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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
    & \$ A& p6 p, b7 C* f5 h" v/ @* j* g. o7 a2 X
    解释的不错
    $ ]4 K' E. D& c6 y0 d+ R" c4 d" B& p- [* e2 l. g. z3 M) 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。
    ' ~$ }* s- R( I- g2 [, N5 a# X
    ) v3 N( T  W- C9 t* r& F 关键要素* s9 T# `8 t) X  W9 e
    1. **基线条件(Base Case)**
    & U2 j; x; R& R+ N+ ^, ~   - 递归终止的条件,防止无限循环
    # v* l. a, c! h$ Q   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    , i0 _3 E6 T: M7 A: g1 S0 v
    " q/ y' ^# u. n+ W& z; G2. **递归条件(Recursive Case)**; n0 q9 \0 H, y3 |/ w
       - 将原问题分解为更小的子问题7 u! N3 `0 n/ L. Z) N0 {/ @
       - 例如:n! = n × (n-1)!* Q1 K0 D4 v- o1 ^

    8 [2 |2 P: }4 }: B* X 经典示例:计算阶乘
    / c1 M% D$ d8 |6 u/ Mpython
    5 \% X: ?9 u' G# t2 I* N7 cdef factorial(n):& E6 t6 \6 b& y, @; P
        if n == 0:        # 基线条件
    / c# w1 [- N7 {* f$ k! x        return 1
    ; s5 `; l  W: u    else:             # 递归条件
    3 t% V3 z4 h6 S& Z* y$ z        return n * factorial(n-1)9 H. v  b, U0 p: i/ v4 A7 [  ]/ m6 |
    执行过程(以计算 3! 为例):/ E* \6 m, K; r# T
    factorial(3). h3 g. ]7 |1 a/ h, v: ?
    3 * factorial(2)( G1 x" p0 [; t/ h
    3 * (2 * factorial(1))
      n' U) N: }7 e3 t, P) [) T$ E; m' D# e3 * (2 * (1 * factorial(0)))
    3 w) N0 r" f$ _3 {* j; _3 * (2 * (1 * 1)) = 6
    7 N/ c, x' ~4 O6 a7 _
    ' m) S6 Z3 U( d2 {$ \& T+ c3 T1 h! o 递归思维要点% \1 W1 l9 R7 F9 I2 Y% l
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    4 `3 r+ t; o' S- n; a" s2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    1 K; _) c7 c8 L' w3. **递推过程**:不断向下分解问题(递)
    $ k2 A! Z7 o# E# j! e4. **回溯过程**:组合子问题结果返回(归)8 W$ N  j, t) j/ I, R* Z! L

    ) c& M# `* J* \! g4 k 注意事项! M4 ~( U; e5 B5 N1 W2 u1 y
    必须要有终止条件
    : }! f, s8 s/ t递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ) R& K' {5 P( H& K( N1 n" r0 i4 t) e某些问题用递归更直观(如树遍历),但效率可能不如迭代
    : q8 y5 |' T7 m1 c4 c/ E尾递归优化可以提升效率(但Python不支持)1 |! d! b( j8 J0 {, e

    5 @8 o4 F: o8 O 递归 vs 迭代
    8 o; K: v( \2 l. Q: }, B|          | 递归                          | 迭代               |, j$ B- @0 N( c; g2 v% L/ D
    |----------|-----------------------------|------------------|) N9 w1 G# Z, \, \( H" Z$ |
    | 实现方式    | 函数自调用                        | 循环结构            |7 r: V# Y; \" V; t& J8 g' x
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |$ }3 J! l; b# y
    | 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         |
    - f# D/ s6 [  @9 C| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ) A' S# H' e$ G, M" w. u0 x+ i. j" ]$ c' n2 H, D5 Y" O0 u: ]
    经典递归应用场景
    2 x4 n; J7 z( c1 u7 K1. 文件系统遍历(目录树结构)
    7 m2 O& K' q. b5 V2. 快速排序/归并排序算法
    # q: m6 _! v% \5 S3. 汉诺塔问题  X1 L" i; F: Y
    4. 二叉树遍历(前序/中序/后序)% H! \7 J$ |* S3 q% ]; {# M+ v
    5. 生成所有可能的组合(回溯算法)
    4 I: x( M. w3 T; v- y, `
    8 t9 r1 y; ^4 \2 q0 d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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  • TA的每日心情
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    昨天 07:29
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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,
    # I0 c* ^7 p  I# w我推理机的核心算法应该是二叉树遍历的变种。7 q0 N! y% c) h6 L+ T- k
    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:, ?5 w; \5 P+ f1 l& ]8 I6 D
    Key Idea of Recursion/ ?7 }, w" X3 Q) W/ Q1 |" i! D

