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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| 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
    / b2 \9 w' F* r* ~9 ]+ q3 L  g1 Q; A$ p- v9 v) n; ~  I$ e
    解释的不错9 X0 j! k" J9 v3 ?* C6 g

    - l3 C$ |! ?8 m8 S3 }! H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。
    + c3 ?& M' T/ @6 Y# ]7 S! `3 ]# {' q# F) V. S& T
    关键要素
    % z$ R; [4 c5 d4 ]7 ^1. **基线条件(Base Case)**
    ; k3 _) ?  K' T+ m& s   - 递归终止的条件,防止无限循环3 I6 c3 h/ ?+ _2 K* X( K" }% c
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    2 _2 G2 X7 S% Y) {! [2. **递归条件(Recursive Case)**
    0 I& Q) n, k. w* h7 d& g7 e! c   - 将原问题分解为更小的子问题! R& }- n6 Y. l
       - 例如:n! = n × (n-1)!  |3 F+ C$ ?. @/ R$ `( s" r
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    经典示例:计算阶乘4 A# [5 }& h+ a' j6 E
    python) h# t. a8 K9 Y- j
    def factorial(n):
    9 d9 F+ M& d: g$ I    if n == 0:        # 基线条件
    & n5 L$ m# l+ v3 N        return 1, i& O5 B1 S: d8 h: s% \: S
        else:             # 递归条件
    3 e, Z2 k1 r* I! }7 f        return n * factorial(n-1)
    . @( G7 ]  S2 x  q! I$ \3 b3 \执行过程(以计算 3! 为例):
    5 n0 R- M; s6 K% W# Tfactorial(3)
    ) [2 `1 S4 h' f) }3 * factorial(2)6 P) _! F4 j1 S. \& l6 j7 n
    3 * (2 * factorial(1))+ ]% @# b5 W; i, K5 g
    3 * (2 * (1 * factorial(0)))
    : G5 u5 X# h. c) B* d3 * (2 * (1 * 1)) = 65 s) e7 g  t, p( @
    , m  \: S* ?) i7 u8 ?
    递归思维要点
    ; S- c# e$ H4 P; Q  x1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    1 s. a1 J9 }+ c( S$ x% M# P2. **栈结构**:每次调用都会创建新的栈帧(内存空间)  c- c7 I2 x$ ~2 m
    3. **递推过程**:不断向下分解问题(递)9 l5 h' Z+ O5 y9 B/ O- k
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项
    1 j0 m* g2 b8 f5 I3 X4 i7 }; y7 M( f! P" y6 I必须要有终止条件& x" z0 [2 @8 S3 x. X0 i5 T
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)7 O: k) I/ I- T) M# Z* [, c% U# [
    某些问题用递归更直观(如树遍历),但效率可能不如迭代7 z" j" F$ G4 b- I
    尾递归优化可以提升效率(但Python不支持)
      [- x5 @6 g, C2 b% A4 M# j. u( f8 K9 B; w0 @( I) x1 _: \
    递归 vs 迭代) l( M/ y8 i6 D: O- {0 L  R
    |          | 递归                          | 迭代               |
    . r; R# ], ~. }' l' c" U% N% z" o|----------|-----------------------------|------------------|
    2 g0 b2 b( Z$ I/ d| 实现方式    | 函数自调用                        | 循环结构            |
    " I( K! X) D# X* Z/ w/ U# e| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    - u; F4 g3 ?$ t1 ?( X# `% t| 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         |! S6 W6 K$ T5 \. F
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |6 M- X+ F" ~3 d) q* r) L

      ~3 n! [# ^: ` 经典递归应用场景
    / ]0 x6 i$ q9 p9 N- E7 e/ y1. 文件系统遍历(目录树结构)
    5 B3 ?" C: C  Z: d0 [$ m  S/ c2. 快速排序/归并排序算法
    % O* z( m7 D+ u2 ?3. 汉诺塔问题
    1 u* J6 r( j3 Q8 v) y6 ~! D$ b4. 二叉树遍历(前序/中序/后序)
    8 g) G3 Q8 T% f; y7 I+ N" [7 L5. 生成所有可能的组合(回溯算法)
    ! v- e3 s  Z' U% T* L- Y* C0 w
    ( O/ C5 [. x* h, {4 ^7 H! s+ B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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    昨天 11:23
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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,: L& Z( X4 T; f1 o& `* J+ g! ]
    我推理机的核心算法应该是二叉树遍历的变种。
      r# h  |  s& r8 j6 z: d$ N9 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:
    . G3 \9 O+ ?8 G$ `Key Idea of Recursion0 e! \1 x* |3 x# [% S# ^2 s/ @

