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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
  • 签到天数: 3 天

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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
    8 t9 Q( |7 @  x: F/ y4 k. V9 L% E
    2 ^% D$ A7 V  S' w- I解释的不错
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    5 m3 ~, l1 T+ V9 T, y7 p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。8 _" G) L& O2 H; G

    4 I% P4 r6 y; T; d 关键要素2 h  V" F: s6 U  p9 U
    1. **基线条件(Base Case)**# ~& ]) O% h+ t# ~5 R
       - 递归终止的条件,防止无限循环
    4 B. p( L& g/ U2 O/ ~: _   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    & u' |. V" E+ x% {& a" s$ _$ g
    ' q, l( G% v: D/ Z' x* j7 [6 @4 B2. **递归条件(Recursive Case)*** o  p  o0 ^6 h  E2 w6 ^
       - 将原问题分解为更小的子问题: B8 S1 b( Y' l7 W
       - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘/ C6 ^. ?5 R+ O8 d/ v
    python
    0 s* \/ y' m/ U2 e6 Q. Fdef factorial(n):
    9 i# T' w' L# C; q" {5 d; V    if n == 0:        # 基线条件$ z! g' r( v4 K% k  j
            return 1
    # B+ `1 b% O4 [& ]    else:             # 递归条件' w# _3 D! \" m1 S/ C
            return n * factorial(n-1)/ i1 |. {/ ?5 v+ ^4 `- s
    执行过程(以计算 3! 为例):
    ) o; n: I8 c! C: H+ xfactorial(3)
    1 m# n! O4 s; C( M9 ]$ s3 * factorial(2)
    / a) x" ^7 h+ L3 e5 s3 * (2 * factorial(1))! r# W9 |; X$ }/ l
    3 * (2 * (1 * factorial(0)))( E- ~1 u# f. V+ ^* C2 F8 q
    3 * (2 * (1 * 1)) = 63 d' y7 F& p$ U, ]# X
    1 r' K8 s- i$ u/ e
    递归思维要点
    7 _1 ]$ w; |/ o9 a% j" c1. **信任递归**:假设子问题已经解决,专注当前层逻辑. A: _5 |% k9 ~, e
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)/ t6 |- G9 H) O! M$ i. A. T& [, c
    3. **递推过程**:不断向下分解问题(递)( V6 Q* L* O) T. l
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项
    ! S1 N8 r% x5 I7 |9 D必须要有终止条件
    . h3 J5 u) [5 t# I9 H递归深度过大可能导致栈溢出(Python默认递归深度约1000层)1 J. f3 v; ?* a9 l3 N5 E
    某些问题用递归更直观(如树遍历),但效率可能不如迭代  q2 ~% |) D6 f7 e5 S& W, K% R
    尾递归优化可以提升效率(但Python不支持)
    9 [/ I: W) d  D# Q- \- [1 J: W
    9 d# D" g1 V1 R0 B$ D0 G 递归 vs 迭代) {1 B. F1 I" }/ r' U9 }3 ]$ K
    |          | 递归                          | 迭代               |3 p# p% y# L) f/ Z2 d
    |----------|-----------------------------|------------------|3 {/ C$ J& Y1 N5 o: L( G
    | 实现方式    | 函数自调用                        | 循环结构            |
    + b* v* B$ i( p8 Z" l| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |4 A- \% D' N3 q
    | 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         |
    4 J$ I& F, j& [% I% Y7 P0 h| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |" z2 _+ y# r, K8 K) H# c  t/ Q. C6 o
    * R3 r$ D0 T( `+ V! Z% X
    经典递归应用场景. F, \+ k( J) n, R# V  B
    1. 文件系统遍历(目录树结构)2 g# k& a# Q9 b# [  j$ x5 ?
    2. 快速排序/归并排序算法: R2 s) W- p3 k( |, M- S
    3. 汉诺塔问题
    0 @/ E9 }4 y& R) y* j" `" y4. 二叉树遍历(前序/中序/后序), a7 L7 D/ x# c& t& }
    5. 生成所有可能的组合(回溯算法)
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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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    前天 06:41
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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,
    ( Z- T$ U2 s8 M" }我推理机的核心算法应该是二叉树遍历的变种。" n' P. S9 Z, R9 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:% J3 `- g# K, y9 U/ Z# A. ~
    Key Idea of Recursion' N# F6 ^  D; z6 e; V- ?& X
    3 c) g, [. H( S9 i; m* p. j
    A recursive function solves a problem by:5 J" l/ `! _# k

