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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
    ) Y) ^9 z/ a4 Y7 B0 ]9 _4 t8 k5 v4 ?  k* p2 ]" ]& ?3 B5 M
    解释的不错! H6 i& L+ i, M' Z, C

    8 f; N2 m4 r. n9 e( b- [5 }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。
    4 {2 n- }' F2 b4 v4 [/ |
    3 k) B: v  y' c# y! j 关键要素
    ) m/ B# p7 p. V, O3 N1. **基线条件(Base Case)**4 p8 K) C6 D4 E) w3 _2 L! P" O2 O) ^5 @
       - 递归终止的条件,防止无限循环4 g: `5 o5 H" k" s; Z/ _
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    9 p' C0 R( M" [7 U' ~1 U7 \* Y9 G- s& ?, u
    2. **递归条件(Recursive Case)**& K( T5 v3 z$ o5 w3 O# G6 Z7 O- G* k
       - 将原问题分解为更小的子问题
    $ h% Y" P5 g9 K* g% n- |   - 例如:n! = n × (n-1)!5 |" X& Y3 g/ ^( W# U

    " i+ ?2 B: W3 i 经典示例:计算阶乘
    ) S' t. ?) _8 `* B1 _' dpython
    3 V3 r+ ]9 g3 Z( X* t$ _def factorial(n):
    4 T# l4 a; U& ~; ]  C    if n == 0:        # 基线条件
    ! W7 K: }3 j' E% \( Q        return 1
    " U& `( x& ]+ D( V2 \( z. n3 S    else:             # 递归条件1 c, P, ^+ }( v: V; y
            return n * factorial(n-1)
    - _& F5 M; f+ Y) L6 W0 L% @) t执行过程(以计算 3! 为例):
    / R' z- L* x- X  V. @factorial(3)  Y/ D+ i1 m; P% ?# M
    3 * factorial(2)9 B7 _) y8 ]$ U1 Z
    3 * (2 * factorial(1))3 w; L( R' f- x9 O8 M, A
    3 * (2 * (1 * factorial(0)))
    8 e; A/ U; P0 t" h: Y/ b$ e+ B3 * (2 * (1 * 1)) = 6+ _2 |: F0 ]% g

    ' t2 x4 q% n. Q* p$ y' ?9 j5 i 递归思维要点
    % n6 y2 ~3 l. |5 \) {" u1. **信任递归**:假设子问题已经解决,专注当前层逻辑& z& k  a  Z* P) w. [
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)/ _! o- \+ C$ t' H
    3. **递推过程**:不断向下分解问题(递)
    1 v0 w! K3 P" t, R4. **回溯过程**:组合子问题结果返回(归)# `: I8 a/ ^' J( d% Y' u  O
    9 f( P8 B' \" |& Y. y* x
    注意事项
    : h7 ]6 `  P% O必须要有终止条件
    1 ^" g" K6 ~3 q7 f$ m8 }* b递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    5 k# t. {! g2 L某些问题用递归更直观(如树遍历),但效率可能不如迭代. ], A6 E2 @3 q# c7 G
    尾递归优化可以提升效率(但Python不支持); T4 q8 k# p: ^) r: b- k+ t

    & J- K, o$ Q1 w 递归 vs 迭代
    7 @* c" Y4 q7 I|          | 递归                          | 迭代               |& h9 j7 e! n, f
    |----------|-----------------------------|------------------|
    % \& o4 i0 [. a  D: G| 实现方式    | 函数自调用                        | 循环结构            |5 ?9 r5 [1 w0 ]- p/ ^2 E+ b
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    8 E7 X2 S" P& {9 `5 S" K| 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         |
    $ g) e6 f) W- \| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |7 N+ y' \% H# ^6 r1 h% L, y. `7 ~
    ' F4 y7 [" n- T: _- Z* T
    经典递归应用场景2 K- T4 m: Z8 e$ z9 q3 b
    1. 文件系统遍历(目录树结构)2 G4 L5 _! ?9 f. \0 B  X  Q
    2. 快速排序/归并排序算法
    " q* w" l( Y7 L& p. {1 Y8 W3. 汉诺塔问题; e# e  B6 X( z' L
    4. 二叉树遍历(前序/中序/后序)1 B( _( g  w- J! U/ e0 n
    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,
    * h) J3 j6 c+ Y' `5 @. T% X, C我推理机的核心算法应该是二叉树遍历的变种。
    3 |" K! d% _3 @" z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:
    9 h2 f6 e$ `- R& u! c8 |Key Idea of Recursion4 e" V4 f& J6 G! x- E
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    A recursive function solves a problem by:6 R1 A, V% F% ^, S8 F  o
    # N: s' M$ ?4 W0 C0 x
        Breaking the problem into smaller instances of the same problem.* S5 |0 Z4 A, C& Z. u& ]! Z

