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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
    . w  T- H! G8 y3 a9 j* _1 {/ W- A+ @; X- e
    解释的不错; G  i% {& y0 }8 q
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    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。) b4 ?' q" o  L) b# N$ K- U' [3 O
    % m) ?% s* x2 A$ ?6 D
    关键要素- |& _! q0 G& g9 @6 w7 b0 }
    1. **基线条件(Base Case)**
    2 \4 L' n. j9 G" D   - 递归终止的条件,防止无限循环3 b4 @4 e6 P" @
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 19 j; K: W) |; j( _
      j. ~$ Y6 U1 E( R0 e0 f" G1 G
    2. **递归条件(Recursive Case)**
    & s7 H) L5 G4 U" `8 C/ a   - 将原问题分解为更小的子问题# [4 O8 }+ ?% u- V+ s$ e' A# g% p3 k
       - 例如:n! = n × (n-1)!
    % o, `9 ^9 c( D, A- t# N  w
    $ b0 B) ]4 K! w. L3 W, {$ w( U 经典示例:计算阶乘
    7 n' ^& w. E! x$ F8 {* Zpython
    9 ]! u) T. y: |  Udef factorial(n):
    $ s2 E  F+ g1 u" ?% H& A- i    if n == 0:        # 基线条件- M) `: C: j# z- H% V. P" X" Z
            return 1
    0 i/ Y/ F) Q7 X3 C; B    else:             # 递归条件
    1 i" G. b" y- e" l        return n * factorial(n-1)5 d* Q* O5 f# w/ ?9 y+ i( M3 l
    执行过程(以计算 3! 为例):! }. b, b: M$ h- w' L4 |
    factorial(3)
    $ d9 c. J: U6 G/ J3 * factorial(2)  [, Y' p- c3 i
    3 * (2 * factorial(1))
    ; `/ l/ M+ I. k# {! k0 O5 Q3 * (2 * (1 * factorial(0)))
    . f8 M! e$ |+ V! g# ~3 * (2 * (1 * 1)) = 69 S& b9 v" K4 |7 F9 O9 E" H
    1 j) N/ h- j( q% @
    递归思维要点+ Y$ K/ |% H/ Y9 {( k, f" \. B
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    / V! I' |5 k3 t) P. ]2. **栈结构**:每次调用都会创建新的栈帧(内存空间)+ c- M) L8 |( U, H3 Y" A9 E
    3. **递推过程**:不断向下分解问题(递)
    6 V- E' L' j* J8 G2 D" u3 F4. **回溯过程**:组合子问题结果返回(归)" O+ ]: _( b" z# |, X" [: H

    ; t# G6 j4 U& V* o6 N. Q 注意事项
      Q8 l% c! @8 e" ?( \5 p必须要有终止条件
    2 N  P4 i- ]- u, x; q递归深度过大可能导致栈溢出(Python默认递归深度约1000层)! e5 z4 G3 p7 d& A
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    $ @- y* ^& H2 u; T6 p' b尾递归优化可以提升效率(但Python不支持)
    # N8 a8 {( H8 g! x8 U8 X0 q1 X' t; w& T) l' o5 ]$ g+ }7 J$ S1 h7 b
    递归 vs 迭代+ X7 \% P6 I3 E! N! w- X1 T
    |          | 递归                          | 迭代               |
    ' ?- [! z( X. T! w# P|----------|-----------------------------|------------------|6 N9 o8 b$ @" |% ~3 A) n) w5 a
    | 实现方式    | 函数自调用                        | 循环结构            |0 |* D7 z) d6 J; j& r
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    , v/ ^3 \3 s* N| 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         |6 V, j# o" Y' M: [5 G
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |2 I& T2 g3 c" z/ D3 L
    - r3 T* m9 i1 @. T  D" ]* I
    经典递归应用场景
    3 J4 D' O1 P2 i1 I/ y* }0 u8 m1. 文件系统遍历(目录树结构)
    7 _9 n; Y7 P+ _! X4 F8 c5 g2. 快速排序/归并排序算法
    9 V3 m0 ~, U$ J3. 汉诺塔问题3 q+ {  s. Y1 @& F8 e6 R; d
    4. 二叉树遍历(前序/中序/后序)
    . {* t# v- ]) m5 @3 Y1 t7 ~5. 生成所有可能的组合(回溯算法)
    3 h& I. U/ r3 s7 F; R; D9 f$ P1 g- |  s( H3 L8 B2 z; d. D
    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,
    ( e, S( J- T+ Y- R我推理机的核心算法应该是二叉树遍历的变种。5 x3 g+ }$ W+ ?) J% u
    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:
    ) L8 y* P/ s7 u. [. IKey Idea of Recursion
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    ; l& l4 f4 k$ Y1 \0 Q: T; MA recursive function solves a problem by:2 `) j  p, _3 h. B5 i+ I0 q" A
    4 o! G, L9 d+ J0 P: A7 s
        Breaking the problem into smaller instances of the same problem.* {6 m8 I$ e. W0 |: x; d

