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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
    5 N+ \  @2 l8 W8 a3 _; g# _: `! J7 ~7 i% S
    解释的不错7 T/ W8 w, K7 I9 Q5 z/ K7 `6 {
    1 ^, o: t7 Z2 `  ?& g- z# _
    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 s3 {- C+ j1 \& l2 A
    5 n) k3 D5 R, c0 M5 K 关键要素4 v, y+ I1 i" x/ }  i- P  X2 U9 B
    1. **基线条件(Base Case)**, _. l* p7 q' T; K
       - 递归终止的条件,防止无限循环0 r9 {$ U) K3 `# V* d
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 16 ]6 m4 z+ |! l0 S% ?  @

    . r3 @( c4 J" t7 ]7 O( `2. **递归条件(Recursive Case)**
    * v7 m8 a0 v) D9 [: n   - 将原问题分解为更小的子问题# n/ q% B; K5 W/ l' P8 _
       - 例如:n! = n × (n-1)!5 P  g7 y. O$ }2 J1 h

    * x" t  W. f: S" S& G 经典示例:计算阶乘
    ( f7 ^. q" ]1 U( W4 M" kpython, T  j" w0 P! P' V4 f6 i) Q
    def factorial(n):1 h& ]; E) k0 s  ]( m- C
        if n == 0:        # 基线条件
      ]  O! ]8 h4 d8 v5 p- l& W        return 1: o' _3 o$ a, L7 u
        else:             # 递归条件
    1 u2 p; T: l, w. i/ ^0 ~        return n * factorial(n-1)  F( l; {( G4 C. W7 z* i. `
    执行过程(以计算 3! 为例):
    / c2 P9 Y3 Q1 g0 dfactorial(3)- N: q5 f) V: \$ A
    3 * factorial(2), ^. y0 h$ u! a& P& t! C4 p. G5 T
    3 * (2 * factorial(1))
    1 b+ n( G- Y6 [( x1 K6 [3 * (2 * (1 * factorial(0)))0 f$ h2 ?! {9 U/ z0 C
    3 * (2 * (1 * 1)) = 68 I1 H: n/ a$ ~* ~6 W# k9 ~6 K9 v

    & ?6 B8 U* Y  `  Z 递归思维要点+ g5 D- K3 M* E, g5 h. {) B, H* R
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    , m; J7 }0 V" D2 K( |  j2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    5 a1 M! V! v) Q" o1 |3. **递推过程**:不断向下分解问题(递)
    2 K  b' X( O( ^% U; I- l9 _4. **回溯过程**:组合子问题结果返回(归)
    . w5 T$ `+ P9 P- c! J
    8 `7 ~" T+ O+ j2 U# a1 r: Y 注意事项
    * Z, ?6 O& l: K" `8 L必须要有终止条件
    ; t! t( D$ N9 i递归深度过大可能导致栈溢出(Python默认递归深度约1000层)/ y/ h2 _8 f. x2 [. {" U
    某些问题用递归更直观(如树遍历),但效率可能不如迭代# j2 m1 \* Q' U. h
    尾递归优化可以提升效率(但Python不支持)
    5 L" p4 }" d* m+ H  t! H& \
    / g/ X: H9 u/ B+ V0 h6 L 递归 vs 迭代! U5 ~  P2 m/ N5 k4 L4 U! z7 m
    |          | 递归                          | 迭代               |
    . y. j" H8 o; y/ C+ A) c1 n* A|----------|-----------------------------|------------------|
    # o' C- `2 Y1 G! \- w$ \) d8 C, J1 z| 实现方式    | 函数自调用                        | 循环结构            |/ \+ B# p$ U1 y3 F' }( n5 ?
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |: a$ K& ?/ ^7 |( I2 s$ M
    | 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 c3 R4 m3 d9 G$ C3 ?% U! z
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |9 ^8 N' r% k) T

    ' w1 S  m5 V" w9 ]6 r" j9 ~ 经典递归应用场景: u# I7 V% Z/ O: [7 u
    1. 文件系统遍历(目录树结构)
    , }6 m  _: a4 o2. 快速排序/归并排序算法# @$ C) R. L# l
    3. 汉诺塔问题
    5 ]" X9 U. G/ i) r3 a4. 二叉树遍历(前序/中序/后序); ^9 ^. C+ A$ c) ?
    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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    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 [. o$ S, [9 D9 {6 x
    我推理机的核心算法应该是二叉树遍历的变种。6 A, I& p! I1 d! X( n$ u* U
    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:; `8 {* D3 e2 [7 a  `7 X+ R. v
    Key Idea of Recursion+ j, Y% m! Y+ Q' h9 a  I. v" z: v9 c

