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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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  • TA的每日心情
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    2015-11-30 11:11
  • 签到天数: 2 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
    ( v& N) y% {6 ^2 M$ V/ U8 N5 |( O& c# {6 l! Y0 T$ |
    解释的不错/ T/ k) u( S4 \( }" c

    6 O+ S3 C0 r6 q6 Q! C1 G7 x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。0 W/ p  [( ]0 j# L  _' a7 u

    ' f8 P9 A0 D1 ] 关键要素4 K! F/ f: O; o* B* b
    1. **基线条件(Base Case)**
    3 B' g4 X8 a/ \4 _$ l& a( f   - 递归终止的条件,防止无限循环: ]2 f* [5 s0 T( Y7 z5 _
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
      a8 ^- K: ^0 S& `4 P& k; C0 I3 M
      m$ c  r! V+ O' u+ J# o2. **递归条件(Recursive Case)**( S  a+ {- Y5 J. t) i, Y
       - 将原问题分解为更小的子问题
    . c: |9 l  W8 ~  B2 |3 t8 U" }  y   - 例如:n! = n × (n-1)!- a+ o1 S+ _, v4 T1 E

    2 B# B+ l6 P* D: ? 经典示例:计算阶乘3 e% q( w" ~0 ?. k& A  Q& v! X
    python
    ; b% O: m! ^. `2 u! I9 }& K6 kdef factorial(n):
    ( @4 j6 ^6 ~0 V% [7 A    if n == 0:        # 基线条件
    2 s$ Q) Y5 z1 T, U        return 15 |! n' Q" f+ H: y  T/ ^* z
        else:             # 递归条件
    2 u2 y8 ~% q( P# R/ b        return n * factorial(n-1), Q/ y  g9 F: }% I0 O
    执行过程(以计算 3! 为例):
    " ]; G: E6 G0 a0 o% w9 M' ~factorial(3)  `! ^& ~! O5 Y
    3 * factorial(2)& t7 S- t' d+ p- Z7 e% C0 u6 q% Y% S
    3 * (2 * factorial(1))
    ) w5 v  v. ?+ M8 U9 H: ]. O3 * (2 * (1 * factorial(0)))
    8 m7 s# d' E- i+ \6 I% I) ~+ j3 * (2 * (1 * 1)) = 6
    " |. ?; Y8 |( J- D6 }8 w# T- {! x2 E0 v: F: d$ `6 W
    递归思维要点3 B$ v/ \$ q6 `* d
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    0 P$ `! i; J; p1 e7 B1 r9 s2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( y: \: o0 h. k& x5 H
    3. **递推过程**:不断向下分解问题(递)
    " l& l* _5 w+ G2 ?5 t0 Y4. **回溯过程**:组合子问题结果返回(归)
    % O& A. @: i$ w4 c1 H' d" U* _0 G; g" x1 Y
    注意事项9 b2 \  c1 u: g8 V7 Z* p
    必须要有终止条件
    & z9 N4 H/ ^( p# l# P递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( K- P( f6 y( b
    某些问题用递归更直观(如树遍历),但效率可能不如迭代0 e& t, p) L! m! l. R9 B. W7 L8 q
    尾递归优化可以提升效率(但Python不支持)
    ' v" H0 v% u) f4 l  W- _/ [  C/ f1 ~2 }
    递归 vs 迭代$ ^4 g" T9 C9 e( E3 F$ _2 g. R
    |          | 递归                          | 迭代               |
    % a! H' U- p) N% L4 ^" o( o4 R|----------|-----------------------------|------------------|( G- e' ~0 k, q6 Z- E$ b
    | 实现方式    | 函数自调用                        | 循环结构            |0 }6 y0 P; C- x2 l8 \/ v
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 R9 k2 R' x2 O) V' j$ v: e| 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; ?- }5 {3 x, `| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |7 |* L5 u7 l9 n; c

    2 t( {/ M3 w2 w) B* D4 E# [8 B4 u 经典递归应用场景
    # g* d9 [! g3 ^6 f1. 文件系统遍历(目录树结构)
    7 M' e( j7 o, L: s. k2. 快速排序/归并排序算法
    3 S' Z/ u8 z+ ?, u7 o/ \3. 汉诺塔问题9 ]+ t4 m7 k1 T' |  B( N
    4. 二叉树遍历(前序/中序/后序)
    ! l, o/ }8 Z  ]0 p, r5. 生成所有可能的组合(回溯算法)
    1 {. }! z0 m- m" `' y; q. e. a/ |7 V9 F) `- B4 G) \; d, J
    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
    5 小时前
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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,
    + l0 |0 B/ G" h# ?/ a我推理机的核心算法应该是二叉树遍历的变种。
    - z6 i$ O1 \- E4 ]2 U  N% 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:8 E( A8 s8 e5 b, q1 M! H9 L
    Key Idea of Recursion  p% W) Z" [+ M8 j! ?. h