    ' b9 o1 j' U: K3 B1 q& L$ m+ Z# I    Solving the smallest instance directly (base case).
    7 O7 W$ y( R# n" T6 l* ?  e/ p* X# I; d3 ?
        Combining the results of smaller instances to solve the larger problem.! _5 Z' B# M& ]: a; }7 j6 T+ t9 ?
    8 |0 K5 Z$ |1 H5 o3 x! \
    Components of a Recursive Function
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        Base Case:
    ' K$ w4 X4 Y" }  Z3 F2 X. j, O, s7 f
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.$ e5 g4 r; h9 f# V& q
    % M! k% |$ r  e
            It acts as the stopping condition to prevent infinite recursion.! o" c- E# F/ \4 W8 s

    ( w- @& \- N( S: I: R, E        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.. a& N. B, s' e

    ( Y3 n& C. j/ Y& I    Recursive Case:" a1 P6 O& J9 V

    # t2 W  Y4 A! j- I+ h; @, x6 E! ?        This is where the function calls itself with a smaller or simpler version of the problem.
    & }  G3 @2 b: Y4 O+ O/ z
    : {; M* v) l& l, p) h        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    6 Q* D/ Y! ^# P8 ^+ _& H) y, B. ]5 n+ @8 T, \! D& a6 w9 C
    Example: Factorial Calculation
    2 ?1 j2 O- G' {' T9 i% U
    , p; o3 t& O7 U$ c  ?( [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:
    . n4 V. [1 _* z
    ( y( R' b$ p( v+ p4 t# Q* \    Base case: 0! = 1
    . ~' o1 s: u) H& {- p" S8 q; o; K  l$ n# e! k9 Y
        Recursive case: n! = n * (n-1)!/ y& K7 o$ N! t- T

    * |- B/ y- M3 k3 U6 VHere’s how it looks in code (Python):
    % A/ }2 R: C1 d5 C. N8 d$ ?python, [" g6 C* G7 P* @
    : N) K& y8 n. t, }( a5 {0 O
    + d  G" E2 g/ f1 k- ]/ F; s4 ]
    def factorial(n):- i& c* K, K9 t/ g$ b8 m) R6 i
        # Base case) _2 U0 L1 C& U5 K, E7 ^: d
        if n == 0:- F- R7 M8 ~. Z" n5 h3 b; l
            return 1
    5 P& F9 V! R0 v. L% D& E0 P- C4 H0 a6 H    # Recursive case
    ! O, H7 I. o- B# e6 `    else:
    - Y- s0 W  B6 _. D) `        return n * factorial(n - 1)
    ' z! K3 a& i! u6 Q3 n9 d
    - g8 t* m/ n& y7 Q9 X/ o# Example usage4 y3 }8 W( ]. L4 M0 g
    print(factorial(5))  # Output: 120: Q! m# n& B; L  c; [9 L% c

    & h7 R0 a0 e; @. |& oHow Recursion Works
    9 [) z$ V+ C  x& u: @* \' d, L5 Q" b8 }! R! \% A1 n
        The function keeps calling itself with smaller inputs until it reaches the base case.
    4 b# s$ V) Q- s' O2 r+ a' t3 D
    - r% d' K+ L1 s, {0 a( Q8 u6 z7 s7 z7 U    Once the base case is reached, the function starts returning values back up the call stack.& @2 e$ {6 Y% q4 y$ h

    * H2 k% A1 Z* V# z( V( @+ `    These returned values are combined to produce the final result.
    3 L/ ^2 L# ^* S' P+ o5 u
    ' a. _. N, w2 Z5 S4 X1 \, K: O& VFor factorial(5):4 t2 Q9 ]4 t( a% ]" t. @" t2 _2 n# |* C
    , Y+ j( [3 z2 O) P
    7 e# [" Z7 j# f7 a6 ^
    factorial(5) = 5 * factorial(4)
    $ p0 o  u, j# O% Y4 }! l% Y  sfactorial(4) = 4 * factorial(3)! }* Y9 @5 f) l. \6 N, F5 w
    factorial(3) = 3 * factorial(2)% c% ~7 R' K5 f( @
    factorial(2) = 2 * factorial(1). w" h9 y+ y6 d, q0 V
    factorial(1) = 1 * factorial(0)
    ) v- c1 t) B9 F, Nfactorial(0) = 1  # Base case3 k& A! ~9 r/ e: ^% Q: M- t