    9 n6 M/ F* j5 y6 W0 c0 iA recursive function solves a problem by:
    6 t  i  `9 }: Q, n+ A6 ?& H$ i! E: Z1 b' r
        Breaking the problem into smaller instances of the same problem.% e8 h1 K; @8 e" C# l' @( i6 k

    , b$ w& o8 x# f% Z  z- f    Solving the smallest instance directly (base case).# c. K+ U6 ]3 R7 t" U
    - S$ `% |" E0 G3 r. o' S" A. e
        Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function
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    $ G& m4 M5 s: x  ?- b! }, b4 c    Base Case:
    * x) v2 i% h; Y4 _1 Z. L0 X6 x1 r* r6 ?. d9 H  Y7 E+ M9 s$ R& A9 H
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.) ~& K' O- w" Y, F
    : b0 F1 l& L/ L7 X& e, c) {
            It acts as the stopping condition to prevent infinite recursion., z/ J+ d( W; U6 {/ X# l

    * c5 u9 c$ p# b& `' L, H7 |        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.7 n2 R0 G8 V7 }% U( Q7 f
    # n2 a. I# G, O1 s  z6 d
        Recursive Case:2 R6 }( M/ L( d: p7 l

    " w& d$ G+ E  p* x5 \$ y$ z        This is where the function calls itself with a smaller or simpler version of the problem.6 H6 M- z4 g7 A5 W( V# T- v( O
    5 P! N, {4 V7 A6 r
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).+ N$ k/ o, v$ P7 E) t: s

    $ {* H! b% l1 j7 a' J1 @/ t7 t$ R; R5 Q, uExample: Factorial Calculation
      H0 p6 N: y1 o
    0 H4 s  W5 [. Q' U: S9 ]' @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:
    5 f$ L* u3 C8 f6 D' H
    ( M. U8 ], w1 s* u3 U3 u8 @    Base case: 0! = 1
    / ^8 r, Y7 c* p- F, }( S& a3 M" E
        Recursive case: n! = n * (n-1)!. T, ~' q$ ?0 |& b: k  v
    8 ]9 ]4 y( j+ v
    Here’s how it looks in code (Python):# J4 x" |% |% \7 p/ J
    python
    1 E: U7 V2 R8 s3 f! @( H  j
    , k- a7 S% e3 S1 }6 |2 N6 h( h5 b
    def factorial(n):: [/ @: }% p. ^2 P7 N, p
        # Base case0 N# u; Q  L7 u& J" y* P
        if n == 0:
    , f2 p+ Q- @, r; I, P9 `        return 1* c: y7 L; C) T
        # Recursive case
    . V7 I7 p/ ]2 L( n3 x5 ?8 N. v( [    else:
    3 y; G- p# g; A6 j4 d        return n * factorial(n - 1)
    ! @4 p) u3 B1 i% T
    - V* ?1 y. X7 U- M2 _# Example usage0 j) C; e# e7 _8 w0 D) Y# J. W8 o
    print(factorial(5))  # Output: 120
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    How Recursion Works( b. j; q. Q- C2 ~9 v* G9 t/ B
    5 F5 n) L: ?8 l5 {' w
        The function keeps calling itself with smaller inputs until it reaches the base case.
    " f$ C+ {: I& h: Y5 n  o- W) S/ d) {/ g* T2 e6 ?5 Q+ f
        Once the base case is reached, the function starts returning values back up the call stack.
    5 U: @8 }0 G5 A  A5 ?8 C2 o2 R9 ~+ u+ g6 l. n2 s
        These returned values are combined to produce the final result.& N; n: @7 X$ y5 e! b, \1 Z6 R8 m
    ) z1 P0 ^. \& U
    For factorial(5):
    : X' Y& s* z( p2 n" Z
    ( ]5 H: ^' O. E! O1 l: ]* Z
    & `+ q; [! W- \0 |& R5 G7 K2 F8 V; Sfactorial(5) = 5 * factorial(4)! l8 _" Z9 S8 q7 }9 D2 ~( t' O0 E
    factorial(4) = 4 * factorial(3)0 k0 ]( o$ X" T$ K9 L) d( t' ?9 T9 V. J
    factorial(3) = 3 * factorial(2)& r- b, T# g- c* ?
    factorial(2) = 2 * factorial(1)' _4 t* P( {9 ]& \! K' c6 a6 W
    factorial(1) = 1 * factorial(0)
    : h: {3 {, I/ C8 L: A- vfactorial(0) = 1  # Base case2 f$ J7 i. M2 k