    ( ~+ A& R7 U" Q" [# O$ U2 U    Breaking the problem into smaller instances of the same problem.
    % T  W& U7 R+ P" ~( `/ `: I7 l: J
        Solving the smallest instance directly (base case).* d7 ?* v! q+ Q* ]; `+ h0 U

    4 v3 a0 B+ N) q6 L5 e& `    Combining the results of smaller instances to solve the larger problem.4 M: F9 f* o/ J& `
    8 m: ], x6 f& [7 R' F
    Components of a Recursive Function5 r+ W. G  t) P" ?$ K

    " v9 j, o" z3 l" R; N' D    Base Case:
    ) J( a% M; |8 B; D) V2 m0 X% |1 X1 N# r" R! l) q
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    6 z4 Y- j: O8 N# x3 ?
    + a, M* |; s9 w( Z( t        It acts as the stopping condition to prevent infinite recursion.
    6 d' d( w* t1 m- u0 e, X* t& T
    3 Y1 m1 ]1 T( ?  e2 T        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.* c% r" q$ ^& ?8 j- g; y' y. }. Q
    2 V  \$ N" {/ A1 j
        Recursive Case:- k  g, M4 p' q/ V
    2 _* _8 I2 P: o+ t; _, p# C
            This is where the function calls itself with a smaller or simpler version of the problem.2 N3 M4 G5 u% D# N+ e+ o+ u( _8 c, Y
    . R9 R. L8 c2 f
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    1 T& x: c7 L" \. E; l! H& ~6 r* D( s4 @" \% v
    Example: Factorial Calculation$ F+ Z% \. F+ @3 s" e+ R
    5 K; |5 e0 K1 l* B0 I% O
    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:$ H5 j0 r* ?2 b$ f
    ! D& w! J2 E% J% t0 R/ L
        Base case: 0! = 1( p& W) @1 Y9 u2 a& l- h

    7 a/ B1 w; m( Z. @; O. K    Recursive case: n! = n * (n-1)!7 d: K% E$ C4 n* Q/ I! P2 B5 C
    ' Y/ B) O5 V  g0 S' w
    Here’s how it looks in code (Python):: p) P5 c  }7 h
    python
    + e: z& c6 g+ o% A* X
    3 o+ \& O4 O: S, r4 n' Z5 x' z6 J" k+ w9 H# \1 h
    def factorial(n):
    $ `+ N& _* A$ x    # Base case
    % k/ t3 R' z$ O4 z5 [# Y    if n == 0:
    * N2 t+ F" O. d7 l        return 15 C8 |+ W& ^5 ]- W
        # Recursive case
    0 C$ I# E/ L" T8 H) a    else:( o; }3 P% V; g( y$ ~* U+ n
            return n * factorial(n - 1). L/ [$ X, j$ ~

    & d  R0 o# S1 `! G: r6 H# Example usage5 D0 b2 U5 l0 z8 ?6 ~8 c+ A6 W
    print(factorial(5))  # Output: 120
    + |/ d" o6 W" M4 I1 l* U, u0 e
    - v. t1 o/ w. @, g& e! J0 [How Recursion Works
    & A& Y( |1 w4 ?7 h! |
    . ~7 P4 z! y5 ?8 m! m    The function keeps calling itself with smaller inputs until it reaches the base case.
    ' ?2 t5 r0 p! a- [. k- ~% I, n4 `" r! ~) l- D: \
        Once the base case is reached, the function starts returning values back up the call stack.
    3 p: y7 m* _' i; Y0 ]9 [8 u- ^( Y. M1 j( f/ [8 I' X
        These returned values are combined to produce the final result.# X" ~( @3 c6 O, T+ }. E

