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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
     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 % `5 G  m8 e; j+ X% ~

    - k+ c5 j  _- ]$ h解释的不错! T1 t' g( S3 X* Q- b( M
    ( l3 k; m: m/ y( }( D: _
    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。6 I8 N1 r( V: {+ U3 U; @/ n

    4 q) R9 S& X7 ]8 J# B/ B" n# r( ?- f 关键要素
    , ~- U! s( E0 L; U) [1. **基线条件(Base Case)**6 y0 V  V( I' P6 `6 m9 T% E, |6 C
       - 递归终止的条件,防止无限循环
    ) D4 v/ @( K" v) A   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    1 G: S: s! ?0 ^. a* [% R! }/ S" N, _3 \; m# V  i; x8 L7 p
    2. **递归条件(Recursive Case)**
    3 _( Y  |& l# q- I7 ]  Q   - 将原问题分解为更小的子问题. M! ~3 I) z. M* {3 _
       - 例如:n! = n × (n-1)!
    2 Y0 }! I5 H7 T. z' d# Y; ~; J& h8 k# o
    经典示例:计算阶乘; o7 F, w% {. G$ G9 y
    python  ]1 Q! S9 l1 [: U9 Y
    def factorial(n):
    $ f" K# |# V! N% I) c; p' O    if n == 0:        # 基线条件
    7 V; G* d' r) u: d2 h        return 1
    # g1 K& @8 Z# U    else:             # 递归条件7 a7 T7 f" J( g" V& f3 m
            return n * factorial(n-1)/ R% m' L, {9 d
    执行过程(以计算 3! 为例):( t$ f9 _, i5 d/ h( D" o! d
    factorial(3)
    ; p: J' j% }1 v4 P! c3 * factorial(2)
    2 m7 I7 G/ k. ?  f/ W( w3 * (2 * factorial(1))
    2 G) f( L' i7 e. x: s3 * (2 * (1 * factorial(0)))% i; y3 a) @  [$ Z, Q7 S0 k
    3 * (2 * (1 * 1)) = 6
    ; a8 i' g3 F% m) O% J: j" m" E" ]' C2 a
    递归思维要点
    / y5 I! ~) u5 A% L5 c" z1. **信任递归**:假设子问题已经解决,专注当前层逻辑. t% A+ O. z$ I$ n
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    # v9 c" q2 E- B, k; P" k; h+ ~3. **递推过程**:不断向下分解问题(递)
    5 s0 R7 G/ Z5 w5 c+ l4. **回溯过程**:组合子问题结果返回(归)( @6 `: U. A& K; b6 K6 @, I

    9 b: i, d' `& K+ x6 R, e5 a 注意事项7 ]: z- u' Q$ d* b
    必须要有终止条件( M& A+ p7 `; f" e
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ( t4 G$ v4 _, ]( S: J: g) V某些问题用递归更直观(如树遍历),但效率可能不如迭代1 n8 G: B" L& d
    尾递归优化可以提升效率(但Python不支持)/ A2 Y+ q/ s4 L5 H8 E7 F' x! q
    ' i1 H2 D3 ?" i
    递归 vs 迭代
    8 w& i( \, A8 w' M# ~0 b|          | 递归                          | 迭代               |
    & N" w& L1 i  z. V5 K1 I  O' i6 |0 U. ||----------|-----------------------------|------------------|
      u6 U1 ^8 p& z. W4 @6 c| 实现方式    | 函数自调用                        | 循环结构            |
    3 }, \4 U9 D5 }1 t: t# S| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# {, ]! B1 j  R- D" C
    | 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         |
    & d+ S& s: S  i1 ~6 _$ A| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |0 P; X3 k. V! x+ T- d0 O2 m

    7 ]& O! O7 [0 l. X* z 经典递归应用场景
    & B. [- z, h. k/ |! M1. 文件系统遍历(目录树结构)
    , c, r: k( {" N! r& g- Y! v2. 快速排序/归并排序算法
    / H0 M+ l' `- l0 Z- s3. 汉诺塔问题
    6 P* b, W+ [8 A6 x/ c! M4. 二叉树遍历(前序/中序/后序)' B9 X5 v* `$ S; q9 u7 W
    5. 生成所有可能的组合(回溯算法)
    + Y# G/ j: e! t
    9 V- K2 o2 i$ p. N9 E+ X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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    7 小时前
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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,. Q! t  Q- Y% ^( v. `: B
    我推理机的核心算法应该是二叉树遍历的变种。0 W% l' ?. Z0 B1 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:
    ! N# Q6 ^% _" h' X+ P% Q  AKey Idea of Recursion
    " {# G. {! v6 s# p+ J# i$ c9 F/ _" E. }" u3 ?. m
    A recursive function solves a problem by:
    # I9 U! E. S+ u  D8 n) r( S5 w3 @4 ]) |3 p
        Breaking the problem into smaller instances of the same problem.; {: z5 P7 l) m& g; q8 V

