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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
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    " Y6 d+ J, A! W" y( E9 L0 L解释的不错& B' {, \" i" p% E% H  Q

    8 t- T, Q, |/ A. W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。3 Z8 }+ N1 L$ ~2 X" K
    3 S( q% \9 D, M0 z* l5 W0 ]
    关键要素' D$ }2 i& _  G' _( {$ k
    1. **基线条件(Base Case)**( q" G% W9 L4 c& i7 M( @
       - 递归终止的条件,防止无限循环
    ) ^/ t9 i6 W: g/ Q+ W   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    5 I, A% g* E! s, x0 H" k$ Z/ C& K7 |
    2. **递归条件(Recursive Case)**) ]$ t! n( z3 o1 o. I
       - 将原问题分解为更小的子问题1 x) S6 a* O! h7 ]
       - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘
    1 W6 j4 q) w5 }8 l; L: L8 cpython! I5 U" t' y" q( D5 k
    def factorial(n):
    " `" }# B5 e3 Q# n    if n == 0:        # 基线条件) @7 Y- O" W4 p. G
            return 13 g& d2 J$ E8 Z1 P- h" `
        else:             # 递归条件" S9 I7 `) }: Q' i
            return n * factorial(n-1)
    ( E/ S. l" A% K* R( Y执行过程(以计算 3! 为例):2 _; W1 F$ P$ \$ }: \: o
    factorial(3)
      M5 R$ ~( Y; g5 l/ @! F  y1 ^5 b3 * factorial(2)5 |! s9 ^' w9 c5 Q. i4 P
    3 * (2 * factorial(1))% \$ B* y, Z* _& `
    3 * (2 * (1 * factorial(0)))
    " ?& T, v- c1 R. y/ o/ ?1 Z3 * (2 * (1 * 1)) = 6
    ' t9 |& n9 S* C. |- g) ^! W9 Q4 `0 e4 L
    递归思维要点
    $ C1 `6 i! e  J# @1 Q. ]3 R$ }# J! a1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : s. [! ?" ]- @& q: W2. **栈结构**:每次调用都会创建新的栈帧(内存空间)1 T# e. I# j9 \" h" I
    3. **递推过程**:不断向下分解问题(递)9 c: [. s5 c  q  m
    4. **回溯过程**:组合子问题结果返回(归)/ \: J2 r# }5 O# ^$ E# r, E! a" ]$ v

    " G$ F6 P4 N% Z# v( q: c$ R 注意事项- k' A, a+ A# R
    必须要有终止条件
    . w% V$ ~" i5 j, i8 Z8 }; A递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    $ M( B; N6 u2 [3 q' y& P某些问题用递归更直观(如树遍历),但效率可能不如迭代4 x8 K9 V+ C  S: w
    尾递归优化可以提升效率(但Python不支持)
    ( b$ ?+ j# i' {0 @. v# V6 a- E5 j1 t5 B" I  M7 U
    递归 vs 迭代
      {5 K2 D2 O0 A2 _" q! c|          | 递归                          | 迭代               |3 z8 u& E) ?& Y9 r
    |----------|-----------------------------|------------------|
    7 f$ [8 m( Q9 o2 X) j" D% W| 实现方式    | 函数自调用                        | 循环结构            |  G2 Q! e" P! W/ p( W4 R) \' P
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |3 w* Q* Q5 R4 j" H& Q$ S" v& e6 w% G, e
    | 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         |* s3 G$ A' b3 W4 j( [. J% h
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    - N4 d' G' V5 ]2 {5 m( `. k
    0 ?% H, v' C4 P1 i, P 经典递归应用场景
    " @; F* [/ e/ H# t9 B, h1. 文件系统遍历(目录树结构)
    8 l. {1 y+ b" S2. 快速排序/归并排序算法
    ) }) F& O4 A6 |' e0 X4 k3. 汉诺塔问题
    6 r. K0 J0 ^4 F4. 二叉树遍历(前序/中序/后序)  f5 X8 u/ c+ u0 F; d7 u
    5. 生成所有可能的组合(回溯算法)
    ; @5 H0 X, f/ \! e, }, A1 }2 h2 d% X5 d& t0 A8 @# G) r8 e
    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,  l5 u  ]- U' P2 B
    我推理机的核心算法应该是二叉树遍历的变种。( r0 }0 H& n& o' @# D; a9 G; m/ \
    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:
    * S! I) f( |% @$ c. _Key Idea of Recursion
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    A recursive function solves a problem by:
    : J0 E  \( V8 U# ]+ f! e+ R: {; v8 q6 a. Y) h& I6 c7 i* r
        Breaking the problem into smaller instances of the same problem.0 v$ W4 d  F2 q7 i" l
    & Y5 C1 Z) ~, }( c9 F. x  A$ w9 y0 S
        Solving the smallest instance directly (base case).# h* z( c  @, Y3 B/ U$ C# F* C

