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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 / n9 z: H2 n* c: {# {: X
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    解释的不错+ Q. a) \7 _% i

    , E7 |% W2 Q9 O+ K" v- C. ^. W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。
    * t" P5 U0 Y' z* T6 j. F, s3 G4 C
    % C0 x& M6 T  D# v$ } 关键要素: k5 `2 v5 j3 k9 {  w  l" d8 N
    1. **基线条件(Base Case)**) F+ K$ c0 D  g: b
       - 递归终止的条件,防止无限循环
    & J% {& ^/ x; q: K" T, g6 Z   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    & h+ V$ _/ ]4 N3 G, b0 S  {( P1 ^
    2. **递归条件(Recursive Case)**1 x$ ]; @' g6 f% O
       - 将原问题分解为更小的子问题
    ! b3 d: H& v! n7 Z/ }( t/ G( i   - 例如:n! = n × (n-1)!
    3 z/ A( x/ N0 e' N6 [! }
    0 z9 l2 X3 W5 ~0 t; w 经典示例:计算阶乘( N% m8 [* Y1 T5 T+ K6 V; m8 e
    python! x2 k7 w  ^% ?. ]4 p7 ]0 l$ a3 F8 F: X
    def factorial(n):
    ' Y8 Y. \2 L  S( w! L3 H1 v" T* q    if n == 0:        # 基线条件
    & D1 Q+ N- h# O* C3 v# E+ @: w        return 14 R: Z4 C0 z/ O8 |+ U8 H& X
        else:             # 递归条件" t& c2 T/ C7 Q& H. ]* R5 j2 t+ u
            return n * factorial(n-1)
    ( t, O9 r7 }: d4 o0 u1 O执行过程(以计算 3! 为例):3 M6 B6 v4 p+ e7 J
    factorial(3)
    6 m. L% W& u" F' }: W( p2 @3 * factorial(2), T9 D. b8 q7 ]3 d& O3 |- A
    3 * (2 * factorial(1))- h& J, `8 @7 e- s3 }
    3 * (2 * (1 * factorial(0))); H& I3 A2 T$ I! K+ Q% t( b# S$ d
    3 * (2 * (1 * 1)) = 6
    5 |2 r  d; t) s$ N( ^8 D$ _0 e: i: |
    递归思维要点. f5 L/ @3 I- r1 N# }5 H
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    / c$ |' P  _2 z/ }3 e! E2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 j3 d, [  H4 p  |
    3. **递推过程**:不断向下分解问题(递)
    5 d, H1 @# K( H6 X4 Y* N4. **回溯过程**:组合子问题结果返回(归)! W# \0 e) d; i

    9 e  n0 T. j1 A, S6 ] 注意事项! R, o1 F- p6 P3 ^5 }
    必须要有终止条件3 N2 N& H/ G  i7 ~* F
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& @# i. D# ^. _
    某些问题用递归更直观(如树遍历),但效率可能不如迭代( O: W, Y1 L! u
    尾递归优化可以提升效率(但Python不支持)
    + l! f( e/ c; Z8 c1 W  s9 _
    - n. p9 T( T5 v8 v% m6 Z 递归 vs 迭代9 T( w/ {: k7 F# b1 _! P; Z5 T
    |          | 递归                          | 迭代               |$ ?/ z; ^5 Y& F5 H/ u
    |----------|-----------------------------|------------------|
    7 @! K# Y0 Z6 R! R% k$ @| 实现方式    | 函数自调用                        | 循环结构            |
    % h) r* ]5 ]2 Z8 K& C1 w| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    $ N, |  e4 R& W4 ]; X| 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         |
    . P9 _2 S9 b( z+ H1 ]| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |9 N8 b" ]. c: g+ }4 D) }
      O3 T0 v+ ~4 l: D: w
    经典递归应用场景
    4 E% ^7 D! A' Z1. 文件系统遍历(目录树结构). I! ^! b9 v8 X0 i& S0 v
    2. 快速排序/归并排序算法. w# I& ~% w$ B8 t$ e
    3. 汉诺塔问题2 L( I, g& G; W7 Z
    4. 二叉树遍历(前序/中序/后序)
    , Q; w# F5 q) a5 Z7 U5. 生成所有可能的组合(回溯算法): p* S4 x/ o% k6 b