    . [, e: k7 Q, n6 S5 bThen, the results are combined:0 ~3 W& N; z( G+ e- y

    / X0 e7 r' X* _3 x( [0 {0 @! h: n0 u# `3 L5 h  }
    factorial(1) = 1 * 1 = 1
    2 \& u$ g+ N  E4 t" \& ?factorial(2) = 2 * 1 = 2. G. M) W2 M! X, C
    factorial(3) = 3 * 2 = 69 q! C* o# _3 Z6 u
    factorial(4) = 4 * 6 = 246 F% Q( @7 {$ s# a' O
    factorial(5) = 5 * 24 = 1201 U  v' M1 M$ _

    " |) q$ o7 P' I* }4 TAdvantages of Recursion
    ( X- H* T) r: d. {1 `" y# L  [3 S2 Z; v  S# j" q
        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)., W- T3 `- q9 [6 ~3 ~4 Y
    2 Q% g2 x( H* C5 r# x: |$ V
        Readability: Recursive code can be more readable and concise compared to iterative solutions., Q7 Z1 s6 o  @8 O
    + y  a2 e/ ?# {$ l) D6 Z
    Disadvantages of Recursion8 k+ J8 S" p8 N& ^1 N& y

    % r4 O* A$ i0 X9 s* x- k    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.
    8 a# m/ J: l% b# M9 |
    3 T/ y2 t4 S8 T    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).' ]3 G4 V% B3 X% B8 k

    % D: j2 x# @& J* P& {0 `, n% XWhen to Use Recursion& P# _' p& R4 }" }1 F3 v

    ) \$ e: w2 U2 i' f" r. e. |5 E8 R    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).1 a; h$ G" Q$ s( P

    + Q5 L1 W% U) ~) m1 ?    Problems with a clear base case and recursive case.2 y' _+ B4 Q/ O- h# s& z; J

    1 N0 }9 x1 y2 R7 s3 MExample: Fibonacci Sequence5 ?: z/ q( {; Y* x, {- i. A  S9 u

    - E4 }9 ], |( k" nThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:, c0 c) c4 ?8 }2 T0 P& @$ r

    8 A# w+ O# `% c9 y    Base case: fib(0) = 0, fib(1) = 1
    ; m* ?6 K% l! V! g% Z2 ?8 ^& p1 H1 Y
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    5 O' K  @/ w1 c2 @! M! H4 c) O5 @  C& ]: T; p& v  `; L
    python3 r1 B& R$ C; k9 q7 K  }' e) K! C, g

    # I: [, q( B  R7 B+ c# K. R3 e* C
    ( o0 z2 \3 T, c0 J+ W) Vdef fibonacci(n):
    & w7 l3 q' \' i. y% [6 X    # Base cases- B7 [9 W- |3 @+ A" ^+ D
        if n == 0:
    5 g/ M' k. {7 ]" ^6 C        return 0
    ; E5 D4 I+ \2 h" \7 o5 p. t& G    elif n == 1:0 @) t- C( Q/ p$ l2 J, Y: ?
            return 11 T7 [* D0 u- \" C9 M( E
        # Recursive case
    8 X/ O6 v2 H! g! q5 i) ~- E    else:
    $ F* c7 q) _) ^- m4 T0 B% O0 y5 M0 F        return fibonacci(n - 1) + fibonacci(n - 2)
    : s% q, S* j: s$ ]4 e
    & g5 i- v+ e  l; t# Example usage
    % Q% M5 r# I) u: D$ {print(fibonacci(6))  # Output: 8) t8 ?# y* `' \. }
    # [) Q8 J; Y2 q( A
    Tail Recursion
      \' R4 N* d! Z7 Z1 h4 f6 `
    9 W0 o  T2 z& b1 o# z2 \, WTail 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).8 M+ {5 B# h1 O7 C, Q
    5 r$ y. P# Z5 h, F/ }% S+ s
    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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