    尾递归优化可以提升效率(但Python不支持)
    # Y- H) l" A4 |& a
    / a2 F6 }, i- y3 F+ ?- F# C, d 递归 vs 迭代+ Y! q( J9 h4 `
    |          | 递归                          | 迭代               |
    . j3 l# n# H9 ^|----------|-----------------------------|------------------|" y2 S1 v" |1 c6 E& C4 m6 ^( Q6 a
    | 实现方式    | 函数自调用                        | 循环结构            |7 W' @  F+ @+ m$ ]
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |/ r$ [' w% Y3 c, w
    | 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         |
    ! [& B0 Y% u& M| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |% l- [- w$ P: Z# v
    ! V; ?6 j+ S2 H0 r/ T4 \9 E
    经典递归应用场景! B7 F8 m1 e4 s5 k& Z3 g
    1. 文件系统遍历(目录树结构)$ o. i1 N2 [/ d% a' X
    2. 快速排序/归并排序算法
    ! G: V0 C  L" F2 h8 ?. A3. 汉诺塔问题
    . P1 Z7 z& k7 ]) {' e; S9 N4. 二叉树遍历(前序/中序/后序); N, _9 |9 r  ~* K4 P
    5. 生成所有可能的组合(回溯算法)
    9 {/ A: [- ^- |4 c7 ^9 G0 P) Z- n  G0 @' f! B7 ]: i: u
    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,  |& g% V+ e8 {8 |( i) H/ @  b
    我推理机的核心算法应该是二叉树遍历的变种。( _% V, R+ k4 q* L* g# I
    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:& B2 O( k# K) j
    Key Idea of Recursion* Z7 M6 z  ~% Y& M: t8 `

    " `* X/ H2 x3 I5 S6 @9 Z5 R" TA recursive function solves a problem by:4 V( r/ o6 R2 L! X  D
    ) [, ]+ v5 r7 ~/ q
        Breaking the problem into smaller instances of the same problem.
    5 U/ ?- G* h5 j0 e$ N! H$ M- d
    - c& h; @% t& v/ u' d' U    Solving the smallest instance directly (base case).3 K! l" W, f* b, J7 T! g1 q/ E: O

    , n: p4 z% E  A. t    Combining the results of smaller instances to solve the larger problem.
    / X1 N" B+ U8 [& O4 Q
    " o6 V* `4 p; S! f. @: o# Q9 iComponents of a Recursive Function
    / j$ M9 Q1 n. L
    & L- L* _6 `6 Y# {# j' C    Base Case:5 \7 S9 _6 B* Y7 {: y
    7 Q6 M% U. M- R3 ?
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.  K1 ~$ Q6 l2 n3 s/ S) A/ U+ V

    : v1 h8 }- s( q6 n        It acts as the stopping condition to prevent infinite recursion.8 _' x% O  V7 |# H* `
    & Y( A+ ~9 y0 _& _3 J& t
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    4 V* G( }" o$ E+ [7 u& g1 M: E1 q' F" S! W
        Recursive Case:1 K* K- N5 Z- r! I2 u

    & B& u! c2 }- j6 L; t) H' ^5 f' t8 Z        This is where the function calls itself with a smaller or simpler version of the problem.* w( E( L' n: w6 S2 i/ @
    - C7 \% _  {( j$ s7 I* Q# y
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ; |; k/ k; P4 `8 E( W. U7 `( y- N! E2 I; m/ t2 @& O; E% n
    Example: Factorial Calculation
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    ) {* q: W9 W7 k5 a! @- H0 yThe 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:
    + k7 p( p7 }) y
    % O- r3 W4 Y- u) _, P+ h5 Y8 X8 L: v    Base case: 0! = 1  B" p; v% {' q  a- e& u/ D# P
    ! ]2 k9 X( S$ N+ i% s
        Recursive case: n! = n * (n-1)!
    ' O4 u  G7 @# c9 V& d" l5 P- r) G  @: ?
    % ^* W' c3 U% k9 T7 y: x+ p; EHere’s how it looks in code (Python):
    % N6 n& l" b4 L, D5 O+ Fpython
    $ Y6 L+ y& w7 a1 U9 X5 j" _$ m
    ' M9 G. l8 ~3 Y" G- j& g4 D0 Z
    ; I( r0 c/ E7 b& ?; A7 R6 ldef factorial(n):" S! }8 z1 ^9 p7 e2 |) {. ~3 Q3 q7 A1 u
        # Base case, I- q7 i/ i9 R0 x
        if n == 0:! c- x8 G  X) ^5 d
            return 1' I4 E+ K. w* U& v1 J0 C# s
        # Recursive case# }' c: X  l* A2 O8 I2 ?
        else:
    ( h8 `% S- k8 q& e. Z/ l        return n * factorial(n - 1)7 o2 p' n- G% \/ o' z7 p

    ! J5 b& n- _2 s) y) [  P( w  k# Example usage* y# q3 q- |; D9 i0 a  W$ \* Q" y
    print(factorial(5))  # Output: 120- S: D+ B+ Q" K' o4 M: s