    / D# b  _* A9 S( EA recursive function solves a problem by:3 y8 t7 k8 m  M! o5 i9 c

    7 A9 c# b2 E+ u+ m, V1 a1 {    Breaking the problem into smaller instances of the same problem.4 d9 Q. D+ _: A" N  c5 p% x$ S
      H+ L5 W" f2 ~; e* P
        Solving the smallest instance directly (base case).; }) f/ P; ]! ~! l7 }9 g2 y* ^4 u

    ' W; F, ?* b: S& H3 h0 h. U    Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function4 A' Z5 B2 r/ o
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        Base Case:
    ' F0 ~% e: z% b4 O! h3 }* p! y' F' `; D" K0 B) m+ V: R1 t
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion." g0 S; @5 z" G4 c: @) L

    6 `! V/ P6 U  Z" z" K/ Y        It acts as the stopping condition to prevent infinite recursion.
    7 I# Z2 f  w! G. d$ ]* k
    2 ?( }. O  X# U4 V' v2 j7 ]        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ O) Z4 p6 Z0 h7 J
      a; A' e3 w! x- s
        Recursive Case:" Z  T! t9 o6 k0 s
    0 J; h( j7 s& X" V9 h) j, V
            This is where the function calls itself with a smaller or simpler version of the problem.
    " P2 h- C( \4 K0 E
    " i# u! I4 |0 W- E        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    : k; q1 u6 G$ ~6 K7 D. ^+ X8 ]% b
    : v' w' A) D, z& ?1 w/ JExample: Factorial Calculation' ~. r# f7 y% a

    7 L2 A6 B+ _2 m* R  o/ wThe 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:  V, G$ N* ^) I/ E" j

    3 C% B1 F! U* m6 r: Q7 h& ]# g    Base case: 0! = 1) {# P# G& R3 G/ [9 C$ [

    1 ~6 d7 u. p3 ]2 f% N; s    Recursive case: n! = n * (n-1)!5 B7 a7 G* D% {) q8 N' j2 P

      D; Y$ F# {4 VHere’s how it looks in code (Python):% v8 \% B) o; r; K
    python( Y3 N* \1 A0 |! \5 w! ^1 j* R

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    $ H' w3 e# W" y, R; Zdef factorial(n):2 x6 j4 S6 m2 M
        # Base case
    3 [- D( r  `3 ^# F+ C    if n == 0:' b& L% N9 V$ S
            return 1
    - {2 w" Q0 a2 P: K, r    # Recursive case7 Y6 _' w9 Z7 k0 f
        else:% p! p. @7 Z6 o% Z
            return n * factorial(n - 1)$ |/ |8 e2 Q( C' G( w7 P- [
    2 `) y4 w3 s. H2 |
    # Example usage
    $ _) n; Q6 r+ a" T" Mprint(factorial(5))  # Output: 120% t0 V5 K; X# T8 S

    4 x; \: l5 R7 I& n. pHow Recursion Works
    . ^( x) K$ i& y4 ]: z8 s) y5 B! L$ @& \' n: X+ J% p+ t
        The function keeps calling itself with smaller inputs until it reaches the base case.
    ( q: F7 V9 l$ a7 W! R/ B3 d$ @) j/ h7 y8 Z& U" G3 R, `
        Once the base case is reached, the function starts returning values back up the call stack./ l3 x$ n$ X# Y1 g

    / Q9 P' U. O1 \6 f: [7 O    These returned values are combined to produce the final result.+ _4 |9 y# P) s9 J

