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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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    2025-9-8 05:08
  • 签到天数: 3 天

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    楼主
     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
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    6 F1 b. \5 L' V/ z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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    4 W/ s; X0 L* Q, w- e$ g; v 关键要素, i$ T  S0 N4 X9 ~
    1. **基线条件(Base Case)**
    ) Z( ^5 y8 U; M- O0 n4 V$ J. T- S" H   - 递归终止的条件,防止无限循环
    2 |/ T/ L! V" Q: S- w1 S+ u   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    2. **递归条件(Recursive Case)**0 P! W" a( k+ T* y( W
       - 将原问题分解为更小的子问题
    . Q! `8 l/ h( u   - 例如:n! = n × (n-1)!+ _5 |9 i! l. `5 \. w
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    经典示例:计算阶乘
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    def factorial(n):
    : m/ T8 {+ k$ A6 n% `    if n == 0:        # 基线条件( S; X' m- N& r& D5 o
            return 1
    6 v5 u2 Z" z* D6 K. V, v, L- q    else:             # 递归条件( a% b7 E9 G& f7 C: `
            return n * factorial(n-1)+ O7 [/ B% J- B5 H/ m( _
    执行过程(以计算 3! 为例):* n* C! G7 c' h! O
    factorial(3)% P! {/ p4 `( q5 H- }
    3 * factorial(2)& }8 _$ P9 K- r  }! Z6 J
    3 * (2 * factorial(1))
    ( h( N& W( O/ F  w3 * (2 * (1 * factorial(0)))
    : G1 a. @1 I8 G3 * (2 * (1 * 1)) = 6
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    # l* I; a. [8 K 递归思维要点
    : `+ G  E  P- ]& }+ w$ \1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    . E2 [3 C, X1 r' U! B2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 s1 n# K- w. U+ K* M
    3. **递推过程**:不断向下分解问题(递)% N0 ~  t5 k( X4 J* d
    4. **回溯过程**:组合子问题结果返回(归)
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    - X* }! X1 x; |. h5 C 注意事项) }! h1 E) F& H1 o& B
    必须要有终止条件
    0 E- d7 g' w0 }/ J) M8 q  B" r递归深度过大可能导致栈溢出(Python默认递归深度约1000层), K' |4 H. X4 Q$ p
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    - `4 D: t  B, r7 Y8 [尾递归优化可以提升效率(但Python不支持); [2 j0 \; K; n

    9 O2 r$ Q4 S* v% I! v 递归 vs 迭代" ~/ g$ C' K: ], Q
    |          | 递归                          | 迭代               |2 h; I9 P3 T: @/ u  q2 b) J% e
    |----------|-----------------------------|------------------|
    6 O+ P. e  R; f# W$ P- Z9 r| 实现方式    | 函数自调用                        | 循环结构            |
    4 J5 Y" ^% r3 u& ?9 w. |$ `| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ( l) n' B8 g! A% z| 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         |
    3 H+ i0 y7 z  Z/ e| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
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    经典递归应用场景2 T1 @6 X4 ~3 Y- P- L9 \7 r% B2 k: [
    1. 文件系统遍历(目录树结构)
    ! o; Z4 K) C' u! N3 w2. 快速排序/归并排序算法, W- W  j0 ]: p1 y  g* x1 j7 s
    3. 汉诺塔问题2 t" Z6 [' [! G6 `6 ?# |
    4. 二叉树遍历(前序/中序/后序)
    - M0 c4 B/ w* H9 C7 I, D5. 生成所有可能的组合(回溯算法)
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    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 06:31
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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,
    " Y' ], c) Z  o) m- h7 B6 C# f  `6 @我推理机的核心算法应该是二叉树遍历的变种。" y8 D4 ]* [7 f1 z; c
    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:# F6 ?; V( c" U/ [
    Key Idea of Recursion
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    1 W/ n( a% `3 M1 T" ^3 T' }A recursive function solves a problem by:2 D( J. x. R9 t2 h. e) ~# e, l# t4 Y' R
    9 J" G* h, F5 s
        Breaking the problem into smaller instances of the same problem.
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    5 }, D. L) F7 B9 ?% M    Solving the smallest instance directly (base case).& R. F) R& r( \8 R  O" p; ]
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        Combining the results of smaller instances to solve the larger problem.+ b; s$ f% D6 Q

    3 v# o# ~: |" b3 x: UComponents of a Recursive Function
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        Base Case:9 n( @* l% N7 ]; t
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.9 Z3 z  \9 n' O7 ?
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            It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    , T- R9 M9 V- ]    Recursive Case:
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    4 \/ {0 z5 O+ {! f6 x        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).6 N5 [0 k+ G: y% K2 s' s% b

    : Z0 ?$ u0 \0 O! v/ E) X0 _2 @Example: Factorial Calculation8 ]$ e; d" L& V5 n' K8 a. T. c

