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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 . v# E# l7 {' s; w; e
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    解释的不错
    7 e# B* t& \  Z8 i
    / R# X+ P. B" v1 i6 n9 k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。& H# t& C3 G3 y& l

    + I1 P9 t, N3 P0 m5 A 关键要素  g6 j# a* G5 {& ^1 P5 V
    1. **基线条件(Base Case)**
      f2 ]. Z/ j5 }8 p, E/ C   - 递归终止的条件,防止无限循环6 d/ |1 ^! M& I. g
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ( l, G" B2 V& c8 q5 F, W! i" j1 E2 n# [1 W& G
    2. **递归条件(Recursive Case)**
    / I  L- `! W5 i) k- p/ b2 v( S. b5 y   - 将原问题分解为更小的子问题
    0 c4 Q. U$ d8 A- k& N   - 例如:n! = n × (n-1)!9 p: R& q0 p2 z* `. ~; b6 }( G
    % ~/ t7 _$ [' {7 P9 N- P9 o4 g
    经典示例:计算阶乘
    : Q: T( X) g8 q, R* v. m1 Lpython- I. F$ d( R9 d
    def factorial(n):9 J+ l& w2 J& Q: z8 n  q
        if n == 0:        # 基线条件3 [" g  I+ [, O0 _+ C
            return 1
    - J7 _5 _" r" D  D0 b    else:             # 递归条件
    0 X# J- x! o1 g* q; X" v* c: V        return n * factorial(n-1)2 n$ h; p2 \# @/ r1 A
    执行过程(以计算 3! 为例):
      W/ H% z3 V4 y% H2 kfactorial(3)
    4 s0 H/ @* G4 |) y2 \3 * factorial(2)$ D0 A5 G# G0 P  M
    3 * (2 * factorial(1))
    + i8 v  L1 |+ a9 y9 N3 * (2 * (1 * factorial(0)))1 i& V4 V' f1 W
    3 * (2 * (1 * 1)) = 6
    - Q$ R& M. j( H& U9 A8 ^' U! u" o1 j( k
    递归思维要点
    & I" y  }6 |" L: Y6 Y1. **信任递归**:假设子问题已经解决,专注当前层逻辑: q0 {: n4 z, B5 e& I3 m. e
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- f! L: M& K, `
    3. **递推过程**:不断向下分解问题(递)
    6 f- F( o& r7 I- R' N4. **回溯过程**:组合子问题结果返回(归)
    ( U# Y0 i8 H( i% t  a
    % o2 |& ?& Y! ]/ N 注意事项5 ?7 ]' z; K; l
    必须要有终止条件" ^6 g* j4 \9 B3 Y6 I/ T
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( g7 _9 Q. g8 `( Z4 P: b" w
    某些问题用递归更直观(如树遍历),但效率可能不如迭代( B& X  B; B2 H$ O
    尾递归优化可以提升效率(但Python不支持)2 W- J1 A( H2 l& M, B
    - z) @% v) Q7 u4 [5 p
    递归 vs 迭代' l' X5 x7 x( [1 L# H8 X( W% w
    |          | 递归                          | 迭代               |' B( W/ O' l1 g$ [9 a5 b
    |----------|-----------------------------|------------------|* x* ?* K" O" R/ [3 E4 b
    | 实现方式    | 函数自调用                        | 循环结构            |7 _* W4 W: O2 _
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    . q- p3 Z8 v7 w. E1 D& c, Z| 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         |8 b' @2 ?- ^1 d2 Z
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |* L. M% U) A3 f) K

    0 Q4 E+ ~7 u2 L  w( h5 Y 经典递归应用场景% v. X( F6 a) h
    1. 文件系统遍历(目录树结构): Y, l& S! r) E& o
    2. 快速排序/归并排序算法, {% g! i9 v# s+ i" [' V0 ~$ }7 j
    3. 汉诺塔问题$ b1 J4 D9 W2 o+ X
    4. 二叉树遍历(前序/中序/后序)
    1 p& C" b. l; B( E$ K0 M5. 生成所有可能的组合(回溯算法)
    2 r+ U  F+ f  M( F$ {
    0 R( U" f% A. F: ~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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    昨天 14:35
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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,
      t' v* {& A/ m$ _6 d/ K$ X我推理机的核心算法应该是二叉树遍历的变种。
    ' J9 @! [) h) h1 z( {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 k3 ~2 f9 T$ y$ C! G
    Key Idea of Recursion2 i- w3 D/ H8 Y; O! I: M1 e