    0 T1 c3 \2 X3 e, rA recursive function solves a problem by:8 `$ t# f1 R! _: d" b7 l

    : N; {/ H6 z* {: ~: a4 Q' K    Breaking the problem into smaller instances of the same problem./ B$ J' }. r8 O& @5 C( C/ P

    1 X8 Y& I% j/ e( P7 y/ a5 D9 }    Solving the smallest instance directly (base case).0 X  s8 K" P" l2 v& u' v* ~

    : M5 c0 t" }' C9 A# J  G' |0 G: w4 H    Combining the results of smaller instances to solve the larger problem.
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      ^6 r9 k! ~# u; r4 MComponents of a Recursive Function
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    , g: E# ^5 X' m3 H4 w! M: O6 n    Base Case:4 T& I( m1 `  J/ G3 L) C" @! P7 [- q

    . a6 W1 h- b( p5 ?" h% K! w! @$ y        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.3 G# y% w: z( o$ u5 I: O' N
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            It acts as the stopping condition to prevent infinite recursion.& n6 T& a+ F# S. H+ u1 _$ ~- h  v

    2 H) y/ ]0 ]! ]8 f9 Y& u; p        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 p! D1 ]" t4 \2 _  |+ I! w: @

    & K* ?; h, T/ w! |& Q! [    Recursive Case:
    2 F3 b* X9 c  H' ]* a: Z
    ; u9 s- S8 F  W0 |) r2 F+ ]        This is where the function calls itself with a smaller or simpler version of the problem.0 p3 Z* G! \9 l& G7 v

    7 Q! E  Q& A* M" S' B7 u' k+ C        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ! z7 j, m) W. A, i- @; u& e1 W5 [
    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:
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    ; ]" B* l; `1 \    Base case: 0! = 1
    ) \/ Q1 M2 V& f; Q( u4 F' R8 }
    $ h% u' l* `( m0 o1 z  r* q    Recursive case: n! = n * (n-1)!8 R; \+ k: L4 O7 H8 {

    . d! G* `5 U4 S' h, ]4 e) @Here’s how it looks in code (Python):
    ' t; H' y5 e/ v; q& zpython
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    6 i0 g& o7 T7 T9 H. p
    " G, L$ u& L- Q2 a1 q- V+ M, o: P6 |def factorial(n):+ A9 r2 h# w+ e( T0 s
        # Base case
    * `0 [' u" \! W    if n == 0:
    8 s. k! h/ [8 \6 O3 K! l9 @        return 1
    . X% ?  `7 `& _, {/ ^* U, Q    # Recursive case( j6 K! G# E2 X9 p6 V
        else:
    ; A0 v* b) `$ x; U- E" X5 Z        return n * factorial(n - 1)
    # y1 B6 w( q+ i$ T
    5 e' T3 |% k. V( z( f# Example usage
    5 p7 N6 T( e, nprint(factorial(5))  # Output: 120* Q$ \. M1 K& k3 s; ^  e
    6 q  c# l6 ~- p0 [) ]
    How Recursion Works
      |9 `$ B- o/ R7 g& ?8 z, a: [7 z! w3 h" j$ Y
        The function keeps calling itself with smaller inputs until it reaches the base case.
    # p9 g/ Z9 \5 F/ I* f8 ]
    5 a- u1 z1 i( |# v    Once the base case is reached, the function starts returning values back up the call stack.
    7 `% Z: Z$ P3 J* L  R1 |
    & U$ a) t! k4 K/ [0 H/ T( J& W    These returned values are combined to produce the final result.2 v" h. G, Y* h) i- a
      I7 e7 ?- @: J7 y
    For factorial(5):
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    9 G1 S: X- `8 h' z- ?
    % }( k# ^! U# j$ rfactorial(5) = 5 * factorial(4), K  j1 {* m/ |9 s
    factorial(4) = 4 * factorial(3)- s, ~" k# s: z0 e% G
    factorial(3) = 3 * factorial(2)
    ( ]* t) @2 e9 P# m6 |3 l" Ofactorial(2) = 2 * factorial(1)
    $ d5 o4 T- W" x3 ]' ~7 pfactorial(1) = 1 * factorial(0)
    7 n% c8 j/ C# c8 R0 w3 ?6 [factorial(0) = 1  # Base case
    9 U5 h/ A2 {$ J7 z
    8 b* B5 U3 O4 b( @) IThen, the results are combined:
    0 i1 ~  X# H$ S7 k: |
    * Q, G" j, R% i/ \& J, V% h0 E
    7 @+ R/ S$ p" R4 w' s# Ufactorial(1) = 1 * 1 = 1+ n5 P  Z5 ]3 u1 l* L; R) r3 \8 `( f$ S
    factorial(2) = 2 * 1 = 2
    ) `$ l( n; {' M9 a5 t& Xfactorial(3) = 3 * 2 = 6# B% {; H& G# g1 B9 r( k$ t
    factorial(4) = 4 * 6 = 24
      W) w" e/ I% q2 ~1 K& r+ v; Tfactorial(5) = 5 * 24 = 120" E0 q7 c- a. z5 F$ t% w& J
      O/ `. C( L* _5 q0 ?
    Advantages of Recursion
    8 m& T  T3 g. W2 W; B7 y, r4 |2 u& f/ X
        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).
    7 |9 x, T- M8 T0 w# M" ^% z) X# e  U1 T. A
        Readability: Recursive code can be more readable and concise compared to iterative solutions.$ T5 D6 I' y& P9 j: Y7 S! t# W' b

