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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 8 k: i! p- o  `& P* z% r. ]
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    解释的不错0 D2 S% _9 Y0 Q5 }. M+ E* _
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    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。
    9 W4 c" y: `. `; F# j. T/ ^  b' p" }
    关键要素! {/ n3 b- }! @& Q# F
    1. **基线条件(Base Case)**: M! H$ G) D( _1 w; |/ W, S( A
       - 递归终止的条件,防止无限循环# M5 D) J- J) F* \) Q# t; j. ~' e! y
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 10 i; E- Y% y0 G; @

    9 \. o: `7 F% F& K2. **递归条件(Recursive Case)**# c) n6 Y7 P/ y* o
       - 将原问题分解为更小的子问题  e' J8 q6 Y% e9 C2 [, b! D) u8 c
       - 例如:n! = n × (n-1)!
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    : }# _8 ^' {3 D 经典示例:计算阶乘' q4 Z2 M) e2 R$ M+ G
    python
    ) T! W* Y4 e2 ~. sdef factorial(n):& W# ]0 ~# k7 l" h
        if n == 0:        # 基线条件, y# b0 B. F: r
            return 1& N1 S1 g1 y2 l% |' B6 L" C0 z
        else:             # 递归条件
    / H  V; F! b0 q+ z3 o, [  K( b        return n * factorial(n-1)' t" K3 r6 |; g1 p5 d$ E
    执行过程(以计算 3! 为例):: L7 n7 {/ h8 O& Y
    factorial(3)
    / U" k9 P, b& `& G! s# C# @3 * factorial(2)& ]4 Y; t) C: g9 G1 `9 M' m+ C
    3 * (2 * factorial(1))
    2 V+ D2 @3 q2 i9 Q4 g/ D0 b. m3 * (2 * (1 * factorial(0)))" x: E) Q9 S$ y5 |- g
    3 * (2 * (1 * 1)) = 6& G3 Y8 v7 x( G3 i8 \7 O8 y" m! {
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    递归思维要点% P+ j/ L1 @+ X. b/ K( Q/ b
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    9 e" g% {7 ]& _& j) b0 ^2. **栈结构**:每次调用都会创建新的栈帧(内存空间); S  q% Y9 Y! a2 O  `- n
    3. **递推过程**:不断向下分解问题(递)
    7 B3 r- K1 E4 I1 t) V( ]4. **回溯过程**:组合子问题结果返回(归)
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    % @% `4 {# f! v 注意事项! i+ c5 M  f; C$ M+ {+ t
    必须要有终止条件$ o; Z1 f4 O& ~3 @! m
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ' M% G" `9 c# C' H3 n* A某些问题用递归更直观(如树遍历),但效率可能不如迭代4 v  R1 _% K3 D- a
    尾递归优化可以提升效率(但Python不支持)$ @8 l% m6 c0 P, o
    & [/ v# H* s2 r  P+ R4 ^# g
    递归 vs 迭代* K/ _) {3 u/ [. w; F" t
    |          | 递归                          | 迭代               |" G' v4 s5 C+ h7 O' z5 b
    |----------|-----------------------------|------------------|: l3 f/ }7 g  K5 S2 ~3 I
    | 实现方式    | 函数自调用                        | 循环结构            |) I4 {7 G# g* K- P5 T8 u, C
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & y  O/ P- T7 T2 B! n| 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         |: Q8 Q- D8 R# R% z& j% y% n1 }
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |: u. J4 |! ]5 J! ^7 ?2 x4 O
    & C6 ?8 k4 @  C% ]* U1 N
    经典递归应用场景, W/ w8 s3 y, F( p
    1. 文件系统遍历(目录树结构)+ E" G$ ^; _* |
    2. 快速排序/归并排序算法: v/ m8 v& }8 G8 `; E+ i$ M" f$ w
    3. 汉诺塔问题, z7 J6 r  I% R" o" r4 ]* |
    4. 二叉树遍历(前序/中序/后序)7 e! n& T: I- v. s# s
    5. 生成所有可能的组合(回溯算法)
    ) @( w7 [/ k! ]+ Q
    + y4 f: Y4 ?  m% U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
    7 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,4 Y7 G4 L4 i3 E! V' A
    我推理机的核心算法应该是二叉树遍历的变种。
    % G( `# w( ~1 \) f& 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:: a, C# o8 `6 M+ U) D
    Key Idea of Recursion$ \, _  O9 H7 p; }$ k+ X
    % d9 }" N0 O. X7 K8 n* K9 M$ Z8 m  m; g- I
    A recursive function solves a problem by:
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    " E7 _" v" _  _' W# l  g6 v' u/ \    Breaking the problem into smaller instances of the same problem.
    " t) {8 \' a* A- W) [) |$ U
    : g& v7 A& d: H' [    Solving the smallest instance directly (base case).
    & e7 s+ t3 y4 Q% x2 {
    , C( h3 H* ~$ R4 f( H7 `/ F5 g    Combining the results of smaller instances to solve the larger problem., l8 J4 T9 @1 H% \4 ^$ Q

