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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
     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
    8 r+ u( h! e* ~/ ?
    ; @# f+ c! `" b. \/ |3 f2 v解释的不错
    0 N# ?2 j  Z. O% ?/ p! C* X) y; z7 ]& N, E3 p7 T
    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。+ ~9 ?7 I! a. p! `$ V# J- P

    , Y3 h' S: t1 A 关键要素% X4 e& N! v( d1 Y; t4 {' E
    1. **基线条件(Base Case)**4 b' I/ s0 K& Y2 v0 p3 D
       - 递归终止的条件,防止无限循环3 W" X6 d0 `" a
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    : {: G8 L2 B- \" C& d2 G+ J) @5 ]* Y7 C7 ?+ D. I
    2. **递归条件(Recursive Case)**
    - G8 W+ s+ n9 x2 n   - 将原问题分解为更小的子问题
    ; s" l: f2 k3 m6 P; m5 R   - 例如:n! = n × (n-1)!
    . m9 f& _1 ^9 {
    - s0 B- S, U1 x- \- c' j 经典示例:计算阶乘
    6 U( b9 ]/ I7 r! w/ cpython) N- R' P+ q+ V, B
    def factorial(n):
    $ `7 [) [7 _" q* C$ _    if n == 0:        # 基线条件
    . E5 H. N( ]# |6 X        return 1" Z; a, W: [  ]4 U3 a, s. h1 p
        else:             # 递归条件: N/ T0 c1 H+ }7 c0 m, W5 E
            return n * factorial(n-1)! m: z- R( ]) K# H
    执行过程(以计算 3! 为例):
    4 \% ]9 c2 B" ?7 [- wfactorial(3)
    * E: V  p; p9 X( K, y/ u3 * factorial(2)
    # M0 _4 b8 J5 t- I6 [9 E3 * (2 * factorial(1))$ o$ W# G) q4 p
    3 * (2 * (1 * factorial(0)))
    . \7 {& j& s1 W; E9 Z3 ?5 e. i3 * (2 * (1 * 1)) = 6' ~; I! ^/ M* g+ a

    ( W$ }+ a# o2 z' X7 V- q8 z- c 递归思维要点
    7 a1 t8 l1 X) v8 {1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , u) U4 m6 C& e1 C+ K2. **栈结构**:每次调用都会创建新的栈帧(内存空间)# \5 l$ `+ f" v: G9 {& k# H
    3. **递推过程**:不断向下分解问题(递); O4 C, \* e: a/ s+ C# x  p4 ^
    4. **回溯过程**:组合子问题结果返回(归)
    + X* z1 R2 t/ v1 n# ~7 G) Y, o3 x: i% {+ b' r6 C2 m% E: i) a1 S1 E  L
    注意事项; Y8 j' m1 j& L+ y
    必须要有终止条件6 {$ d* _. i% T% U4 e
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ! E' L! o1 [/ G3 X某些问题用递归更直观(如树遍历),但效率可能不如迭代8 ?, B  U) `: Y0 u1 }# J' S: T  p
    尾递归优化可以提升效率(但Python不支持). L1 E: ~+ n. z, X1 e' \

      \) N% R# W$ F' a 递归 vs 迭代/ w0 P! y% |0 z5 h, G9 [
    |          | 递归                          | 迭代               |
    8 v" R  v. x- k! k4 L/ S|----------|-----------------------------|------------------|
    1 t# B% z* E4 M2 p4 E3 K2 n. k) A| 实现方式    | 函数自调用                        | 循环结构            |
    ( k7 k: @# |4 l  ]. V( R| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |0 \/ R2 c* P, x# z6 p
    | 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         |
    $ l$ U. T, ~% q/ o| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |! B$ Q7 L! T) N! N( P
    4 L; Q+ u+ R+ P5 K/ u% F& @' r
    经典递归应用场景* q0 ~3 i  o0 L' I+ M
    1. 文件系统遍历(目录树结构)
    $ q3 j$ S9 G' |2 n1 N  D  t+ X2. 快速排序/归并排序算法
    ; K0 F. n6 ?9 O( _5 U% ]5 e3. 汉诺塔问题8 z" V" [8 ?. n! J) o. t0 L
    4. 二叉树遍历(前序/中序/后序)) a9 P, \3 Y6 \
    5. 生成所有可能的组合(回溯算法)+ U1 d5 C  {: j7 Y1 o
    ( ^1 ^/ B5 U# ^3 ~  G
    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,2 K4 Y% }$ x" A5 X6 M: N7 {: H
    我推理机的核心算法应该是二叉树遍历的变种。9 K# R- T3 I* L& B
    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:
    ' u7 g% y. ]4 V; m) C5 dKey Idea of Recursion
    1 h+ E; [+ o. k( y
    $ m7 H, E; n& ?$ X1 a" uA recursive function solves a problem by:* o( Q. [6 ~# N% t3 w$ U  S) P
    ) L8 Q  ~; h3 Y6 B! i- |# |
        Breaking the problem into smaller instances of the same problem.- Z& m! `! z- y  z/ F$ a+ x

