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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
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    解释的不错+ A( K' ~# b) R  J
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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。
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    0 K8 C  ]# p6 k7 X" O 关键要素9 h8 l, i- ]$ _& D1 q7 \" E% ^
    1. **基线条件(Base Case)**
    / ]: ^( h$ A  \! c   - 递归终止的条件,防止无限循环- t+ }: x8 a8 J  j0 z
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    # J8 W1 z: \0 d' G6 O+ u! D2. **递归条件(Recursive Case)**4 w" s% D) r. T- W
       - 将原问题分解为更小的子问题
    9 a. ~* q; `$ k   - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘. ?  g4 e8 o! k$ y9 h6 z) @
    python. H$ F3 p1 X' K1 I7 u7 Y
    def factorial(n):1 F! ^2 k" {! G8 s" {: T: l2 j+ X" X
        if n == 0:        # 基线条件
    % S$ t6 [$ {* s$ h) C" t) v, i+ J        return 1" ^0 D) y2 v9 `5 _) c0 C  w) N4 e
        else:             # 递归条件8 f9 C5 m7 k/ r. Y: y8 G  o2 i; c
            return n * factorial(n-1): ^1 ^" s& X$ b. ~
    执行过程(以计算 3! 为例):. b8 Z- u  J0 W, I1 h, P" W
    factorial(3), w; K* ]- B5 D9 {9 B# I
    3 * factorial(2)
    # T5 g& P# S! ?  ~/ Y8 K3 * (2 * factorial(1)). u8 |8 c' A0 g% g, [0 i( T: Q
    3 * (2 * (1 * factorial(0)))- v: U' a2 R3 m1 q
    3 * (2 * (1 * 1)) = 6
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    - l9 r) L* k5 Z/ _) {  @1 \ 递归思维要点9 V8 A3 n$ f4 {) k) N4 h0 m
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    + s4 {1 ?/ M1 M9 M2. **栈结构**:每次调用都会创建新的栈帧(内存空间)) h' e6 I  q8 }; R
    3. **递推过程**:不断向下分解问题(递)
    * T7 c9 T/ S% }1 }5 r4. **回溯过程**:组合子问题结果返回(归)6 \0 @, ^9 K) I- E. ?, w2 ~

    1 v# e+ ^9 [; ~: ~' b! H 注意事项
    3 n9 @" f5 w* w( ~- _6 U; h必须要有终止条件9 y1 ]6 l  J: H6 M6 V
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    : s- L) _& _; Z! N) o某些问题用递归更直观(如树遍历),但效率可能不如迭代
    7 D8 f; p  `# z0 X7 w尾递归优化可以提升效率(但Python不支持)2 F; J$ m( Y- T' T/ e1 f7 w/ ]

    7 i' B5 j4 R5 }! N 递归 vs 迭代( _6 ]- ^7 m: c: a7 z+ p! F
    |          | 递归                          | 迭代               |
    4 G* E. y+ ^  T& R|----------|-----------------------------|------------------|6 Z4 A" d% V$ w* X5 B% q2 T0 e
    | 实现方式    | 函数自调用                        | 循环结构            |  J' b! O% n! l" D* u$ ?: C) Q3 l
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    * ^4 d) e! h$ O* q+ r  ~| 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         |# o" D' ~! q1 S3 B
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |. w3 A2 l' y4 P8 K3 Y

    + `) `2 @/ i# W, M! W 经典递归应用场景% S, \, n" m& ~
    1. 文件系统遍历(目录树结构)
    7 o" h9 H1 c# T2 z" @) Z. w2. 快速排序/归并排序算法
    , J$ r, p: ]0 F& p1 D  Z# h, S3. 汉诺塔问题5 L# T2 ], f: B* ], K
    4. 二叉树遍历(前序/中序/后序)
    ! [- u2 J# j& J$ {5 o  R) K) U* ^) L5. 生成所有可能的组合(回溯算法)
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    ' r6 E4 I" V* k7 b* F! N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,* ^$ Y8 y3 t$ \/ B% }+ q4 Y
    我推理机的核心算法应该是二叉树遍历的变种。
    ; f; P2 b2 C* O+ a- w! v  E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:
    5 p. |5 y9 N3 b& [$ dKey Idea of Recursion$ @  `, b& E4 ?  Y) w" A3 K

    ! s: i1 M2 S! }" y% @A recursive function solves a problem by:+ R4 W( n. P3 z6 s6 E. W& ?+ M1 z' z

    1 l8 \) W; C" d. X( X    Breaking the problem into smaller instances of the same problem.4 O& C* n$ U3 x+ ], d+ V& A2 i! B  G0 Z
    $ p! w6 B5 x; e3 l
        Solving the smallest instance directly (base case)./ Y7 b" h/ H" k3 a
    + L0 G6 Q# _* C" n+ N
        Combining the results of smaller instances to solve the larger problem.! I( L; }. _, v4 E6 n

    7 G! B4 B( |0 h7 q, ]* s+ [3 yComponents of a Recursive Function
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    ( m* O) Z+ C% j" F% K    Base Case:9 J4 d$ d- l7 A. I& e* L( O3 F
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            It acts as the stopping condition to prevent infinite recursion.% B# W. c. A6 `0 p
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.6 y1 [$ p/ z6 h. u! [8 ^( g
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        Recursive Case:
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    % c: s/ O) p) }# ^/ N. i        This is where the function calls itself with a smaller or simpler version of the problem.
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            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation
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    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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    . I! w% F, E1 k    Base case: 0! = 1
    " J/ f, c/ Y; M$ s+ _8 N$ b7 `3 b, L3 R
        Recursive case: n! = n * (n-1)!- i0 l: {' S' o1 j& g