    ' f1 i9 R6 C7 E+ V( }; jA recursive function solves a problem by:8 \6 r: u7 \( ~# H) F
    , f7 @# f# |" D5 h
        Breaking the problem into smaller instances of the same problem.
    3 d! P0 Q- z& n" E5 D+ l: L# H7 Q. ~
        Solving the smallest instance directly (base case).3 \+ W, Y, y7 s* \1 d$ D
    ) E) e9 [& n: q
        Combining the results of smaller instances to solve the larger problem.- T  r- H6 w; h% k0 M6 L' q

    2 Z. \0 ?; j8 mComponents of a Recursive Function; _8 v5 y! L: K/ j2 A. `/ {7 @1 y- I$ Z
    8 o. b) A; k6 m# A* I. _
        Base Case:, x' j+ V* l; F, b

    & D; n; p) V! A/ u  R! P0 b% Y        This is the simplest, smallest instance of the problem that can be solved directly without further recursion." @; O% ~6 `' ^2 m" B

    9 _) \' X( r5 e! W4 q, L; u2 X        It acts as the stopping condition to prevent infinite recursion.
    / G& L2 [' j/ A4 K" |* y& v% u
    0 g0 @. |1 w' _. @        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    3 D* D) `% H0 I$ U# X" b8 B  ^1 k5 ?, N" b0 |4 q
        Recursive Case:6 U7 J/ a3 m. j) s9 m, `7 f( b

    & U* G; j5 \) j        This is where the function calls itself with a smaller or simpler version of the problem.0 Z$ a5 c' U6 o1 ^4 M+ F# W

    # @4 J: o2 v' O8 C# k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    + w- f6 G7 y1 j. v* I
    ) y+ a8 ?% e' Z- v2 C9 v% `& _. KExample: Factorial Calculation
    4 U* q8 q, X1 }+ Q8 I: N% W  O  ~* I/ N$ i' v" W2 b, `# r! L: y
    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:: f2 ^8 E1 Y8 b
    2 g4 J2 c0 h6 c3 N. T" ~4 k4 T$ K
        Base case: 0! = 1
    3 J/ G8 d* C  |' U# S# }& f
    ( C( v* w" V' d7 y7 P4 m! @' c    Recursive case: n! = n * (n-1)!1 \( p" f- E( P7 t# R. o
      F6 Q* @5 w: b- ^1 @+ u. v
    Here’s how it looks in code (Python):
    5 w9 k' z% \% m' E; zpython
    1 v" a. v; C# J% h
    * O+ \8 `$ |! V4 O
    8 y  ?% e5 ^# X% L; Q2 j* p- I: mdef factorial(n):9 d4 u; Q- {# ~- e! |/ N8 M; T9 b
        # Base case) O' G  g1 A( v  Z9 K8 ~" Y& g
        if n == 0:0 E( @1 ?7 G* X/ n
            return 1  U6 |8 V) Z% S8 v
        # Recursive case
    : }7 \5 G7 H2 {7 ^* |; Y: K0 d    else:
    ( K! b3 }+ a, Y5 w7 b        return n * factorial(n - 1)0 a; u% z! }( A$ p* R6 E
    ! N% X8 z) {( D
    # Example usage
    3 `' t7 |8 r+ u2 Nprint(factorial(5))  # Output: 120/ _- f/ D+ h* \7 n" f( S3 Z: s
    1 H4 c4 f) Y0 r
    How Recursion Works
    1 b+ o6 e0 B- F. n
    - X) Z( \1 u; z* x    The function keeps calling itself with smaller inputs until it reaches the base case.1 Q& e. l% }3 ]( z' X# E

    2 l, m1 x3 G7 j' S3 ^0 Q) x7 k    Once the base case is reached, the function starts returning values back up the call stack.' s- U, t% v/ X+ A
    & E' p& k5 R) l" s. S6 ^8 y0 P
        These returned values are combined to produce the final result.1 C* H7 a0 p& k5 L5 \  \

