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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
    6 l5 W7 i1 V/ i7 U3 q; K  c" d" @5 x
    解释的不错
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      Q/ U- O, o, M5 l" p5 H1 P5 T+ M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。: P& j: l' B' \
    9 I8 V" R! n" }( M6 E. v2 v6 z
    关键要素& J$ ^- Q7 k0 @
    1. **基线条件(Base Case)**1 O" {3 @) ?; ?2 r; O
       - 递归终止的条件,防止无限循环
    ; r2 n. z, L' Z   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# t5 U2 e) V* G- O- a
    ! H/ o; r% d5 @+ s) m2 g
    2. **递归条件(Recursive Case)**0 S9 R8 g  z3 [5 v3 S( J" F
       - 将原问题分解为更小的子问题! |6 k3 j5 q2 a" T/ m/ j; I
       - 例如:n! = n × (n-1)!. j0 F9 e! w6 O' v

    + X& y9 ~4 e) ]& B! a6 `$ M 经典示例:计算阶乘$ _$ u- ?9 h- }. g- N
    python+ F, O+ q) N, U+ O0 x5 x
    def factorial(n):8 ]0 z% t+ s- A; [1 e
        if n == 0:        # 基线条件: p# F0 ?' b. L8 _- B6 f
            return 17 F, p$ Z; v2 w5 S( I
        else:             # 递归条件1 j: {8 Y. _! `# c, f
            return n * factorial(n-1)
    ( Y( _: m  [( ?执行过程(以计算 3! 为例):
    ; O+ c6 T' G& Xfactorial(3)
    4 Z, w: m, A' |9 e+ O6 N3 * factorial(2)2 o$ N! t5 E; ?' r/ O% K  t
    3 * (2 * factorial(1))
    ; P' S0 t5 X/ B3 * (2 * (1 * factorial(0)))% L# n( x9 [( q: D$ W$ Z
    3 * (2 * (1 * 1)) = 69 e: S. C) u& a0 c+ K3 |

    % t4 g$ u7 N  Q3 n4 o6 s 递归思维要点
    ' b" `$ ^$ k/ `& C1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 n4 W& i0 B' x" s
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    / D3 V# G2 A9 E3. **递推过程**:不断向下分解问题(递)2 h, y0 y: k5 O. l
    4. **回溯过程**:组合子问题结果返回(归)& S7 ]3 |) T4 T( y7 e; S+ W# ]

    - L* x/ Z& h( ?* J4 h" t1 e 注意事项
    6 V: s- K- N+ J必须要有终止条件
    , k, g5 }5 }, I* v$ R7 J& |递归深度过大可能导致栈溢出(Python默认递归深度约1000层)7 F/ C5 @: f5 x* b
    某些问题用递归更直观(如树遍历),但效率可能不如迭代8 k8 i3 X8 E* K2 ~
    尾递归优化可以提升效率(但Python不支持)
    % o# K  W! x  `: d+ y) i8 [% D, ~2 w% U
    递归 vs 迭代
    : p, u7 \# T& \|          | 递归                          | 迭代               |
    / I$ t5 d1 k: W|----------|-----------------------------|------------------|7 h' F' [2 ^4 c7 F3 x9 U3 i
    | 实现方式    | 函数自调用                        | 循环结构            |
    6 [/ Q4 E" M8 y# b9 r+ t| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |! }- N1 D4 l- 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         |- U7 ~' T+ y& D3 {( F
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |. s! F7 o7 x+ J# j5 J/ g' m  l5 g6 F

    2 T. |( i: C' G  B- n 经典递归应用场景" E$ D1 ?% q8 E" Z8 |, ~- U1 a
    1. 文件系统遍历(目录树结构)/ b+ G5 G  P/ |5 d, F/ z* q
    2. 快速排序/归并排序算法& U: s3 h4 O7 ^; \; d) ^- V
    3. 汉诺塔问题+ r# b% B; c$ P' A6 [
    4. 二叉树遍历(前序/中序/后序)5 v, |7 c; z( W
    5. 生成所有可能的组合(回溯算法)  J& t2 s% k6 `+ K# o. }8 |
    5 g9 }: z  B) S# Q: ]( T5 i
    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,
    $ `, b/ _6 B$ ^! ~$ `. S我推理机的核心算法应该是二叉树遍历的变种。
    - ]% |: ]! N. L( 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:/ U) M1 H3 j; |4 S. J& }& @9 u: z& k
    Key Idea of Recursion' q$ H$ n; o4 R) G* d" r  m7 s