    % [& T$ S7 D' H: 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,
    4 z# {' C6 V, z9 N5 R; Y我推理机的核心算法应该是二叉树遍历的变种。7 R( M) i2 }4 f# |5 x
    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:
    + S  C' U2 T% Y. R, aKey Idea of Recursion6 E4 \5 E6 {+ |0 }+ M' d! O
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    A recursive function solves a problem by:, B  C9 w! _: J0 v

    2 d! H/ @- k! w$ p* e% \, X    Breaking the problem into smaller instances of the same problem.+ q2 @+ }* A9 m6 e

    6 `" }# |" z  |, l# k) m! i. B& a    Solving the smallest instance directly (base case).
    3 |' y/ A# b$ {8 Y5 O4 c2 }- y
    ( p" G' \3 P5 o. F0 F    Combining the results of smaller instances to solve the larger problem.
    " h* Y+ B% C) V9 i! a3 w! @
    ; z7 E0 {" u9 i" J1 k4 {: W( Z* ?" M4 eComponents of a Recursive Function
    , D$ j6 i9 |( |5 t: ~
    6 {/ h! G; K4 U! p4 A' c    Base Case:
    - G# Q- f* p' w+ c4 P/ q9 V+ C; ?0 x( [
    9 m: ?+ k+ J& v$ o3 H( d# J' @) j' a        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ! H9 \9 a4 w8 t$ o$ ?
    + ?2 I6 Y9 y0 h+ S$ r& z& U" D# ]7 h        It acts as the stopping condition to prevent infinite recursion.
    4 q( \" L) H. ?: M8 v# v* D+ [. \5 h) t! Q
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.9 V- q, V( q* O& t. B( }* k
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        Recursive Case:! }8 o8 @0 ]% r4 H. b" ?
    9 j. u' x' T. I1 J  V
            This is where the function calls itself with a smaller or simpler version of the problem.
    8 l; Q( t$ I6 N4 O7 c2 K8 _! C3 @9 l/ _' c  C4 C
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ' ?( i3 f  I* \3 q7 V
    ; h$ Q2 E% [2 s0 }5 J7 TExample: Factorial Calculation/ C6 D; m% E# I4 x& g! P) x. r

    6 d- x/ [, L7 d& s3 v: PThe 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:
    7 Q* W) U  Q" }0 `5 K# Y4 w! z
    8 L  m: c% h: D: x5 ?* x$ z    Base case: 0! = 1
    7 q/ A1 T- V" ~
    * N  M! k3 x2 \    Recursive case: n! = n * (n-1)!
    ( T' {7 M+ D( Z% f; k/ T
    2 s/ p* h4 X' b- m1 {1 b6 i8 @. HHere’s how it looks in code (Python):
    9 t8 F! q+ H) R# @  X) ypython# s0 c* j2 B! Y0 u! {) _/ x! y  N

    + T4 O( ?( s: Y1 y+ Z0 R5 k8 `3 g: S) _8 i  W
    def factorial(n):9 o  D! P  U0 ?* D5 H+ y1 z- e& c5 b
        # Base case& v4 k4 n: [" ?4 Z0 ]( ~' ~
        if n == 0:- I  X% P+ a- f
            return 1
    / o; Z% t  z: K" r: w    # Recursive case
    7 e) s9 \+ C+ Y: L7 p# [    else:6 \& P* b. R- M( S) u4 |
            return n * factorial(n - 1)
    ( m* q4 G# `1 G. e2 L0 _7 S# Q9 c. l, j- A4 ~2 |) O
    # Example usage1 V7 }% J& {% s4 u0 A0 l6 h  N
    print(factorial(5))  # Output: 120+ C2 k6 k3 F; p* g) u3 o6 D5 \

    * d( A# z3 S$ v- F# V. UHow Recursion Works
    / A# ?1 Y' O. k( U' A+ t) A0 d
    + O5 W& ^; C- ^! T6 Y0 Z" F" X    The function keeps calling itself with smaller inputs until it reaches the base case.7 F1 D# S# y( j% j: C. G

    1 y% h  C5 y) z+ C# o    Once the base case is reached, the function starts returning values back up the call stack.
      [9 @2 a; u) ]. D: e% v3 D6 @
    4 k0 y( w/ M3 o4 X1 H/ f5 B4 }    These returned values are combined to produce the final result.
    7 `+ u0 k5 Q% D3 V( c8 e
    9 c- V5 v  U2 ^9 m  a1 y: a) JFor factorial(5):* w1 @  K3 E/ y2 z1 G0 u2 }" d$ N
    0 R' ]7 L  Y8 @* N( t5 d

