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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
    + D" d% X4 k3 i4 }( _: B! k8 d& ?/ f4 Z5 X8 O# O. Z: l. S
    解释的不错
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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。
    3 x, H! B9 Y) L2 K. C5 p1 O) T! Q) G" G: C* W# C6 w
    关键要素' B* f! U# w5 y; r  X! z# `6 B9 R
    1. **基线条件(Base Case)**/ @* T0 g8 a2 H: T
       - 递归终止的条件,防止无限循环
    5 c. A2 @) C5 Q) u/ R! R' n   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1+ [5 e! P( i# E$ Z- H- D

    ! j7 {2 J- R% r5 d* z$ P, `2. **递归条件(Recursive Case)**) x- C: J9 G9 H3 z
       - 将原问题分解为更小的子问题
    $ h: H- k% N: _; p# Q4 T   - 例如:n! = n × (n-1)!3 a+ e$ B. ]1 P* I" J
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    经典示例:计算阶乘
    & Z5 }7 c: o: N. |( |3 hpython
    8 T9 S2 O7 X  Tdef factorial(n):9 {- u2 t' H9 Z+ l+ ~$ ?$ ?
        if n == 0:        # 基线条件3 r2 l( G$ I1 L  c& ?
            return 1% P2 K0 q  \; k( p! X
        else:             # 递归条件
    . l3 N- h! M2 z- P0 ^        return n * factorial(n-1)
    ; q5 U0 o! J6 @# O* D7 Z. \执行过程(以计算 3! 为例):' ~  [% r$ [) e% O0 U
    factorial(3)% n+ f# t+ g$ \2 R+ h9 O
    3 * factorial(2)
    " l5 t! ]  {/ G! D3 * (2 * factorial(1))
    , v3 n9 |$ R  _0 S, d9 t+ u+ t1 h3 * (2 * (1 * factorial(0)))$ t$ U& `. b' D
    3 * (2 * (1 * 1)) = 6+ r( ^9 U8 d5 t8 d: F2 Z
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    递归思维要点
    & [% f% {, r- S+ R1. **信任递归**:假设子问题已经解决,专注当前层逻辑( E, I3 |6 z7 S4 W1 E
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间). a* w& b5 |+ d! x7 ^2 H* F
    3. **递推过程**:不断向下分解问题(递)
    5 x, ~: P8 H, Y4. **回溯过程**:组合子问题结果返回(归): X5 S0 L8 X$ c
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    注意事项4 C+ T  r# A7 D6 _% U
    必须要有终止条件2 X% X9 O% \# C) f# j0 V: U
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    2 y$ A9 F7 t" l0 }: [+ {某些问题用递归更直观(如树遍历),但效率可能不如迭代
    * A9 S- m; c" T- y$ g! L尾递归优化可以提升效率(但Python不支持)/ r* H+ X9 M; ^  X% }: |: f- _% `

    2 x) T3 Z: l. @ 递归 vs 迭代
    : H5 }5 B% B6 ]. S9 q|          | 递归                          | 迭代               |
    ! Q4 m4 r/ L  D" c. ?|----------|-----------------------------|------------------|& L; V4 s; R! F( w  X. G/ A2 C
    | 实现方式    | 函数自调用                        | 循环结构            |
    ; w& v9 t7 `+ ?9 C2 ]' Z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |: v5 z% `. h( `" j# l
    | 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         |
    / I* o5 u  W. T- f3 y* j* R  _+ [| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ' T7 L7 }9 C2 ]2 w' k! Y
    & V$ |. d. ]: l$ A2 V; \, S6 C 经典递归应用场景
    3 [( E6 f+ t9 X0 u+ X; w" n+ n+ V+ B1. 文件系统遍历(目录树结构)+ i/ z% k+ R0 }- e; i6 s
    2. 快速排序/归并排序算法0 ]* }: d: c' E  d, t* \  d# P
    3. 汉诺塔问题! K8 g' {8 \2 M# N( S
    4. 二叉树遍历(前序/中序/后序)1 ]) H/ Q: t: y& R
    5. 生成所有可能的组合(回溯算法)
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    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,% p) s; E3 K4 F- P5 {) q
    我推理机的核心算法应该是二叉树遍历的变种。2 I  O+ A, F7 p
    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:# I' A/ H0 f6 W- g- u
    Key Idea of Recursion( ^- B. J1 o* Y+ i
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    A recursive function solves a problem by:8 E) v) @, a; I$ s) x
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        Breaking the problem into smaller instances of the same problem.0 Q, I6 _9 a8 i% d0 k  a, I
    * [3 q8 k  u$ J! R- y
        Solving the smallest instance directly (base case).* j  }& ]7 N! Z" g$ f, u' O