    2 i! O( m1 m4 H) p: wComponents of a Recursive Function: g' U$ ?2 J2 c1 i9 M3 w$ G  i
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        Base Case:
    4 a4 @0 K& y* K+ B4 F4 D
    + W2 x5 l1 n- `, m" x' j; @, K; P        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.+ X! S. B* ?& K! D) @4 ~, s: F$ @

    1 y; y0 y& h( l& j$ l! H: e        It acts as the stopping condition to prevent infinite recursion.- P7 `4 Y+ `2 [7 [. P( B
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:  C+ N5 r0 ~, C. u9 x

    9 [  Z$ k$ A* l1 k3 k! b        This is where the function calls itself with a smaller or simpler version of the problem.+ C5 E7 y' O1 O% S* k
      R/ |5 R. z- J) O( q2 x
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." K* `$ k% u6 ?6 Y3 a2 B3 Q/ n' \

      \6 \5 \& U% J" n! {3 j; Z0 Y1 KExample: Factorial Calculation. t, h! Q7 W$ ?5 I3 m4 C: M! E
      r8 r* U% q' a/ @& m* C/ h
    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:
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        Base case: 0! = 1
    8 ^' |& K) a. w9 _1 n+ i# ~" N( e# L- y& v5 W# k; T3 d# ]* V
        Recursive case: n! = n * (n-1)!/ r+ f5 R+ i# N
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    Here’s how it looks in code (Python):
    * c; Z) ?0 h2 ~; J7 \- s- G9 M- [python) A: p( N3 h* |7 }( j( c) ?

    # N  @8 c: f/ a! ]
    2 F. U5 J6 [9 i# q/ K0 hdef factorial(n):
    " K" z( V% X. f    # Base case8 Q( p* G# F! _1 Z) N" l
        if n == 0:. D3 ^1 N- c! R5 T% h" K' D
            return 1/ c0 c: h- R: i, n( c2 o
        # Recursive case/ V2 n- `0 f0 {) ^
        else:
    ! w1 f4 q% g% ~" N" o        return n * factorial(n - 1)
    4 C8 R3 ?% N* f
    3 g8 b" H2 H+ ~6 S; u# Example usage4 [! ~4 E' S' H
    print(factorial(5))  # Output: 120. k) x) p: D3 h5 n- w8 R

    ' T, z1 V; r6 V( l- ]4 gHow Recursion Works  }( E: K1 U$ k! F
    $ \7 M8 j& j+ I# ?0 V6 J
        The function keeps calling itself with smaller inputs until it reaches the base case.) V, ?. p  S2 M! X$ q$ f
    4 K# [1 g# H) J" p& H3 P% X; ~
        Once the base case is reached, the function starts returning values back up the call stack.0 c/ V1 b/ S7 p# w1 f3 J: l. d
      f# b+ y5 z+ [* L2 z# ~% j. X3 S
        These returned values are combined to produce the final result.
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    For factorial(5):! m* |8 y; w% B) t

