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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 0 W' l8 z7 W* d& i: ?: ~: q& o
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    解释的不错  s, w( h3 x, p( F- w
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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。7 D2 c5 W6 m3 ~: W& h& s

    " J: r2 I! |7 j8 N4 ]2 a5 W 关键要素/ t! |1 k# k$ m/ Q2 C
    1. **基线条件(Base Case)**
    ! K, r4 J* L( s6 A   - 递归终止的条件,防止无限循环* V' x3 [- H, Q+ i9 O
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    8 |" M7 N  O5 Q6 a: T( @" {% V$ x; P* w( M, ~/ j
    2. **递归条件(Recursive Case)**
    5 d" `: M9 `8 ~( g' G) \   - 将原问题分解为更小的子问题+ c0 I+ `/ U; Z3 t5 \& m1 D# ?+ U
       - 例如:n! = n × (n-1)!  Z) m6 X( f& }7 f- h7 W0 \7 X
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    经典示例:计算阶乘: ?: Q( Q) b- e1 x
    python
    & \/ [1 g$ l, G+ u6 I6 K! c% k  idef factorial(n):- A! i, z9 X" k% t
        if n == 0:        # 基线条件5 @1 |9 i4 }; q/ x9 ~$ k3 }6 V2 v  K
            return 1& ~' E- u) I* W# {3 w5 I7 J6 O8 c
        else:             # 递归条件* B" z, \8 }1 _. {3 U: A
            return n * factorial(n-1)# v' }& R: a6 `
    执行过程(以计算 3! 为例):9 m4 X( i4 v: a6 R
    factorial(3)
    : r9 s. F5 M9 R# b. c$ n6 Y3 * factorial(2)& n% N6 s: {. c* L$ H0 [, l
    3 * (2 * factorial(1))1 M% r% n+ T- [0 l& A5 v
    3 * (2 * (1 * factorial(0)))) @9 v: g3 i, E; o% w
    3 * (2 * (1 * 1)) = 6
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    递归思维要点
    / U6 t9 O+ U2 Q8 q$ y, c1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    7 z' y: s" N4 [2 r2. **栈结构**:每次调用都会创建新的栈帧(内存空间), O5 n3 o) Z. `& u7 y
    3. **递推过程**:不断向下分解问题(递)
    ; i4 n5 \% T2 u. m, m3 p8 r4. **回溯过程**:组合子问题结果返回(归)% D/ `% U8 m& `3 g# c" s, O
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    注意事项
    5 S' H, s* U* F  F4 d) P必须要有终止条件
    " q9 ^6 Y/ f8 A& ]& T递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    5 u$ k( h* C. G4 z& ~( L2 A某些问题用递归更直观(如树遍历),但效率可能不如迭代/ w& _; Y6 p" [4 H4 v+ K! V) f
    尾递归优化可以提升效率(但Python不支持)5 F) \# N# D( {( K0 d1 r
    4 p9 v9 U$ K4 T; ~
    递归 vs 迭代
    ( E, C) F) ^, b1 q2 ]$ R- t|          | 递归                          | 迭代               |
    1 n' M5 ~# v3 n7 N8 x|----------|-----------------------------|------------------|
    1 v. [: q' v5 M| 实现方式    | 函数自调用                        | 循环结构            |
    9 i1 h+ D; ~# a# I2 h/ W: b| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 k7 r$ n) R3 n6 D& Z3 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         |1 K6 }, U' f+ s1 P+ F
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    4 a0 R: j/ A% P% k, A' d, t4 E) L  `- g) t/ C8 E0 W# P+ _, S
    经典递归应用场景
    + b! P) o# E. l8 u9 X1. 文件系统遍历(目录树结构)
    . n4 v5 S6 x0 g7 L" p" T. a/ a1 H5 |6 c2. 快速排序/归并排序算法8 R2 ]7 D% l& Y* H
    3. 汉诺塔问题' ]; R0 p9 \3 v
    4. 二叉树遍历(前序/中序/后序)
    : v: {7 r, a. ^- x' A! g5. 生成所有可能的组合(回溯算法)
    # r  B3 W0 I3 l& X5 r
    ! I9 e% O% B& y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,
    1 ?" t' j( w# x; v, @4 |% U3 F我推理机的核心算法应该是二叉树遍历的变种。
    ' s. j0 b* z$ T4 S- O3 J& s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:
    3 p/ G; s! g9 }( E- o4 X! pKey Idea of Recursion
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    A recursive function solves a problem by:  |3 s4 B) h$ G$ h; c" E1 s
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        Breaking the problem into smaller instances of the same problem.
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        Solving the smallest instance directly (base case).* N6 `0 F( G- c# F, s  f. F7 R7 q' M
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        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function1 r, }: H5 M" k  w. P4 ~, }% @

    7 r# m5 z) `) e* h/ J+ v    Base Case:
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    4 N6 f- K+ d9 |        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.9 c7 d% x/ D' h* t

    . I% f( g. Z7 L6 O' m8 X        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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    9 m7 h5 A0 n# T( t8 S        This is where the function calls itself with a smaller or simpler version of the problem.  r2 D) ~# T$ T+ `, `. I
    1 @+ P$ g. I5 X6 P0 b5 J& d) U
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).& L+ T1 S/ b3 I% y  T
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    Example: Factorial Calculation: ~0 c3 W. Y& x
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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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        Base case: 0! = 1
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        Recursive case: n! = n * (n-1)!1 x1 s* m& F! t# r& ?+ A8 w  w

