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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 7 w2 F. V! Y& R. f
    $ D$ j) o3 f5 G: Z7 {) O6 X# c& L* O7 \
    解释的不错
    " X( G# f8 l" e
    # \% g. ~! y4 g  \* S, ?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。' P! [( `& Q  E0 }+ b
    5 D1 R  X# S) t) S$ [7 T
    关键要素. n3 _: n5 A% e: S
    1. **基线条件(Base Case)**- e6 J0 N* Q: ?4 K! M
       - 递归终止的条件,防止无限循环- b. u* {+ a5 M. s
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 13 o$ I- e5 V( o- F: J$ g6 A

    + c% C& I; t3 }* G6 n# J2. **递归条件(Recursive Case)**2 M. M  k& R0 u+ V
       - 将原问题分解为更小的子问题
    " H$ i: n! @! E# D   - 例如:n! = n × (n-1)!2 B) Z7 ~# g5 Q' A% Z& |% W+ u

    / a7 @1 I( u* E* z( Y1 n. R8 W 经典示例:计算阶乘
    3 R9 G' J- U$ D2 H, wpython! r9 x) P( }* t
    def factorial(n):
    3 e- `% Y" h5 F! p7 g, G* i5 w    if n == 0:        # 基线条件
    8 V- z9 W3 w' s& }        return 1
    + p- U  X4 \/ p- u2 l3 Y5 y0 b    else:             # 递归条件
    4 ^+ K+ U" t7 I- S6 H2 F" x        return n * factorial(n-1)* a4 a  G: [" [; P
    执行过程(以计算 3! 为例):/ k" n  N$ S7 u
    factorial(3)3 W0 i4 @+ K8 U3 o( a% L* U0 |
    3 * factorial(2)' `6 v, k: t4 P/ M! _- q5 a
    3 * (2 * factorial(1))# I( t6 O0 W0 `& J* f
    3 * (2 * (1 * factorial(0)))- j; `5 r5 ^7 j$ s
    3 * (2 * (1 * 1)) = 6* L  N3 Q" H' x7 r! r1 m% ^7 g
    6 A* N& K: v: H2 b% c6 v- }
    递归思维要点
    ; x9 ^1 ]3 _, Z. N2 M6 b% U1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ! V& F1 s. h. N0 ?! \+ R" f, n- V2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( T  p, y: o% h. x3 D* O
    3. **递推过程**:不断向下分解问题(递)
      a: g9 W% j; Z3 x4. **回溯过程**:组合子问题结果返回(归)0 I4 j! C( T: f: f
      P. e' h- j% Z$ e5 z9 P0 v
    注意事项
    3 s0 Z+ F+ Q5 q必须要有终止条件9 N3 E; M& g5 y4 x# O
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    5 O6 y' `. `, j- j某些问题用递归更直观(如树遍历),但效率可能不如迭代- q7 r& |7 G; ^: I" t# z. p% n5 R
    尾递归优化可以提升效率(但Python不支持)5 n' T+ t1 _- t2 f. J5 E/ v
    ' K7 A$ z8 f6 ^# l3 i7 u
    递归 vs 迭代
    % a; R! `2 M" l|          | 递归                          | 迭代               |# ^/ G" t6 k' x* u5 w! L6 F
    |----------|-----------------------------|------------------|
    * c9 z: o9 v- l$ r/ X| 实现方式    | 函数自调用                        | 循环结构            |+ p; r: g6 g& p9 s$ w
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |7 _" l+ ?2 u7 i( Y& r: w
    | 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         |2 t7 U7 x5 ^& R/ u2 O' K
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& A% L- K% W* J* P, h: l3 K4 f
    , F+ [& D. j4 {0 _# M7 f6 o
    经典递归应用场景
    ! @( f5 J9 B0 W9 B1. 文件系统遍历(目录树结构)
    # b# R8 U7 ~: d4 |. ^2. 快速排序/归并排序算法
    " F6 }: ?, G% m9 c$ X: W2 x" Z3. 汉诺塔问题
    & z& ~6 Q& m' S4. 二叉树遍历(前序/中序/后序)
    - c" w$ [& q' F& w" G! v5. 生成所有可能的组合(回溯算法)) W* I/ p6 ^- R2 J  T

    % V& [4 T+ L7 o  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,
    2 n/ b, f: Y" z1 D我推理机的核心算法应该是二叉树遍历的变种。! i$ a" g+ f$ a- Y8 f0 H
    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:2 s4 I0 G2 V3 }7 Q
    Key Idea of Recursion1 H2 k2 }: N: O+ m5 I