    ; h, r( I( y5 _! |    Combining the results of smaller instances to solve the larger problem.4 v" r: g$ j. S% ^  ^$ A* }

    7 c1 |; |4 k. \9 g9 @% a- BComponents of a Recursive Function( {0 Y5 w+ b+ L' Q9 u" b
    7 ?7 N' H; u1 M2 K, D
        Base Case:
    0 {8 e  y7 q4 z* _. n5 V  o# N% `5 G) |/ Z* f) I' Y; J! H
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
      k, T# I$ O) S) u6 o  B: X. K' g0 E* l, B" u, [4 T1 F/ g8 ]
            It acts as the stopping condition to prevent infinite recursion.
    ) ^; I& c9 F- `4 p* {  c$ g3 J9 G' a/ g# i
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.. y1 E# o& W- ?5 v+ ?  ^+ G+ [# S) k
    ! b: _& }0 i2 |
        Recursive Case:8 P, s) o* I  `- m( ]
    ( ^8 n5 {; }) V
            This is where the function calls itself with a smaller or simpler version of the problem.
    ; t0 _  p5 ^; X& s! e  M
      \0 [$ `% E- r. N' }) a        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).  b) t( a. n# F  {
    6 z9 m' e4 y- g: L6 Y- M6 ^& v
    Example: Factorial Calculation
    4 g) J# g0 d7 i$ ~4 Z/ G% ~
    . T* M" b7 d# L( S( P( C# |$ [5 TThe 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:/ K5 u- g2 ?+ ]) D# y

    5 U6 j9 P* o# |% r; C/ B7 C# y    Base case: 0! = 1
    # q$ [- K. L4 ]; w- R' L7 i: ?$ K8 Y  V' A. a, n  L$ {3 O
        Recursive case: n! = n * (n-1)!
    : Z* }" t+ x. u( }
    1 m! \" |( \6 T4 Q* aHere’s how it looks in code (Python):  c0 I/ [8 m1 I, V% C5 \, a. g% t
    python
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    ' S3 c% S/ d5 G& O0 L# \' [* Z: h
    def factorial(n):
    3 g  n9 f* I  l) f9 G/ B    # Base case
    7 `* ?  q( n. ]  s" y2 {/ d5 k/ f    if n == 0:
    ' L5 v! ?- b4 Y        return 1
    4 C3 Y1 j8 C0 r. o9 E% N    # Recursive case
    9 {) W( M& A( k) O    else:
      }- y1 N# N# D$ Y8 e) N: m$ I! t        return n * factorial(n - 1)
    + w, o# s8 J( L" x5 \/ l/ k" [
    , t1 p4 L: `& Q2 s# G& `: j0 q# Example usage
    7 \7 q! ^/ @  U9 u1 c( s, R* Uprint(factorial(5))  # Output: 120
    - S) d- O: S% X  n) G* S
    1 t' D( ^7 q0 x+ ]# UHow Recursion Works& `2 \1 f2 k3 g# c/ z$ n0 H
    , h/ Y6 u6 z. l% m( a3 ^/ @3 _
        The function keeps calling itself with smaller inputs until it reaches the base case.; m9 [  l6 [& B7 a! q
    3 l( `4 y0 j$ ~6 u) P, v
        Once the base case is reached, the function starts returning values back up the call stack.( n# N1 Z/ `& R3 g9 V
    ! E: m* s# D- \. ?! ~
        These returned values are combined to produce the final result.0 ?7 G3 A6 b  ]: X: u8 Q