    ( Y- Y$ |& H5 C- N' IA recursive function solves a problem by:2 ?9 R9 H, a/ @  ]  Y( @( ^$ Z& W

    % r( @" o# t- {) }: i; }2 j    Breaking the problem into smaller instances of the same problem.
    4 I6 w3 d8 i* A* l+ Y; _( t7 \' {+ M# G8 N. L8 Q
        Solving the smallest instance directly (base case).0 n& G) Q) a. I* G  j
    6 R0 J, P$ E+ X0 [+ N  J
        Combining the results of smaller instances to solve the larger problem.7 u: m+ p) A* o; G8 W% ~+ J, q+ i
    % B! {) c' W# F4 c
    Components of a Recursive Function& a6 R0 Q+ f; T
    0 p7 V& x0 v* H6 Y
        Base Case:
    7 v$ ]+ X( `# e- A" h5 D  ~  b" c. m$ r- O
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    $ z; E( I) P# _6 N3 W4 w0 b$ |5 ]0 @3 w6 b9 }
            It acts as the stopping condition to prevent infinite recursion., l5 c5 Q1 b8 a- I5 x" M

    1 M$ w, z2 u9 E# g        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.0 m3 {3 B6 G6 h- A0 T& N

    . s! @, P& z8 o  L    Recursive Case:' s3 \9 S  b( n6 o" i
    % e8 Y5 I  O: r
            This is where the function calls itself with a smaller or simpler version of the problem.
    4 S* y7 n1 N" H5 a  M, m- S, U; t7 ]
    4 u! b) r! v* g' r% j        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    2 q' {; q5 G7 A0 w" j3 G1 ?3 J! }  u, z2 t% E- r2 H# N
    Example: Factorial Calculation
    * K9 e+ W/ Y) d1 w' T# I" S1 d, y  g! y/ r" l; E
    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:$ e5 e, d/ u- ]2 \, y7 d
    . q, _5 t4 B5 c7 z, V) b0 o" G5 i
        Base case: 0! = 1- ^3 M( P* ?' j: j
    1 W" B' j- C3 P: D7 S; p3 Z2 ?3 I. L
        Recursive case: n! = n * (n-1)!
    8 n0 b2 x& R( A9 ~1 }
    9 }4 Z0 {. Q6 O, {% MHere’s how it looks in code (Python):* B; B4 ?* F& ?0 g- f$ z& u( B
    python
    6 w8 M& e2 \* Z+ w
    : X$ g; m/ `) Y+ ]& i) ^, O6 c
    " [# g  ?! P" t3 edef factorial(n):( [& V  n. |' l# Y/ p, r0 ]. J+ n
        # Base case
    6 k" R9 `9 f# F# c    if n == 0:
    " E5 y" q/ {3 F! O2 c        return 1
    1 ?: K: p1 X4 ]# a    # Recursive case2 M" C. h2 m1 u+ [3 {
        else:
    9 o# Y& T& _2 g( L, m! f6 v$ L: x        return n * factorial(n - 1)
    2 J% P% Y0 f% N3 K6 [/ u9 h) \4 A' C
    # Example usage7 p0 p  ^1 o# e! U" m9 @. e
    print(factorial(5))  # Output: 120# ~6 v) m8 I- M6 t8 T) p

