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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
  • 签到天数: 3 天

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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
    % I* ^5 q( d+ T2 X1 b" K! M7 D/ N  E! g8 ?8 G/ O" k7 {! d, x
    解释的不错5 P; I4 R. l/ r5 x7 S

    : F6 E$ m1 W1 G" {' y! B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。7 L/ _) h' I9 w$ d" }7 z

    8 _7 I4 R1 D/ n; t4 ` 关键要素7 d* J' F& [9 x9 o; i. z# K4 |" K+ \
    1. **基线条件(Base Case)**
    1 R( k& ?0 _5 I" w# A8 Y1 j- a, k" z   - 递归终止的条件,防止无限循环
    / t# O4 F+ ~# u3 c8 j$ ?+ }   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ; M9 i9 @8 M$ {9 m4 @5 \# M. f' C) f: v2 _1 M
    2. **递归条件(Recursive Case)**8 a* h: T- U; H: Q
       - 将原问题分解为更小的子问题7 ]* ?, L; |' r
       - 例如:n! = n × (n-1)!0 g* \5 v+ m: p6 k- R0 ?/ @
    9 ^* h" P2 a9 b: l2 A5 k# K
    经典示例:计算阶乘
    ; R, l" i( V/ D0 I, |; Q: h0 qpython
    4 y/ i9 g! m3 [5 r8 [* {def factorial(n):  }. {4 M- W  x
        if n == 0:        # 基线条件" k6 @  |7 |3 [$ ]. Y- Q5 i
            return 1, u# }4 g0 O. w1 @( B7 w, W$ Z' X
        else:             # 递归条件8 O8 _0 v- F- B& J
            return n * factorial(n-1)
    - r# Z2 z! `! N. L执行过程(以计算 3! 为例):7 i0 j1 A- J/ j2 u  J1 W1 Y3 Q3 N
    factorial(3)
    + n+ p% d  M& ~" i5 v3 * factorial(2)
    ' _7 _9 w6 ]4 D4 ~3 * (2 * factorial(1))6 v0 \1 s. @3 ?0 C8 k
    3 * (2 * (1 * factorial(0))), D' r+ Y/ T7 k8 y
    3 * (2 * (1 * 1)) = 65 P  P, p4 E+ k- U8 h, l, G7 Q

    % B) }' v' ^: j, m  p9 R' E9 y! f 递归思维要点
    3 R+ i( L( y8 H5 d$ d' k7 a1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 I- n+ Y3 s0 h$ O, b' o  i
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- Y0 v) z5 ]! V/ V6 k
    3. **递推过程**:不断向下分解问题(递)
    1 A8 T4 p3 K/ V4. **回溯过程**:组合子问题结果返回(归)) z: q; W( f$ \

    8 c! D/ O; J2 k# g8 {4 t9 s7 n: \  X. V 注意事项0 B% J2 P. T# O1 N" [( B  Y6 G1 f
    必须要有终止条件
    . b* Z( Z) A; [" g; d递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    # |) a. Z3 x$ n! z+ v某些问题用递归更直观(如树遍历),但效率可能不如迭代
    : V- |6 I2 J3 G4 w5 Z尾递归优化可以提升效率(但Python不支持)
    ' j; q$ K4 V: p: P
    9 Y* w! z, J8 F9 l/ Z 递归 vs 迭代
      ~% V. S" x6 [% h) ~- `( ?0 T|          | 递归                          | 迭代               |
    " \  g. ?) g4 U9 `|----------|-----------------------------|------------------|5 l* p( H: l0 l  B  M4 {" v
    | 实现方式    | 函数自调用                        | 循环结构            |
    8 @6 N- ^% X! ^/ h| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ; }. v7 J9 i+ _$ A# i; v8 ]* x| 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         |
    4 K# D, {2 I+ j0 }2 Q6 \6 p| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    $ j* f3 I# U9 Z
    , w0 \. C" c4 Y! N, V, d 经典递归应用场景
    . r. H9 r) Y& u( R0 C" v1. 文件系统遍历(目录树结构)$ u* \) |1 G5 W  K
    2. 快速排序/归并排序算法4 w* J  G4 l/ n* g: d# o
    3. 汉诺塔问题
    # D/ x  ~1 [1 p4. 二叉树遍历(前序/中序/后序)! V: n* c) J2 ?; n; _
    5. 生成所有可能的组合(回溯算法)5 E6 g0 ~9 F2 O2 K7 T- z4 T2 k# ?

