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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
    ' ?( T+ p) ?  d( ~, X3 s7 U6 }) U
    1 i9 j% c0 h, L* B; e解释的不错$ J. E7 c& [! z. a; H$ q" e
    5 G' P: U: m9 \% U: ~
    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。
    1 Q( p4 @7 Z4 g( v; B0 Y& ^- g
    ) m7 w* a- f/ W 关键要素
    & \5 G* G0 b- _1. **基线条件(Base Case)**
    5 D- H2 i! A# P! ~. W  y   - 递归终止的条件,防止无限循环
    / M9 h) y. u$ D! Q3 Y   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    2 o; G9 g% l# ]# d; H
    " D  f( Z3 L( C1 K9 U! z# ?' E& k2. **递归条件(Recursive Case)**; X0 T0 s' ]; [2 P0 n" X' v2 c* c- k7 S
       - 将原问题分解为更小的子问题
    - A+ q/ l3 o8 N' j   - 例如:n! = n × (n-1)!% d2 G  P8 N+ _, T( t) t

    & x2 g; H) ^8 J: k' z, p) Y. g 经典示例:计算阶乘
    * [3 w4 n) k" ?, g- Q) d4 [python
    / d9 a+ c# i& c: q4 Rdef factorial(n):
    ( }9 I2 z: B8 w$ Q5 _0 a9 u, ?: X& Q1 X    if n == 0:        # 基线条件
    / o7 }! ?  ~5 B6 u0 X2 }  f        return 1. J! g4 J8 c$ n- N* p' C
        else:             # 递归条件
    & O! T# ~* G/ G6 Q& J' W5 s        return n * factorial(n-1)
    # F, c. J! \! R: C3 ^执行过程(以计算 3! 为例):7 I1 p- N' v* ]# E* c$ P2 L
    factorial(3)5 n! T! k! U, D1 G
    3 * factorial(2): O/ N, [* Y* V5 S3 c
    3 * (2 * factorial(1)): t8 f6 p7 s+ I1 X8 W! U: Q
    3 * (2 * (1 * factorial(0)))
    # T0 G# y9 c+ Z$ N4 \8 L3 * (2 * (1 * 1)) = 6
    0 H$ J# p- `( e2 W# ^' ?9 X; v4 h1 ~+ O% {
    递归思维要点. e/ j7 B! a$ j0 W) B, U' _7 f5 n  `
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    5 o1 l; w$ i0 ~  K+ b$ d2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    & o, h1 R6 Q5 x- L# v3. **递推过程**:不断向下分解问题(递)# }7 Z2 k0 `; Y7 d
    4. **回溯过程**:组合子问题结果返回(归)
    5 z. _) _# j+ Q6 g
    " q# l: ]  j/ l/ \ 注意事项( p, S7 ~( G  `1 Q# X4 C+ Q- J  v
    必须要有终止条件: h8 W# `/ P) L9 V% l+ R, u
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    : e* [2 Z6 L% A6 d/ Q6 E  `某些问题用递归更直观(如树遍历),但效率可能不如迭代6 P4 b- t- i# J( g5 \$ `$ ]6 \" y
    尾递归优化可以提升效率(但Python不支持)
    # f* c, C- t$ X, o( A; ?+ l; q
    " u( O3 z5 {' }6 a5 p4 S* |. u 递归 vs 迭代
    # C% q! L* B, A3 W' K! x- r( b, }|          | 递归                          | 迭代               |
    2 z, r5 p4 c/ j/ f|----------|-----------------------------|------------------|, X5 X* |* r" T0 G
    | 实现方式    | 函数自调用                        | 循环结构            |8 O2 q4 W* S- Z# t+ P5 y! W
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ) A$ \4 Q/ M2 b0 L$ g5 G6 G, ]| 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         |, ~2 R7 r5 P- \: k
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    5 `4 a, d: s, o. z
    ! Q& @5 J, j, T1 I/ U: H 经典递归应用场景
    : v/ p& {' p" j6 E4 s( g! a1 J1. 文件系统遍历(目录树结构)
    ( x' T# ^# h2 l0 E2. 快速排序/归并排序算法
    + _+ A/ N, S; W/ O. @( P3. 汉诺塔问题" i8 M& ]6 F7 ^) P- t% X
    4. 二叉树遍历(前序/中序/后序)
    * O$ ^, @/ r4 U5. 生成所有可能的组合(回溯算法)1 K. J* h* J* D4 x1 ?2 w
    9 b" l5 T; p& ~, u9 N8 _4 P
    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,4 h: ]5 k- c7 W1 r) R
    我推理机的核心算法应该是二叉树遍历的变种。
    # C8 k5 S4 {) o. l; M+ Z$ P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:( s4 _: L1 R* G" Z% t9 \$ M* _
    Key Idea of Recursion
    " P2 k; Q" n( G! B, f$ \3 z, d8 Q# @  z6 y& |8 ?  X1 N! a
    A recursive function solves a problem by:& i* R1 h  s5 Q5 [6 U$ g
    3 P$ k4 O* |' y
        Breaking the problem into smaller instances of the same problem.
    , u" G/ a, u6 @! r1 c# ?1 c5 G0 X+ A( H7 U
        Solving the smallest instance directly (base case).
    4 R2 K3 p9 ^& j5 b4 l' K4 T' H+ |. W( e
        Combining the results of smaller instances to solve the larger problem.. E* J, g& w; |! t# u

