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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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    3 t3 S! V# x. x解释的不错3 o) ?8 w7 C, d. Q& k, ~& }( K

    # `) w& s; z* V' t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。
    4 [# I, P' j4 J
    ; b/ }0 h: o: ]1 q 关键要素
    . o% q! |) g, y$ b) o1. **基线条件(Base Case)*** H4 p& S7 x8 t) F
       - 递归终止的条件,防止无限循环
    4 j& @. }' i5 [8 \1 |   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    7 Q! x6 d9 M- q* e+ r
    ' A% `8 S3 C) @( _2. **递归条件(Recursive Case)**# |! y; A. ~% Y$ \% F# @; ]) E
       - 将原问题分解为更小的子问题4 r! r. w! X: H" n0 E
       - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘5 }! i% T/ B' _% D5 y. p
    python; S9 j1 R" T  V
    def factorial(n):
    " _  e& J6 y( h7 m) O- C    if n == 0:        # 基线条件
    ) _7 v* j; G; H% ^        return 1
    8 O9 H$ J4 X+ _1 G4 G    else:             # 递归条件  y# d& B6 C" k) o! ^  x
            return n * factorial(n-1)/ ^, t: W  _; P6 }! @! a6 X" g  `
    执行过程(以计算 3! 为例):, S) ?& M, D8 ]4 ]9 P. U. p5 [
    factorial(3)' N% ~3 s& w; m9 @( i8 h
    3 * factorial(2)% a! R5 B( k6 ?& W8 S
    3 * (2 * factorial(1)); \; x: V( Y, M$ K1 N$ u
    3 * (2 * (1 * factorial(0))); Q% z. j: X) f6 a5 \9 D2 {' t
    3 * (2 * (1 * 1)) = 6
    0 a9 K; G, i8 {) Q5 `+ N6 t4 M
    ! V2 Z1 M, j) y1 L/ y$ o 递归思维要点2 ^' f) k9 n; D* A- t& [
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    $ q7 D" d6 D% p' s2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    1 d, E( J+ L; `) y3 Y8 y3. **递推过程**:不断向下分解问题(递)3 c9 |& t# v- b1 O3 @8 _# v, b
    4. **回溯过程**:组合子问题结果返回(归)$ }7 P* J7 z* x1 K5 a% g3 u0 m
    % s* t) O2 }# V# I: d
    注意事项4 `7 ], \/ K- L2 {  _8 X
    必须要有终止条件; C6 q! f2 z" W1 ~! |( X% k
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)- \: ?$ E! h) Z
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    0 M! q( w( D: |& X+ J/ s尾递归优化可以提升效率(但Python不支持)
    7 Z) {" p. O" ?3 l1 R' Q5 h3 O( G9 L  Y0 C1 N2 [
    递归 vs 迭代# ~$ ^( B+ n. K" a$ l
    |          | 递归                          | 迭代               |
    : {0 _; _6 K: v0 R|----------|-----------------------------|------------------|' D: n9 _! u& G3 W# K; o) k
    | 实现方式    | 函数自调用                        | 循环结构            |
    . L, ^. H9 J" _2 o| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & h6 l; e% j! 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         |' }" @+ i: O% ^5 r' a/ X1 I
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |, k+ K' }% I  C& s6 G" M: `# n+ C' ^7 Q

    & @, U  Z" U9 }' Y8 ^# u3 h 经典递归应用场景
    # K! S. _+ c5 z' \# g1. 文件系统遍历(目录树结构)4 e7 v8 X* E* c. e0 S- m) m
    2. 快速排序/归并排序算法) {6 ]6 q8 _' \
    3. 汉诺塔问题
    7 }4 e1 l- H8 V8 I$ U; \# C( F4. 二叉树遍历(前序/中序/后序)
    # l" [- \# W/ h  L* I, `/ B5. 生成所有可能的组合(回溯算法)
    & n2 ]4 r- ]8 _8 f
    , D! F7 Y+ `, w. M7 A5 m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,
    7 ]7 g, u2 V/ G. U我推理机的核心算法应该是二叉树遍历的变种。
    5 `/ Z! ~& s, _# Z% O. f& w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:& {/ l  \* V- _- ^- }
    Key Idea of Recursion
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    A recursive function solves a problem by:
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    ' n" ~: v* X7 K' f3 O) d& m/ W/ f    Breaking the problem into smaller instances of the same problem." R4 `% W" d; f/ {5 X+ R  O

