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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
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    解释的不错+ E8 X. S9 p- O4 ^6 Y8 Q: @

    * x0 p, n- z6 V% E0 k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。
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    $ o1 G" n" s! s* x, W$ b* _ 关键要素0 ^: {4 \9 s6 H7 H2 t  z* E
    1. **基线条件(Base Case)**
    6 }, ~5 J# K1 C4 x: J   - 递归终止的条件,防止无限循环
    ( @: v* K7 Z! z+ U/ c   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 15 R" i0 c$ a' Y6 h. w

    3 C% w) B7 u; ~4 M# X  A' R1 |2. **递归条件(Recursive Case)**
    : X  V/ L" }3 k$ `) O& W   - 将原问题分解为更小的子问题7 a& Q6 z! ]8 z0 O/ _4 d1 ^
       - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘
    ) n: r/ V# e6 ~python
    8 Q& ]( P# R4 S1 S! t# Z8 H' sdef factorial(n):
    + ]6 W% L) D& P4 ~  y6 l    if n == 0:        # 基线条件' Y( Z* ?2 q3 W: b* D- a
            return 1
    # e- M, y; m: {4 Z, R- a    else:             # 递归条件
    1 @4 N/ d+ N8 ]. L4 X9 c& P5 ]        return n * factorial(n-1)7 {3 j" S' ?8 l* I; C
    执行过程(以计算 3! 为例):: E  o* U* b/ v" E9 ]! ]
    factorial(3)
    ' E& `7 y, g7 P$ _! G0 g* T3 * factorial(2)( Z9 E" K, I  @4 r- ?4 b
    3 * (2 * factorial(1))
      R$ @, }- c: D# _2 ^8 n) l3 * (2 * (1 * factorial(0)))
    / x1 E* V( n; ?0 ?" i! K, K3 * (2 * (1 * 1)) = 6
    . X5 K5 N. u4 N3 v% ^
    & N: ]  }; h  @) x 递归思维要点
    : x) u) w; u# S+ a! K8 Z1. **信任递归**:假设子问题已经解决,专注当前层逻辑- k/ ~6 ~+ v2 [' _* d: j* j
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    # q% f8 |, B7 B: P* ]. U0 e4 V, Q3. **递推过程**:不断向下分解问题(递)
    1 @6 C6 g# {0 p5 `3 s0 [/ D4. **回溯过程**:组合子问题结果返回(归)
    ) m. _1 F: m$ Q5 ]8 ~6 J! a
    ; n5 @! p; Y. D1 J2 _0 J 注意事项3 P. ~# W* X0 y  Q
    必须要有终止条件
    5 u6 C1 G- u/ ^% |1 P) [% Y9 r递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    $ J6 w0 b3 S5 O0 p7 [# F# Z4 e, o1 }某些问题用递归更直观(如树遍历),但效率可能不如迭代( H; g- y8 c3 [: L
    尾递归优化可以提升效率(但Python不支持)( Y5 s# S. T) b& j7 x! F/ O5 }
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    递归 vs 迭代
    + g, W# ]6 i' Y, ?9 K|          | 递归                          | 迭代               |
    ! I% k$ U) l9 I, Z9 ^|----------|-----------------------------|------------------|; C7 s8 f$ I" ^0 H4 G2 _
    | 实现方式    | 函数自调用                        | 循环结构            |; V( r$ M. K, i( U
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# m) B% e5 Q9 V- A! J3 d5 {
    | 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         |
    4 i: i: B  }  Y& w* x| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |+ l7 Z. S) c0 z# D1 l
    # `2 \9 a9 \% J, W' ~/ a% O5 O
    经典递归应用场景
    " @3 ?& H+ k. P1. 文件系统遍历(目录树结构)
    4 y5 D0 S8 R) c6 t, m- z2. 快速排序/归并排序算法" G+ p& r1 W4 f
    3. 汉诺塔问题: @& \; S3 Q' d  U% w, P
    4. 二叉树遍历(前序/中序/后序)
    2 t3 b1 m3 u% G% j5. 生成所有可能的组合(回溯算法)6 X2 \& D8 ~( [$ M1 Q2 d8 s

