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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
    ) r) o) q2 I3 Y% o7 }  K3 `
    " ^! P! {2 H* m, L解释的不错+ B4 a, O" x6 Q/ ~7 V* h1 ^
    ) x7 R- I5 C1 f1 r  ]; @
    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。
    ! U  N0 A1 h1 @3 ?3 I: o2 `7 z' K+ f& M
    关键要素" \( [. v' l& M' T8 H- E
    1. **基线条件(Base Case)**
    - ~- x4 O" \- g9 E   - 递归终止的条件,防止无限循环5 x! g9 z: w4 S) S/ q; @
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& M5 V+ R/ l! R+ C; j$ G; B& B

    + c/ i% `( c  x: f, C2 R1 W2. **递归条件(Recursive Case)**
    ) J5 V4 m: z6 Y1 T8 N   - 将原问题分解为更小的子问题- Y' b( y2 x- L  F: A+ E
       - 例如:n! = n × (n-1)!' |/ L5 ^& [/ j
    3 X$ C8 _  m, E# s/ f+ d) n
    经典示例:计算阶乘
    ' n# r+ H* r" ~+ {6 g# x/ n& epython) I7 Y6 M, P2 F0 e, u9 _
    def factorial(n):! b/ W: i$ i' _& ]  v) [8 W
        if n == 0:        # 基线条件3 v7 z  I. t! B' W8 B. P9 p
            return 1
    ! K/ V0 ~' V7 |9 Z2 A/ h7 H    else:             # 递归条件
    3 j  x  X: t! m( ]2 |, ]2 y        return n * factorial(n-1)* R' e7 e: E4 t) ?$ E) t$ H* [
    执行过程(以计算 3! 为例):
    5 S; J$ v* ~3 P8 ^- @- Efactorial(3)
    9 O9 n1 H; E4 m5 ^2 U, m- ?" e" @+ u3 * factorial(2)0 ~. W4 J  h$ T2 n! F8 a1 @
    3 * (2 * factorial(1))
    8 o7 V$ ^2 U; A9 O$ Z/ X3 * (2 * (1 * factorial(0)))
    ; R+ E& y. H' C1 F/ @3 M0 G3 * (2 * (1 * 1)) = 6! Q% z* u; F% C/ x; C) r+ b/ l" e

    # C: k, D- ?3 j) P 递归思维要点+ i' y* Q0 Z; p' a0 s5 o
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    1 r& X0 |4 P6 z2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    # Z8 ~, A0 @/ k6 }* i& \3. **递推过程**:不断向下分解问题(递)6 Z/ D/ U! Q0 J) q" r
    4. **回溯过程**:组合子问题结果返回(归)" C6 n( t& F; V  G7 N( Y6 l: ]
    6 n: U( ~6 c5 V. k8 U2 Y
    注意事项$ ^' w6 V+ r* q3 R7 n
    必须要有终止条件6 z  o  o4 C& Z( M3 J
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    / N2 n- l; Q- M7 J2 u某些问题用递归更直观(如树遍历),但效率可能不如迭代0 d- T2 `  u3 X* p+ S- I# b7 w, W
    尾递归优化可以提升效率(但Python不支持)9 @. t7 G3 P) T% m  h! |5 L$ X) ^
    6 e! d3 t, j/ @* D  [3 }
    递归 vs 迭代
    - ]2 m3 A$ Z8 s8 r) ]|          | 递归                          | 迭代               |7 ?* S7 @. w' q' n6 x
    |----------|-----------------------------|------------------|2 V) r5 @  b0 @: g
    | 实现方式    | 函数自调用                        | 循环结构            |
    " ?! D! _, Z! [) |: P| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |: a9 {' ?; 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         |
    , J1 D$ Z/ D. B/ F  U- v$ o- Z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |8 E" x/ ~6 ?& w; n
    7 _: R3 |; i  q! ~
    经典递归应用场景3 d' F# [5 W& a' e
    1. 文件系统遍历(目录树结构)) n4 I1 s# v2 b) X" S+ R
    2. 快速排序/归并排序算法3 i. }5 G, V6 B8 n- M' a( K; K
    3. 汉诺塔问题; p$ I0 i3 x" ^- A5 x, m1 K
    4. 二叉树遍历(前序/中序/后序)) B, X, L4 P5 T
    5. 生成所有可能的组合(回溯算法)  y7 M0 o" p& g9 o- q
    ! I9 E* |5 `( j! C0 M& z
    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,, h' |; i! ?" ~# D( i
    我推理机的核心算法应该是二叉树遍历的变种。
    + @: I* v# F, o; `0 T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 G+ `* j; a9 C8 t+ M, W. tKey Idea of Recursion
    $ F" q  s4 M5 u# n9 f' K4 E. L% j$ k5 U* b5 P. a
    A recursive function solves a problem by:
    + H+ j7 ]7 T. k; B8 v5 j* n6 @+ ]+ Q0 v2 d- f4 X  R& s7 c
        Breaking the problem into smaller instances of the same problem.( n4 k8 k2 M/ C