    + G& `+ `& ]/ I- `0 WA recursive function solves a problem by:) s. _5 F& v4 L# w& M
    6 w8 g/ N2 C7 O9 N3 ?( j+ x0 c
        Breaking the problem into smaller instances of the same problem.
    ( ~; [8 r2 u  g( Z2 c. A. P
    / R- b' U* A' e5 C: V6 b! i    Solving the smallest instance directly (base case).# ]- D9 b" P3 A2 W  H
    8 w" F; S4 Q' Y% y# z6 V
        Combining the results of smaller instances to solve the larger problem., ^' @4 T; |& C2 h0 G  R6 A0 N4 O

    5 \* {! E: c- y4 D; D% o/ K+ }" ]Components of a Recursive Function
    0 k* n4 o7 U# Y2 G9 I$ F/ G+ _: J. @; Z2 G1 \1 L2 d' h, b0 f
        Base Case:8 F& Z$ @) O3 B  O8 l

    # a" T; `! r4 ~! _' h1 n* k4 G        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; x" J% O, o  \& J9 k$ f+ g
    : G4 z- y; ^- u' R  o        It acts as the stopping condition to prevent infinite recursion.: k( N- T: f* w8 o+ |& d

    6 L6 e% j! U$ k1 x6 M1 k/ U4 y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.7 y+ c* m1 X" \8 \) @* O# e& o2 c$ R
    - B, h8 N  U0 l6 X) S
        Recursive Case:" v7 v' |! m( S: I8 O" v
    ; y! h  n, c& t
            This is where the function calls itself with a smaller or simpler version of the problem.
    ; l2 n) P4 K! ]- C: O: _
    3 D; I, x9 w8 G( d8 x        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ( r  u1 L2 f+ u3 [0 j) M& ]$ b. j6 [
    Example: Factorial Calculation6 |, u' b2 `+ e# Y
    ( Z# ]; a! p$ t( G
    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:
      n9 E# r0 p) E9 U; F" G, c& ]* u, p1 Y: k2 ~0 U
        Base case: 0! = 1
    $ r/ U) O) z, P! _/ A& ^1 ^( t) P
    ' u5 |; [' h$ f  S  p( i8 f    Recursive case: n! = n * (n-1)!3 l* F7 ^9 d. E
    * R# `; D/ @, k% p  |3 n
    Here’s how it looks in code (Python):; m9 r& f6 u6 k- B
    python
    0 k' B" q, h/ M& S0 u5 l' W) {6 S
    7 K; f4 J& |+ a6 m" k4 M, i
      C6 T" Y0 I; m! d" H  b: k+ Wdef factorial(n):
    " s3 E; l4 Y, z2 t    # Base case# G& u5 ^4 b4 H) M4 b. V. q4 w
        if n == 0:+ A1 }" S6 O3 H/ F' W& v& }1 X) q
            return 13 ^. f$ ^' p* {/ F% j' d
        # Recursive case0 ~( R- _* r/ S$ l) t5 `( `
        else:
    + i* ^/ B) j) O" Z7 d        return n * factorial(n - 1)1 t2 X$ N; ?3 M3 R/ B
    . V2 n1 g1 Q. F2 Q( V, c4 q
    # Example usage' b1 v1 T9 w1 ~! A3 e7 v1 T" T
    print(factorial(5))  # Output: 120
    ' L6 J& i% P# J* ?2 |" D
    " T( ]' u; q9 B, c' t2 [8 u0 IHow Recursion Works
    0 j7 p' a- ]& M4 O) i& I& Y
      k. Q+ F) E5 u! U4 E    The function keeps calling itself with smaller inputs until it reaches the base case.
    1 p& W8 W4 D! c, O) \: W8 E% B$ H5 }4 S. k  U
        Once the base case is reached, the function starts returning values back up the call stack.
    . R: `2 R' v; x2 f1 T' p! D5 @0 k* f8 ~
        These returned values are combined to produce the final result.
    . w6 r2 {* O! I+ F$ U$ g
    4 k5 O! N7 [5 l0 Z% z# [For factorial(5):
    4 s$ ~+ `7 V# E% c9 q( N% s
    $ ?1 K3 {2 r: J8 n* x9 e4 v1 c6 A* ]) K9 Z
    factorial(5) = 5 * factorial(4)
    1 Y. f! e0 N& G6 T% yfactorial(4) = 4 * factorial(3). z* k' y) ^/ j/ f8 l) k$ t+ @
    factorial(3) = 3 * factorial(2)* L' n+ k- M5 n5 S3 Z& l5 G
    factorial(2) = 2 * factorial(1)
    3 G6 v. b/ N" ?6 Y/ Lfactorial(1) = 1 * factorial(0)
    ; a! V5 Q/ A- M& }3 o. F( p. `0 Yfactorial(0) = 1  # Base case2 ?& K. {$ t9 ?: K; q
    5 C+ u; f  [' ~5 {# P
    Then, the results are combined:
    ( I! X+ _1 i5 H5 x( q8 T
    1 x. R2 W4 l: _7 f  O* _; N8 t7 P6 {% P
    factorial(1) = 1 * 1 = 12 }* X* Y9 p$ z8 S! ]
    factorial(2) = 2 * 1 = 29 Y# f3 U/ a5 B+ Z$ @" y
    factorial(3) = 3 * 2 = 6
    ! R" Q# u# D* K" Nfactorial(4) = 4 * 6 = 248 w1 J. X5 H2 x9 U2 N: z9 @
    factorial(5) = 5 * 24 = 120$ D( I$ v5 n4 S& `% T$ i
    3 Z, ~* H3 g, `4 {5 p8 X8 q: M
    Advantages of Recursion
    2 w- n7 z" U$ A5 `+ M
    ) k6 w9 Q+ N8 Q% v9 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).
    , t9 w1 b! w: |5 C( l" v0 `
    9 p# C$ z  f) m9 }1 y, d2 G    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    5 Q9 E* b% x3 H$ q+ z, H" y
    8 K3 E5 p- \2 \Disadvantages of Recursion
    6 J6 [/ ]2 o. I% g/ }% [% t0 N( h- N: \
        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.. ]1 Q2 Q0 [. C7 [/ n8 d" [
    ' J! O( E8 G0 \8 g
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).& R  t5 B$ ~; S# s& O

