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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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  • TA的每日心情
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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 6 z( A; K' \/ _  n

    ! j5 r' o+ b" H: W5 D( ^8 Y( x解释的不错  s/ @7 \: F! l+ R$ m
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    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。
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    1 }1 h8 d  }: }. Q, q 关键要素$ |# S) [# f8 u. Q. b5 R" A- T0 r
    1. **基线条件(Base Case)**5 H5 X$ X4 m5 d7 P" K0 ]
       - 递归终止的条件,防止无限循环
    , G% V- v: I4 @- V   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 16 S. g6 ?  v/ K8 ?, v

    " g  N* b3 S7 G9 r2. **递归条件(Recursive Case)**
    3 t# }; ]. s, T* a0 G2 G   - 将原问题分解为更小的子问题, I& @1 G+ n/ B9 l* _
       - 例如:n! = n × (n-1)!
    % O( }0 a) K5 d% Z! n# K; X. }$ X  C  j$ [# c0 n
    经典示例:计算阶乘
    : o, A- W0 O2 h% kpython
    # y. c6 {  \' x  V8 ~) P" K  kdef factorial(n):8 p' w8 B( Z4 Z4 a. F9 N3 |1 S
        if n == 0:        # 基线条件
    / p) P- r# |/ `; q6 Y$ H        return 1
    8 M. G) {0 r; d* a/ c6 g    else:             # 递归条件4 H2 @" H6 K) P. m! w6 h" g
            return n * factorial(n-1)# ^5 n0 F; W9 K. P% ~* L8 Q5 h* V" h; `
    执行过程(以计算 3! 为例):, A$ ~, K% O& U8 b7 I$ J$ }
    factorial(3)9 E( I. ^8 m( u  a' O
    3 * factorial(2)
    , B4 Y3 c1 V6 l1 X8 X3 * (2 * factorial(1))
    6 Q" ]1 d0 Y% ~/ _6 Z: ]1 [3 * (2 * (1 * factorial(0)))
    - N7 t  v  y2 P: d8 d0 r3 h3 * (2 * (1 * 1)) = 6& V9 w; K: [; E& b8 a# U( ^; Z
    2 E) S$ Y0 w, H5 N3 A
    递归思维要点
    $ S1 I/ [- T* p, P9 I$ z! l- b1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    / M* G6 ?3 B: `& ?/ G3 R2. **栈结构**:每次调用都会创建新的栈帧(内存空间); K7 b8 L+ R* d+ T$ \) D" ?
    3. **递推过程**:不断向下分解问题(递)1 h: q2 p/ ?3 C) {
    4. **回溯过程**:组合子问题结果返回(归)
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    # {6 o! N8 _5 Z/ C& ?& L4 [! D 注意事项
    3 w8 P4 s" A( k* Z( ]- ]1 K' p必须要有终止条件
    3 p* n! p+ [  Y. K( J递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    $ U, B( E& x5 Z) V  O: B! u* k" A& t某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ) \7 h+ G3 Z: U# r6 Z8 n% _( D尾递归优化可以提升效率(但Python不支持)8 A  H( A" V9 H' p  g/ P5 y7 e
    . b( z; w* p/ U. Q: {8 c
    递归 vs 迭代
    / X; p, `8 S  f1 U|          | 递归                          | 迭代               |8 |$ x9 Z/ ^6 U! v3 z* s
    |----------|-----------------------------|------------------|
    8 z: p- k8 n) Q9 C# z| 实现方式    | 函数自调用                        | 循环结构            |9 L3 y4 i5 a0 H+ \/ x
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    1 g) v* J) @. R) 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         |  h4 v  e' ^% R. b+ s
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |" C3 n# _# I/ w
    8 Y+ @% K( N. G, J! t4 {
    经典递归应用场景
    + Z0 y) {0 m9 N4 V0 f. [1. 文件系统遍历(目录树结构)% s) s" q, L; }" z8 _7 R0 m# M
    2. 快速排序/归并排序算法
    ; f2 I& S: y) X) _3. 汉诺塔问题: U+ `* B2 R: y$ H6 Q; C0 |
    4. 二叉树遍历(前序/中序/后序)
    3 E5 S9 q7 w/ \( F' ?/ q' S5. 生成所有可能的组合(回溯算法). k6 H7 U( a; z+ r  G/ n, i
    ) n9 ?! Y+ [, T7 J( c
    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,
    # M( \' i6 D/ P0 D8 `  p$ Q我推理机的核心算法应该是二叉树遍历的变种。$ Z$ ~8 W8 J6 M' 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:1 m, o( V  c1 h2 y3 X6 I  F- L
    Key Idea of Recursion
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    A recursive function solves a problem by:/ m$ u- q+ P& `) o; c

