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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 ' B# G# V, S+ D$ G) e8 p6 |4 ~

    2 a/ o# a# J1 l9 b& F7 J" p解释的不错1 `% b" L  v3 q+ |5 j/ r
    ) H) r" E0 t  o& f6 F
    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。
    / H5 J6 B$ ~* ^, U  Q% D* o' e1 p8 I- f7 N' o8 F& Q
    关键要素
    : \9 x6 _; Y6 b' i) w2 K' }6 }1. **基线条件(Base Case)**
    ) K, H( L4 R9 I" c   - 递归终止的条件,防止无限循环
    5 s* x/ o8 C8 o2 r$ z2 [9 E2 N   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    , A/ ]) G7 K. g
    " @# ?1 Y5 G+ M; _2. **递归条件(Recursive Case)**$ D  T4 X9 B+ \6 z, U, T
       - 将原问题分解为更小的子问题
    1 w  x& q/ ]  c7 I   - 例如:n! = n × (n-1)!$ F, L8 q7 K# h$ z" `3 M

      C/ {+ r) y( Z# c/ P; v$ R0 r 经典示例:计算阶乘6 ]; R5 p$ c( V' y! u% k% D9 h
    python' B/ k! K( ?3 p9 D( m
    def factorial(n):
    3 u1 J3 O/ S* l( U8 o    if n == 0:        # 基线条件
    , g9 X9 M& `2 ?7 n        return 1. n4 F+ A( r) Z& Q( j! T# _
        else:             # 递归条件" v! B+ S  W! Z' U' Q. x
            return n * factorial(n-1)6 y8 s! w/ ?1 a
    执行过程(以计算 3! 为例):
    8 ~0 P' Y5 P0 H7 K* M' B% `factorial(3)1 F9 S$ |& T* S& T8 E5 l0 v
    3 * factorial(2)! V2 ~- w+ W( Z8 @+ A
    3 * (2 * factorial(1))' S) J8 C1 w5 a$ e& z" {
    3 * (2 * (1 * factorial(0)))+ J& e- _  ~4 t" O+ v
    3 * (2 * (1 * 1)) = 6
    " c# d2 K" A- x; b  u4 }, g% I
    + `5 l  E8 O/ L, s5 Y 递归思维要点
    ; V% Q: @5 z9 u" o: S; _6 h1. **信任递归**:假设子问题已经解决,专注当前层逻辑5 ?9 p/ ?0 }# }2 s/ Y' ^  O2 C
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ( ^6 T9 w* M; `) w3 k3. **递推过程**:不断向下分解问题(递)
    # a# ^2 K9 X( ^7 U4 \4. **回溯过程**:组合子问题结果返回(归)
    % u6 a$ X# F+ x3 m
    " X' _( R+ l7 q* d7 V 注意事项
    - |, u( c% p) @必须要有终止条件5 s* t  F: H# u2 k
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    % P4 A8 C5 N4 r某些问题用递归更直观(如树遍历),但效率可能不如迭代
    $ @! D- e: ]6 n# x* I8 Q" x尾递归优化可以提升效率(但Python不支持)
    , f7 ~5 Y( g  q( _/ o' }
    % s. z' A, u% x# ]6 Q2 [ 递归 vs 迭代
    ) Y4 \% W8 H4 d$ t* _|          | 递归                          | 迭代               |
    7 s1 F  W2 Q* X6 W6 s: G# M|----------|-----------------------------|------------------|
    . }+ Z5 x8 ^! l  s: p| 实现方式    | 函数自调用                        | 循环结构            |9 A3 o4 i: O7 i( j5 P
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    8 O5 f- C. u- I| 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         |
    " T. \9 S& w6 B7 t6 J| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    & e7 C- p1 M* \3 H: g$ J; D, s% E* {1 H3 L
    经典递归应用场景
    ( p# e3 Q! k% ~5 @1. 文件系统遍历(目录树结构)% q  l' }, d5 [/ t
    2. 快速排序/归并排序算法
    6 D/ v) V' w0 R" h6 Y3. 汉诺塔问题9 |! K2 W1 `/ ], P) x$ F: [6 \
    4. 二叉树遍历(前序/中序/后序)
    1 \' q( v- ?, N% k5 p. k+ T) H) n5. 生成所有可能的组合(回溯算法)
    $ E1 c2 \% ~( D5 z# n# A. l" L, S6 h7 ]0 x% a( Q
    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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    前天 01:20
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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,$ V# a" K9 d: i" G, H9 |: K
    我推理机的核心算法应该是二叉树遍历的变种。/ [" Q* s- l6 L+ @4 n3 \
    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:; B0 s2 B) n3 H" x( ?. F% R8 E
    Key Idea of Recursion
    8 _, r7 f* l1 f) N9 f" Q8 e- j, u7 T+ k+ l% Y/ J- j
    A recursive function solves a problem by:  X7 _0 N! |# ~

