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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 ) R! Z9 M; a+ s4 L- `

    ) _3 u5 a$ p5 b6 C, ^解释的不错
    + i; c" u2 S; b6 K" f. S5 j9 y) @9 B
    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。
    * R8 t+ B" c, B' u0 M" x* ^* G2 f3 e! O! ^1 A* u
    关键要素! ~( D1 ]1 x4 b! n
    1. **基线条件(Base Case)**8 B5 Q: Y. H2 J3 O
       - 递归终止的条件,防止无限循环3 u; {" i: g7 E) Q; M
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    3 y4 W5 w+ ^" Y
    9 Y- d6 \& R. {% z- c" u2. **递归条件(Recursive Case)**5 S% ]4 G) K5 z! H/ j) l
       - 将原问题分解为更小的子问题
    9 L& Q# @* G" r; \0 s   - 例如:n! = n × (n-1)!
    , T) u! c0 w4 |% q, x, g4 z( X" U$ q  u1 }; R2 r
    经典示例:计算阶乘9 U7 d$ c; Z8 ^# u9 O
    python! T) t3 l( W% K3 E: G
    def factorial(n):" q! E8 T2 J1 A5 O
        if n == 0:        # 基线条件
    1 u, F- J( |8 V" V' w, N$ j        return 1( ~$ G9 N3 L* P. a
        else:             # 递归条件
    ; o" V. X* J3 `% s* j1 T9 @9 h        return n * factorial(n-1)) u- |! x: _" \5 Q
    执行过程(以计算 3! 为例):
    8 M" Z% A+ q( J% I- xfactorial(3)  h6 p' \& {& Q( F: g8 m6 o
    3 * factorial(2)
    ( E$ j0 j9 v) T2 D+ B: U$ ^3 * (2 * factorial(1))5 s" R% d! m2 [% A( U
    3 * (2 * (1 * factorial(0)))
    1 h: V3 C3 C; v- u- i% v4 k! ?5 g3 * (2 * (1 * 1)) = 6
    " G' X7 w' v1 j. ~5 H2 k+ s- ?* [7 [+ `( x
    递归思维要点+ q, s9 y5 I' t. o  \0 t
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    2 @/ D. W- d9 G3 `. M1 ~0 u: t7 t2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    6 a  v* f1 H  v- g) T: X! x4 W3. **递推过程**:不断向下分解问题(递): q, m/ x% M( x+ E2 L0 a
    4. **回溯过程**:组合子问题结果返回(归)& x8 g! p- E+ u' [& ]

    * ]4 H0 I0 m+ Q6 n- e: g 注意事项) W2 O2 q% b$ m# ^" V7 I
    必须要有终止条件
    2 n! t" C% s, E* k: ~2 p' Y" j0 c递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    6 `1 q5 ^4 y8 b) o: [  n, J某些问题用递归更直观(如树遍历),但效率可能不如迭代! b; A2 r) p5 |4 o
    尾递归优化可以提升效率(但Python不支持)& K9 r5 H) e8 N) q- c4 Q2 N  I! ?

    ( j& Y* f3 h0 C5 }; X  } 递归 vs 迭代+ J  u5 a' v- z" y# ]# W2 q' N$ T. r6 r, h
    |          | 递归                          | 迭代               |8 x* }3 U% e$ q  T! j
    |----------|-----------------------------|------------------|
      x( r4 E; p; X+ _7 z| 实现方式    | 函数自调用                        | 循环结构            |2 I5 D2 b. b7 F. B6 v' c
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 t, k+ L, ^& u/ l  u- }| 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         |1 [1 g! W# ~( E# m
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    4 ^# V2 P: `& y* b7 k/ T* ?! }+ C8 [8 E4 ?4 K9 n
    经典递归应用场景
    1 P( p! U$ W7 {1. 文件系统遍历(目录树结构)8 n' o. w: `5 S$ p
    2. 快速排序/归并排序算法
    - W" ?( s/ E1 H+ A3. 汉诺塔问题- O, P% U5 a* R6 r! ?- u* i
    4. 二叉树遍历(前序/中序/后序)
    9 K2 Z8 O1 y# v3 H2 ?5. 生成所有可能的组合(回溯算法)6 K7 F5 z6 Z8 d/ f( x
    * z* [! f/ L" P% z* G
    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,
    0 j7 }/ k- J' c- f我推理机的核心算法应该是二叉树遍历的变种。6 ^. C8 c3 A4 I5 C& _- B! 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:# q+ {' l, h  c3 R, a) l
    Key Idea of Recursion
    9 t8 N+ I& W! m: o# \( _% d: `( B/ I- D/ k0 O
    A recursive function solves a problem by:
    # U- x- u! U" J$ A" T. F/ H- S
    9 L2 B5 e/ F4 G  W* i    Breaking the problem into smaller instances of the same problem./ Z9 L  w! k2 y& J6 z
    1 M. u$ L, L! z1 B2 s
        Solving the smallest instance directly (base case).
    " x: ^  Y# K6 t7 c4 f8 E4 U9 S  v2 G) m) \0 G; X5 s( l" D
        Combining the results of smaller instances to solve the larger problem.6 Y4 b, B) H7 L- K3 T" [7 s

