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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 - s" G; V/ h' ^- ]2 E; j8 X8 V+ O
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    解释的不错" }  Y, q' p1 c: A3 \4 o

    ' S' T6 Z/ Z, t( R/ M, Y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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    , c' o0 l8 v# j7 g( w' ^0 ^: F 关键要素: s4 U$ k: m/ ?2 ?% h  L
    1. **基线条件(Base Case)**
    5 _3 A4 c) f; ?, I( E, o( }   - 递归终止的条件,防止无限循环( b4 n4 \, M7 r) h3 z5 @
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# |# q6 u8 ^" S- `# V. F+ @$ n

    4 y3 v% k' {# _. i/ `3 L6 S2. **递归条件(Recursive Case)**
    + U. i' p4 J: x* l$ p$ I   - 将原问题分解为更小的子问题
    & y! P! u9 }: M1 y/ ?   - 例如:n! = n × (n-1)!/ l; g  z5 {- q# @8 i6 m, h- i% i' Q
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    经典示例:计算阶乘  ~: h/ V; @: e
    python0 W7 p$ `( z$ H- y; |
    def factorial(n):) A. T% o. c& _& b, R7 l# h
        if n == 0:        # 基线条件) o$ g+ n9 h) v9 m0 y: T
            return 1
    + q0 W, P/ w8 g5 d& \, [% W1 j    else:             # 递归条件# _1 Y5 x' S, j0 a
            return n * factorial(n-1)
    ; j' Z; r5 A1 o3 `" q执行过程(以计算 3! 为例):0 S1 \: Q( U: n" |. s0 v" A! C  A
    factorial(3)
    5 C8 A6 \0 ]( R# C# Q, C# O3 * factorial(2)7 F( h# E; b1 P, W4 v# e' U' ~
    3 * (2 * factorial(1)); C5 A2 R- x7 Z1 d8 Q2 D! Q- r7 C
    3 * (2 * (1 * factorial(0)))
    & g0 A# ]  N  A2 d$ g$ u$ Z7 g3 * (2 * (1 * 1)) = 6! Y' t* F2 h0 @* a. v; U
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    递归思维要点) }3 C% b* ]. u5 W! R# E
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑& s2 M5 w' ~% K# v' a& L
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- Y: R4 W# F7 a6 q
    3. **递推过程**:不断向下分解问题(递)
    6 W% e& h, U  \4. **回溯过程**:组合子问题结果返回(归)1 A  \( @- ^2 m1 _( {

    7 W- E3 v% r1 R9 [ 注意事项
    ; `$ t7 E( D- b- s9 A2 M. i! d必须要有终止条件$ W, B9 ~% i. u* w- o# `
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)- t/ b+ ^+ n, z) H
    某些问题用递归更直观(如树遍历),但效率可能不如迭代0 L( R6 e* {9 V6 Y6 N
    尾递归优化可以提升效率(但Python不支持)
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    1 M; ?  D) C' [! p4 D 递归 vs 迭代
    ) D! v" s+ [8 f4 C% Z% L|          | 递归                          | 迭代               |
    - c. U& P* V6 z|----------|-----------------------------|------------------|2 V: `. C% o6 |
    | 实现方式    | 函数自调用                        | 循环结构            |
    / m$ S1 U7 h' \| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 {$ ~5 J' l7 K| 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         |
    4 {2 h# y  ~, l9 L2 l| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |4 o9 h* O/ V2 o
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    经典递归应用场景/ v4 l$ _% _- E8 {0 p# O
    1. 文件系统遍历(目录树结构)
    ! ?% X0 v3 O4 y( O' W+ f2. 快速排序/归并排序算法
    + a6 H7 U5 Q+ m. ]3 n2 C3. 汉诺塔问题
    ( ^3 i; @. N$ K2 U4. 二叉树遍历(前序/中序/后序)
    4 T, B/ n6 h' t+ n5. 生成所有可能的组合(回溯算法)! a0 ^; d: A* L* t0 P
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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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    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,. ], S% {7 U0 Y6 P. o  r: q
    我推理机的核心算法应该是二叉树遍历的变种。
    5 U: j( \& ~; Z  u' o1 y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:
    % V  U1 L# Q. Q1 q6 n: }Key Idea of Recursion
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    ! T* Z$ E+ @! N) E+ F1 VA recursive function solves a problem by:7 A5 |, c* M) J" o' F0 x% t+ C& J
    7 K' ]$ S' C8 H# ?5 r
        Breaking the problem into smaller instances of the same problem.3 e+ T( t+ [8 Y  l0 N9 }
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        Solving the smallest instance directly (base case).
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    ! P! M% F3 i0 c7 f$ u0 V2 }    Combining the results of smaller instances to solve the larger problem.) O2 w1 B; W6 O  J( b3 s

    . R; I) E' T6 e& P7 F7 Y, b" qComponents of a Recursive Function
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        Base Case:
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.5 M* M+ G$ n1 P( R
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            It acts as the stopping condition to prevent infinite recursion.
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    9 o- x4 Y# l: T5 a1 j; m# B        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:
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            This is where the function calls itself with a smaller or simpler version of the problem.* V6 T$ W8 S! q$ M$ u% ~' T# f) y

