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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! K$ P4 {/ j5 O3 e

    9 M& T+ }0 \! I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。
    / I. c$ y+ Z  R( y. W  u$ d! a, W
    " A, L2 h% N$ g" w( I 关键要素
    6 d: b% E! S2 _5 z0 a& R1. **基线条件(Base Case)**
    9 z' x1 _, v; X% l( k% z   - 递归终止的条件,防止无限循环8 p" c+ `1 q! ^% b+ x0 g3 a
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    $ B1 }4 w6 V5 n% j& L1 _
    ! f* }- z0 b/ d% i7 K1 V/ l2. **递归条件(Recursive Case)**
      r7 G, x$ t- h; _4 `" }   - 将原问题分解为更小的子问题
    & _2 ^  H( R2 \% C# S7 o$ K0 o' _   - 例如:n! = n × (n-1)!( @( a% H% w2 V6 f- Z$ V1 `
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    经典示例:计算阶乘
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    def factorial(n):3 m% I/ P; A! x. ~& ^. _
        if n == 0:        # 基线条件- c$ n' C% {* Y1 G
            return 1
    9 D) Z8 g# w6 s4 A- j    else:             # 递归条件. z3 Z/ l: \$ n1 ~
            return n * factorial(n-1)8 Y- S+ T. w3 _  J) b
    执行过程(以计算 3! 为例):. ~6 Z0 ?3 V  @. i4 t3 C
    factorial(3)
      w* ?( {( ?3 J  N  E& W2 e/ x& y* g3 * factorial(2)' x, J8 q& b! O% v3 W; W0 `  S
    3 * (2 * factorial(1))
    ! \- t) o- f# H; n  l3 * (2 * (1 * factorial(0)))5 P$ X, E+ c/ }, U
    3 * (2 * (1 * 1)) = 6
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    递归思维要点* Y+ |* n; R6 [( X9 t3 v. R8 k1 g' F
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑2 r, h+ ^/ ]' P% z! R" P
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    2 U: Y5 @: e' R5 v2 }' _8 ~1 B9 c3. **递推过程**:不断向下分解问题(递)
    ( B8 a4 ]- n6 }' n7 L. N4. **回溯过程**:组合子问题结果返回(归)
    : D: M, K/ H0 S& U) l4 \$ b7 k( v& @
    # N1 ~  G1 ]! @$ N5 q0 R 注意事项6 F9 k1 O3 R4 U, X- a4 Z
    必须要有终止条件/ S& t+ m# c) ~8 c( ]/ g" b4 D, v8 Y* w
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& [( `  [( y) k; K+ |' g6 T; D+ K
    某些问题用递归更直观(如树遍历),但效率可能不如迭代0 m! ~8 u! j8 @) x: e% Q+ f+ E: x
    尾递归优化可以提升效率(但Python不支持)
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    9 O" _5 f  J+ n1 g. I 递归 vs 迭代
    . h5 a" s4 K  p- K7 R- P& T9 G|          | 递归                          | 迭代               |
    ; a2 F, ?$ c$ w3 {|----------|-----------------------------|------------------|
    / K# u8 A* q: C; E# N| 实现方式    | 函数自调用                        | 循环结构            |
    * T" d. Z0 I' l  \| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    % o' H  ~$ k2 x- N! G  A$ H| 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         |
    ; d9 M& w* |3 o2 a+ h5 s* ^$ Z7 @| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    9 k. r- k* `. q+ S% W; ?
    . V- f* J4 h; g9 Q: q 经典递归应用场景/ \* s+ ~. L( o1 f
    1. 文件系统遍历(目录树结构)
    / r5 i$ C8 g! _+ c/ ^) q2. 快速排序/归并排序算法4 O" a0 s5 \' y
    3. 汉诺塔问题
    / y; }3 k% u0 s5 l$ B3 U4. 二叉树遍历(前序/中序/后序)* F& {. b: W0 @# L) G
    5. 生成所有可能的组合(回溯算法)/ E* L- z. L7 t4 \
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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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    昨天 06:39
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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,
    : }; h/ n8 i+ j/ `4 K我推理机的核心算法应该是二叉树遍历的变种。# [1 ~+ ]  e+ f* }! K' I
    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:! b0 @8 C2 l2 X# d7 K# ^  h" g; s
    Key Idea of Recursion
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    / l8 K' E+ d9 ~* o2 cA recursive function solves a problem by:9 ?9 ^3 u( o: H9 }$ z$ O7 n