    1 M* w5 U) M# U- B0 H  F" BComponents of a Recursive Function
    : w8 C* i* W7 p2 z4 {  n4 G
    & e" G4 `& ]6 `4 ~' [$ b, d1 N    Base Case:
    7 A/ a) [% V- m! ]/ @2 N) w/ h9 `+ T9 B2 K3 t5 y
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.0 b4 D5 x+ d4 x! [* B9 }0 r
    5 w( M" D4 s# h- z/ W
            It acts as the stopping condition to prevent infinite recursion.9 a3 H3 d! Q# q# ^5 F
    ! T: l' N$ A: v; @9 F
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    : m% `9 m2 `" x$ G. d* M) Q1 G
    # }* O# J2 i7 r    Recursive Case:4 H% `5 ^% n! r% Z4 W" X% j. V

    ! p1 B" V- w3 s, w* R        This is where the function calls itself with a smaller or simpler version of the problem.
    ; ^4 r! E$ a( N8 f% W! r
    3 {2 j4 x: ?! q2 f. \( R8 U6 k# O        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).  ~0 o! U& k& F) C+ ~
    + b( s7 {6 q: r: c# N7 m
    Example: Factorial Calculation8 @5 ]! y& q; B' i) H. O6 H

    - X4 b" F% U) C/ kThe 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:
    $ j5 F- I( ~9 M, ^" {4 g9 k0 p
    % j$ V$ }! y6 V5 [6 R1 n    Base case: 0! = 1
    5 g1 ^7 s$ G( B. Q) h! C/ H8 P& h, f: m6 O6 F6 x
        Recursive case: n! = n * (n-1)!
    & i' I. F. @( e, K- m, T3 W6 _5 p, o) q
    Here’s how it looks in code (Python):1 m7 j6 g9 @" |! y9 |& n
    python2 x4 Q. y+ u' c

    ) ~$ v9 y+ F, w2 H+ A/ U/ C' c# ]/ p3 ~* x1 _
    def factorial(n):
    ) o# h# B' C, E* Q7 D0 }# o6 p/ @% x* l    # Base case, _+ {! x8 g- e) Z! h1 G
        if n == 0:/ p& O4 C  m5 x+ J1 a# @' B
            return 1
    + Q& Q' p1 k- ?/ C    # Recursive case1 M7 E% Y6 @3 d
        else:
    : v1 m  v/ D# c% ]9 M4 D6 w        return n * factorial(n - 1)
    & U; ?) Z0 m0 Q! A! b/ a
    + l% I. Y9 N2 `( f# Example usage
    6 G) `0 P$ S4 P' ]/ ?% z+ pprint(factorial(5))  # Output: 120( f, o6 S* w8 [+ I. b# r
    ( I- ~3 g! R, F  i8 a5 {; t. [
    How Recursion Works
    4 I9 L. Z/ \# r! ]- v5 `, v
    & N) a$ q, _, a: U: J    The function keeps calling itself with smaller inputs until it reaches the base case.) r: m) i, F7 ~7 i% d3 `0 s& H

    ; U+ d$ N2 k. N    Once the base case is reached, the function starts returning values back up the call stack.4 J; a0 w0 z& l( t( v3 e