    1 L3 {/ B5 @( K$ i% yHere’s how it looks in code (Python):
    # x3 O9 j9 C0 b( v" F- A, Ipython
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    # _) F) c+ C% y- I5 A0 d- x6 Wdef factorial(n):
      F- B/ R8 Y" o    # Base case. ~3 B8 B; c  U4 K5 N3 J
        if n == 0:
    ( K* L! i) `8 _& F2 {        return 1
    5 C* d3 i( x+ }8 U7 @    # Recursive case
    6 k) ]: W5 f* I* C' d    else:
    , W1 P6 L$ L* \6 z, U        return n * factorial(n - 1)
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    # Example usage
    2 Q# |" Y4 w4 M% yprint(factorial(5))  # Output: 120& n8 j! w# R9 ~6 q: M, i* V
    + |/ d0 c8 C0 N. M2 `8 [7 ]
    How Recursion Works( ~2 O' A  \) G! P% j

    " l3 a7 Q/ M" D4 N    The function keeps calling itself with smaller inputs until it reaches the base case., [( Y* L9 q1 v: v; t: v5 y
    ) k. {, x- Z) ~5 v8 ~8 R
        Once the base case is reached, the function starts returning values back up the call stack.
    7 Q- I/ F$ I! z3 s
    9 C5 m6 V& A. E! j    These returned values are combined to produce the final result.- K5 r/ P% g- H) L: z" {2 I

    0 |0 y1 {/ }* a3 G6 a, TFor factorial(5):
    ! }, w1 t* E' ^
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    5 I! t, z% x: s& r( B# v: j. D4 ~factorial(5) = 5 * factorial(4)
    ! E; u/ H! w! ^) Q6 Y4 |factorial(4) = 4 * factorial(3)
    & U9 g  z' \5 O/ z* Jfactorial(3) = 3 * factorial(2)1 H* r0 [0 Y1 D1 i. _5 O
    factorial(2) = 2 * factorial(1)  K3 W. [. L6 M! V9 f+ w
    factorial(1) = 1 * factorial(0)
    1 G" q: i/ M( Y  G7 ffactorial(0) = 1  # Base case3 t, X) s: @' B) ?% `1 A( I

    " U. _' T8 v7 ~Then, the results are combined:7 d: N4 x9 l4 ~( W0 j/ I. ?' b
    ( D3 [' _" O0 g  t- \2 `

    9 R( s3 T+ O6 h' |( @1 i2 U/ Hfactorial(1) = 1 * 1 = 1. W# x* e8 t! F3 ]6 k- Q
    factorial(2) = 2 * 1 = 2
    ! o8 |1 f( L4 g4 v$ Hfactorial(3) = 3 * 2 = 61 b* {3 p- q3 G; T% W
    factorial(4) = 4 * 6 = 24# m/ ?" H) T5 @1 R1 o
    factorial(5) = 5 * 24 = 1201 v4 L! u  F6 O; y+ g( i8 v
    7 ?/ H6 J4 |- ^/ I  z8 B
    Advantages of Recursion
    - |' h: C% d! q7 E, N/ X# u0 O9 M  K
        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).3 p! w# m. ?& j" b) `

    & k- j; K7 h5 b7 ]" J) K& n    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    7 {$ _3 l; \1 p# ?2 g
    6 K5 {7 F! c! t: aDisadvantages of Recursion
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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( Q2 O+ B- V9 K4 |7 f3 {  w; I* N
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ! y  s3 v' ?3 |6 t7 S1 }  |& k
    5 z. p9 D; D# _! ^- `9 wWhen to Use Recursion( X9 I' M# u$ Z8 {) R0 g" k

    ! o- g6 [- t- I, X3 w2 V    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).! n+ K0 f0 M8 e; X! U

    & V4 l4 f% y- d2 E! h    Problems with a clear base case and recursive case.
      u. q! z. p# L% x3 U& Z' R/ J5 W% N0 ?8 c$ E; Y
    Example: Fibonacci Sequence
    * L0 j5 o$ k) \! `+ V1 X. x  [
    3 d7 }3 n' h1 O8 K- p5 A% p( aThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
      U! U: Q0 m) V4 D! f5 m- w; P% p
    , l- B* O* N9 f" A3 ]    Base case: fib(0) = 0, fib(1) = 1) l1 {$ m: j- Z1 W$ u# L
    ! G  q" x" X3 E& I; t
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    ; _; O8 d0 q4 rpython
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    def fibonacci(n):/ C! a- p: H# r! |
        # Base cases9 ^- Q0 J, T, {: o& T1 E9 v3 N1 }7 I  V
        if n == 0:! H4 U$ e& M" R
            return 0
      @4 L: J" K* k7 s    elif n == 1:1 e: X' D( c5 O* U
            return 1
    ( O2 `' }5 f, l1 s& P7 V; Q2 v; {2 X! N    # Recursive case
    8 B1 y. r- |3 ^) G    else:
    $ w3 F" i5 W! s1 I$ C        return fibonacci(n - 1) + fibonacci(n - 2)
    : M' a5 t3 `* ~6 B+ a
    0 `! ~. \$ L: n# Example usage& E! |' F$ W8 J% r) y* T4 ^
    print(fibonacci(6))  # Output: 8+ |# {/ E1 k- s0 k$ f
    , m4 z8 U1 ^$ Y/ m/ p6 H6 G+ l
    Tail Recursion
    ) e) q/ y! b) r# t5 s& p- |$ m/ O8 s0 A6 y$ u
    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).
    # Y" v7 }8 A7 u% Z. o6 v  f9 p; m% X
    In 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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