    $ z4 O( i/ |) i0 k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,, |: G% C+ `$ [* g7 w4 z  w
    我推理机的核心算法应该是二叉树遍历的变种。8 m3 g3 f+ T! L. m0 J# @4 ]
    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:- R* B2 O3 W. U
    Key Idea of Recursion
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    A recursive function solves a problem by:
    ' x! Y  ~: L; ~3 |" M) T/ L3 y- Z: s) N
        Breaking the problem into smaller instances of the same problem.: s- e8 e# x% `0 K

    ) K7 h+ e/ p# p& O/ [) \% G4 y    Solving the smallest instance directly (base case).. s+ O* d/ d( d5 e

    - @' n0 n6 p" ~" F8 v    Combining the results of smaller instances to solve the larger problem.
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    $ X/ B/ F2 d3 x9 sComponents of a Recursive Function
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    . Q2 F, S4 C& b5 J# D    Base Case:
    : f4 p% L7 P- c" n9 o+ Y) C
    + e, `3 e4 J  D* v        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    + X0 q* c" P9 ], m' C# J! n) h- H- }1 a2 q# A2 B
            It acts as the stopping condition to prevent infinite recursion.
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    5 ~$ |; J+ T+ T9 t. {        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    4 j$ P2 Z$ e" L
    ' i  c4 z9 x7 L) C    Recursive Case:
    5 D' c- O- t$ L; k. |, `( r- O. ~; h! I" q7 y$ G* I
            This is where the function calls itself with a smaller or simpler version of the problem.7 h; v8 |. V2 G

    ) [# c& g, T& }  v; m        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).0 ?. @  `% n- w: S" [- S
    7 G# g# D$ o% u3 N4 t# J) L
    Example: Factorial Calculation
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    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:# X. J( h. g# k* t% f: J
    * d8 K0 A) A4 T. P9 I8 u1 A, r, a2 S' k
        Base case: 0! = 1
    * t! w: h) Z9 L7 d& ^* _  p5 e2 d5 W# _* J, @1 k" i
        Recursive case: n! = n * (n-1)!
    + \( H0 s6 O7 t, M  ?+ m7 a1 F; ~7 X) l7 |$ _
    Here’s how it looks in code (Python):
    4 w# r" Q: G7 W# }$ V' Hpython
      m6 Y0 h6 Z0 B# A& ~: q5 Y9 d* b$ b

    # i6 h* g4 \5 v3 D$ a. Q2 ldef factorial(n):
    / K/ H4 g8 K* w, T    # Base case& N" O! n: d: S& [* o# w7 n: y
        if n == 0:* w5 `. F9 g% v  ?
            return 1. ~# Y& T/ [% J) x: W
        # Recursive case
    7 [, @6 K, \! w3 H    else:. \4 d: d# E7 ]. X
            return n * factorial(n - 1)* j2 P6 G# s$ x( W- P, u. p- o
    6 D! l% o# A1 G+ P! Y
    # Example usage8 T7 x" \; a$ h8 b5 f
    print(factorial(5))  # Output: 120
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    / f3 [" g* X- B0 dHow Recursion Works# Z" f- q7 v6 a; Q

    8 [# n, x' M; |* H    The function keeps calling itself with smaller inputs until it reaches the base case.+ g+ d$ v- d2 T4 X4 ?% u4 i- J3 X0 `/ Q
    # ?- L1 V2 D" v5 q4 M1 k2 y& a
        Once the base case is reached, the function starts returning values back up the call stack.5 P6 S- N/ @7 S! W% V4 T