    ( d: Q# ^3 I2 U; o: k9 w* I    Solving the smallest instance directly (base case)." z) f6 n3 f" O/ ]; G  {9 b6 b/ @

    ' Y# C0 }" T  u. x% D    Combining the results of smaller instances to solve the larger problem.
    4 ]' p" F3 X7 ]# ~0 d
    3 u' m( O6 e! M% _Components of a Recursive Function) y. m3 Y; }1 o( @

    $ s2 n% m9 S" q4 Q6 C" j    Base Case:
    $ v# J8 x9 y0 B1 w/ i7 A( r! x% X& |) u: f2 i: k. k* e/ _! O
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ) T; q5 |$ \: S" D9 `6 c3 L' t0 U% _) L3 e+ ~
            It acts as the stopping condition to prevent infinite recursion., @! z5 P5 I- A4 C; V6 M" I

    ; ^& M" H# c1 W0 m1 l' c( q        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.  C4 ~5 n; U- {" M7 E$ g" N2 B
    ) A& N! e& B7 U0 g5 h5 a
        Recursive Case:
    % V: |2 l/ C: d# O) n# @4 l
    ! d' M, `! w' S% i4 [+ l3 M        This is where the function calls itself with a smaller or simpler version of the problem.% e, |- x$ w9 k* T  L

    5 }7 u" P* V  X  u* G        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).. T! a1 u1 c) y- C' I

    $ F: x. h2 k- ^; }8 Z0 s! }$ ~Example: Factorial Calculation
      n1 H3 @! ^+ F, i! f$ b; w1 C$ M/ z* _4 m$ V; y8 ^
    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:0 Q$ K, K$ f5 W- @* u. L

      x$ R1 `# |0 M3 X7 m9 ^' i9 \. t    Base case: 0! = 1, i. V8 r; t! w; F7 v# j- p
    7 j: a5 p* {0 {% v* N6 [) ~1 L
        Recursive case: n! = n * (n-1)!2 I$ o6 e% n9 O1 q( a

    ' G  k9 T4 V9 y) ]* Z4 |$ \5 \Here’s how it looks in code (Python):
    ) O% P& P* Q. a  ?& b: A% g* ?1 Zpython
    9 K5 ~. y5 y6 \! D0 x. O
    8 w" A0 U/ s$ p9 Y# c' F/ N. M2 V) C4 s
    def factorial(n):
    : d  L# q5 Z3 y7 W' K" ~  E    # Base case
    2 F2 I. l, f7 W% @; S& F    if n == 0:' ?" E2 }9 l1 o4 _2 p* E
            return 1
    ; S6 k6 x6 b+ ~5 v    # Recursive case
    & O- S% \! i( j    else:
    3 E1 g/ t9 M6 N- T3 s& V9 V        return n * factorial(n - 1)
    $ a, d( I# A+ d  [6 Y3 h- u7 E, c8 }. ]
    # Example usage3 D. g: v1 v" Z( ]; V
    print(factorial(5))  # Output: 1209 [$ s. |) S5 H) o, o# e% ~* l+ P

    : _( Q2 F: s( p& y' V# e' k: Z; zHow Recursion Works4 {+ U( e2 h6 S( ?# [1 V

    9 }( f2 K# K- G& d7 U" \    The function keeps calling itself with smaller inputs until it reaches the base case.
    1 v, s8 h! X6 \. v( h
    ( O% _, F, F: B& x) f    Once the base case is reached, the function starts returning values back up the call stack.
    ; ]4 `4 @7 O3 y$ @' G( A3 X. w  d! {( O+ K* r5 q. \
        These returned values are combined to produce the final result.
    / V9 [* @, O) t- J, E" n8 t
    ' @/ \  C/ a& |# X7 |! T* MFor factorial(5):
    ' i( s6 y$ v: q8 w0 v' E- A* Z! X2 A3 m  P* r+ x2 V' y  Y  S* J