    6 w4 s7 @7 }1 g+ s    Breaking the problem into smaller instances of the same problem.3 _1 P: x; Q/ \9 [$ Z- B1 ^8 z  L
    ' I/ d- L6 j. w
        Solving the smallest instance directly (base case).
    1 K$ H% o2 V1 \1 d" k- v
    3 L$ R6 V6 A& N6 W0 m) N    Combining the results of smaller instances to solve the larger problem." s4 y8 f0 X6 D
    + |$ B9 n5 p6 ^3 e9 l: q
    Components of a Recursive Function5 D9 c1 d8 X1 a( [

    / E; I7 b* t  z* f$ @    Base Case:
    4 G( H. |. H4 }
      Z3 P  [0 |9 x) l4 j; A6 S% v        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.8 a  Q% V) I" I0 _% O% d# z
    / }/ y$ s8 L4 P8 |) K( ^8 M
            It acts as the stopping condition to prevent infinite recursion.
    $ R/ M" r; P" N8 }
    . b0 W4 f1 \- g        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    & z! b& q$ L2 Z6 J. d/ L4 D6 F; M2 t& a3 R
        Recursive Case:
    ; f* n  e+ Y( F, }/ F6 s6 u. ~5 b0 [7 ~4 i; u! W
            This is where the function calls itself with a smaller or simpler version of the problem.
    1 n/ |  ^5 z; f7 ^( u2 b+ n5 m' Z: q4 k2 a
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).) z' r: |) z+ d+ W) l/ B2 {
    / e3 ]  t5 j7 j( Z5 @& p
    Example: Factorial Calculation
    # f6 d1 C) v; {7 o+ \; O6 l- u2 c/ u/ J- ]
    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:( e4 Y( |- [+ S0 M. t5 ?! _
    - R7 d1 l9 c7 r5 ?5 l
        Base case: 0! = 1
    # g/ d& _  z6 U
    ' A# u- l2 Y4 V  H5 ~2 w; N$ I7 {    Recursive case: n! = n * (n-1)!  D; h' \5 ~4 A( {

    / x( c: r. |9 Q5 ]6 MHere’s how it looks in code (Python):: P. c& [, u1 ^, S
    python- e! r5 u3 r- {+ q- S; G8 p+ w
    * @6 Z. |  J& z7 H" v7 s

    2 H$ A% \# I, `def factorial(n):0 K* Q; Q* Q8 l8 _
        # Base case7 [# i+ M/ o1 r7 W
        if n == 0:' y$ b5 o( `: `
            return 13 O9 C/ G; u0 M
        # Recursive case
    * y0 W, w/ ~7 X8 M. D5 }9 ^; O! x. M: j    else:. E  A8 k3 a% u
            return n * factorial(n - 1). b4 B& A0 o5 h* ?

    # @) q1 _9 H6 Q$ E8 i0 n8 ]7 r) s# Example usage
    * Y$ i; s. l+ [$ ~1 D9 \( Rprint(factorial(5))  # Output: 120
    & L" {5 }) f% A- j; l2 H# X8 @4 o' r: e. E) u& B
    How Recursion Works5 y; G7 J  W+ A. ^- |

    3 F, z" I8 }; [& g    The function keeps calling itself with smaller inputs until it reaches the base case.
    8 [: v+ j& n6 t, G/ ?7 x! D8 x' B" p. T! t& S
        Once the base case is reached, the function starts returning values back up the call stack.
      e, V9 Q: a# ]* e! u9 E" B6 D* c
        These returned values are combined to produce the final result.
    7 o/ ~6 c5 P$ w6 K: Y1 N7 O( P% _( v4 W2 k7 c0 s
    For factorial(5):
    ! b& n: J2 C* k5 l9 f
    " O+ j- H) M" T& T6 l6 T7 a) V/ u: _4 _# Q
    factorial(5) = 5 * factorial(4); @: U# k, T: \+ O3 [+ Q# B
    factorial(4) = 4 * factorial(3)* {' A; C+ F- x3 z1 P
    factorial(3) = 3 * factorial(2)
    + e( ?' e# u* J/ p, O4 cfactorial(2) = 2 * factorial(1)2 o1 w* }# g% ]$ O- E4 m$ F
    factorial(1) = 1 * factorial(0)9 ?- d4 f) p4 Q4 H9 q
    factorial(0) = 1  # Base case6 V# {( {! a5 n8 H- {2 H
    # Z" i; E8 W1 g/ u  r! a7 b
    Then, the results are combined:# Z: Q, c% @; K0 v% V
      a- N5 U3 y  o  H3 i0 a