    # Q- e) |5 u1 M" X    Solving the smallest instance directly (base case).
    7 Y& ^) x; m* Z. \, ?0 Z1 Y6 {7 {4 Y1 s0 r2 o
        Combining the results of smaller instances to solve the larger problem.' e/ k+ n6 ^/ h' r( g! R$ ^. X+ w
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    Components of a Recursive Function, o' I" R2 x8 U$ l0 J' d
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        Base Case:
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    + q% [! h' b2 @' I& u2 _        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.5 V' U0 e0 i; x* g

    9 m' b+ G- r7 W5 [; g        It acts as the stopping condition to prevent infinite recursion.4 I2 `/ u! R! J$ {+ l4 t& V. }
    ; U: y' B6 x; C  e/ [5 o. \3 P
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    $ b( w7 Z9 n* [  T    Recursive Case:
    ! ]1 v/ z2 s6 t8 f& s
    7 @0 e! H6 e7 k! R        This is where the function calls itself with a smaller or simpler version of the problem.6 l0 h  _. x8 g* W
    ; p( T) {* B. n2 g: M) X9 {- u
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    0 V9 D- n+ a0 |' a6 pExample: Factorial Calculation, A* e5 [+ P$ t& Y$ u7 n
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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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    7 i, U- O& g) @$ n/ Q    Base case: 0! = 1: x/ D& B$ c; n6 O
      F6 o! I% _6 u9 A) u
        Recursive case: n! = n * (n-1)!
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    ( L, W+ ^( o, aHere’s how it looks in code (Python):
    : j$ O  J1 d& D/ p" x6 npython) n7 }# X- t0 I4 g/ y0 G
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    0 E3 u  X/ @  ?+ k/ C; {* x
    def factorial(n):
    4 a. a, U/ S0 j8 y0 Z5 C    # Base case
    4 ?) e6 I% I5 g  x. w    if n == 0:
    - E. q* E2 |4 |& f        return 1
    " }8 a' C# w: G8 X    # Recursive case
    $ A8 R/ F/ O6 H7 M! z% B) S* ?    else:& O3 @3 {, T/ [) c
            return n * factorial(n - 1)
    : _. W3 \6 S) o; g. v1 o# h6 n- g' X+ b6 Y: f
    # Example usage
      p2 q5 F9 B; U, W8 W/ mprint(factorial(5))  # Output: 1207 ]+ n1 B2 ~$ V. c

    * ^$ @; L) ]# j! r. EHow Recursion Works8 T; q6 i% Y" `9 p

      o; N& D. C- ?6 P3 k    The function keeps calling itself with smaller inputs until it reaches the base case.. U- ^/ m2 a& z! H  C8 Q
    7 ^) i+ i4 H+ M+ Y6 U. Y
        Once the base case is reached, the function starts returning values back up the call stack.7 N% H4 |; M' F, v7 H- K4 [9 V! z

    $ Q" c! P2 g+ J    These returned values are combined to produce the final result.
    $ W; @" @9 b4 D$ X
    ( c4 j0 V& c* z' ^" x1 c% k9 L0 OFor factorial(5):$ a7 p- p; |5 Z: b* i) w+ m