    : ]" g& w3 t1 F6 l! ?A recursive function solves a problem by:  d4 f8 I" w( s2 s, b- ~9 R3 X

    $ @& M. \6 `) S& M    Breaking the problem into smaller instances of the same problem.0 ?' f" ~: h) h0 Z5 {# c' w' D$ S. Z

    + h, a- `8 h- B$ K- d+ @/ c6 g( a6 P    Solving the smallest instance directly (base case).
    * q; e6 b, L' }; D" ^. s; p/ U; ]
        Combining the results of smaller instances to solve the larger problem.4 W; O8 ~6 y" [' O

      d' p( S( N/ i+ ^( }- n3 z# v4 n- k7 QComponents of a Recursive Function3 }) a* r  p6 k7 e5 y& |, }; n

    - z/ _1 P8 x1 i% ]    Base Case:! V0 {6 J/ j9 d. M5 I
    ! c) s. _" x: d& Z' p% }
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.. F0 e' D- F' A- `/ z
    3 V" E) Y8 P0 T: R, v5 m& |
            It acts as the stopping condition to prevent infinite recursion.' _8 c% x" |3 B; |# \' N; Z/ F+ z

    5 d3 L5 T* f6 ]% k1 r0 X        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    7 r: S( r2 E8 v- H! f1 p1 m+ K) q+ G* N/ v3 W( ]) }
        Recursive Case:. k# _4 y1 F0 a4 I

    ' Z0 o0 Y7 u' P  q        This is where the function calls itself with a smaller or simpler version of the problem.* C& m' v4 @" a0 j

    / M- E0 [* o* `+ g5 f; r- g+ [- u        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    3 X. s8 x; p$ J8 Q9 r  ~0 w& P3 U* ?0 U
    Example: Factorial Calculation9 T, V: u1 i6 y7 H0 J$ R& a( k

    ! q. X6 N7 C; V% n- f. n6 H! y" vThe 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:
    ! i) L2 Y% N: e1 O
    & {' \0 D; m. t8 r+ T    Base case: 0! = 1
    + v- O0 `2 L7 t/ j' o! H; G: M/ s9 d
        Recursive case: n! = n * (n-1)!
    ; A7 Y$ c, ]# b5 L, B; {9 h' h$ G; V, h
    Here’s how it looks in code (Python):, f8 k8 \) H3 C/ P. {7 p  m; A5 i
    python
    ' [; u6 U- V9 T$ C$ Q) `
    3 C5 l" O$ R5 G, c4 f3 A" ?7 s8 G, X% p
    def factorial(n):: F9 x0 [$ u. w& I, e9 j6 [1 v4 `1 k
        # Base case
    $ e2 l% U' E* Q! D) a    if n == 0:
    / V1 I3 {# v7 E3 g: |% A        return 1/ l5 ~4 t  R/ m3 X
        # Recursive case
    . G$ k# ^+ t6 ~9 I# c( Z5 X% F8 e    else:1 g2 a3 g0 c/ h+ U" q. G+ s6 |# _
            return n * factorial(n - 1)
    5 Z- G1 g* ^! u
    7 J$ y  [$ h2 z4 q* x% ?# Example usage- z/ h2 D1 K- C1 k
    print(factorial(5))  # Output: 120
    + }1 {- R1 i0 ?" h1 f4 `7 Z. l( X
    7 O2 D/ y& ~7 h) n) r! EHow Recursion Works
    ) D) _4 @# S' y! y! X0 H% F
    9 }1 |$ P7 ^5 i9 W+ k' s% j( R) j  A    The function keeps calling itself with smaller inputs until it reaches the base case.
    9 @% J# V3 |- F) E/ N! N" X3 A# ?! ]$ ~" M
        Once the base case is reached, the function starts returning values back up the call stack.
    : E: C- k, |- n9 h& G) w& H
    / W8 C0 L9 b5 Q# ^9 i    These returned values are combined to produce the final result.5 c* x  h  J$ l8 J# h
    1 ?; i( w" M& G+ a) Z" b* U
    For factorial(5):
    - }/ @% a* C1 q' d" ?$ e7 `" y" U2 y2 \6 c% z' z
    : ~+ p) c! y( R
    factorial(5) = 5 * factorial(4)
    ( b7 L4 g- e* Q. x  [3 ?factorial(4) = 4 * factorial(3)! M% `# A$ x9 A1 x" F: @. o
    factorial(3) = 3 * factorial(2)
    8 q4 O7 j7 ~3 P. ffactorial(2) = 2 * factorial(1)
    3 s* s7 a0 Z8 p5 h. B1 Jfactorial(1) = 1 * factorial(0)4 S; `  X' @0 ]/ v# H" ~7 Y
    factorial(0) = 1  # Base case
    ) \3 }$ H% a$ U9 i
    , H1 [6 s: [5 i2 |9 E% w% s" kThen, the results are combined:$ I" d( Y! V% y
    ; }5 f/ l" R( C6 Y$ q  ~& e2 o