    ( \0 j2 r5 V6 m    Combining the results of smaller instances to solve the larger problem.* y- r' }) a: [0 z1 s
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    Components of a Recursive Function9 I& |' g8 z; d' J( @, f
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        Base Case:, c1 e! U+ \* E- n( U4 @

    0 X2 w: L* R+ L- q) E* |        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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            It acts as the stopping condition to prevent infinite recursion.3 K* C9 R1 {- e+ l: u) Y' c
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% `. A' i& @/ H) E" G

    ) u) r4 ~3 G. a. c  @    Recursive Case:2 A; K0 m  U6 \: g
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            This is where the function calls itself with a smaller or simpler version of the problem./ o! n- F* t! k/ I3 C- d

    ) e8 a; Z+ k9 ?* B5 ?  ~; y        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* q& k( ^) Y/ p6 w* o9 J
    2 S! u% h( l- J( _
    Example: Factorial Calculation
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    : q# i7 N+ T1 V5 O# K8 BThe 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:
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        Base case: 0! = 1; n" j( F2 g$ Y, f: m% g& X* E& q

    7 a8 K, s+ z! p* R    Recursive case: n! = n * (n-1)!
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    Here’s how it looks in code (Python):8 f/ h  K$ N' Y3 ~8 \
    python) D+ M( j8 T( u5 s
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    " F9 o, D5 o4 W, idef factorial(n):  B$ B9 O5 `' c6 c9 P' y
        # Base case! d, y; {2 o. M/ x! R, O# p" ?
        if n == 0:- |  Y- }6 {/ K0 }* _
            return 1
    + p+ z$ g- w9 B% V) ^( G    # Recursive case# G7 B  k" ?% @# Q9 s( ^
        else:
    # D7 V" {! _8 ~" i: ]% M% k# C, w4 z        return n * factorial(n - 1)
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    % h7 L. W- R& z2 {9 r# Example usage
    ( a' D- a$ O5 i+ v9 ^$ Bprint(factorial(5))  # Output: 1200 p: f6 E: ?: y$ F8 O: B: ]

    5 C! @1 ]1 n$ ]/ K& Q% S& \+ AHow Recursion Works; }8 |6 j% M- `. e3 V

    4 \" a0 q; P* s( p    The function keeps calling itself with smaller inputs until it reaches the base case.. g0 m% O6 u  A/ m# z) u0 T# {9 i
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        Once the base case is reached, the function starts returning values back up the call stack.
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        These returned values are combined to produce the final result.# I" O% i9 {  N; P+ B/ a! l- p