    # @; }  F: K# f: M; D. sFor factorial(5):* H+ T4 k2 j. ]
    8 _- n& }! E# d6 b) Q

    4 B, W+ ~( P: w5 jfactorial(5) = 5 * factorial(4)
    % u3 ^2 M- J+ [8 e9 m/ I2 yfactorial(4) = 4 * factorial(3)- D! q4 c$ r; }% ~/ A# C
    factorial(3) = 3 * factorial(2)
    8 L9 E; Y1 `5 O4 O5 Dfactorial(2) = 2 * factorial(1)
    + e7 s' ~8 C, ~7 r1 ?) B' h8 d8 ^% ~factorial(1) = 1 * factorial(0)
    1 ~& R+ L1 V- N" _# K8 y( ]/ {8 kfactorial(0) = 1  # Base case! T  l$ ^5 F2 D0 g( N

    , N; S+ Z3 |6 @  q. T& oThen, the results are combined:
    ' u9 N0 a) f! @
    / x0 `7 q% H7 U- m, F
    - @( J9 i6 \9 `- }. Hfactorial(1) = 1 * 1 = 1
    % h& t1 J+ @5 t. O1 T; Vfactorial(2) = 2 * 1 = 22 u8 {5 G/ j& V# o8 c
    factorial(3) = 3 * 2 = 67 X7 N- T8 e4 J6 a, U
    factorial(4) = 4 * 6 = 24
    , }/ w; q( c5 M. X3 v/ z8 D3 Y& b& Pfactorial(5) = 5 * 24 = 120" Y: |9 n) C' `/ F1 E/ [  W) X
    0 b5 d6 ]) q# s% b& B
    Advantages of Recursion
    1 p5 i# {4 X8 L+ ^. b& L; U9 x& n- {: @
        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).
    * w! ]6 J2 E7 I& o( y7 x" ~" G
    3 |  K/ R" Z5 r1 d) i    Readability: Recursive code can be more readable and concise compared to iterative solutions.( `; F6 ~4 A' B  \( P/ C
    9 r* q2 Q2 H! R' u- m8 Y$ K4 J
    Disadvantages of Recursion' }4 v4 |  G' _5 I* ?) J+ a

    ; B( _8 \( p  ~    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.' E  X8 N. G6 I8 R- z/ B# m" x  Q
    % W( @4 V4 W+ A" Y
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).* J" q, ~. q9 j0 q+ e% C
      T: [! O# F/ B& t( m! a! F
    When to Use Recursion- @5 O" b" L) Y) {6 P
    1 T5 }4 F; A& Z. u' W- k5 p
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    % x2 G" |( F" r0 }( [' R0 K8 t2 m; N* j* Z0 A8 k* J6 o; Z
        Problems with a clear base case and recursive case.
    6 Q# [6 A, @. ~$ v% a$ m$ J( D9 w& Z  ~
    Example: Fibonacci Sequence
    " z% A: Y% I8 ?7 V8 T: ?3 H$ }8 d# i7 h0 {
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:" d. g! E& Q# B: T- J
    3 z8 {: q% V5 [
        Base case: fib(0) = 0, fib(1) = 1
    7 e* y! G- ]+ g; e% H, N! s5 R0 w; e, _& A+ U
        Recursive case: fib(n) = fib(n-1) + fib(n-2)8 x$ }4 x1 {$ f- u- l
    , [! @1 X. a' Q( l6 S9 z
    python
    # G3 X! V4 E# k3 Q4 |5 l& Z" s. y! q! @9 A  t, E) V6 f) _: r2 O

    1 B: Z% C* ]1 T) K) pdef fibonacci(n):1 d4 _/ M# r) g8 W% V! z+ _" a
        # Base cases
    # l$ _) ~- u2 y$ a2 {    if n == 0:
    ) q! h; [- w2 f* G) D        return 0
    4 }: T2 w0 O8 j9 D    elif n == 1:6 V' |! J# Q" x! x  P  E  ^, f
            return 1
    & [! T: O+ l6 j' U4 |8 n* C2 t2 `    # Recursive case: Y$ A$ Q5 }& ?2 f& V
        else:- Z7 n. ?; U4 u4 `! g1 i
            return fibonacci(n - 1) + fibonacci(n - 2)
    1 u+ v& h; F) M0 a9 R/ q3 ]* |# h* P6 T! }7 \7 I4 t% w8 p' J
    # Example usage
    / A, z- ]9 X! b3 x/ O7 Yprint(fibonacci(6))  # Output: 8$ g9 e. G- r, [

    " U2 U! J9 i' S3 S+ D' kTail Recursion4 l4 t& ~/ K0 X8 x4 U( v
      M- v' c) H5 _9 P7 |8 f
    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).. E/ L: |; @+ m) G# ]8 i7 ?" `2 F' O2 H

    0 ]3 |1 C* m! y8 h1 l( JIn 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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