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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 7 V- o5 @( c2 E

    3 z7 }! E! |: ]4 D- S解释的不错
    & d* n1 }* O& \% S3 r8 D, _+ s/ B* D
    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。( ^, t: ]/ Q( o

    % k: Z8 u+ x2 h) Q6 a% s0 _ 关键要素9 ~, E4 I2 L" O2 u7 C% ~
    1. **基线条件(Base Case)**
    # i- }0 i" E  J+ X+ t) S: O   - 递归终止的条件,防止无限循环/ h5 O) U+ I% U1 T# L: y
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 13 e7 W* R+ A/ J  _4 i

    2 J0 S9 h; g. R" c1 a& R2. **递归条件(Recursive Case)**
    ; `1 |* S( U% A$ N: v   - 将原问题分解为更小的子问题( C  B. i- K- |% k
       - 例如:n! = n × (n-1)!& v: g' p1 M! Q) P4 G
    7 u$ ^  T( {0 L2 ^3 j3 F$ F# I, r/ ^
    经典示例:计算阶乘
    ; n2 B. T5 W3 ?5 vpython# b4 a( q$ V3 W+ y! y, q3 M- K
    def factorial(n):" c9 m; B0 M8 R7 t: \: ^$ v% }
        if n == 0:        # 基线条件1 E- D# w" I2 A
            return 1% l% {" k8 e8 k
        else:             # 递归条件
    8 E# V2 a* l& {- N( i& a% _" e        return n * factorial(n-1)+ |6 n' x& b7 {3 V( L/ Z" u3 {/ r
    执行过程(以计算 3! 为例):
    8 |% i, v' b; o% a1 V: t* v# s; H2 {factorial(3)8 g  ?* O+ d( q9 N7 [
    3 * factorial(2)
    6 |2 q9 _$ ]* r' d2 d3 * (2 * factorial(1))
    " ?3 F+ F) z8 S0 F3 T3 * (2 * (1 * factorial(0)))
    # _3 r3 ~9 T5 j! y& }5 o7 u" _3 * (2 * (1 * 1)) = 6
    ' V- U/ n$ [2 I0 m" y& |  e. i1 c( o" C! [* b5 l3 r7 L
    递归思维要点; y# f; ^% X* ^2 V0 P) n; F
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    5 m1 y; A: R0 v2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    % r) z! }; ~9 |& O7 g+ B/ f" O3. **递推过程**:不断向下分解问题(递)
    2 C% a7 B( p. k! L! ~/ i7 b, y4. **回溯过程**:组合子问题结果返回(归)0 Z# A1 C$ v% V; A) s1 y) U3 A) m

    $ d  K8 _. m: z6 t  u 注意事项
    5 i: g# G. z$ Z  T9 l$ k必须要有终止条件5 b+ i8 U; F. n
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( o1 T: I+ i" ]! r- y% l) o, F( Y
    某些问题用递归更直观(如树遍历),但效率可能不如迭代0 T, \2 Z' V* w# [! I
    尾递归优化可以提升效率(但Python不支持)) Z4 C3 _7 z7 Z4 ?$ q

    . J- t6 I! N% X- E 递归 vs 迭代5 O* S0 M8 l4 a3 q# A/ }
    |          | 递归                          | 迭代               |+ U* @* B2 }0 g) f' k  j# f: }7 N6 [
    |----------|-----------------------------|------------------|: e+ i3 I. e0 T- z1 m0 n
    | 实现方式    | 函数自调用                        | 循环结构            |* {( D/ R. s; ?1 q& X
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    $ j" y3 Y7 `: k, q* O+ m& ]8 o| 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 S& Y* g  H* L" K& x+ n4 U| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    $ V* R3 m0 w4 d+ `
    # K7 i' G. K# r 经典递归应用场景
    3 Q  U3 G% |& g+ v( x6 A1. 文件系统遍历(目录树结构)1 v4 X6 r6 q  u, t* `# L
    2. 快速排序/归并排序算法2 C  c/ O) U! d
    3. 汉诺塔问题
    ' \8 g$ Y1 X/ S5 m; U2 q, T4. 二叉树遍历(前序/中序/后序)% g0 w, `" G2 J. j5 b' ~
    5. 生成所有可能的组合(回溯算法)
    2 V3 o% p% k3 x& Y( Y% Q: d6 W/ \% o$ E
    ! g$ ~6 l; j7 X" l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,2 `, y& L" T+ e' n5 R
    我推理机的核心算法应该是二叉树遍历的变种。
    / k3 ^: e) K9 B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:
    9 I% X$ z* l; G8 }- S) I7 ^: wKey Idea of Recursion) w/ c2 ?8 O+ v  I, s
    1 l* ^9 ~1 n. L: h* _3 }( P) u2 ]
    A recursive function solves a problem by:
    & h- g: W/ B& g/ ~, c; ^) W
    9 A, ?. b4 M+ x& A/ M) ^    Breaking the problem into smaller instances of the same problem.
    2 E6 ~( |2 Y- S( c& }$ V9 |" A1 H: r2 q7 _7 q1 n/ |
        Solving the smallest instance directly (base case).# A1 @( m* w, \6 i- y- v2 f