    : Z) Q! t3 a' A/ O' E/ d( ~' @For factorial(5):$ I7 e& N- |6 m9 O. a' U. n

    ! k, O/ y) Q2 k) o. X& A# p
    / I/ i0 ?% S4 i8 b" c* p7 k+ A! rfactorial(5) = 5 * factorial(4)
    ) U3 S. E3 l. x- i& _! K9 sfactorial(4) = 4 * factorial(3)3 T! f. j/ t* `% o
    factorial(3) = 3 * factorial(2)
    . W3 y6 e: C+ \9 d7 R2 _factorial(2) = 2 * factorial(1)
    ) Z3 _3 k3 N+ nfactorial(1) = 1 * factorial(0)
    9 J% I9 W' O0 x2 O$ z( `  Vfactorial(0) = 1  # Base case
    6 K" l6 i: c0 b  k! {) R1 J2 u( n: w6 Y5 ]! W
    Then, the results are combined:# h6 N; |3 Q/ A/ f- A0 J/ K

      A( W% E( ]& I8 x4 H2 ~  J; `2 W
    . H8 m  U: o4 B- rfactorial(1) = 1 * 1 = 1: U; C: f3 p; W
    factorial(2) = 2 * 1 = 2
    / v- O3 J& i, rfactorial(3) = 3 * 2 = 69 q5 u8 ]/ T- g
    factorial(4) = 4 * 6 = 24& A: b) U+ s  T  z0 d$ l# Z8 L
    factorial(5) = 5 * 24 = 120+ {/ H" c9 p7 p4 D0 C
    0 ^6 e7 f& |/ @+ K8 B
    Advantages of Recursion
    / u$ b; _* k* G: ]0 m/ {/ p/ T
    " F, p+ I1 v$ 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).
    + W1 ]8 z5 ]- A7 }1 P
    6 `8 N& ?+ _0 t; i7 i7 H    Readability: Recursive code can be more readable and concise compared to iterative solutions.: C4 v5 w6 v6 R' A1 `0 n1 l

    : L( l- \7 L* R( _- T% YDisadvantages of Recursion
    / M. ~. p4 N8 m! P+ C3 g
    4 [# s1 v' H1 {7 x, P7 ]    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.
    - m) ~: w6 t* }, c  b% P4 w  V# I( U6 x! ^' H: Q4 K+ `
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).9 A  Q" ?. y; {3 u7 l( p6 U

    5 y  e# i3 p0 D9 t* l+ q0 qWhen to Use Recursion
    ) q1 n0 {; O( b3 ^- a2 j6 u0 d
    8 i" d: y3 m, B7 y# G3 ?4 v    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 m7 N+ D6 R) ]

    " J( v" H0 B3 n1 n. A    Problems with a clear base case and recursive case.4 w# l4 ]& N9 \: B0 I  w! V8 X
    7 {7 j' v( O* y& D
    Example: Fibonacci Sequence& b" W8 J" \  p8 Z* w5 x8 v  `

    ) v& H- h6 `" B6 ZThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    5 @+ t0 J' A! |) o: D, |0 z6 C1 g2 ^; p: a+ t8 `; k
        Base case: fib(0) = 0, fib(1) = 1
    : m  P2 f! @4 E  h- W' r+ Y) P8 }0 p, n4 y- A6 q
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    7 t. Z$ E& h+ e. t3 l2 M' j1 V# n2 P( z6 @
    python8 Y7 f* I/ e# d+ k, Z! b7 W+ }9 w

    0 G' V1 O1 n7 M. C0 r. I2 z; M( y1 Z) d4 @% |7 {
    def fibonacci(n):& b; @% k+ N: `: P; Z; U
        # Base cases$ o/ u4 k, Z8 e" `) p
        if n == 0:% Q9 V. x7 F; X2 v5 u
            return 0: H' F" O6 r' n- W9 e# E
        elif n == 1:
    3 B5 s! L5 A6 i4 F! c        return 1
    ( K; Z; [  B. }5 b3 J9 e    # Recursive case& R% d4 Z( y2 ^2 h9 M! R$ y
        else:, |1 S( A' z4 ^+ R9 o/ \$ X' V
            return fibonacci(n - 1) + fibonacci(n - 2)# i- S! H  y9 V3 p! r. |% J# W5 f. F

    8 W1 G& r2 p8 z! D* _# W% B. G# Example usage
    2 y! Z' ^2 c. w% [+ l  K% Xprint(fibonacci(6))  # Output: 8
    " o9 s+ L2 _" ]( Q  }0 O8 `
    ) u) @" D+ b1 rTail Recursion3 }1 {) g/ J- Q) @4 N( E

    & N5 f) W* _( q0 VTail 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).
    & N0 S- `- v1 w& h. }# e- x% Z: x' Q
    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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