    , O9 B7 E* C1 A! h: cDisadvantages of Recursion' Z( M6 a+ y% n$ F: q
      r+ E2 Y% \( M$ d5 {
        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.
    0 Z$ m8 Q7 }1 g/ L, m& t1 R( a0 Y) G0 e. Y8 f
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ! A3 O% g- A8 l1 ]6 U. w, C0 w4 r6 n7 w; s* s% s7 \
    When to Use Recursion" V, @% d1 k+ M- M

    8 t' k8 `! \# M; G7 J: _    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).: Q7 o" R9 A/ _7 T; G# ~) w+ ^1 }6 E
    : r5 m. V0 l( c, r3 x  `$ E5 O
        Problems with a clear base case and recursive case.! I% K% Y! t8 T& v% F
    ! d+ a  ]: |$ R
    Example: Fibonacci Sequence
    2 c- k" M2 j1 h& U; j3 x( K8 _0 O- E9 r; `
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:, ?$ a* r& u9 f. N9 \, \
    . H1 H6 R3 R, S2 y* L# M$ ?: f7 p* g
        Base case: fib(0) = 0, fib(1) = 1( s/ ?0 \+ S8 Y6 W% `2 j; c
    6 i8 x. d" q( F# [8 x2 P% I1 g
        Recursive case: fib(n) = fib(n-1) + fib(n-2)! ^' A" i6 q! E1 w9 w, d8 ]
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    python
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    5 o7 S* \; ~1 M& f' l( Y
    def fibonacci(n):
    - D2 O0 X% Y/ \8 [    # Base cases6 M( P' p% M5 f, a6 A$ L
        if n == 0:
    " r" u9 Q1 A% Z8 p5 b3 [        return 0
    ( u) v0 ]' T& f! u1 O$ m+ i    elif n == 1:
    6 A" Y3 J1 k5 i, A. u; P/ n        return 15 o/ h5 z* M5 N/ r- Y
        # Recursive case+ F; w1 _, n& @, G
        else:% F! b5 K8 Z1 g# g# r* M. h  \
            return fibonacci(n - 1) + fibonacci(n - 2)2 ~! X! f2 K8 d

    1 l2 J) E+ [% w" C# Example usage
    * ]- W) n1 {5 k5 p$ E- }2 zprint(fibonacci(6))  # Output: 86 e- u. R4 X8 v

    ; Z, `. M4 H& Y0 Y! v! K8 kTail Recursion% v( L# U3 J/ \( G
      F, p( J& ?, w
    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).
    0 [& W& c# K1 q
    # }) W4 b$ n, q7 E: TIn 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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