    " z; L: P: C& S9 \- Bfactorial(1) = 1 * 1 = 1
    ( ]# y3 ~" T4 u. {. Sfactorial(2) = 2 * 1 = 2
    2 X) L( [* c1 o' @5 lfactorial(3) = 3 * 2 = 6
    + D: @0 U" v; d- cfactorial(4) = 4 * 6 = 248 g$ E( S2 Y# ?: D7 }; G/ [' K
    factorial(5) = 5 * 24 = 1205 J2 o& R1 A6 j% P. k: y4 R
    8 Y/ I. h; f# D/ U. j- A
    Advantages of Recursion; O: B( o' g" [

    - |. [# D: h2 S9 L    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).
    . M+ {7 ~; P8 Z9 p  S9 J! e6 r. a, }* I5 h1 O; Z1 I+ V
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    + v) u* V+ H. j6 j, h; p9 N: f2 _$ U
    # m3 k- ~8 f8 m0 `Disadvantages of Recursion
    , F6 K6 Q4 D% _6 c( D
    . |( G& ]7 ?7 q7 {1 o& M* [$ O    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." a/ G$ L% S$ p2 m

    8 I3 V. U2 W1 m    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# _8 t) M9 ]3 d) v; i

    . |/ s, A& c- hWhen to Use Recursion- _( D: ^2 s( y% [0 ]- ]) k2 t

    / x- G/ l5 M9 O0 K5 X) R, k% ?# Z* J    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    " z/ Y9 l; ?* _4 R# O# U: a5 G/ T! R  o; e8 k5 E: P7 J6 y' i5 ~
        Problems with a clear base case and recursive case.( I2 b3 f; O7 n7 i
    ; ~2 j2 ~; D, I* k
    Example: Fibonacci Sequence- i: n7 a+ X& L3 ~$ V7 U  ?
    ; X7 H3 i/ x; H- F% T" j/ m
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 X5 E: ]8 r, E) w! O
    / O5 |) r- y+ P/ z/ n) R' M
        Base case: fib(0) = 0, fib(1) = 1
    % t+ v( Q: z4 e' w1 }! \/ b3 {. P9 l, S$ ^7 e6 }; U
        Recursive case: fib(n) = fib(n-1) + fib(n-2)% F% m0 ~3 {4 F  o' l' ^* T

    5 b/ K: i9 S) M* t+ \python
    0 e) U, K% ?( U5 C% D8 [' \5 q
    2 b3 }- U& d' A; r2 r- i% Q2 x+ A% l7 ]% e0 R. i( [7 F5 H) H8 S
    def fibonacci(n):
    6 P; U- {0 f* U* K6 _" M1 T( T    # Base cases: d  m# s# f0 n4 D/ G: F
        if n == 0:* F3 K. p) G3 y8 p- r" l4 B
            return 0: _% h5 {7 y+ G# @0 O7 N9 P( }. n
        elif n == 1:1 d. P1 |3 u& _) g
            return 1
    $ Z: S$ G; ?- x7 S( c1 o/ @    # Recursive case0 K$ K( h5 L' q6 I; b: N! X9 |
        else:9 ]! X$ l! K' q, a% l( ?* r# G& A
            return fibonacci(n - 1) + fibonacci(n - 2)
    - P% T5 [' Z. l$ V& i8 }# w7 m' s* G0 F
    # Example usage0 n! q: K: O9 [
    print(fibonacci(6))  # Output: 8
    + q7 M& q! d" n8 a# K( b0 v
    2 x/ c" f2 l( ZTail Recursion& {2 M* Q* U: {0 y  S) L$ n9 ^

    : @. q4 }! n% ]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).# ?2 {' z" x2 Z4 s3 g  {
    , U# I! |  ?2 C6 v; b' i
    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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