    ; ^% V' i5 A2 t# x* _Components of a Recursive Function, W$ B% o! ~; y" g2 U0 m
    # N  A- q  F  }) O! _
        Base Case:
    ; ]0 T2 W+ Q# q, J  q& g" X4 U9 d5 o, v3 ]8 t: Y5 y8 E
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.+ w9 o& R6 X3 O) T

    - N9 s: x7 y) j+ T        It acts as the stopping condition to prevent infinite recursion.0 `/ T5 ]: V  H2 p
    8 V+ a1 I# T1 i* B( i
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.  _. E. O+ W3 C/ {; M7 G
    - B1 q0 Y0 L4 P7 \$ g
        Recursive Case:/ t( ?8 n: h" F6 |. n* d

    3 t5 }# z) ]5 N( |# r        This is where the function calls itself with a smaller or simpler version of the problem.; o& u/ V2 w1 ]7 w8 P+ c
    , |, Q/ w9 i7 c' P+ \
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
      p" B0 Z9 c) w/ @0 t8 ~
    & ~7 y& t% J& }' X. @$ {6 Z6 XExample: Factorial Calculation2 n: N* G. v0 J

    : z' G9 S0 ~8 R4 Z+ F% BThe 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:
    / e! c+ N- B9 a( j% J+ s! y; c" K* c2 _" ^
        Base case: 0! = 16 u% I& d4 x) N

    ' U, P* z: W/ S8 u; K- W    Recursive case: n! = n * (n-1)!, z5 u( ]; L0 S3 P! @
    0 D  L. E+ E1 ]$ N+ C
    Here’s how it looks in code (Python):9 N( Y1 Q: X: f% F8 o, X6 s
    python, P$ G; [1 e4 z+ o

    " j2 l$ H7 C+ I* G( e
    ( A6 v2 ^+ F3 ]  F6 Ndef factorial(n):5 F: z- l* H( o: f! ^
        # Base case
    ' Z8 e' ]: c7 ^  N2 [7 K& ~& d    if n == 0:9 E( F& s9 m! D. S' B& s5 w
            return 1
    ! M5 ^# `& r" U! M/ e* v9 @3 s1 F    # Recursive case! o" |1 @: Y. o4 \
        else:& M7 i  M# w8 _6 _* G
            return n * factorial(n - 1)
    2 Q! j% T4 O, @5 R' ^2 X$ @6 Z$ @7 p/ @- |
    # Example usage2 N- [, Z. b9 N1 {+ Q
    print(factorial(5))  # Output: 120" S8 D6 o& d- ~5 A% X. |
    2 c: b# i3 a6 X! ^$ D# p
    How Recursion Works, p/ q$ e9 C, C$ [6 {6 p1 d& k

    $ @7 h5 _  x: `# r6 {/ x    The function keeps calling itself with smaller inputs until it reaches the base case.0 c/ I2 F7 l& W, q

    * h6 I) V- b+ z    Once the base case is reached, the function starts returning values back up the call stack.
    ' Z6 y6 `  |) u8 Z; L
    7 K: h1 O* z5 g) L4 E* }    These returned values are combined to produce the final result.' u3 i. J- z9 U8 a# T9 g( R6 O
    1 e2 P: t2 x" d! I2 x
    For factorial(5):" u. Q# G* W' {& B( {9 q1 x/ C' G$ x