    * @3 D" ]" a; U  N+ k" j    Solving the smallest instance directly (base case).
    6 `/ I- T. u2 B  G' }* X* ~; ^* K! p
        Combining the results of smaller instances to solve the larger problem.5 ?/ h7 ]+ K9 B& P* E
    ) f) I0 y* Q4 }& z
    Components of a Recursive Function7 @) u  `- a" c. n  h+ B

    ! @' U3 r2 X( _5 b    Base Case:
    / M+ C2 ^% G6 u9 _+ {1 c
    , H" O1 Y* L% z# y+ i        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.3 g: H) @% D' b+ }3 v4 |
    : W5 ^* E5 ]0 T2 J7 b. T
            It acts as the stopping condition to prevent infinite recursion.- M& D3 D! ]; I" T

    6 _4 C/ h' m# B        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.) Q* T7 l+ \% \% I4 l

    / S" W$ |+ [# D    Recursive Case:2 g$ i/ c* n( _) N& N, Q7 S! V
    ; U* e! @# G5 h5 ?( B
            This is where the function calls itself with a smaller or simpler version of the problem.
    : ~. J; R& B) A1 {
    . Z% F, g7 T; I& \        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).! [7 I! u! j3 i& S  s

    2 |8 |) t- `$ H# x8 W& DExample: Factorial Calculation
    1 O  S: g# w; U! ], D8 v  Z: ^4 x. x1 G6 L7 W: b6 a
    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:. l8 X, s; W9 T5 C! f7 \" e- s! T

    & d  t6 o. J7 s" o4 v    Base case: 0! = 1
    % i# d! P" o# {- h- f. r% H7 M9 x% s6 z
        Recursive case: n! = n * (n-1)!; n' Z7 n% v! y2 b" _! F: P8 V& a
    " A3 N0 C1 q! M; n7 k; P; w
    Here’s how it looks in code (Python):
    " O6 `+ f: l  q" m; f. Y! jpython7 o# A. L( q; D! ]. }  t1 g' C
    ; h- D1 S$ U5 \, A) L& Q
      ?# u" f( D' K- \+ z0 x" y  s1 c
    def factorial(n):9 [" `  R1 A# |  Z2 a$ ~1 c
        # Base case* n! p  E4 t. ]3 w
        if n == 0:) |& G$ p7 p( d0 f' A
            return 1& U5 w$ P/ o6 S& m- e+ o
        # Recursive case6 l0 M! d9 p! d, f* I
        else:
    7 B2 C4 o% U& |, v* v, ~        return n * factorial(n - 1)2 ~7 k) o! q, l1 J8 f1 c
    - g7 g  i( }3 T; F
    # Example usage
    ( w4 P  {) X: i0 R. c$ [% R4 v! ^( c1 `print(factorial(5))  # Output: 120; t0 r; j+ z9 }1 b1 W" u' v; C