    & N1 d0 o% s! e  Y9 H0 Ufactorial(5) = 5 * factorial(4)* R% ~6 T# T3 n- q& W# F% g
    factorial(4) = 4 * factorial(3)+ L2 u. j7 P; `# A5 ]
    factorial(3) = 3 * factorial(2)
      S. R; l9 K4 \2 w  |/ Yfactorial(2) = 2 * factorial(1)( d% S" a0 m+ C% s' p( b
    factorial(1) = 1 * factorial(0)9 K0 Q- I+ j' e  k  Z
    factorial(0) = 1  # Base case
    8 W* c. f2 T6 C
    1 C7 Z/ H! Y: J$ l$ j9 x. t% E1 bThen, the results are combined:
    : ^5 r. i& u0 m7 t0 |2 X& ?5 n1 R& }0 S# ]. h+ |

    0 d, @1 R: U$ Y/ {factorial(1) = 1 * 1 = 1
    6 `* S2 u- I3 D3 Dfactorial(2) = 2 * 1 = 2" g3 U2 X3 @# T8 m
    factorial(3) = 3 * 2 = 6$ ^8 E5 Q1 e7 ]! l: t0 ]9 w0 Y: m
    factorial(4) = 4 * 6 = 24
    7 K% r+ b0 B( m3 bfactorial(5) = 5 * 24 = 120$ X. I2 T2 F1 R1 K. u' n+ x  L- m

    2 ]0 L, N; ]- L4 b) ?4 iAdvantages of Recursion
    - ~+ K. B7 B9 W- l# n- T2 K" j4 c3 t% G; U- L  O5 u7 b
        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).
    ; V: j2 t# u8 d  u
    1 o# K6 T8 k( ?3 V7 O0 V) j    Readability: Recursive code can be more readable and concise compared to iterative solutions.' [" D" g3 J" O/ L
    ) T+ d: b* c1 ]6 H
    Disadvantages of Recursion
    : g6 ^$ a. k: `" |/ T0 c; L4 h# D" v) `7 ]1 k" i, V( i" w2 x
        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.2 T# _- w7 }3 ]. Q. H
    1 ~) q* m' `$ r1 y0 s( v
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).9 t, D# B6 D. r* B0 F% `% Y9 {

    ; W! ?3 L# E8 X) b' ~3 e3 X0 |When to Use Recursion/ d  ^* B7 n0 D6 n, `8 F0 D' j- A- J
    % T9 s& c$ b1 V9 }- j3 [
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).% ?$ R" G1 I5 Z! A+ p: _

    " V5 h: R' F( B5 [  w    Problems with a clear base case and recursive case.
    4 \, v' j+ c- E, p" K6 b3 G% Q
    4 o9 O1 p) Q" _4 Y2 t( U% i9 ]8 d( {Example: Fibonacci Sequence9 L, N: |+ `1 A5 o4 H$ B2 @
    & W% A# y: V" N2 U: o; q
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . h& C( v. n. q: U" G2 {& s
    ' O5 H4 y+ I3 r5 Y* q    Base case: fib(0) = 0, fib(1) = 1
    # F( D3 B% h- F$ K8 ^& }3 w* t; D% p& F4 }& m: E( m
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    7 S. L! J/ D8 b9 ~) v8 k
    + Q# c% |( `: _3 U8 _python
    5 |4 S" u# Q9 E" k
    * n1 I! a" \) Z; h
    : R! A3 `: ?9 K6 J' z0 Adef fibonacci(n):
    5 b+ f' b" V8 l6 n    # Base cases2 V0 G* w; Y. z: q% I8 `& |
        if n == 0:
    0 o4 }- k) K5 U        return 0! a! @6 M+ j  Q# b. Q
        elif n == 1:* {! l8 u$ Q* n( L
            return 11 K& e  U3 f2 Y+ F1 Z4 P5 O
        # Recursive case2 Q1 X; W. O% n& O6 L8 u
        else:
    0 b; q! B  C3 |9 ]9 M. \" v/ d        return fibonacci(n - 1) + fibonacci(n - 2): T# V+ b0 s" g2 {
    7 c6 Y- O; p& v/ r( N: m
    # Example usage
    ; A; ?+ |7 S2 t! {% B# V7 X" h. rprint(fibonacci(6))  # Output: 8
    ; n% s+ m9 z* \" E3 M. [) g8 K
      t  @9 x5 }! L3 C3 t5 V8 Q+ F( n) PTail Recursion
    ( B5 h, x, b* U& X- p7 {
    * N: F0 l8 r3 ~( }& l1 B5 d; M) STail 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! F5 B+ Q! \! v$ D7 F# b; N$ \
    4 x9 T9 {1 j# ?/ j
    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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