    . K4 `6 r1 m7 ^$ G! I( E    Breaking the problem into smaller instances of the same problem.3 ~+ o+ X- n" ?, k* q* v

    $ L1 q& t9 t/ Z" N( h3 }    Solving the smallest instance directly (base case).4 h& j! O4 ]3 T" b
    8 K- F; v. z8 u# x0 @2 t& x$ X
        Combining the results of smaller instances to solve the larger problem.8 D( M# J+ f% r

    # s1 O( @. A0 l) k; JComponents of a Recursive Function
    4 {6 w+ Z! o* {8 F" D- D; C1 N, H# l
        Base Case:
    / P6 b% _4 a: w, p* K* D) J( e( \
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.0 i4 I& H  p, z4 T* d; e
    $ e; E/ C1 ^. {" ?% v2 i
            It acts as the stopping condition to prevent infinite recursion.8 I% Y, l  D( j  @2 X
    8 |: I" H3 {9 L3 S/ t
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1." [5 g( L  G: ~( x0 L/ W& F& O

    5 Q, z3 b! h, r" t* n$ j  k    Recursive Case:
    4 d6 h8 E7 B* O
    ( n( f) K0 ~5 n5 u: m1 T  k        This is where the function calls itself with a smaller or simpler version of the problem.
    2 z# `: i- e0 Q
    * c. Q+ v( L+ ~8 D1 X. |& H        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    & D7 }# R1 j+ d) Z5 c/ J3 w$ v1 x" }' o& q+ L+ E/ }
    Example: Factorial Calculation
    , c- K9 H: K0 V7 q6 e. V! I0 F
    0 b5 F! S5 _. l. xThe 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:6 F7 F, D; Y6 b, R0 z) i
    - J- M6 Z& r7 Z: x7 f1 V9 c  S
        Base case: 0! = 1
    4 T' v9 o6 |, {$ U( ~3 h3 y: r: e0 ^/ u4 L
        Recursive case: n! = n * (n-1)!! ~( f1 U2 I7 y4 }' M/ I( a

    : a- Y: H# q5 Y5 B7 c) H. oHere’s how it looks in code (Python):
    ! T: H: X; r  W7 opython* I' N7 K/ |6 f  g; X5 O* S7 h- Y+ A/ Q$ \

    ) T+ l' S: j+ V2 R7 j" o
    / i& u1 N- a6 q) L& q5 ldef factorial(n):
    2 Y" ?. w# }- G& k3 E8 R2 j    # Base case5 K* B2 j7 J1 X/ _6 |. [7 o
        if n == 0:
    # J) k# g1 e7 x( _3 _, `4 M8 ]: h        return 1
    & x" e1 d4 A3 N# b. _5 V+ D3 U9 n! ?- C    # Recursive case
    ( S$ U' K: x1 V    else:
    2 `/ N# V3 w3 ~0 f4 t5 J        return n * factorial(n - 1)
    0 X* w4 T, B" v* \! i
    $ {; `/ t' P  Q6 C# Example usage
    8 b( O# \$ P0 a; Uprint(factorial(5))  # Output: 120
    + l4 d  F6 ^$ {% k5 P1 r
    5 ]. j& ^: q9 z) ]1 R+ G! O- _How Recursion Works. ~: u* c$ R0 `0 R7 _3 r

    2 w7 ^# O9 d+ g    The function keeps calling itself with smaller inputs until it reaches the base case.$ y0 J' C: y1 o* n* r5 Y  k

    ; [0 k4 b" ~+ x    Once the base case is reached, the function starts returning values back up the call stack.
    5 T% S$ c7 @) d7 }( i. [. I5 G' i/ {% Y$ ~) l
        These returned values are combined to produce the final result.
    ) P% o9 t# m- X* u3 X, e) y% J0 L6 E6 w( @- ?' [8 m, T
    For factorial(5):" }7 T4 F: v5 u& q8 A4 i