    - ~3 f& V0 o6 a  @For factorial(5):
    8 C  n/ }6 l6 L9 r3 q$ D- B5 ], p  ]- R! {
    : R0 T8 ]: b+ ~7 a) v" R
    factorial(5) = 5 * factorial(4)
    , y9 h) Q; G8 dfactorial(4) = 4 * factorial(3)/ u% |: `' [2 }* G7 D8 S3 H5 R( c
    factorial(3) = 3 * factorial(2): A/ o6 l! X2 L" q# L$ o/ B
    factorial(2) = 2 * factorial(1)
    ( }0 q- W. W( ?3 U) z3 ]' f9 `factorial(1) = 1 * factorial(0)
    5 q8 I; e" F! ^* z9 W: t, ofactorial(0) = 1  # Base case3 ]( F+ p/ W  a4 X( b4 S
    ( K7 i/ `% n& v* G# t8 y
    Then, the results are combined:6 r: \( M0 E7 `' P, @
    : [2 ]' a. p% n' B% e# {
    , n+ o" P1 b; b  {
    factorial(1) = 1 * 1 = 1
    : W" r( w. ^4 M& F0 e; T5 F( I4 U. d" yfactorial(2) = 2 * 1 = 28 p' a2 ]$ e4 Q% m
    factorial(3) = 3 * 2 = 6
    1 Z) x4 L8 F+ w2 ?8 G( ffactorial(4) = 4 * 6 = 24
    $ b" ]: F& _5 B6 A3 `2 I; Lfactorial(5) = 5 * 24 = 120: T! R3 `4 W7 A# g- }$ o' _/ o

    " {! ?5 A& E. v' vAdvantages of Recursion
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        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)., j+ V( M% n2 t& }/ X5 M

    8 I* K; E' m7 y    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion
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    8 R3 p2 u9 Q" r$ Z. l, [    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.( B2 I1 E- U, c
    . ?- h# x# T; `' c
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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    ! C* a* L# A9 F7 C# M' OWhen to Use Recursion, }* s  P( z6 P

    % ~  d: [! e& t2 L. h    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).7 e) ^) G6 h: o" Y

    . n, X0 J* W, f    Problems with a clear base case and recursive case.% g, X; l$ ]/ I+ \
    ) ?) C1 s  Z* T& I8 Y  I
    Example: Fibonacci Sequence  Q  ~7 j5 f8 c# ~2 z4 m' l
    5 \* M& a' O4 m  {7 K5 Q) y* i$ O
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    6 V% m$ @( q0 z/ U& ?2 y3 h( c( S( X
    ) X8 i+ o7 y  r1 a& B9 s3 c    Base case: fib(0) = 0, fib(1) = 1
    2 K! }0 ?/ _3 Q, \; A/ d$ N! L9 c: \% z- a( T- L% @8 q
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    python/ W/ l$ _# L2 }; t' U, Y. Y1 f" ]5 C0 {
    + [, y. X# p! j  A+ M( B

    . @* m8 @8 `( S+ y6 n2 L' [& gdef fibonacci(n):
    " b5 d; Z! U2 B- Z& A7 r/ H' B6 T! ~    # Base cases
    + h, o% p7 Q  _- S9 g, w    if n == 0:7 A$ U$ R6 b8 }6 G0 i
            return 0
      K5 K- L' q  e9 n& Q9 D0 t    elif n == 1:  Q! Y; e7 t8 i+ M$ u
            return 1. M& i7 b$ g+ ?+ |/ c+ v
        # Recursive case
    8 u& r/ c/ s  f3 M' l; `6 ]    else:
    # d& p  M5 O4 ~" y5 d3 E        return fibonacci(n - 1) + fibonacci(n - 2)! O! l' I9 `2 j9 U% B5 ^% X' L
    8 }" V( \( O: s
    # Example usage2 E* j+ `7 k4 X9 e; W
    print(fibonacci(6))  # Output: 8
    ) x# [/ t6 _& H1 }1 P% g' z9 O1 X
    8 ^. _$ v5 w8 w0 kTail Recursion) o/ j  m' b- G; @, Z% o7 _
      V2 \" x+ u8 f
    Tail recursion is a special case of recursion where the recursive call is the last operation in the function. Some programming languages optimize tail-recursive functions to avoid stack overflow, but not all languages (e.g., Python does not optimize tail recursion).6 v' x! ?( \) M; w2 P4 `

    9 j% ~) l1 m6 i0 @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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