    , Z( }0 X, N# i( z& l! M# E% Z& mHere’s how it looks in code (Python):) j- H, v8 ]( s9 L/ R+ H' l
    python
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    + C% [" k$ r+ Zdef factorial(n):
    & J: q5 m1 g( V3 \# P, ^: [6 z, U# N    # Base case: C3 Q' r! i, i: Q
        if n == 0:
    9 B- @  r$ A9 S, V2 J        return 1
    : e, Y$ B" c; u* m5 }+ Q( q    # Recursive case. a/ P7 ]& d' k2 T$ i# y5 H
        else:- Z7 j* Q3 n) k2 n) u5 B% W1 t
            return n * factorial(n - 1)/ @6 @3 r  c* c) J3 f
    9 U/ z! I; J7 m9 F3 l- T, Q
    # Example usage4 S) v5 k( B( J! I9 P3 j, y" j
    print(factorial(5))  # Output: 120
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    How Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.5 m  D2 o8 f: k! R, _$ n7 i
    % S1 B9 k) }8 {  q& i* P" V2 l
        Once the base case is reached, the function starts returning values back up the call stack.
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        These returned values are combined to produce the final result.
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    For factorial(5):2 K1 o7 B) b, R- @( Y# S
    " G) q+ O, G4 b) S' T+ h' _
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    factorial(5) = 5 * factorial(4)
    + R: K; M6 @' c$ l6 mfactorial(4) = 4 * factorial(3); n( J2 g* Y+ _+ C
    factorial(3) = 3 * factorial(2)+ r2 D8 y. J2 {2 ?
    factorial(2) = 2 * factorial(1): s2 T) M0 j7 W2 `
    factorial(1) = 1 * factorial(0)* y' Z6 J+ P$ ]: ]
    factorial(0) = 1  # Base case2 V7 j8 X. e8 u

    9 D) P5 D% R5 w2 LThen, the results are combined:
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    + o0 \, C; q9 {5 o( ^" W6 w, H/ M' d" I8 h
    factorial(1) = 1 * 1 = 1
    " t% B7 d1 h( kfactorial(2) = 2 * 1 = 2: w* `! y6 |9 A* M
    factorial(3) = 3 * 2 = 60 d6 {9 G* Q. r# O0 v2 O9 F' z
    factorial(4) = 4 * 6 = 24# a' }7 g+ V: q
    factorial(5) = 5 * 24 = 1203 }# b- I5 }2 ?) X$ T
    8 p4 n7 R, J. @' _% C. q9 e, z; P
    Advantages of Recursion' Y7 F' |- Q1 E7 G

    ; N3 f, e/ q% ]    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).+ \  d/ t7 V; ~. }* k

    , U7 T4 E# M! f    Readability: Recursive code can be more readable and concise compared to iterative solutions.0 L: p1 Z5 J# z  _6 F6 s0 J1 g
    & n% H9 I) x5 m# _8 b) x9 d
    Disadvantages of Recursion+ M: [0 P( e6 J6 _

      X1 l  |1 \  C8 |' q    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.
      m! Y0 d  D" y; \6 Z6 @2 J3 o& c8 v; T5 A
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).8 _7 n8 Z8 z3 N2 ^7 [8 t& ]5 V
    ) C1 q) X& v' a/ h5 {+ O
    When to Use Recursion9 ]2 w2 [' }1 u" w9 |9 n, r
    & q$ i& x# _' f$ Q$ D# b
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    - J: t* W. z" d/ r  Q7 ?% g- l4 g3 F% C$ d: G. }' ?
        Problems with a clear base case and recursive case.! f) G: Q/ I5 }3 M
    8 `0 u  D4 \' A  u" N, k9 y
    Example: Fibonacci Sequence
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    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:4 T; p" d& s3 a
    . ]7 v- X% V7 i  ?; u
        Base case: fib(0) = 0, fib(1) = 1
    . t  D! E( L, c8 n4 ]4 v: Y/ O; G8 G9 T
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python
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    3 E/ M2 D" L3 j/ {
    def fibonacci(n):
    2 e( a' Z8 v/ R6 T6 W    # Base cases
    3 b8 g9 V) |$ g8 D! v- L6 n    if n == 0:
    7 Z- X  k& a5 {& Z        return 0
    " |; q5 z1 w9 o' o6 x8 F    elif n == 1:0 T! g, k3 Q. \4 d; b  t
            return 1
    ! N& j9 M# V4 M( z6 h) c$ J$ o    # Recursive case  y: v9 e- v+ R4 |; |' x
        else:3 x9 V) R/ t& a
            return fibonacci(n - 1) + fibonacci(n - 2)
    . R9 x$ B, B. O4 u% [( R9 W! ]% }9 g1 J: w* y& `5 h! s
    # Example usage- |$ }7 ]) P1 R( n
    print(fibonacci(6))  # Output: 8! `) U7 a! I) ^* A+ u
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    Tail Recursion
    7 J5 A/ ~% |% H; }" [2 ^- x6 V" e' w1 y" s) B: j8 d% U! z+ R
    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).& V* ^! r' s8 p' u9 ~3 x

    . V: k! l8 e9 ]0 `9 MIn 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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