    " L/ n8 `' t! c8 J6 z7 d* |( c1 Q" b9 B, a6 }( O3 q2 s
    factorial(5) = 5 * factorial(4)
    9 L+ Q/ H/ V% q8 ?0 Hfactorial(4) = 4 * factorial(3)
    2 V% z) Q3 o7 f# h+ I. d4 Mfactorial(3) = 3 * factorial(2)
    . u' v' K8 F) f7 A, n6 Y8 _factorial(2) = 2 * factorial(1)$ z0 e6 `7 N- R' {0 w0 i& r
    factorial(1) = 1 * factorial(0)
    8 W$ P* P. N. Hfactorial(0) = 1  # Base case
    , q# H  F& h2 q- D2 p* o
    / k$ q! \5 n& e; H& b1 V( ]4 A; ]Then, the results are combined:
    % C8 R7 m8 X5 b4 U* x9 s: T: X! b7 i! ], ]5 h
    / V2 F' @- K' }" s- J' `% ~; z! Z
    factorial(1) = 1 * 1 = 1
    1 F! |% R5 j  Y8 s; j8 Pfactorial(2) = 2 * 1 = 2" o: N- k7 W5 i  a+ K4 X. t
    factorial(3) = 3 * 2 = 6
    * H3 f3 J' r- z. hfactorial(4) = 4 * 6 = 24- F/ e0 C/ {2 S9 ?2 j6 [# v6 k
    factorial(5) = 5 * 24 = 1208 G& ~8 c/ }3 X/ }
    7 v6 q% p# I  F) I4 C. q
    Advantages of Recursion8 i. S+ B: }7 y0 ~! |: X

    3 L& h$ \& s9 Q) S    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).
    9 N' z  g/ k6 ?* d2 m
    . s3 T8 `  E6 U5 U( ?    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    , `3 n' P  y; T' o7 b2 i3 U, e1 g( j: T+ d8 c/ o4 C
    Disadvantages of Recursion
    $ ?/ p5 X: O9 v
    0 S" f7 S$ R% ?4 L! l. p4 ?$ r  l/ P    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.- N* a% x+ n! V5 X# \
    , i  ~, B3 U( w" k3 ~7 w3 s
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).% y- i) w/ K6 r4 I, m) h
    $ k) A$ V: v) Z$ B/ q( Y% S
    When to Use Recursion7 `$ e) {1 {1 S+ n7 b% p9 F

    # }. ^5 M! m8 L' y' h  M( w7 P    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    1 Z! t: S/ G* }2 U. |2 S0 f8 |! a( y+ _4 M6 Y2 U
        Problems with a clear base case and recursive case.
    ' c' g2 b4 t1 ?6 S* O0 ?& A9 g3 l% [' z
    Example: Fibonacci Sequence; e( a! V1 e( k4 R& x' P& X. Z  r

      u) e( m* l; R7 o/ gThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:6 l4 w6 T/ n3 b+ C+ l3 l
    0 `8 t& z4 Y- `/ [7 O  a% d
        Base case: fib(0) = 0, fib(1) = 1- m% S' T6 R" c0 ?( W' V
    # A2 H+ v( N+ j4 H% n  G
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    * M4 I9 K1 z* S; c0 L) T8 H1 k% T0 t/ f/ G
    python+ o# z; E' G: k1 N0 C" {

    & d5 G* b$ a& Z6 t1 X( M3 X- J( x7 [. d7 J  V- K
    def fibonacci(n):
    0 G( K* X: l9 D; D1 _" U+ f    # Base cases1 [' ~% _- o4 h/ J7 q1 R
        if n == 0:
    ( @5 ?2 J. o$ b; d# B        return 0; b) X$ O) E" M1 F. J" F# V
        elif n == 1:5 u' |- ], m& i# e1 b2 Z1 i' {. k
            return 1: [2 p3 F8 W  `. i7 Z
        # Recursive case
    ( L; x9 U2 Y' `; C( q: {9 K    else:
    - t. S: x# i/ A( y        return fibonacci(n - 1) + fibonacci(n - 2)
    9 |4 U. t! {, G+ c' m
    & S4 }+ T6 D$ q2 g7 L# Example usage+ W, `2 D: W; Z7 P( ~
    print(fibonacci(6))  # Output: 8
    ; Y+ h4 p  E. m; x. `6 h+ ]' T/ b1 I) I! t3 T- ?
    Tail Recursion, ?/ |5 Y2 G8 \0 m9 [1 R- [
    9 @5 T$ k+ |+ s0 R* x
    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)." ?$ P3 [. R3 J. b5 \! H3 i

    ) u. i1 }4 I7 g! Q! 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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