      ^+ T2 r) z! L2 v) ~* W; b    Breaking the problem into smaller instances of the same problem.
    3 z( z3 H* \: B+ I7 J9 c: b5 Z2 `7 j4 S/ r9 @) n
        Solving the smallest instance directly (base case).
    $ y7 D$ [( }3 |0 J' Q8 u( J& o! h$ q; ]* @0 V
        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function
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    0 R2 {1 G# w9 {5 d; c    Base Case:: B2 i, h; w+ b8 N1 b( [

    / a# h/ q; k1 ~* z1 y5 _8 [        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.9 R- F) \: Y8 {  V( P7 `

    5 |$ I9 }  Z3 o7 F; d; ]: H9 }* |        It acts as the stopping condition to prevent infinite recursion.
    / n( H& i. E( U6 G  J; d$ O
    2 H7 K3 m/ Q. e; T8 \2 @8 G        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    0 h* n+ [2 @6 W8 `# ~    Recursive Case:
    * u1 J+ h8 u+ \5 R+ M. s+ A1 L& B, e7 T" ]& B0 X7 c) {
            This is where the function calls itself with a smaller or simpler version of the problem.! G  I( g& j6 l7 k9 E/ F

    & @3 q: w: n2 v* X* Y7 u/ Z        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    ' o1 R+ c. J; f4 X# ZExample: Factorial Calculation( H& ?4 V$ Y* {/ y9 K

    1 f9 O9 b+ [( V8 AThe 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- g4 R' H% F4 ]! ~  S$ M
    9 F2 ]$ `; y2 ?3 T- {9 v9 X
        Recursive case: n! = n * (n-1)!
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    Here’s how it looks in code (Python):; e4 ~, ^; r5 R+ ~! m/ c3 c$ k/ t
    python
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    7 b% Z7 w( x3 c( u
    def factorial(n):
    5 w4 X) w" f8 ?/ ~  t    # Base case
    + ^6 d2 d1 y3 l7 Q' s    if n == 0:/ ?2 N$ z$ r) d; m
            return 18 ~* M/ o! y! J* g' [
        # Recursive case! [3 Y( P; w/ A0 U% A- f
        else:
    / |9 ?, X' b( c+ m) u8 `        return n * factorial(n - 1)
    ( @. n3 R  k3 \+ I) z' H7 e9 X: G2 U& U3 h4 w0 X
    # Example usage
    ) _- q( N8 A* wprint(factorial(5))  # Output: 120* f0 K6 e( X5 k
    ' U" F8 C4 E5 s/ L
    How Recursion Works4 l, o$ c  n, r

      l' p# v& o% j) s) \1 }1 U    The function keeps calling itself with smaller inputs until it reaches the base case.
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    5 q) f# F- I+ u# u+ }% v: }# ^    Once the base case is reached, the function starts returning values back up the call stack.. m& T+ w# b! I6 A% h

    , n" m0 n- |0 ~    These returned values are combined to produce the final result.
    # _$ s. @5 S0 }# k# M; E& H" g" d: ^) S( Q/ R6 S, Z9 z8 Q6 q
    For factorial(5):! r$ v! L* S+ D4 [; w& P

