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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
    3 y. ]5 m  H/ s2 P, C* Z* k) y' y4 t: H# i, \: y' C4 ^
    解释的不错
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    8 L6 g8 {9 G  n  J2 g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。
    ) V' N0 A' D# B; ?; l0 a' L; ~
    $ j+ _( M" |& ?: Q& n6 L! V" `2 T 关键要素: w# N. q/ l" G: R$ O
    1. **基线条件(Base Case)**
    : F* j9 J6 w5 ]. c   - 递归终止的条件,防止无限循环, B' h& p% G) W5 _9 W  P
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    / t" x4 Z" w+ U- _( y) b
    ( Q1 P4 X+ n, n2. **递归条件(Recursive Case)**8 I5 l9 p& r! C" o) C: U/ o) i
       - 将原问题分解为更小的子问题
    6 q/ t0 d+ y' n   - 例如:n! = n × (n-1)!6 p. H: P2 g) M- G

    # N8 G/ K; v' t- _7 P! c7 u+ q 经典示例:计算阶乘
    1 g9 ?: k& ?( N: j% o" [& zpython9 T9 z8 e5 i0 X- D2 y9 P/ {
    def factorial(n):
    / i- E! ]9 `9 z- W/ I5 l    if n == 0:        # 基线条件
    & I7 ~' U0 {7 j        return 1
    ) J# u" |' m! P% \: c, @    else:             # 递归条件
    + C& P; n  V; L1 e- L        return n * factorial(n-1)* j  _5 N" T$ \. J
    执行过程(以计算 3! 为例):
    ! X2 P: g) s+ sfactorial(3)! R" ?1 \) z7 n! g
    3 * factorial(2)5 ?2 s% }$ X/ W4 v
    3 * (2 * factorial(1))
    + e& c3 C4 Y6 L* p3 * (2 * (1 * factorial(0)))
    3 s, U+ [, N1 V4 Y2 d0 V5 W3 * (2 * (1 * 1)) = 6
    8 }/ w" D' P$ @6 M# X9 Z1 f
    8 Q5 q+ e" f4 m5 g1 m 递归思维要点) ]" L- o2 g8 b
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 U0 @0 K# b3 }7 ?
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)' ]' J* A, Z1 V
    3. **递推过程**:不断向下分解问题(递)$ P4 m2 {- ?7 w1 k# U0 |
    4. **回溯过程**:组合子问题结果返回(归)5 Z2 o5 ~" |+ f9 p3 e2 `
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    注意事项2 l# U/ ]% A$ L3 ?; m7 g$ K
    必须要有终止条件: n7 f; _/ \2 y, P' m# ?
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    % ~% K$ R: M0 n- l; J某些问题用递归更直观(如树遍历),但效率可能不如迭代  j7 I$ H) _4 v3 T" U3 K4 a- P. k
    尾递归优化可以提升效率(但Python不支持)
    * L6 s9 R' Y6 G0 a/ I) L
    ! M- E8 q; m5 S% ?( K 递归 vs 迭代9 U6 z  P. L" ^, b' Y
    |          | 递归                          | 迭代               |
    # T/ P$ L; W& _% t, G|----------|-----------------------------|------------------|/ p  b6 z, U' y; }+ a9 i5 ~+ f
    | 实现方式    | 函数自调用                        | 循环结构            |
    0 q; ]& ]" v1 `  F' U| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |; M0 q6 x% T' D
    | 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         |. z, |3 r( X) F7 `* J( V) l$ z
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    6 k- a6 [7 x- P/ B8 ?3 J) i0 M; F' a* K- o, p1 \9 G; r) Q
    经典递归应用场景
    # y, _3 q0 P% p+ F3 y1. 文件系统遍历(目录树结构): G8 ]9 @# x* b) k! @* U) d
    2. 快速排序/归并排序算法  o: @0 R' u2 b7 y0 p6 Z1 E" U
    3. 汉诺塔问题, W' U1 k* B4 K: N' j0 Q
    4. 二叉树遍历(前序/中序/后序)9 g  U; V1 W1 F3 g% P
    5. 生成所有可能的组合(回溯算法)3 g- N* o7 k/ H4 `