    0 u2 @: r& ~6 N' k0 Y    Combining the results of smaller instances to solve the larger problem.% ?; ?; w7 a! s% e0 v; S% l

    6 g% i3 I7 l# K  y2 ^Components of a Recursive Function* p! D- B. G0 D* Q# u# H
    5 V9 l% O# C5 u( Q
        Base Case:
    ; T- [4 l2 d: q- K/ t9 O- o9 s) @* r  |8 E0 c. c. j4 o3 v
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    % A8 l* {, u1 A+ y# P: \! q9 I  w9 G8 N
            It acts as the stopping condition to prevent infinite recursion.
    * w' t7 y3 p2 F, G) O- u  P: m+ N: C. r
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    : y# l  n4 H% _$ {# \) _% E3 H5 q, d4 d
        Recursive Case:
    ' X) e# Y& o7 Q. W* m& @
    # o0 I- s# V- [9 Q) z* q6 `" r        This is where the function calls itself with a smaller or simpler version of the problem.0 o. z# Q5 L0 [

    6 o! ]! b$ X) i4 |0 m        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 ?. C5 i0 i7 q2 y% w4 E; v
    5 E, o# k% P  x! {/ r1 ?  G2 N7 U
    Example: Factorial Calculation
    / F+ U" _' F2 [( W' U' i
    $ Y& Z7 g( I- }) R3 p+ y! uThe 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:
    ' }7 z0 R& j5 b& m( p0 c
    1 r' m! Z2 V2 X  q    Base case: 0! = 11 T; |# e3 s/ |4 V+ n

    $ i; c0 W/ x$ n- z" g% K1 }    Recursive case: n! = n * (n-1)!# w+ q& a* i2 _( l( B4 B

    1 f3 I0 F9 ^2 A% j% H4 s3 JHere’s how it looks in code (Python):: n1 I/ m1 ^2 m) d$ P1 c
    python% o% n6 q' i/ o( b% d( Y4 K. K
    9 b! i. W2 p. C8 ]) L' F
    ' I, ^3 d7 Y% x$ z, ~# G. {9 q% O
    def factorial(n):
    / X/ c; m0 r( }+ ^2 x( z; S0 o    # Base case
    % _( {/ W' F, w( m6 I( j    if n == 0:+ `, f' l$ o7 ^$ x( @8 U
            return 1
    : d! [# F2 F1 o( n    # Recursive case
    $ W; ]/ R- q7 P8 p% |    else:0 d  c% k+ V3 U# c3 X* M. s, i
            return n * factorial(n - 1)& H: Y1 z8 k  x" t, s9 C
    ) n/ l! w+ O5 E* {0 p7 H2 b
    # Example usage$ {1 u( v# T6 g. S# o) P
    print(factorial(5))  # Output: 120
    & s% b3 d, T( p* n; Z" `+ T& O
    . Z5 O8 n8 ^+ ]How Recursion Works
    1 c6 L$ S; T5 r# h$ K! K9 P- t4 ^; ]- Q/ }; I# \, K
        The function keeps calling itself with smaller inputs until it reaches the base case./ i' Y  q4 {; \, L- x
    0 e7 a) d8 W( \' g! W5 h5 T  j
        Once the base case is reached, the function starts returning values back up the call stack.
    ( U: t- x3 e9 Y/ r) X  l( n- H1 _; O4 T& M+ W# q! J  \2 `
        These returned values are combined to produce the final result.
    . `) U5 C% b+ D1 I/ x' `) ?5 a8 O$ \. c1 [0 C  f
    For factorial(5):
    1 {; U" m$ g2 g) c" n  O' S  M: h# ]& P. L! P