    " Q; {7 v* p/ V$ }( @# L3 D0 _- H8 [2 S  l* ]5 J5 @
    factorial(5) = 5 * factorial(4)3 z2 m8 s3 T/ v( w
    factorial(4) = 4 * factorial(3)+ p0 F% x9 r: L  y5 N$ {
    factorial(3) = 3 * factorial(2)# @' c% z7 V. M: A+ A5 h" F0 }
    factorial(2) = 2 * factorial(1): ?4 x& B* @) R6 Q8 _. M. |8 V
    factorial(1) = 1 * factorial(0)3 I; g% J+ Q4 A- E! l
    factorial(0) = 1  # Base case
    1 Z0 p! Z/ `9 ~# a$ o7 h0 N3 A2 i7 Q- C' m9 k* z, i8 o& B) g
    Then, the results are combined:/ q: Y9 Y6 p% |* o. t0 n9 M

    8 |" d" S( @, f( S, `$ Z5 G9 W( @% T( {/ @* M9 V) u- [$ F
    factorial(1) = 1 * 1 = 1/ m# o4 k6 k7 @) E
    factorial(2) = 2 * 1 = 27 r/ y9 \+ c* m9 B- C1 m- S$ I
    factorial(3) = 3 * 2 = 61 t3 [( e) H' }6 I/ Q, ^5 c* t
    factorial(4) = 4 * 6 = 24
    2 s% i/ s0 T2 Vfactorial(5) = 5 * 24 = 120+ Y! G# t4 v/ N$ s" i3 B
    1 @6 J7 \$ Q0 v' o' J" n) w
    Advantages of Recursion
    " @, f' |+ y/ w+ H% c
    . N6 l! H- s% K7 h" O8 d1 [    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).- U8 n  X6 a, p4 ?; _- F  w
    7 j1 Y3 t" j+ ?* x' H5 f
        Readability: Recursive code can be more readable and concise compared to iterative solutions., Z3 L& u- I4 Y7 l3 g
    $ b7 f9 d7 f! R' D; B* p8 p8 a
    Disadvantages of Recursion
    / x/ c5 Y6 g! |  L; v  o( H& e: }2 @
        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.- k6 B8 U! E" |, f6 Q

    ( t! H6 e* o" _3 x  V    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    6 Y2 x8 o' e# @: ]5 q
    7 k( @7 D4 k* l: e! c- ^When to Use Recursion
    8 h& o1 S" X; u* D. Y. i, E" v9 X% \! l7 N0 o
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).% O* I( m  h. H' m2 h

    0 j: g% j1 [% Q$ x4 X2 c    Problems with a clear base case and recursive case.
    6 l. M; H9 ]2 L& P& @3 [# ?; t! f5 n8 J0 A
    Example: Fibonacci Sequence9 j# e$ Z/ C7 J" p

      W1 j% U/ ~% Q$ D+ j9 d# x2 B( }The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    $ J# R; p: L' G& \  `( _* U7 P. L4 y# B: v; h* \( e' V
        Base case: fib(0) = 0, fib(1) = 1
    9 C4 G4 A4 K5 n' {4 H' p( V  a
    4 I' O) c: d6 g. y    Recursive case: fib(n) = fib(n-1) + fib(n-2)/ k# B( F: P9 a- R, _! @
    1 x6 a7 M1 s7 L1 }9 o
    python
    8 `+ a; a! J3 l3 w" a8 u# w# P( R# X) U  [- U' z, Z9 @7 B6 s5 ~
    1 D6 ]; [0 m3 i# }$ z
    def fibonacci(n):- R/ f5 f8 G1 V
        # Base cases
    ) V- n% Y: g0 z    if n == 0:
    ; g/ }" A! E, _        return 0' O# d* }2 P- I; Y/ m
        elif n == 1:8 ?* D1 `- n' P
            return 1
    ' w8 d2 r3 Y: l    # Recursive case+ q! b0 H9 ?/ L- A! h
        else:9 K% X8 ~, O$ k/ ]7 P* X
            return fibonacci(n - 1) + fibonacci(n - 2)6 n+ g$ O& x. ?  b6 X. h

    ; i0 j  Y  ~. w( P9 Z. m# Example usage3 G2 a( v$ R1 p+ l' ]. Q$ k
    print(fibonacci(6))  # Output: 8
    + O, A! q! |7 b/ R  }* o: _3 ?  T$ e: J$ c. P( X
    Tail Recursion
    & V+ Z5 Z) e& {+ f$ A
    / p1 y' Q8 S6 T9 ^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).
    ' v4 d7 O  L1 r# ]
    8 e( G2 h! a( X1 j7 L% MIn 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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