    ' ^2 A8 ]7 [: a) N! J% u( Z* k) R1 }3 J1 a" P
    factorial(5) = 5 * factorial(4)! x: Z& c3 f. j, C5 P. l8 a
    factorial(4) = 4 * factorial(3)
      A3 E' U9 z6 M) j1 z8 `. {' ]factorial(3) = 3 * factorial(2)' J! Y$ F# P/ Y; P
    factorial(2) = 2 * factorial(1)
    # A) Z3 {, T4 Z) v/ @factorial(1) = 1 * factorial(0)
    2 I' y" {, I: @4 ^factorial(0) = 1  # Base case7 D  m& y4 w' _* l
    % t6 _2 q1 @( c
    Then, the results are combined:3 F* o% {9 W# J2 N

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    factorial(1) = 1 * 1 = 1
    # G; o! q3 W+ I7 Q. Z. ]factorial(2) = 2 * 1 = 2- _$ o: j  O* V3 R" {
    factorial(3) = 3 * 2 = 6
    / O9 j/ `$ a5 m' t- H& j3 q! Wfactorial(4) = 4 * 6 = 24' l. n  B% Z$ j( H" O( D6 j9 F
    factorial(5) = 5 * 24 = 120
    % P: A% k! O( F6 O+ ^3 N3 f
    # }( D& V8 K, }6 _/ v( P  C% U" k/ ?Advantages of Recursion* E) ?# K- C! |8 k
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        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).
    , f- e$ A3 T$ G( u( u& {' g0 k6 G! \+ j) `! X5 |$ {) o
        Readability: Recursive code can be more readable and concise compared to iterative solutions./ S  B% U! |2 S
    ( {) Z7 @) x! _! T: j& F/ w' I
    Disadvantages of Recursion6 v7 W- Q! I$ A

    9 o+ C; Q. Y( t0 `- z    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 D$ l" l5 d3 f' m9 |8 Q4 n& U6 f
    ; z0 w9 Z. K( i1 i    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).& [* `6 A% o, E' _: J
    : A3 e0 t* U2 d& G7 X
    When to Use Recursion/ i1 j9 q4 k' ]. H" B3 l$ a3 A

      S$ \/ O7 L5 }6 w0 s6 Q    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    & Q$ e* y. G# `8 ]) b+ r: V5 D6 Q  _9 p' J/ ], i4 B# U
        Problems with a clear base case and recursive case.* V+ J- E  L* F! C
    8 ~- l0 s) y2 T+ r6 g/ ^' }
    Example: Fibonacci Sequence
    , A4 m8 c; N6 q% j2 F+ R& y9 l  C% M5 E, O( Q+ Y$ f* Y
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    + T6 U8 a& W2 C  t( y3 j7 P
    & _) O: R' p  e7 t. X0 b    Base case: fib(0) = 0, fib(1) = 1/ Y+ n1 z* D/ Q& w

    & A3 [% _  y; m* B* O; ?  t    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    - E' u. Q/ X! d, n0 }1 J/ B! Y7 m# D/ u' ^& \
    python
    ! _& P9 M2 k+ ?7 \& q" a, G" A4 a! n

    + I0 F# \& V( J5 {# W) T5 ydef fibonacci(n):; D3 e8 b. K: }  @# d4 M
        # Base cases
    3 }7 W0 V2 v9 U$ j5 e6 P- H# h9 k    if n == 0:7 u$ P: V2 H  T" x
            return 0- A" [( ^  @1 n$ m4 W3 a/ J( f# M# P
        elif n == 1:
    * m" O2 r8 V$ T1 m        return 15 D$ r& j+ m7 i6 ^  k
        # Recursive case6 ~! d, n; o# V" c- A3 g
        else:
    - ?& u8 J3 [1 C8 J        return fibonacci(n - 1) + fibonacci(n - 2)( O5 Z! F! O% u* g' c: p. I

    # N& ^# r/ p9 l8 K$ u" b# Example usage
    7 L2 r% P7 ~& a$ T) `, f% Aprint(fibonacci(6))  # Output: 81 I  H8 \+ W% h4 h% |7 _

    ; w& Z* N. h, r9 A, O) ]8 {9 xTail Recursion
    1 S* ?7 R/ X- _6 L" P+ Q2 y! |( K* c1 S  L0 e4 {& c9 [4 |7 f5 @
    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).
    # G7 K: z7 x6 S' ^, L7 U+ f
    9 t, t% f0 ^6 D+ RIn 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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