    6 |& k& |4 w+ B5 d9 z    Solving the smallest instance directly (base case).
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        Combining the results of smaller instances to solve the larger problem./ t% C8 S2 y# x' x' p6 m0 _& Y

    & X3 F& N( @( O9 Q. OComponents of a Recursive Function
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        Base Case:0 t% v7 \/ l$ ~  s7 z
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            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.  l! W( F( M3 |0 @, V& i

    : E, D+ m8 B1 n: P        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.( j$ J  ?2 L& Y( L; L
    ) {( T: u$ {! @
        Recursive Case:0 E0 T# S; K* b1 W8 {9 l2 f
    # b1 O1 a+ j1 g- J: x' A
            This is where the function calls itself with a smaller or simpler version of the problem.
    % {3 X' A, p5 ^, B
    9 ]+ ?0 v( y# S! B: V        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    9 s  A+ T2 X2 r5 P( y6 H
    + U& F1 Z7 Y9 E, J0 H8 yExample: Factorial Calculation
    + s: i. L% `* H/ C+ c' Y
    6 Y! l# J2 \. E8 q2 b8 o9 @7 yThe 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:
    ! a5 T5 u: I- k1 @: a, T
    9 Q) p" V7 h" K$ z0 ^" o1 k. N    Base case: 0! = 1* B( _* @1 a3 H# G4 s7 B+ S

    $ D9 b8 O* W: m% M: V    Recursive case: n! = n * (n-1)!
    1 Y- g. N- I) k+ g! R0 e
    ) A* ~3 \0 @( x0 J- h, r/ eHere’s how it looks in code (Python):+ w9 |4 x8 I/ X
    python
    " S+ R* ?; H: Z5 f  Z+ g, J8 T8 ^* P; z# P/ q9 o) I
    9 K1 N$ \5 t6 S; ~
    def factorial(n):9 v5 X0 z% f. U/ T% P
        # Base case
    ' K+ ?2 R$ P) r! p. E: I    if n == 0:% J/ a- g! T1 R: |  J
            return 1- B# T7 {, u1 P5 r0 n+ t# V
        # Recursive case
      e3 p4 k4 X" i    else:. e& M* n2 l% f8 {2 i! Z
            return n * factorial(n - 1)
    / a1 m2 K( m& F8 w, R7 Z
    " a4 r( Q" Y/ i: I1 \2 u# Example usage
    6 ]3 D( z! e* x0 L" ]print(factorial(5))  # Output: 1207 C$ x) }* d1 M9 U3 @
    2 G: W" S! l" D  y
    How Recursion Works; H" B  e! Z. ]9 X
      ]2 }9 p2 {$ b! C- w9 s$ U7 q
        The function keeps calling itself with smaller inputs until it reaches the base case.
    . F$ ~" l5 j/ W& \6 _4 \6 v% z" l2 C# K; z  j5 B
        Once the base case is reached, the function starts returning values back up the call stack.
    . C) P5 ]! D: C3 k3 \* \( k$ @4 `  f  p" l( V
        These returned values are combined to produce the final result.
    4 X1 |( W* z1 f0 K) g4 J* f2 f- l" E- J# x: O$ A
    For factorial(5):
      c. z! K/ o1 }
    0 z; @2 q8 E( e! ]
    5 a( \) g8 @5 Yfactorial(5) = 5 * factorial(4)
    ) [3 l6 J, C0 ^factorial(4) = 4 * factorial(3)- Z6 z( t- F) a  d- g) R
    factorial(3) = 3 * factorial(2)
    & x7 m* J% ?8 ]" F' }factorial(2) = 2 * factorial(1)
    : ^' D: V$ t4 t% A/ ]; y( {0 Sfactorial(1) = 1 * factorial(0)1 T& H  K. p2 c( @$ V+ ?% q7 K
    factorial(0) = 1  # Base case
    ; p. [7 ~2 H% z- s; o8 T* C
    - @/ Z9 b. ^, WThen, the results are combined:
    6 b% A* a& N; \6 f# y( C* m" }$ g" L: t0 |! a- ]( h# X+ L0 W