    : S, H- B5 _2 a' A# c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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    郁闷
    昨天 07:21
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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,
    % r" Z+ i9 E) Y3 X" ]! F8 I我推理机的核心算法应该是二叉树遍历的变种。
    2 ^5 H* f9 U& }+ _% Z; B- l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 [9 ^9 X, I: J* e# V- w6 [
    Key Idea of Recursion4 j$ \1 C4 C) E& Q8 d
    # ^. x2 J% o6 e* g
    A recursive function solves a problem by:
    . D& V+ z2 n' R. n, I3 W
    9 d* J8 _: j% |' f    Breaking the problem into smaller instances of the same problem.
    5 Q2 r% `/ q( E5 J' M7 |6 N1 E( c2 C4 U) m) N! n9 q
        Solving the smallest instance directly (base case).
    9 y4 h0 Q: `2 `0 S
    1 Y, k$ W8 ^4 q) I( }    Combining the results of smaller instances to solve the larger problem.; M: g3 G; d* x9 l; K

    3 z; l  Z$ {! }7 A% }2 Q$ RComponents of a Recursive Function% x9 Z3 l, I/ t  b* w3 K% f* y) H1 ]
      p* ~0 p( p6 R% N& W* i
        Base Case:
      \% b0 |1 q4 O" t& ~) x
    . N& d- P3 W; ^2 f        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.6 _# r- L( W8 d1 i
    0 {+ n6 h( Q" |
            It acts as the stopping condition to prevent infinite recursion.0 o/ O. m# E# C9 p( z) o! l+ e0 `
    % m+ e  u* t$ \
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.6 l* S" q( I: N8 T% t
    & ?! L( [6 I$ k" s: ?
        Recursive Case:
    1 `1 e# o' c: W2 M; U$ z6 j, o' c2 A6 Z1 O2 G" f; @
            This is where the function calls itself with a smaller or simpler version of the problem.6 y) u5 ^" \$ b/ b4 _
    , ]5 }( e/ m/ X; V+ e. `1 {: j: k
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)./ r: S5 ], `. H$ e7 N

    $ o' h% M% U( H2 l! ?Example: Factorial Calculation
    9 Z' g0 h4 A5 T+ B0 @7 ~& L# h. q3 H  E1 v% m8 V! x9 q
    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:, d6 r  F4 t* {0 |& R9 j

    1 A" @4 l" h9 j0 l7 H) l  J7 \* Z: W    Base case: 0! = 1% X4 t5 |5 }9 a1 t9 ~
    % p  Z3 B' I/ Z
        Recursive case: n! = n * (n-1)!
    4 D3 d0 l) [$ c: a. z' r# z  u" l1 A$ r) w  y
    Here’s how it looks in code (Python):
    6 {4 @5 f, y. Q! gpython6 z- r* S4 q0 g  L4 j6 ]

    0 k; f) }) C. Q* d& J
    4 H7 G9 L2 j: I9 v) ^def factorial(n):! K& H: f" ?; U- X0 f
        # Base case$ |- T7 |, K( ]9 w- Q- F5 ]
        if n == 0:! A1 p( C4 h2 c9 Y3 ?- _+ G
            return 1% l/ g% ?, e1 ^4 D. c3 s! J
        # Recursive case
    ' c/ l1 _: O$ j. u) B8 T! n5 H    else:
    + W, E0 Z" ?0 A/ ]; a$ N        return n * factorial(n - 1)$ _* L7 p6 H/ |9 l5 T

    6 t1 U( J: y  ~: b" `# Example usage) V( y7 r! u0 T" M% j1 N1 r4 Q; e
    print(factorial(5))  # Output: 120* @" ~+ H% J5 G
    % A2 S+ O  i& @. S) x
    How Recursion Works  r# @$ o8 b9 \6 }. _8 j5 o1 Y8 R
    ' |/ I7 a2 }/ }8 d
        The function keeps calling itself with smaller inputs until it reaches the base case.
    & M2 r+ y5 t. H, r7 S5 y
    3 a. Q9 ^0 w7 K$ |9 U# u1 m( ~    Once the base case is reached, the function starts returning values back up the call stack.
    + i8 t) y0 ?' d7 I$ R5 A
    2 U( N5 q3 Q, n1 J; r/ Q    These returned values are combined to produce the final result." z; v7 p" ^/ n: ^1 K, j( @