    ; ?5 W+ L$ K5 t) }2 f% rHow Recursion Works
    ) y7 O" i" X& z1 g3 v) Q0 H1 l( [; Y* A$ ^
        The function keeps calling itself with smaller inputs until it reaches the base case.
    # O7 I( k/ o( r. ~3 T& e8 y0 C4 w: ?9 T1 a: \
        Once the base case is reached, the function starts returning values back up the call stack.- B( Q$ }1 U. V6 o1 V5 U
    1 l7 ^8 f5 U8 z0 a: p' t
        These returned values are combined to produce the final result.- y/ G) f3 _! d3 M1 H
    , I+ \! U3 a4 b1 D- Z' }# a' @
    For factorial(5):
    0 v) K  a% e& T6 Z" M( O  y
    % m. X, `( a/ m  W1 D) `2 G* x* C0 _  `: P8 t4 {' t
    factorial(5) = 5 * factorial(4)
    + m2 T/ p* _1 _, S* F. M( }6 P& Ofactorial(4) = 4 * factorial(3)% J/ x( a& R7 X2 J% T. {, |
    factorial(3) = 3 * factorial(2)2 r/ r7 z% P- B+ g& F( d- ~
    factorial(2) = 2 * factorial(1)
    , W5 l" g3 M  l: w1 Lfactorial(1) = 1 * factorial(0)
    7 K. v% s9 a4 v. l0 S! Z" Q6 _+ nfactorial(0) = 1  # Base case
    - ]5 v- ~9 J* X6 z) g8 u/ X* h5 C+ A/ D/ Z
    Then, the results are combined:
    / |' B$ @# L" L- m1 q; A6 ]# z# s- V; [  S
    ; i* `* Z( ~5 ?, j
    factorial(1) = 1 * 1 = 16 B/ d: w" x: b6 g
    factorial(2) = 2 * 1 = 2: D; G' K" O% G; E' p- k2 _* o
    factorial(3) = 3 * 2 = 6
    , e/ r% g( H. k* G2 J4 |5 Gfactorial(4) = 4 * 6 = 24
    ; K. w" Q  S% c3 M7 \& B* A/ Qfactorial(5) = 5 * 24 = 120* I3 Q/ H% |$ k
    ( i( x- }5 d2 j  a
    Advantages of Recursion( s$ t) F) d/ J% s0 `  i+ E
    5 C6 |; W/ e# j, O9 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).
    / B' H' [+ y3 _& u  _$ [+ C# F8 b
        Readability: Recursive code can be more readable and concise compared to iterative solutions., N) w* c$ u. Q
    3 G7 j) I7 K' d$ M9 R
    Disadvantages of Recursion
    7 C+ I' G/ a. T) E3 C
    + k: ]  a( i% R4 L) F( U, }: A: W    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.
    ) \, S& }0 e; t3 \/ e7 V! @' P% B' g( ]
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    " P- U6 N3 T( `6 H* K3 ?) I8 U1 k! D' w
    When to Use Recursion
    / J: e& }! ?8 p' N
    - i( z* v% M  Z. |" o, B: s    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).3 n* w* y# o1 p' u: s  e
    / K2 ~6 |$ `3 a! B( j
        Problems with a clear base case and recursive case.8 W0 g2 \1 I5 w

    " K# p4 S# F8 L& A6 EExample: Fibonacci Sequence% o" ^& c$ o; l/ ?, U* p

      D, i7 ~1 S9 l2 ~' BThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:! C7 }; [5 N, Y  O6 A
    # v9 x  c( u. _6 A
        Base case: fib(0) = 0, fib(1) = 1
    1 m. j/ u5 L; R
    % p5 M- G* b9 M' s  N    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! g! F1 I. e6 E* |3 {
    / z$ y- l" q  P, y0 `& E# g1 V; Xpython& g) L. z2 d  W9 ?
    , U# m3 |: R6 m7 a

    ! b3 h& n' F" C; i1 a& vdef fibonacci(n):. T9 z/ {  |' o  `- N4 a6 @
        # Base cases2 g) T# W4 f! b
        if n == 0:. E' W7 ~' O$ r; G
            return 0
    ) K. y. W$ {  I    elif n == 1:; M9 j+ Y5 ?- X# X9 o5 h
            return 1
    4 R8 L. K4 H8 S+ u  J2 g% E  n' ?0 W    # Recursive case
    " r9 b* L9 J8 t0 ~7 j    else:) v1 ]) N1 l# [4 v% t( C3 r+ G7 v
            return fibonacci(n - 1) + fibonacci(n - 2)
    6 p" H2 E# r; r! H9 X5 M
    : t7 }3 B3 x( ~" }# Example usage
    , ~" n3 b% C2 w4 |6 j% k" W  hprint(fibonacci(6))  # Output: 8
    # x- h! @/ h$ m& r6 T  E
    # h8 D. z/ c2 K6 lTail Recursion: q9 |- |. Q6 U7 o/ t
    ! Q; s) o. [% T2 K/ ]% f& q+ N
    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).
    0 N, C0 T! @6 B. P+ o0 G& Z4 p" |* h/ N6 h' G: k
    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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