    & K3 ~! x# o; l7 f/ S% A$ F3 }' o' I$ g- |8 w& x! R. |
    factorial(5) = 5 * factorial(4). T8 D1 W. S, x3 d( |$ h+ ]4 E
    factorial(4) = 4 * factorial(3)( y7 b4 t: |1 z9 C) b; r' }# d
    factorial(3) = 3 * factorial(2)8 Q" `$ ~5 S8 ~# h0 u( ^% O2 f
    factorial(2) = 2 * factorial(1)2 f& s1 [* r( o4 r# |
    factorial(1) = 1 * factorial(0)8 J4 B3 y. B  u- q4 @- @
    factorial(0) = 1  # Base case2 t5 E2 T! R, s* P# c! K
    * y4 T( ], J+ C$ |! u
    Then, the results are combined:
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    ! c" O& j5 C! {+ A5 @& d2 T' Q% D  [$ m# p5 f: |. _2 }+ D# k
    factorial(1) = 1 * 1 = 19 f" O8 P/ q  A8 p1 G
    factorial(2) = 2 * 1 = 2
    , j6 I9 b7 f5 w$ F: z, ^factorial(3) = 3 * 2 = 6" C! d' w3 y/ V/ k8 y; T% ~
    factorial(4) = 4 * 6 = 243 |8 p' y$ M, J
    factorial(5) = 5 * 24 = 120# ]7 h+ x3 J4 p2 Y9 J

    ( C6 a. _( Q* M& ]( QAdvantages of Recursion! H/ K, t' ]0 I# t& I

    3 H" V( k9 z# ~. }    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)." k! @/ q7 e5 b! y5 S: {( A
    " _( Q# U5 \# y4 a6 ~2 S
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    6 ?  }8 _6 g! }* K+ A0 b  ^' i1 `
    0 |5 o4 r5 I$ ?. v; p' X( UDisadvantages of Recursion
    9 B8 ]- Z2 t# Y7 e% u0 x/ w- I. d1 ]! a! u/ B
        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.
    5 j  u* v6 C1 y  L$ `3 _. y" P1 v& f6 E
    , n- `% Q- O$ |- ^  u2 u- V( _. {    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).6 X6 B6 @" n" O3 h: O8 V- q
    / ?" Y  g; j# G, S& H: w. c
    When to Use Recursion
    ) F. w* U3 j  B4 x3 H0 d/ h- t& ^( Y7 S1 B+ i! a* @1 @
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).- o9 l2 \6 }& w: P" H8 q% @

    / x8 L  F- g9 {6 X    Problems with a clear base case and recursive case.
    & ^- B! `; H+ J3 [' I4 F# T6 }1 j- F$ g
    Example: Fibonacci Sequence# s% \* _# |$ t1 j$ s
    $ b/ e2 ~% `/ B- b- C4 p6 a5 u
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    # o/ \+ D1 x* y+ G& M$ [3 A. \; c0 i, Y
        Base case: fib(0) = 0, fib(1) = 1  B# ~% P+ {6 D5 F
    # g6 L7 A& s+ P; Y+ k# G
        Recursive case: fib(n) = fib(n-1) + fib(n-2)5 d( ^4 X% y: P0 H4 N
    ; T9 [. `4 U# |1 x) _
    python
    " H  R% X. w- t. \0 `% m0 @: ?9 A* J3 g( u* y* H
    8 Z, f, O" d% Q6 n+ B$ x
    def fibonacci(n):$ E4 N9 x1 f  b0 H
        # Base cases
    / e% g6 _4 U( z6 Y2 [    if n == 0:9 ~6 i% l3 s  H! a, u" F% _, j+ {
            return 0
    / k. c- [( s7 q7 n/ o4 [    elif n == 1:( @. j# [) Y6 }
            return 1
    : P! X4 W) f0 P5 R* ^3 N# }    # Recursive case2 i  K5 a" {7 g' V9 i. y; s2 z0 I
        else:! \5 ^2 V5 q2 O. y. a% @* U
            return fibonacci(n - 1) + fibonacci(n - 2)7 U% y7 B0 e0 P- l, [% Y
    , t& ]; C+ _5 G
    # Example usage
    4 j, `# ?: B+ }) i" n9 U7 Q$ _# ?0 ^; Oprint(fibonacci(6))  # Output: 85 C- Z) \/ z7 D4 ~7 |( }* x* z

    : y, ?1 D6 }2 S; GTail Recursion5 R, G; s0 [9 P- d
      g+ T; [  _% c. \0 _: 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).
    : s2 F/ E; Q" z! k' W
    - D8 ^: U! J( h2 G  r( T+ f0 m' ~$ QIn 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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