    9 G: f! d4 o0 M/ v, K# o7 Z! {For factorial(5):
    4 k" @! \0 @' k: t# ]
    8 A& H# N5 S0 Z' u% T
    2 P: c7 l1 v7 ]' Y# ?5 b" w) Afactorial(5) = 5 * factorial(4)
    2 P- B6 _; F, E) y0 j- y* B" Rfactorial(4) = 4 * factorial(3)' k6 X8 u# X3 S. V; B* l
    factorial(3) = 3 * factorial(2)) E- f; B" X& M* s
    factorial(2) = 2 * factorial(1)
    6 \5 h/ Q* I8 h6 |5 f' l  {$ m* E5 Mfactorial(1) = 1 * factorial(0)
    $ O% S  b/ e8 U( Z+ H9 wfactorial(0) = 1  # Base case) D) ^* K3 D: B' w! Z
    . L; p, Y/ H6 z7 ^( ^) ?
    Then, the results are combined:
    + \3 k( Z5 {+ J7 c& @3 T
    / w4 {, l$ H3 f* P( l9 s  F# S9 W+ G. k( K
    factorial(1) = 1 * 1 = 1" p) @8 I7 |! Y8 I
    factorial(2) = 2 * 1 = 2
    , o# e1 F) j" ufactorial(3) = 3 * 2 = 68 o( e; p# T% f; m
    factorial(4) = 4 * 6 = 246 ]: [: H! M% A5 M% |
    factorial(5) = 5 * 24 = 120# |+ O* |, l4 E0 t- a* b/ C
    & _4 R' _3 M6 c& x2 G
    Advantages of Recursion
    2 l4 O" G) Q1 R  s& u: G8 \9 k$ Z) `" T& C& x2 M3 H
        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 U! Q4 q$ ~5 c( P; n5 m) t5 ^

    : W+ `+ B* L2 U- _( |/ m    Readability: Recursive code can be more readable and concise compared to iterative solutions.4 \1 n& Y6 D# i) x4 s0 P$ G0 o

    8 {1 B7 L3 Z0 T2 y. SDisadvantages of Recursion
    ) k  D$ j7 E% f0 R  C2 ?/ J1 D0 @$ q4 n3 c, ~4 V) Y
        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.9 h2 r1 l3 Q' P/ L+ ]3 }- h! v! o- f, J
    6 f5 h/ m' c. T/ c) ~
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ( n& s* k* D, \2 W: z2 n$ t+ U! V( t
    % {# t( W- q8 `7 v+ q! VWhen to Use Recursion
    ( Q5 m  B. r! I7 b  ^  e; v8 Y0 Q8 E3 ]5 m2 E
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    8 ?# ?; E+ S/ D5 `; E: W- S; _8 T) ]5 j
    ) Y9 m0 |0 I2 b    Problems with a clear base case and recursive case.
    8 g. }# q( G! i% b6 ~' d  ^% i. r& H7 b$ v- ]8 b
    Example: Fibonacci Sequence
    " g; [  j4 }( u+ K6 n( j/ w" s
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    # [8 H1 P) Z" p: {4 I8 t5 |6 n% l: g. ?  c6 ^
        Base case: fib(0) = 0, fib(1) = 1$ i. `8 f& H" V- x4 g/ @  q
    + R- ^5 {' p: G# D$ V$ r
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    + i6 C5 m5 u3 f$ I' H( ~1 p) S6 J$ M. A# _3 h& v6 p; E8 V! u* x
    python) T( J9 B0 ?7 a( {
    * A* X. \1 H1 d$ b' j" X

    2 @) b) U8 a3 z+ g: Z" T2 O8 W" kdef fibonacci(n):# G! Z, \$ d# ^7 `
        # Base cases
    3 c, ?) q3 f# m5 V/ U- x    if n == 0:2 K4 e6 D. b. F; e
            return 0+ C4 Q4 w+ H+ L
        elif n == 1:2 H" E9 a) x! G  ?! }# R
            return 1
    5 C, l: L1 W  S7 E  C5 l9 j    # Recursive case% _$ L; M1 g9 Y9 F+ v' ]* t! r
        else:
      Y0 x! l3 f+ X" ]& j        return fibonacci(n - 1) + fibonacci(n - 2)
    + g) N- D, X9 V& V: z
    9 a% d: S* ^# L# Example usage
    , r0 P7 R" d$ W& _- ^( Hprint(fibonacci(6))  # Output: 8
    % o2 r- t2 r; g/ Y* _; a3 I8 _: f  D, m8 P0 A$ b4 ?! O& S
    Tail Recursion+ @; N$ i0 T$ |1 {
    8 {: a# w9 O- l; f- d
    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).6 Q' ~4 R/ P  e

    . x; ^  @6 t# Y' Q# q% _5 ]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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