    3 w3 x5 [7 i7 j& D- _" G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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    昨天 08:02
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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,+ [( O; {" N, X6 A3 Y
    我推理机的核心算法应该是二叉树遍历的变种。% U: v5 ]$ Y" w# L5 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:
    2 \6 h4 J. }9 ?. k6 C4 hKey Idea of Recursion
    4 M1 B4 x" N: N, Q) A6 C) `
    8 v0 L. V: p0 p- [1 ]! jA recursive function solves a problem by:, h, p7 N) l5 |; w
    3 k# K) C2 L! Y
        Breaking the problem into smaller instances of the same problem.! S/ x2 r) y/ Q) Z& ~9 k* ?( K

    " U3 R4 L" i: V) c/ A    Solving the smallest instance directly (base case).
    : w2 q1 i0 [+ P( {: F- k+ x- _: v8 o; k0 c5 a& s
        Combining the results of smaller instances to solve the larger problem.
    ! ~# |/ B. [; u
    - j8 H9 {+ Z3 `/ wComponents of a Recursive Function& T# _1 u1 g/ u' @1 I

    9 \, P* l6 M3 p0 z& W    Base Case:
    ; q) G7 u( e; l& W6 a/ Y# t: x3 {8 v3 J
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    $ E, r$ h% M. ^4 `" q. S: N, d1 n5 E' |- t; c% e8 u
            It acts as the stopping condition to prevent infinite recursion.' I) U9 {9 C. Q7 S4 c1 X

    : f7 n' v4 x% c  I- z1 w. |, a" \0 X' m        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.7 r# I6 j9 `3 }
    5 }. }6 p3 P0 ]2 S3 a1 b
        Recursive Case:3 \0 H1 Y+ V2 r' Z

    - S  C  j" v7 N# r/ X: E- F        This is where the function calls itself with a smaller or simpler version of the problem.7 }# q2 m& Y/ Q! t
    , w, E* F; G6 @. X- S" C
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    * W& j! s) n( {/ c( b" w5 a" b) j& g3 U2 n: f" N$ Q( B
    Example: Factorial Calculation* E; u0 z2 v& g0 Y! F% L6 {. J( q
    * x2 T6 o9 \7 j! [+ ~* 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:; N2 h; {  N# a9 n' _. w" m8 c
    5 r( m. `5 U0 `- u
        Base case: 0! = 1
    4 u2 k7 W' J8 X' b& u- k2 ?& Z9 n1 i$ p, H2 W+ Q/ S6 }- Z
        Recursive case: n! = n * (n-1)!2 ~: Y/ k% ~$ x8 s: @4 r

    # D: |4 |; B! R6 aHere’s how it looks in code (Python):* _% G0 B/ c  b5 r5 |1 @
    python: k8 o6 T4 m! E: n# a% A4 L, |( y
    5 e) _9 ^% \; v
    9 ^& O+ S% ]8 O6 |, Q2 q
    def factorial(n):
    $ y& J$ L( H4 c1 K" J/ ^    # Base case5 B) R& T% p% F& D' r4 [* q
        if n == 0:' _7 y. J! d% p  E, N
            return 1( t& A' w- V! m5 D4 c( H+ a
        # Recursive case, G  c5 Z2 t. g1 K/ B. b0 ~
        else:" S2 E: p' _8 a! h3 h- u# b$ E
            return n * factorial(n - 1)2 R" j1 q, ^" u4 ?& l- K
    9 w- c% ?4 r- A4 @$ G
    # Example usage
    ; g6 h  X) z: Z# oprint(factorial(5))  # Output: 1203 y- [) w' Y' G' b0 g) @& Q

    ! S( _$ N8 I8 E6 _$ c8 CHow Recursion Works% S! Z) M+ ~2 ?0 c% A7 t

    4 r; ]$ X: {0 F" G  n    The function keeps calling itself with smaller inputs until it reaches the base case.6 N4 j, t* ]" [3 W
    ; V9 z  {) O# y, I/ j8 X0 p
        Once the base case is reached, the function starts returning values back up the call stack.0 D4 `3 E2 D- x$ G

    / ?7 c( u1 j  M8 x7 c# o, m    These returned values are combined to produce the final result.0 ?1 H. Y! G+ z( L