    + {& e) N* X/ U4 m1 [1 M0 Lfactorial(1) = 1 * 1 = 1# P* D& R1 y0 X3 r+ j
    factorial(2) = 2 * 1 = 21 Z& m6 I2 O8 p+ s9 W; H
    factorial(3) = 3 * 2 = 6
    " s4 G. Z" |  M5 G% bfactorial(4) = 4 * 6 = 24
      n5 t7 l& ^6 q. _) h3 sfactorial(5) = 5 * 24 = 1208 w# e: N& ?. v% U% F9 l4 c

    + I, t) C4 s, T& O: k: d, U' e1 hAdvantages of Recursion
    : @2 P0 ?' H9 j+ E( ]/ G- i* O4 K6 b
    2 _4 a9 x' \; @+ b4 M% ?    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).# ~( K8 b7 S" u6 J( w& m
    $ c  r% z' C: ~, E9 T3 y
        Readability: Recursive code can be more readable and concise compared to iterative solutions.* W0 h: m% R: e' W( W

    6 ^( Z/ ^" k' Y3 [Disadvantages of Recursion" |' G8 I# H8 T
    * D8 F) o: O# Y
        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.
    9 P3 B6 n; r/ M. t$ Z1 a; l1 K1 h$ M% `( N
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    - ^1 a* p% J9 t* |2 L8 C" t% \. ~4 T3 K, D
    When to Use Recursion
    . e& `" o) f4 M/ J
    : {% ~6 z$ S) W$ J    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).8 G3 \2 r1 a/ g5 B) m

    ) r, l1 i3 H& ~8 k% ~    Problems with a clear base case and recursive case.& v. T/ n: V7 f2 a+ T( B2 h
    " [; |' @6 Q8 I0 _0 m4 ]
    Example: Fibonacci Sequence
    - o# }5 v  N* a6 ^7 ?4 ~) _
    7 S5 U# e0 J" d& vThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    " `( @& i) w4 k: c- ?! L# y+ N/ b; a+ D
        Base case: fib(0) = 0, fib(1) = 1
    * w  t( b0 \9 f) v0 V' D
    * ~# L3 I: s, E) k" f( \    Recursive case: fib(n) = fib(n-1) + fib(n-2)2 w4 H7 \4 k' B3 v. B, k& F
    : l( V# b- ]& ?& h
    python. A3 o/ D( n  Y& ?7 i5 f
    7 |- O" q! a$ a8 p& S. }
    ; W7 ?! O5 |; G
    def fibonacci(n):" A( o- S) p$ d
        # Base cases
    : G. W( W" M/ \  d2 X! U    if n == 0:/ K* N' W9 F! G9 v
            return 04 Z+ s$ ?' l, q. c6 D
        elif n == 1:
    # e0 ?) H5 d$ c& c! I* C        return 15 s1 Y( p( f. ]) `9 W+ Z+ Y
        # Recursive case
    # z$ |3 a' W$ u( U    else:8 B  L, y( j5 G, F3 ^
            return fibonacci(n - 1) + fibonacci(n - 2)
    7 C; w" e3 }" {% _9 _1 b) t3 F; n/ B- z' S
    # Example usage+ e: V3 e! ?0 e% E2 s; E
    print(fibonacci(6))  # Output: 8, j( D6 ?, F8 e( ]* s

    3 O2 q, ]8 g; T; }9 PTail Recursion
    # D/ j6 W4 }0 N  S7 w9 g# y0 c+ E, Y# ]* z' l8 e1 w8 u
    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).7 R4 C0 H) [( G! B! k0 Q* v% F
    + J* W0 T) u7 n, k$ d" 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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