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    factorial(5) = 5 * factorial(4)4 ?4 a/ H+ {3 ~. Q7 f3 p
    factorial(4) = 4 * factorial(3)
    ; C/ _% o0 \( p: S. x( x" a; ?) s1 Tfactorial(3) = 3 * factorial(2)
    1 m+ K9 j1 R4 _( t' k( `factorial(2) = 2 * factorial(1)
    . ^( A, @( x7 r5 j$ s/ vfactorial(1) = 1 * factorial(0)
    ; L, |# i; Z: J; tfactorial(0) = 1  # Base case
    9 J2 Y" N8 a+ N5 j5 i1 i; Q" C! ^) A8 J0 y  [
    Then, the results are combined:& H! V  @2 _  A; ^8 S: O
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    + |) Y) ?& \0 i; N
    factorial(1) = 1 * 1 = 1
    . r4 {9 c' n1 F; Gfactorial(2) = 2 * 1 = 2& Q  ?6 X9 L% g2 s- x$ [5 m+ Y
    factorial(3) = 3 * 2 = 6# ]" u0 i- i+ O0 q
    factorial(4) = 4 * 6 = 24, u5 I* @' B( |% B9 \
    factorial(5) = 5 * 24 = 120
    3 }& {) {7 y" w. h; b) o# C
    : o9 B2 H% k+ ^9 r0 ], n: AAdvantages of Recursion  X2 r" @, i  c" i& y$ v9 ]! P

    ( `6 u& E3 s5 F1 i" Q1 L- Y/ Z    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).
    - g9 J2 R7 k( D3 O0 g( g3 s& A% }) R$ b) P/ s
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ' m; O. I, R5 I) w* z2 o. ]
    1 _, `+ q' Q' ^: y% i" ^9 m6 @Disadvantages of Recursion3 }5 p! x$ Q! D+ ?2 u8 f
      x$ _4 p( h' ?! C- N7 `2 ?5 Y& h7 s
        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.
    * j9 m" ]2 g5 U, _$ X- U" g: F1 q3 Q0 t
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ) j2 J2 {# G3 W8 i) G
    5 R  b; b8 c  C+ D( p0 \( _When to Use Recursion
    2 J- D  ^/ c- f$ f/ x$ K" J, Z" _% I
      {* q  C. F' [, w2 A9 b1 L    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    : m3 K" [! Z/ X" F
    ! f  t) F. R! X" ?/ y! ~; y( G8 X# U1 W1 @    Problems with a clear base case and recursive case.7 j; }4 j7 Y+ A& K0 j

    8 L* [( U0 t5 Q- `Example: Fibonacci Sequence
    ! t* T2 u) x3 V( j; T/ w, U; b- H8 E8 U1 [' T, `8 {: g
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    , V7 t" [* p/ Q& S/ n6 _) x  Y
    / ?  L. y& `' h* ]1 i6 j    Base case: fib(0) = 0, fib(1) = 1
    + M$ i! O  H' x# M) V1 s5 d8 V6 @, x3 b+ a$ H( E1 X
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    1 G: F! i; c/ \% n1 j* E* ?python
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    6 ^# F0 ~* b% [6 y9 G! Y6 f7 S2 a+ r7 ^* u, M6 p% g" K
    def fibonacci(n):: r' Q9 g% U+ R1 y' a/ [
        # Base cases
    , _: p7 x8 Z- D" ]6 ~) _- u+ E$ Y    if n == 0:% O. Z9 y% [( V7 N1 K8 m* O% v" x* H
            return 0
    0 g" T5 Q% C& }; ^2 h    elif n == 1:& Q, r& z. b& F( v% K
            return 1
    6 f8 l1 h# d6 R9 G  `8 F1 H: Z7 G    # Recursive case
    + r) {  H" K6 A  E: T    else:
    ; r. `4 Y9 L& E7 f3 x3 w        return fibonacci(n - 1) + fibonacci(n - 2)
    7 p! [' C# D% V' b" O0 ^: x9 Q$ C. K  u) t+ C" N
    # Example usage
    & g; m+ f* L. j% z6 U' Jprint(fibonacci(6))  # Output: 8
    : X; k0 s0 p8 O0 H! y  U+ `/ b0 f4 ?) y
    Tail Recursion
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    5 l* g5 z  K: \; h" G/ e. }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)." i" \4 C7 r4 ?1 O9 L7 o2 ^

    " I4 K6 f  O  B& RIn 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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