    1 T0 D: y- s' r$ N0 H; DHow Recursion Works
    0 d. N6 H$ `9 b/ n% I, d& B. o( ?/ a! {, W  B. H" n
        The function keeps calling itself with smaller inputs until it reaches the base case.
    - L) X. I3 a7 j- G, r, ^+ e! K% F( E; z& I7 y8 H- i9 M( v8 ~- W
        Once the base case is reached, the function starts returning values back up the call stack.
    ) p. g- a% k8 n$ U* X" k
    / L0 p; e/ m/ ^1 M, `2 Z6 Z% B" m    These returned values are combined to produce the final result.
    " l. j8 u0 A9 h. g0 U9 v! s% {& e  R# R& @  J1 E! ]
    For factorial(5):
    ' l$ I5 x7 k: y+ v- Z3 b
    ! L" l" ]+ U: J0 P# g0 W  w
    + h# m. _- j# s- @6 {! wfactorial(5) = 5 * factorial(4)- u6 M( O5 H8 q, n% q. m# F, L
    factorial(4) = 4 * factorial(3)
    4 w- T6 }5 o  e0 {3 m' K3 jfactorial(3) = 3 * factorial(2)
    8 Y8 C1 }% J% S7 k+ Wfactorial(2) = 2 * factorial(1)
    4 x* _& h3 b/ z+ `( _2 _7 j' @factorial(1) = 1 * factorial(0)
    5 t) U6 T8 {  o! {* u3 [1 g; vfactorial(0) = 1  # Base case1 s0 S7 }  d& U3 h3 D, W

    2 o7 c3 A( j1 n% S% [3 |Then, the results are combined:% e6 k$ h, u, _% B, |

      U1 A# K. y3 I* h
    6 Z! g; ]; V8 ]: b! V8 t2 m: x% |factorial(1) = 1 * 1 = 1
    8 w; a. ^1 r% j8 M7 Wfactorial(2) = 2 * 1 = 2
    5 I; e5 g: w  q7 z% w8 Zfactorial(3) = 3 * 2 = 6
    : o6 z$ Z0 b& d9 d- k4 wfactorial(4) = 4 * 6 = 243 ?# Q* t# P' Q3 \% c. v2 j& y
    factorial(5) = 5 * 24 = 1208 c% ~# Q  M  H0 D
    5 x$ Y1 \+ N! f2 K, H4 l& p
    Advantages of Recursion6 N4 V6 F- N, K& K
    ( q/ @* B2 }& p( U+ {6 R
        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).( u& B/ c( V3 {$ W& j
    ; |2 J8 V: X. m' K
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    7 G7 G' G0 @8 V$ C) g) O! {' x2 }4 ^% K2 v& h+ b
    Disadvantages of Recursion
    0 i2 J7 c4 j1 L" v0 B- d
    9 w4 B, {+ i' G4 g" 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.! r2 P: `) e) v0 {

    : m2 N5 ?, i& X& z! X    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).+ H9 T, C& C4 D6 ?0 U6 R5 {
    " A9 ]+ G) D/ [& k9 X6 N2 u6 T
    When to Use Recursion. g3 G: c0 v. B$ h) r
    & h8 U& ?: n2 {( t
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    " I2 E4 f, {2 L2 u+ b. N$ [, m7 ^* s' p9 o0 v4 f% t, W2 a: K9 v
        Problems with a clear base case and recursive case.* ?2 w) g' L) V' y8 m

    1 G2 F8 f  L" WExample: Fibonacci Sequence
    + |# Y" L8 f! e% r- [1 J! N" g# u  d/ ?5 n- S( Q5 I' z/ O  p: u$ \
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:: q! O* m0 g0 z) I% I
    ) y5 m0 |7 v7 ^1 a
        Base case: fib(0) = 0, fib(1) = 12 N2 N9 D3 H' s; X
    8 _- L7 E- o# X2 o. F
        Recursive case: fib(n) = fib(n-1) + fib(n-2)% n1 l8 N* D8 O) L! ^
    * `7 _2 W' m( g3 V1 y
    python! i5 t9 h/ A4 r, [4 [
    1 b% G5 h: w) A- z! Q

    % s3 R& T, K+ {$ H$ a4 ]. ]def fibonacci(n):
    * ~4 V- R$ N6 l- u2 b8 U    # Base cases
    % {( A& y4 B" A- j$ {4 V- g    if n == 0:
    ) ^1 x0 _5 P# E0 h# j) n/ H        return 09 M0 P. x# y# x$ J  Y& E
        elif n == 1:
    $ a) L4 o& @: F$ c/ W& b5 U0 H% t        return 1
    # w  k! Z# y5 w( A) g    # Recursive case
    9 @# w  T9 f- t* B; D" U    else:5 Q8 G5 M+ G8 X6 j  r6 A6 y
            return fibonacci(n - 1) + fibonacci(n - 2)' A: G7 m0 o5 _8 S

    , d) w- b% I( n; z# Example usage# Q8 }! i0 k) j" _" T- t* a0 k7 ~
    print(fibonacci(6))  # Output: 8
    ) d5 e: i( h; n+ h5 d* r6 M0 e4 n% w0 @
    Tail Recursion
    : M! C# ]" O3 w) @" S/ P2 T( @/ W
    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).
    # F& L  v" j. h
    1 b( W) J% c1 b6 X8 G% g# G' lIn 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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