      W6 ~' y3 E  K: L5 |2 }How Recursion Works8 @( v% L5 q8 f# w# E2 r4 I

    * s9 S* a' s9 }% L7 H7 k    The function keeps calling itself with smaller inputs until it reaches the base case.
    & B# |; o6 C7 `; c& O( ~! R  D6 `& p1 w) D/ s
        Once the base case is reached, the function starts returning values back up the call stack.
    % }( [7 E  K$ y$ V7 C+ e8 ~  ?& ?& W" ]! o
        These returned values are combined to produce the final result.- ^1 U" \$ H# r
    / s4 @: f6 e/ _& Z& _' o
    For factorial(5):
    & m, J- y$ l2 T
    4 Q% R0 d5 C- @, R% R2 k! A$ |0 `' u4 h2 G# z, a; n
    factorial(5) = 5 * factorial(4)# h* F  [# b* \) q6 [
    factorial(4) = 4 * factorial(3)
    5 A- O- n+ i1 `factorial(3) = 3 * factorial(2)
    " [* m$ \: d/ a& L1 A8 D& gfactorial(2) = 2 * factorial(1)- O" K9 j* u* J! X1 |2 `% D$ w
    factorial(1) = 1 * factorial(0)+ K/ {/ U: v$ D& s1 o3 j
    factorial(0) = 1  # Base case$ H( ^: ?5 V) |$ Y3 F' w
    ' K  {/ n, d9 G; u' K  y/ h7 t
    Then, the results are combined:6 |/ t$ t6 Q5 F* F( a8 N8 k; }
    % q' V: e5 B* r( U
    8 y/ e) y2 R$ t
    factorial(1) = 1 * 1 = 1
    3 F- V9 u) o# I  q+ F; Ufactorial(2) = 2 * 1 = 27 y7 S' v5 z# @" p7 t
    factorial(3) = 3 * 2 = 6% y( W4 H8 v  F5 P9 B' w
    factorial(4) = 4 * 6 = 24) K8 W6 p3 X8 T  t2 {# U# `6 A
    factorial(5) = 5 * 24 = 120
    % L0 n! L+ y' X8 ?. x, ?2 i
    " v( s6 U" }( o8 U* M0 i0 E4 VAdvantages of Recursion
    . W$ z, p, U7 l
    ) X% @# M- r2 }9 O$ R    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)." {' M2 G% h% z1 I3 s& q* [7 n. ^' _/ A+ z

    3 \5 y5 [7 p: |& ~; O& a    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ) ?& A3 l* i( `- _6 l6 s- T2 s3 ?* n3 z+ k7 w, S# ]
    Disadvantages of Recursion
    : `/ N. E" O- k
    8 ]0 F! `% h9 T% |/ \* l9 n! n    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.
    ) B1 ^5 B5 y' p( z5 o/ g) N0 w% }: t8 x! i* B
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).- e" U; h* H' h- l0 ~$ m

    7 G: u! s: M* D: GWhen to Use Recursion6 H9 I' P) p& p7 l

    3 c/ o1 x2 c8 g4 ?) V- c    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . Y7 O  U& O# p4 Z& T
      P9 @$ L+ y* i6 b1 ^    Problems with a clear base case and recursive case.
    0 k7 e5 L& Z, [+ a2 |6 c) `) w7 o& z) Z- V& S2 C
    Example: Fibonacci Sequence9 [/ _5 o) b( V3 e4 ?. R- _

    1 F1 K! G3 h* R" n4 zThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    * C( V9 ?/ ~- U0 p
    ! e6 a; h* y" q5 \4 G    Base case: fib(0) = 0, fib(1) = 1
    9 }/ f7 t( h# w6 f% z1 Z7 j, a4 J: e+ O: P' t' U4 {
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ) F  i3 L. g# _( Y7 V! ]
    : i0 S) ?; c. x. H% F) q: ]; ppython
    " j: v1 V/ _" c/ d( h: {7 Q
    % g5 U9 S: F" x( p5 V* S8 g1 Z  K. v3 i
    def fibonacci(n):/ k$ }) _& w/ v* {  R, j) y' j8 F& g
        # Base cases# {/ \8 I8 I/ x3 C
        if n == 0:- i+ e9 h- Y2 r7 a
            return 0! T* P: p4 f' n% s0 W
        elif n == 1:9 ]) I5 d" D' |
            return 1  m% ^. {+ s! m4 V  P; w
        # Recursive case- I$ [1 ?4 D7 x9 @
        else:# Z) a0 |+ n; I
            return fibonacci(n - 1) + fibonacci(n - 2)* Z5 V' P2 o: A4 S# b% m

    4 r6 j8 Q4 w# r  U# Example usage2 I! [+ j. l1 L5 D  O$ a4 t
    print(fibonacci(6))  # Output: 8
    ! Q/ O, s0 d( R! m# [
    ) }9 E1 P' W/ f& q) mTail Recursion
    % m; M, |# T% E# E% a$ o( p; D  g9 H/ N! n
    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).7 v( R) L3 i  q) w  ^

    . V0 u% }# C0 @% z0 SIn 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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