    * _* g8 d. O; X9 H( q/ \3 o, Ffactorial(5) = 5 * factorial(4)! |- n) W( G( R& y' D" G
    factorial(4) = 4 * factorial(3)
    " k, A7 s% ^- N) k8 X8 `# rfactorial(3) = 3 * factorial(2)
    : ~/ O; U( r3 i2 X8 tfactorial(2) = 2 * factorial(1)2 ]0 O( b* W. [" U
    factorial(1) = 1 * factorial(0)5 y1 O9 J7 y5 l
    factorial(0) = 1  # Base case6 q' Q6 o3 j5 |" u0 R9 H& G

    ! K2 l6 W. C/ J( w5 H! `Then, the results are combined:
    ; L9 D* o- j6 z) Z
    0 V- x$ ^' B' k1 L3 [
    ' W" J2 o+ ]; n- j" nfactorial(1) = 1 * 1 = 1
    ) p1 o2 w+ Q% e! \factorial(2) = 2 * 1 = 2
    : J& [- R/ ?( d# nfactorial(3) = 3 * 2 = 6( R, n# z1 ^1 E( z
    factorial(4) = 4 * 6 = 24
    ( O, l7 I# g9 I, q$ pfactorial(5) = 5 * 24 = 120
    6 l: `1 n, F9 G5 S6 |9 b8 P: q) O6 }+ Q% |  F- C
    Advantages of Recursion
    ' Y0 J7 B( e* B, i) ^' }, ~( Y1 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).4 [, Z4 i$ R. c( n7 I0 y
    ; ^0 F6 l/ l& Z" M0 ^) U3 G& o
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    $ C. z0 p: [6 o9 u1 Q) p/ Y" {3 S$ ?5 r( V  C, `
    Disadvantages of Recursion
    ' E( s4 M; v: z/ s3 j! S  F/ L4 g. {' G1 J, \  D! `
        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.
    ) G% _* r, x! I" {1 Y
    2 W  p' d; }2 k    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).' s1 b% Q- \1 S  ?- v; g
    : W  g0 P% P% M& y
    When to Use Recursion' n5 ]) I7 z! B; N$ b/ O2 i& j; n
      q7 v0 t9 a* m  Z
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).: g$ r, ^& ?8 z  x& b

    - K- L' i% r+ O, Z3 }9 |! N    Problems with a clear base case and recursive case.. c4 k) s' }( Q3 G; E
    % [7 @0 Q) r+ x0 j, ~1 G* s
    Example: Fibonacci Sequence* Q3 k: K8 H3 v; K
    ) w2 M3 l1 o7 R) y( S+ z
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    0 G  u8 j: b/ t8 l5 ~
    : t/ |% E, H/ t3 L: ~- b( v  M    Base case: fib(0) = 0, fib(1) = 1: O3 U, n% p, g$ n) j6 H
    : f; G" C2 Y' }1 w8 U* _+ f' S
        Recursive case: fib(n) = fib(n-1) + fib(n-2)) a$ i: n, d: r$ D* e! R& i$ e
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    python* B' ?' Z7 e* E+ V9 @
    , j7 V, P# T% X1 z

    # B) U! t! ?7 bdef fibonacci(n):
    ; m& \- k0 P$ ~/ C/ b/ h3 @* H    # Base cases
    $ U% G8 D) V6 _! N: l: z% x    if n == 0:
    : b6 m# U8 M& s: b! h* Q2 ^        return 0
    ; M. h$ K) L, e/ l9 V    elif n == 1:
    & c2 Z2 N( S2 G( T. c5 _        return 1/ j( c# p$ t! p, e1 _2 U% b
        # Recursive case
    + \2 g, a1 M/ {% d    else:
    5 B: ]7 O" y( f6 i+ \        return fibonacci(n - 1) + fibonacci(n - 2)
    2 N8 p- ^: N2 x: _) i" _$ |# ^  [7 b7 r
    8 Y8 v1 p6 x, y1 o" X$ [+ E  [' Y# Example usage
    ' b# }( ~& I* \- J- Nprint(fibonacci(6))  # Output: 8% D/ Y5 M$ D0 ~5 B( K: m
    , a& H2 s# i$ u1 h, P; J3 Q
    Tail Recursion
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    ( o  j; i6 D2 Q% J* tTail 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).
    / H. P7 v5 Y6 _( k0 `# \
      }& B* l  ?: E2 B6 LIn 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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