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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
  • 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
    2 ~2 `. j, M, x; J: j
    # ?7 l5 u5 N; w$ Q! \解释的不错7 X: d: N6 F$ O& d2 c

    5 w+ k: v, `5 R7 l' Q* F. I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 s+ m/ Y. c: s% P( n# ~; A' _$ ^- \4 _& ?6 z
    关键要素+ h$ t2 a  M, D- K6 T
    1. **基线条件(Base Case)**# l6 I0 e9 m1 s/ T/ |# Z8 E
       - 递归终止的条件,防止无限循环
    ( a- W9 M4 ?2 o1 C  j   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 19 G! ?0 J2 Z; m& P
    3 }' P* X/ U+ L- d- k* h& v
    2. **递归条件(Recursive Case)**
    - H& H- v" ?: _  W8 n& E% m$ t   - 将原问题分解为更小的子问题; p/ t8 Y3 Q8 `" E" {
       - 例如:n! = n × (n-1)!1 c4 d; W- I+ n& A7 S
    7 z0 i9 Y) ~0 a* B
    经典示例:计算阶乘5 @% r+ _+ Y- k/ b
    python. t( ^* G  g3 z- P
    def factorial(n):
    0 T  g& O9 W  d6 ?0 a    if n == 0:        # 基线条件
    , d& l3 I+ z: y        return 1
    0 ?2 H% ?# `' R# r0 P% O    else:             # 递归条件
    6 b& [+ [# p9 D; h# c        return n * factorial(n-1)9 S: n5 T( \; ~1 t; O
    执行过程(以计算 3! 为例):; g& C1 c7 c  g9 J$ V. g
    factorial(3)$ S0 Q( C  m2 D2 x4 _5 X$ O
    3 * factorial(2)
    8 {9 k4 f3 a" K& e0 N3 * (2 * factorial(1))
    : r' h/ h1 Y) v' d+ \+ g3 * (2 * (1 * factorial(0)))
    ' f" g: X) H) X5 P3 k, k" s2 R# e3 * (2 * (1 * 1)) = 6% T9 a  {" O  X. k) i( I0 G/ X4 A
    4 t$ s# N- g  X
    递归思维要点
    : _2 o+ B5 l8 B! K* \% t, i1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : F: R" ~+ i6 J0 h: A& U2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    % D% x+ N4 I' o7 D9 p( f6 p3. **递推过程**:不断向下分解问题(递)
    ) m- i" }3 t. f" u4. **回溯过程**:组合子问题结果返回(归)
    . V# b: p( r3 o! V
    " `( k0 o) O: A# n. n& m, ?( Y; e3 ^ 注意事项
    3 a+ z( q% N2 k1 `必须要有终止条件2 Q8 [( q) U& P$ a$ }
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)0 M, w. E" C1 c9 y1 x
    某些问题用递归更直观(如树遍历),但效率可能不如迭代5 f" O5 N" a, l% ^
    尾递归优化可以提升效率(但Python不支持); J$ L7 H. E8 w4 O" ~: B0 R# i2 K$ Y

    : v+ q( ?6 o. x0 }2 N' p3 K 递归 vs 迭代- q( U  g9 o# Y  C7 [
    |          | 递归                          | 迭代               |
    ' ]/ w+ F! o/ `! Y|----------|-----------------------------|------------------|
    8 r/ [7 P& E9 b! P9 W8 u- ?- P| 实现方式    | 函数自调用                        | 循环结构            |7 K+ P5 ?  s4 \1 o! `5 {) w
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    2 B5 t( T( h: Z% p0 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         |% O7 Q, n8 u. V, B/ X
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    9 H/ y: r/ p) q1 T
    2 R$ n, m+ |$ B& N/ r5 w 经典递归应用场景
    1 }" W1 f* l1 w' ^( B1. 文件系统遍历(目录树结构)1 a- ~  c7 N  r9 V' L. _! {
    2. 快速排序/归并排序算法4 h+ F: \# c; ~3 }. p
    3. 汉诺塔问题  d- r0 h9 W1 }
    4. 二叉树遍历(前序/中序/后序)
    0 U& T, \. _1 M5. 生成所有可能的组合(回溯算法)
    5 O0 J% `# {+ j7 p
    2 A* ~7 o7 U0 ~2 y: ~% J  H. @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 08:45
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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 {$ `4 K5 j' M( ?0 X我推理机的核心算法应该是二叉树遍历的变种。
    ; m# [" A) b6 t# 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:+ Y" R  w8 D7 }/ W) j9 m9 K
    Key Idea of Recursion2 j, ?6 V( O8 F/ s; X8 q+ e
    4 _+ M1 v9 Q" E: o1 S8 \' i$ C5 c
    A recursive function solves a problem by:( @; O, k! A8 I, b
    . L7 {6 f* c7 c1 r9 U* u
        Breaking the problem into smaller instances of the same problem.
    1 z$ _" P* Q+ V& |  F* L. y# W4 C" o& g& P
        Solving the smallest instance directly (base case).: J$ k2 f- y3 @+ N
    9 a/ o, T6 l8 \+ g
        Combining the results of smaller instances to solve the larger problem.  M3 R  R6 y3 k