    : D! p( P+ X: T2 H( X7 }' Z    Solving the smallest instance directly (base case).
    - z" d0 @9 p0 h  }4 ?! }+ H& p; D; ~. ]: N7 R9 [1 Z
        Combining the results of smaller instances to solve the larger problem.
    ' H& u8 Y- J, }% E) B5 n) q
    # ?2 E: O9 E( Y5 t+ q. VComponents of a Recursive Function8 ~7 {7 K+ n" @, P! i" F

    ; ?& H6 I$ R! r4 @9 ?2 x. d9 x    Base Case:
    . A- n" k$ ~' a) g  G% @, I& q: z3 R4 ?! c$ ]
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* ?6 V4 z& |1 X' i& [1 ^

    / u3 @. o$ x" c        It acts as the stopping condition to prevent infinite recursion.; o* F. _1 G+ U6 y) i4 r+ Y# P
    8 F/ g% `* T$ |6 W  R. }
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    4 A) z+ b: J5 E/ f2 C+ {. m
    * m# w8 d* B5 v6 T    Recursive Case:
    5 p$ ~, R% O" [0 Z) T. j
    ! y+ B' Y( K* G; A        This is where the function calls itself with a smaller or simpler version of the problem.. n% D7 Z, C% l; k7 Q6 `
    ; n2 A- _6 r$ \# ^  F! n4 {7 G" I' C/ z
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    # h' H, Z7 m( |8 K" F2 j# u8 G6 X- L7 I, b' R
    Example: Factorial Calculation
    3 W0 u4 v1 u) v2 p; P. X8 Y( J5 V, ^) B9 S% n* ~$ Z4 z
    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:
    " z0 K0 z' k+ c& d6 _$ F$ @  }3 @) Q8 D  ~% T3 `) H
        Base case: 0! = 1
    . r2 f2 `8 I. T( Z9 M) q8 x& t. {' ]; ?$ a8 b
        Recursive case: n! = n * (n-1)!
    : E9 [* w. M" Z4 P- K  R1 M% B
    - |9 S4 m; m/ rHere’s how it looks in code (Python):
    ! U- H3 o  g6 `& Z) t0 Ipython
    " h! t, w# x  n) d& B# v+ f! q* s( y* F; i1 H9 j

    4 p( q+ M; h2 I4 U' |! D/ v% Pdef factorial(n):
    ' S+ n7 O, k0 A' @( p    # Base case
    2 a3 W- k" ^) p    if n == 0:
    " a1 ?* ~* V0 K0 _        return 1
    1 X; ]+ b/ S5 p# _0 W, q$ k    # Recursive case5 t9 f7 O+ W) t7 p  O  Q7 j
        else:
    & C0 X# i% |, z        return n * factorial(n - 1)3 l3 p9 o9 W3 ^* y* W9 t" X, z

    & q" S. x! F8 }. z. b7 N# Example usage) [3 E7 N3 l# u- p$ y' o5 u8 ~
    print(factorial(5))  # Output: 120$ l: ~$ x- q) U9 f5 t

    ( r9 C" I: d, L: {" k" c. U7 o7 T" E( WHow Recursion Works4 X: k' l" u! b3 z* z6 O1 v3 C
    " @- z& ~! w# ]0 q$ w2 L1 d
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ) N5 N0 f( N4 E
    1 @2 W+ J; D% n8 b    Once the base case is reached, the function starts returning values back up the call stack.
    1 E; M+ c, T! w* ]0 ^1 v6 Y( q. W5 t" J% E- l0 ~
        These returned values are combined to produce the final result.8 v# H! O; t4 u4 f

    + i5 O/ X8 a8 k6 LFor factorial(5):" h: a/ y" Y& D; Y4 ~0 S7 L. \: T7 K
    * L( Y, I, Y6 B% z9 l. b