    # V1 S* ~$ A- v4 _/ C( Nfactorial(5) = 5 * factorial(4)
      S4 Z  i5 R& r4 |8 `6 Ffactorial(4) = 4 * factorial(3)
    0 R7 [" I5 v  zfactorial(3) = 3 * factorial(2)
    ; m5 P2 E. i+ gfactorial(2) = 2 * factorial(1)
    2 I. h; t0 `# @. _( N' t9 sfactorial(1) = 1 * factorial(0)
    . C) N! ~$ f4 R2 Kfactorial(0) = 1  # Base case, z6 }5 t# s/ \
    + z0 F" @1 w* S/ h& A
    Then, the results are combined:4 M2 ^) H1 e' R5 V) ~; B0 ]0 z

    $ I: M3 M: O; W8 L* z' H  D
    ) `+ l0 x* m$ sfactorial(1) = 1 * 1 = 1
    : M  [; |: ]5 l, Wfactorial(2) = 2 * 1 = 2
    % B5 [9 `9 _" }9 e5 V4 m' b. ifactorial(3) = 3 * 2 = 61 U3 B3 {& l4 b3 _6 {: D
    factorial(4) = 4 * 6 = 24. y' y9 ?; w( k& ~& P4 i
    factorial(5) = 5 * 24 = 120
    ( G* [8 B+ U7 u- n5 Y# ~5 x8 T$ Z
    2 I4 k8 ]' W. J1 M' o# [Advantages of Recursion
    $ Q' s5 @3 S5 R! T! }! Q9 l6 ]
    ) g5 x/ u. s! |2 w5 l4 @& @4 |    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).3 x9 J4 U7 Z2 r/ h- n7 {% r7 l
    ' e: Q8 U( {' L' U; {+ Q2 I
        Readability: Recursive code can be more readable and concise compared to iterative solutions.( `7 {, i; D, ^4 w; I. T1 g! g9 O
    3 g. n: t; X0 \. S4 t2 W5 \9 }
    Disadvantages of Recursion
    7 P5 V0 g( r4 O8 Y  j% i* w8 `: X$ \6 G) J
        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.
    5 T4 s4 c! z! [4 X, J
    0 @2 l6 [$ L& p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)." {1 R- `/ b6 V' c/ i
    " |% P6 O! Q; L* R5 C0 p7 e
    When to Use Recursion
      v" y1 F0 @6 Q/ B* T' q
    3 o3 O9 ^" L- b4 b; l    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).) B4 W0 j  M0 Q( w" Y

    ( N  v1 c5 ]& V6 _7 l    Problems with a clear base case and recursive case.
    ) W3 q& I9 b# w( t" G% ]. P% Q) T" U! u2 H, J
    Example: Fibonacci Sequence
    9 T0 d$ p% g4 A) p
    3 y) t6 s0 n+ w* g! ]The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:/ z4 W' ?, C  t1 Q- C+ M6 R
    & {$ v: y6 o/ P) W
        Base case: fib(0) = 0, fib(1) = 1; `2 ]1 d- S1 D" F$ F

    3 p: L2 k: m4 h7 E! c; H( z' K    Recursive case: fib(n) = fib(n-1) + fib(n-2)1 M+ w) L. X5 K0 |

    2 S7 Q) m, Q2 z) R* h, i+ L' [python
    . x4 O' i7 I3 g6 ^0 }4 ~
    3 T4 I+ F( f5 \# B+ ]
    ( q" _! L2 Q5 K$ @+ `8 v9 o4 {def fibonacci(n):9 }/ t0 m8 `; K" p# n
        # Base cases7 S4 m" G2 K! ]8 F) s1 x$ u
        if n == 0:6 B. d, X7 B# K1 Q/ T3 t4 D4 f: _* U
            return 03 k) T* h1 l: F9 b# H# G9 V
        elif n == 1:
    . n) y+ @7 K  y' O  e% l        return 1# _$ t1 _+ y2 L' |0 \! x
        # Recursive case
    ! L" p0 ^1 m& ]% _8 x    else:9 x# c4 X; l& t8 k
            return fibonacci(n - 1) + fibonacci(n - 2)
    / s) R% E/ u. U4 G4 \* i% {1 S# _& C3 p0 v/ g; B
    # Example usage
    4 v: A# m4 C- i3 [print(fibonacci(6))  # Output: 8% S/ f7 d8 D9 A' \9 e7 i

    , w6 V! v1 z0 i# \Tail Recursion
    ! u' v5 E8 V; |, z# x3 D: a
    ! z, f3 b  W  n( j( n- @" cTail 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).
    $ d/ L! {: s$ J/ M$ V0 B( v! i5 R( `, Q" A
    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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