      _4 x, A% e1 z% E+ H, {- lComponents of a Recursive Function
    5 p* n) t+ m' l8 q
    % ~# W3 _! _; g2 ]' N0 X) S" y    Base Case:
    ( z$ s. t6 h  `0 w2 Z3 L2 l, q: a: y( A- C& p% m3 _
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
      ]  d3 ~: X$ v5 p( X
    : h$ G$ |# @4 C) m; |        It acts as the stopping condition to prevent infinite recursion.
    * O) ?& q2 z; o8 s
    9 L  Y7 T& c/ ]0 d7 J        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    3 Z; z6 M2 u. c3 Q4 K! Z+ t
    ! k4 O, [0 H& W8 D7 A1 F    Recursive Case:2 Q# ]5 K! r. q" n6 p
    7 m, K7 g% _" @3 d& \  o
            This is where the function calls itself with a smaller or simpler version of the problem.* x3 |; @' y6 ~/ j* j
    ) ^& I9 ]% I/ i: w; [; h7 s* I
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., [0 Z/ b5 h' u& Q6 K3 F+ u+ O5 i4 c
    ; @" C) q: T! I) A$ h; ], z
    Example: Factorial Calculation
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    % Y" d; i3 P# h" g, W' jThe 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:/ [" j  d! z# s6 Y' j$ ?

    + _- F- G5 {, Z' d6 }    Base case: 0! = 1) [! e7 T0 A6 {" ?6 v( J+ N

    8 t/ X& q. c4 [. @    Recursive case: n! = n * (n-1)!
    6 \" M$ F3 x, t2 h- W6 U/ |( i1 f1 Q
    . n4 @9 Z6 R1 Q& D0 g5 WHere’s how it looks in code (Python):) J% H8 S. C- }5 k* ?
    python
    3 N$ j5 V0 }  U6 N; ~, [0 Z$ D+ Y0 V0 ?6 q8 i. l
    + `9 J) [6 O4 V- I( s/ W
    def factorial(n):; D- y1 p9 l5 r! s
        # Base case2 V, A1 x/ S# G% u
        if n == 0:* P' Y* ^  W9 a- O0 X
            return 14 d1 X4 _! v2 \' w
        # Recursive case
    5 e- o+ ?! c' z8 [    else:7 [4 t+ j+ x3 v& I7 z/ o
            return n * factorial(n - 1)
    / a% `6 Y' d2 j9 V4 \
    ; _7 a/ \8 Z: W& Y7 n- `# Example usage
    - ?0 U" Q: y8 Z9 N$ xprint(factorial(5))  # Output: 120
    8 t- g9 ?+ K+ i5 Q) p
    2 h3 H5 V7 {: S" ~$ B4 fHow Recursion Works' u9 D1 {8 W6 I- s$ J2 Q2 k
    ; }, a  x, h5 B0 I( f" P
        The function keeps calling itself with smaller inputs until it reaches the base case.+ ]# U1 K% k/ ?