    - }( P/ l" }5 R% p% F6 x1 \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  h; x/ F5 J$ C: i- n; F
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        Recursive case: n! = n * (n-1)!6 @  q2 R2 G" ^7 i8 r* {5 m+ \- |
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    Here’s how it looks in code (Python):5 S+ c# \, x- _8 U
    python
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    def factorial(n):$ |! Q. k% x0 F0 D. u- [+ ?- A2 r! Q
        # Base case$ P: M0 b$ J* t5 J  A  |. _# |
        if n == 0:
    1 K3 O# V5 Y3 @* z; c$ a        return 1/ k6 V; l; s  g1 S  m* e' A
        # Recursive case* U4 J/ j% x, M5 R  P! {% c
        else:
    1 ?  h2 P- j) x" @- i        return n * factorial(n - 1)
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    # Example usage
    % N0 p, O5 m$ e# j' F  l, L& V5 m. ~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.
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    - M2 }- }% {4 y6 U' W1 n# P, m( Z4 I    Once the base case is reached, the function starts returning values back up the call stack.
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    1 ^* {! _$ V9 Q9 \$ z    These returned values are combined to produce the final result.7 t9 K, J( W- d' E. u9 ?" S0 T' h
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    For factorial(5):+ M5 H& B) b, I+ Y: r( M! c
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    factorial(5) = 5 * factorial(4)
    & O! I+ \+ w- d( l! f. ]5 p3 Tfactorial(4) = 4 * factorial(3)
      |: R' Y5 o. _factorial(3) = 3 * factorial(2)
    6 C! T1 z) F! ofactorial(2) = 2 * factorial(1)4 M0 G5 A) K9 P5 f2 U
    factorial(1) = 1 * factorial(0)4 O% }) d/ o; N3 l) X' W7 j9 c
    factorial(0) = 1  # Base case: w* a  ?- Y; \' D
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    Then, the results are combined:
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    5 T/ ]7 k/ g- C) a0 sfactorial(1) = 1 * 1 = 1
    ; x8 }- i5 f  W' @factorial(2) = 2 * 1 = 2
    8 A+ x& n3 U0 E& C' W& E2 q4 Q+ Hfactorial(3) = 3 * 2 = 6
    9 E. p2 c! }  v8 ifactorial(4) = 4 * 6 = 24+ E0 ]* T' Z- o5 e1 y
    factorial(5) = 5 * 24 = 120* n6 y& A9 g" h0 _% y
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    Advantages of Recursion% k* B3 [2 M( |7 S

    ! n1 H/ W6 M/ f% r# T- ?    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).
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        Readability: Recursive code can be more readable and concise compared to iterative solutions.8 g, L: l5 R, v  ^7 m- x2 W2 M

    6 f! J2 @/ J* l) K/ \) n5 c* w  BDisadvantages of Recursion6 R6 y9 @( M1 X, o9 k

    , i3 v" v# v% D5 x9 n& R7 g4 D    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.
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        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).3 S$ ^. T' ?2 {

    4 b$ }) t9 e6 P' f. Y3 k. ?0 bWhen to Use Recursion5 ?& I) V+ x. M9 F

    1 @* \+ X& E0 c# Q$ W    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).! _# \9 @2 j4 m; g1 G

    & ?+ \: s; c- z    Problems with a clear base case and recursive case.
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    : _! x* A* |# h5 y% T/ v: g! pExample: Fibonacci Sequence6 m& m# }) d9 V& O9 v; u1 [& L

    ) r$ L' K0 P. O7 l/ d0 e5 r  rThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
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    : T& M) J5 l3 d    Base case: fib(0) = 0, fib(1) = 1
    $ y, v' h' V* ^2 ]% U. l
    - t6 j1 @) L. Q: w: [% C3 ]    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python
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    ' n* \) |+ R5 U( J3 f
    def fibonacci(n):: \( @+ [2 ?/ M/ X# _
        # Base cases
    ! ^/ F6 K% W3 j, q. b8 t9 O    if n == 0:
    $ u, y. L- e5 l9 g( C1 r2 p        return 0
    5 B2 A/ c* p# o+ d6 v4 B% A    elif n == 1:, y: g8 G8 d6 u* i, E. s
            return 1' \0 p, V9 |2 K% g1 H* ?- Q* Q$ a
        # Recursive case
    * @/ S6 S, i. @* x    else:
    # V" f* H8 d( n. E. m5 B        return fibonacci(n - 1) + fibonacci(n - 2)
    ( [* Y$ T: O. _- j
    , }" @( f7 V( o' e: }  z' L# Example usage
    + J7 B0 [4 ^, f, q0 E' W- }print(fibonacci(6))  # Output: 8
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    Tail Recursion+ @2 f5 J$ v  C' x

    9 T( D) \* J' z( Q7 yTail 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)., Q' _% }! Z7 x/ \+ d+ o  U

    8 v/ z" n  t) C. {9 J, \" DIn 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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