    8 x" X4 ^% \( r" P+ e# O  BFor factorial(5):% N( V6 N: c1 F. h1 q* _4 w- c

    : o$ q2 J, V5 k( I# c9 k) i& ~% a  L  _/ r% J# R) b2 F( \* ^
    factorial(5) = 5 * factorial(4)% a- [, T$ \7 ?' G4 s6 g
    factorial(4) = 4 * factorial(3)
    : O8 G, K+ R5 kfactorial(3) = 3 * factorial(2)
    + p0 S: [* e- y: `8 ?9 vfactorial(2) = 2 * factorial(1)5 m! d# b# |; ?+ l: V& N$ v
    factorial(1) = 1 * factorial(0)7 O* o% {$ S5 _* V8 {9 F- C' t9 @: c
    factorial(0) = 1  # Base case1 }3 }6 f- R$ Z/ D3 T" L
      l& K$ {2 x: B, B" V9 ?
    Then, the results are combined:0 x- Z' X0 }/ q

    6 {1 d: K. Y! E* o; d, n; E8 ?
    4 @5 r& h5 |/ c& s% A5 ^5 m& x8 B) Lfactorial(1) = 1 * 1 = 1' ]; w' U" s5 T8 E& \
    factorial(2) = 2 * 1 = 25 N3 T4 u" j  u: U
    factorial(3) = 3 * 2 = 6
    " J1 @/ B/ B! N/ b9 g- F" Cfactorial(4) = 4 * 6 = 24& W% O( f0 [; ?3 V  A7 g! u% M
    factorial(5) = 5 * 24 = 120. o- |4 P" K  c3 k) \1 [6 X/ p
    3 P; K' ]7 ~) y) J# y' J* N
    Advantages of Recursion. g# z0 G. r. U/ g4 j: X- L* j

      X3 Q6 V* W  m; z, E    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/ c4 E! \: m% e( `% y. n6 @9 A0 m
    + G1 q: g, `8 l: b8 W  c    Readability: Recursive code can be more readable and concise compared to iterative solutions.; P0 o. E0 `, c8 M1 h& M% _
    7 N6 H$ z6 x5 t/ B4 ?' \
    Disadvantages of Recursion
    4 v6 \) w+ E& H" h! S! B9 N' @6 h6 j! @0 K/ L. z8 _
        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.
    $ C/ f+ s) J: s. ^/ z! F$ E& i2 ?; n* v: K
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).$ I$ x+ p0 q* Z6 ?7 h

    & ?1 y3 C% `0 f; kWhen to Use Recursion
    % K+ T% m/ a, @) o* x. X1 m0 ]& J' s; e& W, O. q
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).8 S  g  T1 a' j" S, K

    ; f- B( R7 H8 ?5 `1 I    Problems with a clear base case and recursive case.$ b; @1 ]$ V( u3 v

    & B# H4 J& C$ V# {+ Y* uExample: Fibonacci Sequence
    2 W& K9 y$ F' j$ z$ `& t0 b
    " H" K# {1 |# a1 Q+ _The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    % S, `+ ?$ p1 M# D! s- e& u. V# t3 D, l$ x0 y3 |
        Base case: fib(0) = 0, fib(1) = 1
    6 d% G! u+ N- L
    : L+ G, m& x* |0 p: m" ?    Recursive case: fib(n) = fib(n-1) + fib(n-2)% J- f1 K4 U2 n1 S+ Q+ ?8 A3 M5 K

    * H) H9 r0 Z% ~! Y' y3 ppython& B# ^7 B7 d1 j
      ~/ H4 T) P- Z& I% O& w
    7 {; A6 y2 ]* U. g8 M) F
    def fibonacci(n):
    # ?7 X9 W2 N$ O) W9 q7 G    # Base cases( K3 |4 @# ^' Z7 h9 i2 L
        if n == 0:- Y' B" |7 X- Y
            return 0# R& s6 @( y* _" O: y
        elif n == 1:
    / g/ x* R+ X. l        return 18 ^% D) G7 e  {9 Y
        # Recursive case
    + W: \+ \+ |+ V2 B/ t/ ?5 y) E    else:! U* F3 y1 ]& y. n' u6 q8 e) W5 n
            return fibonacci(n - 1) + fibonacci(n - 2)
    + A0 d  B" y, V0 _
    $ |, c5 V6 ]' i# Example usage
    ! t" u4 n7 \/ o3 x( r4 B# J+ ]; tprint(fibonacci(6))  # Output: 8# K& F6 b, |2 J- x  _
    - k) F7 e6 f* L8 |# C) Z
    Tail Recursion$ f) f5 X$ P3 _

    9 g; W- y+ S6 n- Y0 ?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).3 t0 S  q  h5 K5 e  g/ z; b1 C
    ) ]3 h: l* u" v$ Z
    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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