    / Y" R5 c2 `( t# o! M) gFor factorial(5):/ r; ?& F, Y! u. T- a  F

    % i( o/ @. K' V, a- @
    3 M. }# _1 s5 @/ p* ], [factorial(5) = 5 * factorial(4)
    / x  b; E7 f" y/ q3 Jfactorial(4) = 4 * factorial(3)
    6 W1 v6 }$ X  S: vfactorial(3) = 3 * factorial(2)
    7 Y0 F( |7 n& \0 J' ^6 |factorial(2) = 2 * factorial(1)
    " d+ G" @" t4 T( kfactorial(1) = 1 * factorial(0), o  l; T' F2 K% U
    factorial(0) = 1  # Base case
    3 R; v, \# t# ]
    ( T: C# @7 F. I/ L' ?Then, the results are combined:
    # o7 ]! Y. H) z8 G6 r* X% H# ^4 Y( q2 ^# C. y; M' N* W

    ) N/ r' x2 K7 P4 p6 T4 ?4 D' nfactorial(1) = 1 * 1 = 1/ P. U- Z% [* p5 Z) s4 \( g
    factorial(2) = 2 * 1 = 2
    ) a6 S' ]  H4 ^, h3 ~& rfactorial(3) = 3 * 2 = 6
    ! Y9 _& x& l3 ^; i7 S- o6 x3 ifactorial(4) = 4 * 6 = 24
    , S+ N8 R# L  D7 r$ F4 u/ I  R4 vfactorial(5) = 5 * 24 = 120
    5 v, ^: {! U% R9 O# H: P6 r# f7 W
    ' O4 t+ H, I' a/ @- zAdvantages of Recursion* N& m1 r/ \$ U

    & U& v+ D& R  L$ G5 v    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).7 N3 W& R/ ?' A9 Q/ O" z" T
    ) ~- {6 s& x$ u4 u' q
        Readability: Recursive code can be more readable and concise compared to iterative solutions.4 P/ M) u0 R4 [  X- O8 t( L
    3 U$ z' z2 r/ _# S' N
    Disadvantages of Recursion* s2 G& s; }1 A5 X# v

    $ ]1 a+ i0 U7 F! U& C    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.+ I% |( @* W, S$ C

    7 A" \4 M! g# r+ |    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).& D5 M, z! U7 t" D8 N8 l6 p) J

    0 Y; |1 [$ R9 @% hWhen to Use Recursion
    9 G- t# O% L% o  g) v) u, S$ T! v% j2 p% @& \3 n0 K
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    # O5 t: D$ T0 S
    ( c; L9 u. ?$ G2 w6 J5 `/ @2 m    Problems with a clear base case and recursive case.
    & h$ f+ P# m# S+ [  _, b6 c# Y2 ?4 x0 I, }
    Example: Fibonacci Sequence2 d7 P( a- s. ]/ _/ Q

    - B* y. v( I. m" R0 e3 JThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    1 s  W  l7 S* z; q0 d: q) f  P% w6 D
        Base case: fib(0) = 0, fib(1) = 1
    . C# ]$ s3 E, z) g5 R. N5 r4 U  S7 l) s7 |1 J/ y+ z. `
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    " Q3 r4 c, P5 T- m5 c/ d% y* e) bpython1 g! ?" K/ k9 M
    : C! A/ m+ c5 t6 X4 H; x8 [
    * M9 o) p2 i( Q; t! i) B+ {! ?! Z
    def fibonacci(n):
    - ~7 t, J$ n1 ^) N    # Base cases
    ' Z! A* r  ?0 F9 N    if n == 0:
    # j9 w3 S5 k2 d3 ^3 I* t  D; x        return 0+ c& S. `% j# Y$ F: h- {' Y
        elif n == 1:
    + o' H6 |- j& A9 F/ ]3 \' L        return 1
    & `: s; v; r4 E# T2 r# ]1 T    # Recursive case- s# u4 m2 B, l, `
        else:( M% ^0 a" X8 W& E& ~- J" t
            return fibonacci(n - 1) + fibonacci(n - 2)7 c; _) W7 g6 N* m
    # ?/ E3 M/ L2 [2 T
    # Example usage$ @* y; q; |/ i( [5 y; N
    print(fibonacci(6))  # Output: 8
    ( E8 C+ ~* I) g) U5 d8 \. x! k  }
    + s. c& C1 K9 H, r2 @* _  _Tail Recursion
    ' K: l: N# c4 O3 T% b0 a
    , R- O; z* T# J5 F7 ATail 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).
    3 T  }: j) k4 {: d( |& x
    * D. h4 w9 C# u5 eIn 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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