    " V3 ]  ?& H$ b0 T, mThen, the results are combined:
    * @- D5 ~7 |- v/ i$ s; c- W+ R% r$ c0 ^" w1 w

    * K2 e; G9 k/ ~/ f0 efactorial(1) = 1 * 1 = 1- O/ r' e: c; Y3 t( S
    factorial(2) = 2 * 1 = 2' c: I6 m: F: M  v3 L
    factorial(3) = 3 * 2 = 6
    ( l- Y; K2 l. h( Nfactorial(4) = 4 * 6 = 249 T' a# e  A# m3 ?' k
    factorial(5) = 5 * 24 = 120
    ) A4 e4 d8 a4 x6 @8 o( h/ C8 Y1 T" s7 V$ K9 l
    Advantages of Recursion
    ' o  b8 m$ U: w/ H4 k% a) t: t
    / {% C: ^* m# t1 K% }    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).; x; @( U" c. A) V

    0 |4 x7 ]/ i3 M' }6 [$ c    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    2 r0 l  H5 P; B+ p$ f3 \, |: c9 N9 D" V# z, ^& f
    Disadvantages of Recursion
    ; K4 m" o0 h& K/ b" C' O- m; A. G( j, v3 e2 ~5 w  M! D+ Q: K
        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./ D6 A  v" X0 R
    5 S! Q# V5 v2 {( w8 P7 W% r9 }
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    $ t5 I; c/ u; r. ~+ ^# S  V
    0 M2 N3 c( o6 c; r+ b3 I5 e6 OWhen to Use Recursion* e9 h; S: D" q/ w- V$ ~# B

    8 Z$ B7 d: {' z; I    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    # _. R0 `& e4 g. l/ [, V8 S* E; @6 w' n) `# r( N& {8 A
        Problems with a clear base case and recursive case.$ _6 h7 A3 f4 M! y2 F8 t0 j* S- h+ l

    $ `1 W3 C& _. @/ c. U# s0 PExample: Fibonacci Sequence
    # ?* r, M9 Q. z$ K$ Y5 v' ]% |! d' l8 r6 U
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    1 {& l3 B1 W  ?" M, H- n, }2 w( N, D& e. k% A- Z- k: `: X! }5 \
        Base case: fib(0) = 0, fib(1) = 1' H/ N1 J) q3 C. |  N7 D: N

    2 c% e/ p! D/ |2 M+ g" }, O    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    / b/ S! s2 o. g4 |2 S% y# _4 s+ x, [- q* L! B8 n( p' X# d. x
    python
    ; P' I5 p' B. `' H7 _
    2 @0 A  x6 y- @; E+ i
    8 ~8 q, C3 r( e3 r1 V+ K: Qdef fibonacci(n):; i4 j9 y, J9 `5 T0 y
        # Base cases) Q; a9 ~- r4 r
        if n == 0:$ ]% g, O8 d/ M7 z7 o  f9 Y: k
            return 0
    5 a+ O& d& c0 U" }1 m, _    elif n == 1:
    4 s' X/ f; e5 |) }  z        return 1
    ! V# n" [# f5 f) V5 m& ~    # Recursive case0 H! P" n- s! x" p; y( a
        else:9 |. ?6 \. J- o
            return fibonacci(n - 1) + fibonacci(n - 2)
    ' c. q2 Q! E' N
    . Q  L6 p! V: T6 P  j1 T- g* K# Example usage8 i" \8 y1 X7 d6 B+ P$ S9 I2 b, r( j
    print(fibonacci(6))  # Output: 8
      ?. D* R. N* ?/ @0 _+ J2 l' x, f3 a' J2 e2 y1 @! l+ F4 s+ f% Z2 B7 H
    Tail Recursion
    * [0 B1 o" t- ~" o. f) ~7 h' W& d1 r- F% F1 M0 x: c% T" L
    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).# M2 c6 ~. S9 i( W1 D0 u
    1 |1 \: |; E- f$ D& F
    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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