    ' w. v) _# I  _( u0 h* H; G* e/ Dfactorial(1) = 1 * 1 = 15 R2 q+ S4 C5 _- z
    factorial(2) = 2 * 1 = 2$ N$ s: ?5 v6 S0 F! ~8 Z+ ?+ r' z
    factorial(3) = 3 * 2 = 6
    0 ^5 U7 ]# [# x, \2 wfactorial(4) = 4 * 6 = 241 B% `! @, ~$ Q  F, K
    factorial(5) = 5 * 24 = 1208 b- f% t! y/ d: h7 C$ L) x4 k

    ' o3 I1 e8 @  ~+ A8 qAdvantages of Recursion
    9 G/ R- b9 O0 p; z
    9 _/ y+ U" G+ C8 Q* ~3 T( 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).
    1 O1 o+ x6 e( i8 l! ~2 L
    * S! @8 ?, U/ g) A+ B; h    Readability: Recursive code can be more readable and concise compared to iterative solutions.. y1 E* `/ N5 i3 U. g
    # Z. o& l( [! c% n+ d5 @3 N
    Disadvantages of Recursion2 B+ x  D; a7 y. [8 X
    5 z3 y$ T- s( ~8 j' e  j
        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.3 S7 d  Z+ ^; ^% T% S% U" m

    ( [. e2 z$ w- `0 _% `    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ( l$ }1 U5 y. L! C
    9 t4 X5 A* N5 i8 n+ kWhen to Use Recursion* ~1 P& V$ G5 X3 }+ S. K; I/ i
    3 u$ K' z$ V$ a/ V5 U4 a9 a8 h. j. I
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ' W1 V; y8 t% T- @9 G9 U; a( Z/ ]
    8 ?# }; w: k% f6 `7 `2 J, I9 t    Problems with a clear base case and recursive case.
    " i. J8 b% K4 B9 K( U1 _8 }
    ) X. e9 W8 U1 t+ ]1 j( \Example: Fibonacci Sequence% K: j' R! F0 `9 k9 M

    : U! e9 ~* H, xThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ; R: T" E4 G% n, l+ L
    ) |5 s2 ?: _. _& l# x    Base case: fib(0) = 0, fib(1) = 1
    # y  K+ X& n" ~5 \7 f) Z+ V: ~3 }; h4 v0 _  j6 G/ E8 Q
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ( L' b% l% p( T/ W* [
    ! H! @* }& e6 B3 i0 ^python
    0 b& K* C# ~! ~2 D) d) ?2 R. N! |
    . \( n, F5 R( K5 e9 k7 ^3 C- f  N/ j% T& L9 b
    def fibonacci(n):+ z$ v' P9 _7 ~$ Z9 n6 ]( b
        # Base cases, @5 S( b% @; L* E
        if n == 0:
    # G, D* i% u/ p; e        return 0
    2 y& E  G4 u! ~+ `$ ]    elif n == 1:
    6 D- k2 a) t$ S! u2 w" r9 u        return 1
    8 h6 W% K# O' D$ b, U" t    # Recursive case
    . r2 i1 k3 R) v! F, _    else:6 g5 `' x9 w/ W7 y$ j2 G
            return fibonacci(n - 1) + fibonacci(n - 2)
    6 S; f' B# F& n- e2 ^) [
    ; z7 W$ N4 C# `5 [5 n# Example usage
    ( j. d, D, K5 s, U+ o1 Z- M8 Kprint(fibonacci(6))  # Output: 8
    ' ?" P/ |: o2 `) M
    9 S, Q& s5 p$ B" HTail Recursion9 k# X, w4 s4 \

    2 e- |6 f7 ], d$ n* Z( v) U& l3 o  D2 wTail 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).& ^* P% q9 D" \# {

    9 V# F$ e; B/ a7 J9 L! o, G% p- uIn 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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