    0 R1 n' w  R  `# }" GWhen to Use Recursion4 n% U' m( ]9 a% V/ J

    3 U0 m7 G$ z7 Z2 d) b1 j8 z: Q    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    # ~" {0 }1 q# g
    2 S' x  d! E# H* J, E' O& J) I3 p    Problems with a clear base case and recursive case.
    / _/ B, q  v& I
    ' p! z# B6 X! `8 k' f* G9 |Example: Fibonacci Sequence
    . k6 D  F& r8 w: V! i! A' O8 o' f1 p7 D- L: `+ G
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:6 D; F4 R2 @  E% O; l4 i

    ' U6 M9 d/ X- }# d4 M; ]    Base case: fib(0) = 0, fib(1) = 1
    $ `/ u- z( Q( a9 l/ f" @1 `2 X1 N, D' F* y2 D' H; {
        Recursive case: fib(n) = fib(n-1) + fib(n-2)$ D& a9 S' F/ g2 B
    / \2 |" V1 P9 y  m3 R* [
    python
    5 k! N& W" X, W. r! r# m& `3 i  w. m9 [/ i  U! Y

    5 Q, B, a" V4 Q5 o9 t8 Edef fibonacci(n):
    . S3 e0 k0 c- J9 w% j    # Base cases
    3 S. i% [  N4 V* {1 i) T    if n == 0:% O  J2 m# `9 E. |. J2 P
            return 0( G) L) ^. A7 g$ c% C5 ~6 ~2 D
        elif n == 1:- x7 I  v! N4 d
            return 1' E& _* `$ [9 x5 `  V% M
        # Recursive case# H( j5 X4 N7 H, R7 X1 n3 a. M
        else:
    ) x% u; Y1 f( V# B  S+ F$ }        return fibonacci(n - 1) + fibonacci(n - 2)
    9 g5 O0 ~3 \: k4 w* T; h! Y
    + x# A$ x0 ~- m) p# Example usage
    9 I1 R) R) a4 B; s* iprint(fibonacci(6))  # Output: 85 [- H& d* z4 E1 C" J) E
    ' t* {: I- V  Y, J5 D, D3 n
    Tail Recursion! d9 X; t7 e1 t" p

    % g- M5 ^) T$ W' Y( \1 Q9 \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).' W* j+ k) o' Y

    $ b" e% y. d! }! p4 [  ?' S8 lIn 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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