    * R7 V! b6 h9 `3 V. F" M# c; m" x        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).! W, d0 h) a5 N$ q' t
      c2 T9 t+ R* T- `
    Example: Factorial Calculation
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    % s$ ?/ y8 S* _. |7 r9 n8 qThe 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:3 M! z& D4 R, g' U4 P, Z6 y  J
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        Base case: 0! = 1+ m7 L  h" W' B
    " u. n& p; h2 o8 L7 Z  H
        Recursive case: n! = n * (n-1)!# X( U1 ]7 l* `2 M. z0 _

    & I: ?( z& H6 e, |5 d" K% J0 qHere’s how it looks in code (Python):' h5 n9 k" W: y+ @
    python
    - ]+ s3 s$ t% O5 B+ Q5 v  v0 t& x  z5 |& c3 J  {- a  l

    ( ]; j1 b/ `1 V, L) _& tdef factorial(n):
    8 p; a, x7 F; F) b7 A  G    # Base case! K( j+ f& _1 g& b# Q
        if n == 0:8 i- I  M) q5 _- N2 F
            return 1  P; ^/ A9 C* J0 U$ n
        # Recursive case8 \; \; A2 y+ ^2 B9 F4 ]
        else:
    $ s- S8 @( ^; v8 a) h        return n * factorial(n - 1)+ W% e6 G: }1 m' J0 p  X1 e: Y
    2 i( V4 C2 e; U+ _
    # Example usage4 }+ @7 g" o2 ]/ ^6 n
    print(factorial(5))  # Output: 120: ~8 {2 c4 f3 y; S7 z8 i3 T  ~
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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.- \2 d1 I" t0 Z4 K5 _. I' r$ M5 u

    ( K: T6 ]9 L- ]( K( o    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):; E8 S& W- C9 f; M
    $ @6 ]5 e5 J+ L7 x7 ]
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    factorial(5) = 5 * factorial(4)
    * h9 B% ]$ P9 H. U" Pfactorial(4) = 4 * factorial(3)5 r4 P# K1 g2 Y! b# k  o$ f
    factorial(3) = 3 * factorial(2)- b2 t4 ?+ j. _% y5 j1 }5 q6 N4 g
    factorial(2) = 2 * factorial(1)3 d" ]) C& E8 J, z
    factorial(1) = 1 * factorial(0)
    $ q8 k3 T* C/ ?- K1 F& r9 W0 ?factorial(0) = 1  # Base case
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    Then, the results are combined:
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    ' J4 @1 Z& ]. L3 }* a
    factorial(1) = 1 * 1 = 1
    3 C1 @  V# V1 `/ y; vfactorial(2) = 2 * 1 = 2! w7 G! D( v  Y! ^0 E( I
    factorial(3) = 3 * 2 = 6
      ]0 F  |/ F  [0 c7 Mfactorial(4) = 4 * 6 = 24/ ~- c" i/ v( w  T2 E+ y
    factorial(5) = 5 * 24 = 120- o, D& r- a6 N7 q* m2 d! r+ k
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    Advantages of Recursion1 p5 P& w) I4 X6 ~3 V# `

    % P! M! n# u% e    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).$ U( {& R! t" [% A
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        Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion* B" d+ m5 d7 B. r
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        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.& g$ G2 g9 O. J. O/ Q0 B5 d2 g

    0 h! Z, Q0 K# q" n1 N" `    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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    # S+ B' ~1 a* _3 G! s- ^; X8 \When to Use Recursion! B9 _! H0 o3 ]( i- h8 @% ]! q
    ' {; j  l0 F  G& o5 d6 r' H
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).: E" r$ {% \; m/ C
    7 A- R9 ~' \. U" e
        Problems with a clear base case and recursive case.
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    % W# f( O$ m+ ?7 UExample: Fibonacci Sequence
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    ! }, ], |- i/ U& LThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:& @, s- I8 p4 R' E
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        Base case: fib(0) = 0, fib(1) = 1
    ' U( |7 i/ M# [: l8 |7 G8 h1 b4 Y4 r* t# D: P' A
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python9 z, U. ^0 P. W3 \$ {0 \

    2 Z0 S0 X1 w" P( `6 J6 R* Y$ L7 O( N& u; ^
    def fibonacci(n):
    0 [# z% S2 q- q9 ^1 O$ k    # Base cases. G9 g) k1 L; Q% `
        if n == 0:
    - s9 e( W0 c) A5 [! _4 v        return 0
    : @- L+ W: U+ `* ?    elif n == 1:. p; w$ L4 p/ n8 `( I3 L" d
            return 1
    # O8 a: D& T) D7 \    # Recursive case
    ; i+ ^( B* q2 Q) e    else:
    $ \& N" }+ f6 m8 h  q9 ?$ n, K        return fibonacci(n - 1) + fibonacci(n - 2)
    3 F# w* r9 `. l) b  s+ D
    0 W. \# H% g5 ]) `! V# i; |8 }# Example usage
    : B+ Q3 A; h( N4 H0 ~print(fibonacci(6))  # Output: 80 `1 |) d9 ^0 w4 c# o

    # U# @' A5 d6 N) ~$ F% lTail Recursion
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    2 _5 ]1 \7 |) u2 ], RTail 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).
    7 j3 v/ F# y; r; l/ n/ @# }! U
    ; ?- z8 B& H1 F: d( E; PIn 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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