    5 X7 H" ]8 D9 y4 i    These returned values are combined to produce the final result.: r% t( u; l; b. k$ k; E
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    For factorial(5):
    2 ^( C+ G) t! X. a2 q" Z
    - S5 V0 R& c. {2 l8 u2 `$ }: n
    7 {# f0 r: l/ {+ V" b- hfactorial(5) = 5 * factorial(4)' k2 O4 B( _) o* z5 M& w
    factorial(4) = 4 * factorial(3)
    * {3 n" S. @. r. y8 ?factorial(3) = 3 * factorial(2)0 O$ J: B- y% `
    factorial(2) = 2 * factorial(1)
    $ F# I/ Q* s4 U2 V% z% M9 p  ^factorial(1) = 1 * factorial(0)
    ; n( r+ J0 m2 f/ K! W' Lfactorial(0) = 1  # Base case
    3 M: Q- U( _* K% ^/ O$ S2 w. h: k& H  i4 z
    Then, the results are combined:: H0 O$ W% K0 m- \3 C" f0 n
    # d* G/ m; v+ J: p8 }0 T6 \
    4 g" a4 e5 `5 G- T6 v
    factorial(1) = 1 * 1 = 1
    7 X1 _- a" F8 w" e% H" qfactorial(2) = 2 * 1 = 2
    : q; H9 e1 P& E. N1 sfactorial(3) = 3 * 2 = 6$ d( ]3 A2 e/ X8 o3 V
    factorial(4) = 4 * 6 = 24
    1 @) l" w  M! @: {# mfactorial(5) = 5 * 24 = 120
    ) ?2 M  h% s. k) l, z  B" L0 [
    , T; e, O) e) D5 P4 lAdvantages of Recursion% z, y) p! v: Z$ n+ Q
    2 k* D! L0 [$ B) s, H) z
        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).( H+ u) h6 u/ g  H: d8 b3 E& {
    . l/ o: p" d9 q& U0 ^# S
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    " N# F9 M/ s. `: c; |- ^4 e1 U* c
      m% C* E) O) k. L% I% [: S7 n' J7 uDisadvantages of Recursion4 z. A0 u2 h1 P- v6 l

    ( e9 n( k0 U( s4 d: Z; a    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.
    7 Y/ c1 T" S& n5 {4 m, B% b. D2 }- w4 T: \
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ! n& _- m* `' C) d" V
    % m% o- I! B1 O9 Z8 K" `( H7 JWhen to Use Recursion
    7 S, q3 ^$ y$ i
    $ w5 B& C- E; q) j    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    0 p$ y$ T# r9 n+ Y& ]$ ^3 ~4 n1 I; y$ K+ M. C  j" O" y
        Problems with a clear base case and recursive case.
    * y, B; c( }0 u3 g  \. h* A. Q6 R4 r9 L2 ^1 W. o& G. L
    Example: Fibonacci Sequence0 @. `0 w! A7 h2 p+ a! q
    , g4 i, ]' y- F0 b$ Q
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    . M- i& u! I- Z8 S5 W
    : u/ Z# N6 U9 u2 o7 ^) o, B    Base case: fib(0) = 0, fib(1) = 1# l  R& e$ n/ j) f

    & m. G( Q0 Z) _. P+ x    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ( l& A' U( h* n! t" T' q" ^$ i5 o7 ]% m0 V4 a* ?" {
    python
    * D$ D5 f/ Z) f0 \* V( @" s  t" J& V6 \9 p7 x( i+ a

    ; }* t6 Z5 @0 mdef fibonacci(n):" @2 a3 \3 Y( I6 t7 J1 `
        # Base cases8 v$ H" Q+ r, X- E7 s  `: v  }
        if n == 0:
    1 K0 L; `' ^( j4 e        return 0
    ( k! X" z! c6 U* ], [+ D! b    elif n == 1:
    , q+ v5 m- T/ R! E, ]' i        return 1
    8 Z/ t+ U# U: f2 D    # Recursive case
    ) M" O& I1 T  F/ M* Q  P/ C$ h- r8 L    else:- V$ |! u. h2 W# N
            return fibonacci(n - 1) + fibonacci(n - 2)- s0 ^0 X2 f0 G

    + P7 e$ L2 i' @/ L: D: P# Example usage, ^. {0 ?1 m& M! {1 D' A9 M
    print(fibonacci(6))  # Output: 8
    ' x* Z0 S) N- X0 V6 Y, Y4 X/ U
    : E: j- B. x/ s7 z. T7 mTail Recursion
    & d, s- a6 e- t# c! f2 K+ @3 t0 V& _7 x/ ^% |
    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).- X2 [/ w7 R6 b
    ( A: W) |7 a5 t; q
    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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