    ; n9 Y3 x  B  \' b2 r7 X4 Kfactorial(5) = 5 * factorial(4)
    . D6 {+ F0 F8 @% H: x% gfactorial(4) = 4 * factorial(3)
    1 _# f& @4 {# I7 E6 ^( [5 Jfactorial(3) = 3 * factorial(2)( N7 A% _: q# M- }$ j
    factorial(2) = 2 * factorial(1)% i. y, h% L" k3 E
    factorial(1) = 1 * factorial(0)
    / U% I7 w$ z- Qfactorial(0) = 1  # Base case
    3 J7 S% A5 Y) L# g3 _' E/ ?
    - g! s; q  a8 U. UThen, the results are combined:
    0 d  E1 J( ?+ R( N& N8 H+ N0 y- @. m# _$ W* k* o
    6 v8 P* R' X/ Q' b, d
    factorial(1) = 1 * 1 = 1+ e; p. u0 S# w6 ^/ C/ d) T: K" w  ~. S
    factorial(2) = 2 * 1 = 2
    ! O$ b$ Z  J+ C6 c2 Wfactorial(3) = 3 * 2 = 6' Z3 Q, @" `: A; B9 r. i+ I
    factorial(4) = 4 * 6 = 24
    2 T8 v! H6 Z/ ifactorial(5) = 5 * 24 = 120
    ( \7 R- v. o: v) ]+ V* A/ V; }
    $ t8 j4 `9 I2 }8 N: D  v  ]5 w$ UAdvantages of Recursion0 c4 y. l& x* S
    $ A; q% R4 L! i$ F) X
        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).  {$ q1 [3 E* ~: b* Q$ y( t+ D

    6 \, Q& L& z- C" |, o! c3 s9 \    Readability: Recursive code can be more readable and concise compared to iterative solutions.: D% p9 T# @" z# Q
    9 O: K8 J# n4 @, Y4 y/ j
    Disadvantages of Recursion) \/ C% N! z7 a9 F+ e

    8 N' |1 n% B0 f! U8 |( \    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.
    4 H' W$ l0 y* _% D4 ?2 K! E( J1 d' t! }+ R+ I. j9 I  r4 t
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    " |  M$ o2 E  V0 V: W1 z
    9 `. ~7 C, `6 R( W6 PWhen to Use Recursion9 K( e2 G" N2 N, p& u% M7 E
    ! F4 Q8 {% o' v9 ~/ p, j
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)., t' {0 K" L+ X  c) X

    : j& U" X& \8 k/ c    Problems with a clear base case and recursive case.$ h  @  }; c# N- w  F

    0 i5 v5 [9 a7 t) ZExample: Fibonacci Sequence
    4 J9 ]3 P2 ~% d$ }
    & b* B$ y, G& g9 G8 X% M9 U3 C8 tThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 k0 K2 ~" w/ |7 O1 N

    , V7 O; r7 B# c1 N1 J* K+ m5 l    Base case: fib(0) = 0, fib(1) = 1# @: v  n5 A* o! p0 @

    $ x$ A( ]* K2 ~2 I, }    Recursive case: fib(n) = fib(n-1) + fib(n-2); B$ x6 C5 u3 \8 L  f/ J

    8 |# ?& Y) x$ b2 V- r2 Wpython0 G4 @( E) g  I8 }# W
    ) m4 {& R+ s% {, }; X; P

    ) {( a( y5 Q6 O. ?8 [5 o3 |  ydef fibonacci(n):
    ( A6 V* @$ j  J- ^* R% b% X; v    # Base cases
    : r; {8 \7 a( j. H2 o5 o- J    if n == 0:
    9 X* z8 l3 n9 P/ q( H* J2 z- O/ F3 ^        return 0
    # y. u3 [  k, G" w    elif n == 1:
    4 d. k7 _) ~+ P; A        return 13 Z  P% `5 v' F3 X( A2 P
        # Recursive case# T+ k! F: e) ^; ~$ H- Z0 ^
        else:; v7 \- A; M- F) X
            return fibonacci(n - 1) + fibonacci(n - 2)" `8 K4 F1 }6 T% G+ ?

    8 m/ D7 b$ Q# a  _+ u6 q5 I# Example usage
    * E2 `, d  ?' u* q+ L: h7 |1 lprint(fibonacci(6))  # Output: 80 S) ]/ E, ~& ?% o
    ; m; G, }  o# z, ?. ^% p
    Tail Recursion
    . o! Y( C* j" I0 g* ^6 F( u4 y  z! E! s1 ]
    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).
    * R3 X& Y9 t) y2 g. U# m* W# D
    6 E* b7 Z/ _$ ]9 C2 _1 tIn 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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