    0 ?1 P$ n9 ~0 Z8 f; {$ L    These returned values are combined to produce the final result.
    ) {3 }9 b" W( q5 ?
    ' N. `( V% {- @1 _  m! jFor factorial(5):
    1 `0 S! D$ O' T. S: G1 H! r; E. N/ c6 @1 C+ i
    ' J5 e) J9 D1 M. ?2 U3 o' L
    factorial(5) = 5 * factorial(4)1 O  e! f( S% v' T) A
    factorial(4) = 4 * factorial(3)
      Y9 X8 v: C9 C& w. w7 y" kfactorial(3) = 3 * factorial(2)0 m' e( y2 O+ _6 y+ q# n: e
    factorial(2) = 2 * factorial(1)
    % X: n  y& Q! b8 _4 o7 vfactorial(1) = 1 * factorial(0)) x& v# t/ G! ]0 g
    factorial(0) = 1  # Base case! b& F& i8 R& y4 r& g/ |
    ( a5 y8 ~5 Y9 r/ J
    Then, the results are combined:: u9 h( z( Q' j9 A* q2 D4 x7 H
    ( d- p* U3 [/ m5 h* j& ^0 `( L6 S% f; b

    8 M" x, [- j! e  Lfactorial(1) = 1 * 1 = 11 x; |, m1 Z; |1 C% Z* ]
    factorial(2) = 2 * 1 = 20 Y8 N7 g" `/ C' q
    factorial(3) = 3 * 2 = 6
    7 y: z; z7 N; e& |factorial(4) = 4 * 6 = 24
    * H" B9 F$ [9 i1 d% C! {factorial(5) = 5 * 24 = 120
    / U4 P8 N% y( W3 w; N
    5 e% X- u/ W" JAdvantages of Recursion
    + N6 C  [. `/ V7 [/ n+ k! }  r
    . g/ Y& k: o* r$ q6 U    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).
    ) u1 J; }" O  H! B+ H! N- d  E- H" `" [2 O5 z' G6 v' ?/ A  R3 d
        Readability: Recursive code can be more readable and concise compared to iterative solutions.7 e# F- S9 ]- l
    ' \' t" w4 u1 H" D- L4 J$ ?
    Disadvantages of Recursion
    % t& p# z  W6 V3 ?9 V& y4 Z: b( t* @8 C/ {0 T3 M+ w  [
        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.
    ! V3 d2 r2 K0 `  I
    " i9 `3 z9 g, U9 [: {& h" {& ^+ j9 b3 w    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).) E9 m& |5 y' J- l* I
    # x; `& Q$ v2 O7 Y6 b, l
    When to Use Recursion
    . _; w+ y' q  A0 W7 J4 g" T  q
    $ P- M! h7 p3 n( a5 D    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).7 |; V! }' c1 [4 j* z4 F

    + X$ N5 P0 o" D) V" f    Problems with a clear base case and recursive case.* o0 W) p6 h1 y& m2 W! T

    1 p; m& s, k- i2 i3 R4 U% R/ eExample: Fibonacci Sequence+ }0 I3 X2 j. X- |

    ) g1 B% x9 J  |% Z- fThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:  V& C+ Y. r; H9 Q

    2 V! G7 |  q4 [4 ^  c2 r    Base case: fib(0) = 0, fib(1) = 1
    % ]  {( @, b! X5 t' B! B9 [% G" H
    1 g$ z/ p! R+ l  Z* o) J2 h    Recursive case: fib(n) = fib(n-1) + fib(n-2)' i) K( P) V. E
    6 ~" u# h. n8 ?* F2 P. W* O
    python9 ^3 a3 g! r* w; v

    * c. j  ?8 C& g8 W$ i" h& y
    - {! v4 J4 o5 `def fibonacci(n):, t0 T' n4 E6 B$ {+ O$ a2 B% @
        # Base cases: X' t1 b( Z+ `# [
        if n == 0:
      Z6 O8 m0 Y& X2 M. F        return 0# }1 L) s) y7 f& J: {+ e$ J/ H% k
        elif n == 1:' n) e5 X1 J9 ?" R2 N. }& Z* }
            return 1
    / d' h+ a  g+ g    # Recursive case# d% t$ D& }$ W  B
        else:( e: F& w* ~1 y4 I  Q5 M9 Z
            return fibonacci(n - 1) + fibonacci(n - 2)
    ) M; r0 y6 z/ t3 x+ x, P+ A4 [  Q; G3 N0 Q% ]
    # Example usage
    3 j7 [7 ]2 ~& d  P2 Bprint(fibonacci(6))  # Output: 8
    6 R/ w0 `" |# i) k" R" Q& Z/ r4 p/ @/ W% p: [
    Tail Recursion; f1 t1 \) Y. H$ m8 c& ^5 f6 D
    * M, T' ?0 o6 ]5 r1 {
    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).) d, f! }% c, n

    ( m5 i7 ]5 j; VIn 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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