    # J4 L* e8 e$ N7 F! G+ ]. B    Once the base case is reached, the function starts returning values back up the call stack.
    " z. b8 q( E7 P$ D- M. {, y' X% \$ d- L
        These returned values are combined to produce the final result., x8 L/ d# D, K  R$ A( W6 ~8 G

    # s( V4 H" ~& X* @For factorial(5):
    9 a$ a4 K3 g5 h- W8 p; w
    - R5 x" Z% }) h% `2 z  o, u- n. ?, |$ E2 W
    factorial(5) = 5 * factorial(4)  }) z7 I6 O. o0 c
    factorial(4) = 4 * factorial(3)
    : H3 Z8 f6 C6 X& `" cfactorial(3) = 3 * factorial(2)
    7 ]8 V" H( g& x7 x  K8 Ifactorial(2) = 2 * factorial(1)1 p0 ~* e. p3 ~! @* ]9 B& H
    factorial(1) = 1 * factorial(0). M8 U- e1 i1 m  d* A6 b
    factorial(0) = 1  # Base case+ j3 d# P; j7 I1 ~8 }
    , d9 z2 Q' Z: ?2 l# u
    Then, the results are combined:
    8 {1 E& E. r  @  p# @# r+ x7 s8 @, R$ X" `& K# {5 W/ i, [
    / Z' o" }4 E2 E% R4 d! f
    factorial(1) = 1 * 1 = 11 Z: H- V7 F1 m
    factorial(2) = 2 * 1 = 2
    ( A/ _  n8 k$ Mfactorial(3) = 3 * 2 = 6( P' f1 H9 K$ I4 \5 j9 m2 c/ o$ t
    factorial(4) = 4 * 6 = 24
    9 B- M5 [! Q. }factorial(5) = 5 * 24 = 120
    # b# y" y' }) p. [9 `1 `9 H8 N# o  E0 o
    Advantages of Recursion$ D! x4 J% A# S" X& ?
    0 x( o. C7 K3 F
        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).3 |1 ~& ~% v# b0 i- B/ Y9 j

    # w0 k0 x( n. x3 U: c" y    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    * V( N( j0 z& }* r, [6 t$ Z* L# P4 Q" E
    Disadvantages of Recursion" D2 f6 U+ Q! t  W& v# b

    , f* A8 V9 b% `0 w  P$ 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.- \4 D3 _) U6 p9 a; e4 D
    2 f% p! K9 l: A! v- a; n7 H
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).: P" I9 M7 P7 R6 y" {' P+ m) u6 E

    ' @2 u! S, l3 ?' PWhen to Use Recursion
    ; P0 g" S. m* Y( o! C1 c, S4 a# C
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).6 u7 R* @( W8 `9 Z

      Q' [. d- k+ `1 H5 K& P6 c: q2 `    Problems with a clear base case and recursive case.( _" Y9 Y0 m% u) B! [
    7 H& o9 k9 G# n$ I
    Example: Fibonacci Sequence
    2 p- O; E3 f# ]' V6 Z' c) p! K, j) P8 X
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    2 z- j2 I3 V3 v+ Z0 f- z" Z4 H- c' o  k4 X
        Base case: fib(0) = 0, fib(1) = 1
    . @/ B9 F# H5 B  N, a
    7 @' U! S& j3 {    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    - Z7 v, f+ `( f; ^1 U- P; U/ w& G: n5 e3 _8 c2 I# e; I1 d) p
    python4 K1 T6 P0 a7 S$ w0 ~
    , ~" J$ W7 O5 N  B, `/ }
    ! U% V& U3 X% @: a; q
    def fibonacci(n):
    $ Q2 }- H/ o9 Q( N5 u9 i3 K    # Base cases
    4 I& d6 L6 {0 J' ]) X    if n == 0:1 E. v" W$ y+ F( B& p% M! t, V/ N1 L
            return 0
    * G7 [! o% a8 T6 I4 G# [. T    elif n == 1:
    & W0 z7 {" n* F7 H) Q! K; @        return 1/ h( N' f6 b0 O1 I3 }
        # Recursive case: |' y. c! D( ?4 ~% S) H0 H- l! @
        else:$ K- X3 J6 _1 F& o2 g6 d
            return fibonacci(n - 1) + fibonacci(n - 2)
    5 H; W- g: x0 s; w
    . ]/ d  ]7 W9 z- O& F# Example usage
    * Z; S1 c" T$ |) p) yprint(fibonacci(6))  # Output: 8
    - I. e- X3 L) C+ R3 T* e3 U% A! ^9 K, S" n5 u
    Tail Recursion
    8 i' e! z, T& F' o6 p2 J  n* L: f& A
    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).4 V. Z; h% k. }; d$ e2 v
    3 Q2 M! s% b6 p- {
    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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