    ; ^# T  F- n$ S. K3 T+ UFor factorial(5):
    - T1 \: Z0 n3 s8 S
    ( C& T; }' i: z- @$ {
    4 O( s3 Y2 S7 K+ \) n( i' L8 Bfactorial(5) = 5 * factorial(4): r$ T) l% Y7 q( C  k- C9 I
    factorial(4) = 4 * factorial(3)
    6 A" a) U# l" pfactorial(3) = 3 * factorial(2)
    * n3 {5 j( @3 i. C  h! z$ S! Efactorial(2) = 2 * factorial(1)
    4 ]9 R( o. e! r1 Efactorial(1) = 1 * factorial(0): ^  u  j) l/ Y, j
    factorial(0) = 1  # Base case
    ; x! V/ P- p* D) ^& ^3 t3 Q4 N, T1 S( P+ C
    Then, the results are combined:" S; M! E! ?" ~5 I% v8 b
      ?7 \+ m/ q+ U( L

    8 A  \3 C9 y) B: [; D; L" U% Efactorial(1) = 1 * 1 = 1
    / n+ w+ Q: c* H3 E. G, Wfactorial(2) = 2 * 1 = 29 h# B1 X& }7 F5 h; W
    factorial(3) = 3 * 2 = 6+ _; s; s3 z$ P6 ?! I
    factorial(4) = 4 * 6 = 24
    3 u8 h/ C2 u1 \6 L, F8 D+ yfactorial(5) = 5 * 24 = 1209 y( P: j+ v0 D% o% C# n6 P7 e
    & L5 y1 ^5 p0 p# ?! g; X
    Advantages of Recursion
    1 N  v( M& }) E! x4 l: H
    + ~2 H, r: z) E  K# `    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).( x! K) M& C' @

    % p( M* h) z4 T    Readability: Recursive code can be more readable and concise compared to iterative solutions.9 v' C& R! p3 n5 o$ g
    ) K/ h5 N3 h5 B5 b! n
    Disadvantages of Recursion9 w4 d( D* Q& K( g8 b$ H

    . }+ N* W2 F( n    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.9 I: P: w$ u6 d4 g

    5 A- e2 ~3 W7 u, f4 {; _3 X    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 @0 D* ^* ^  W; O/ i0 P( \

    0 D/ |4 y8 X% |0 B/ F) u% L+ tWhen to Use Recursion$ Z) t: \% t7 q- L
    0 \* K+ @6 s; m- C' y
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . Z- l. j$ c3 N: g
    5 y1 x) a" K- x3 x3 W    Problems with a clear base case and recursive case.0 {: H+ F- r2 ^

    $ G0 R( S! {% `9 c" o! `Example: Fibonacci Sequence9 g; `, ~4 T& K

    4 `5 a; e9 }# Z, M  u2 DThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:1 h+ O$ a1 n% Z- e1 E) Q

      S3 B( Z. H% M3 n3 X    Base case: fib(0) = 0, fib(1) = 1! t  V8 t( y  H' V$ u5 Z

    + F; ?, N. z4 {" Q9 o. f    Recursive case: fib(n) = fib(n-1) + fib(n-2): b) D3 F. }6 v  @( L6 S% T

    2 M9 k/ g6 Y" ]# ppython% Q2 ]1 Q8 l9 W
    : A- L8 E) z1 \
    1 c) c* H$ E. x0 n( p+ ^
    def fibonacci(n):
    . G, ^4 m! y* d) `2 a+ j& X& x. b    # Base cases
    ; _/ C8 ?1 @, q" H# B8 O) V- a    if n == 0:0 Q5 N# l3 }- @+ w# Z# z0 b9 F. \
            return 0
    " Z9 }2 m# g5 i, V4 m" T4 V    elif n == 1:
    5 {& ^0 Q" Z1 H3 S! g! Q        return 1; k4 W+ M* X/ f2 a- Y! a, M
        # Recursive case
    * [/ G$ e; Q$ J' m) P    else:
    : J4 [/ U5 ]& q+ l* P3 _9 S2 e2 R        return fibonacci(n - 1) + fibonacci(n - 2)
    6 |9 c+ s9 e2 b$ v7 s
    $ k4 i# t0 I; @) y. ~# Example usage
    3 H1 G# a6 H/ J) U8 @print(fibonacci(6))  # Output: 8
      g0 q& t  _2 @
    3 ]3 I) s  u' i, r/ k9 d- q1 e# ^- dTail Recursion
    ) b1 R/ \$ t, ^& e5 E! i- |/ R$ u, M& o
    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# O) Z1 K- _, Z
    , b" s( Y7 w7 Z& }6 m. [4 [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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