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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 , W- ?/ w% `$ v( M7 l1 w2 T

    1 J7 o  D3 P3 l) c3 e, P6 ?解释的不错2 r' E; P  Y7 y/ P9 q& }
    ( J0 }7 ^8 f, i6 O- ]
    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。+ u; S& e8 J# J  i

    * s, P5 H0 [' Z0 S' r 关键要素
    ) n1 L& o/ |: e  a2 B1. **基线条件(Base Case)**
    & B# r: D: `' \5 w+ r- Q   - 递归终止的条件,防止无限循环
    . o  v6 e! F3 n0 U% P* H0 k   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1. Y/ H, Z) m5 y
    - I7 V/ F, [& G+ m: k, N0 E
    2. **递归条件(Recursive Case)**
    + M3 `1 f1 }. y4 Z5 M) K/ o   - 将原问题分解为更小的子问题
    / F  G! |' o7 U   - 例如:n! = n × (n-1)!
    ' Q$ l* Y0 N0 N: E! h) a
    3 I$ V+ K- V: t  y, w 经典示例:计算阶乘
    9 l- N& {4 l6 P9 O* J! |8 M! v, R+ Spython) ?( D3 g# n( x5 I" U, l; z6 k
    def factorial(n):% d2 Y$ A, ?6 k4 i; V
        if n == 0:        # 基线条件4 _, Q% P4 s8 K, ~9 q! X# s* {
            return 1$ l2 z9 w2 x, b2 x1 \
        else:             # 递归条件3 Z# |" r3 d  x/ f! ]
            return n * factorial(n-1)0 t# P' R3 o; e. ]6 G1 d, v* C
    执行过程(以计算 3! 为例):% z* m: v/ q2 w3 N0 S
    factorial(3)! u0 z  Z& v7 N. ~* d8 ^0 g
    3 * factorial(2)
    & ?8 D8 y+ V9 p3 * (2 * factorial(1))
    + {$ X; O9 b# s- ?0 L9 m3 * (2 * (1 * factorial(0)))( q. I+ s+ i: \. ~7 Y
    3 * (2 * (1 * 1)) = 6  d; D/ r7 X- a- m. p
    5 j7 V: `1 c# b% j
    递归思维要点% d( H1 ?( p, ^' f4 u. q- ^0 Y
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    8 i- O5 X) T& [, I2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ; F6 ^0 X6 C+ d& }& z& E* ]( {3. **递推过程**:不断向下分解问题(递)
    3 E0 ~( r; e4 D1 }" N4. **回溯过程**:组合子问题结果返回(归)3 d0 n! Q# ^- y" U( y, n/ m5 e
    5 A% K2 v9 ]5 v9 h6 i. Z9 s. {
    注意事项
    + t1 ]( ^. Y! O+ c% y必须要有终止条件
    & G/ B* \, E0 p) [3 x9 i递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    1 _* G3 n6 |3 h3 I8 J某些问题用递归更直观(如树遍历),但效率可能不如迭代
    3 L7 A8 A  X' V/ h尾递归优化可以提升效率(但Python不支持)6 g% H8 f% i& ^
    2 L7 i4 }2 T" }/ R  y8 v' p
    递归 vs 迭代- y) i. O! a2 A
    |          | 递归                          | 迭代               |
    4 h4 r) G  v2 F+ P|----------|-----------------------------|------------------|2 q/ E1 U8 ~7 [. @+ d  d0 R: i
    | 实现方式    | 函数自调用                        | 循环结构            |$ W" ~8 B% L' G7 a+ N; z, A: ?
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    $ ^$ }2 b2 s3 A0 C  ~| 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         |
    . _; o* F1 M+ T6 a9 Q4 o( w4 B| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    . z0 n% c7 X4 a+ v
    % d( _% t) l3 z+ ^3 E 经典递归应用场景
    : H  S; z, B# z" W1. 文件系统遍历(目录树结构)
    , P2 B* q$ I9 H! i+ R$ v2. 快速排序/归并排序算法
    / k/ h! i4 B/ y, g/ x; T- e5 t- f3. 汉诺塔问题& L! E9 \: X, T1 e% p
    4. 二叉树遍历(前序/中序/后序)
    2 X5 T. m9 M. q2 C$ z5. 生成所有可能的组合(回溯算法)
    # @3 T6 g: t( x2 V1 r5 X5 n
    / i5 U9 T& \8 p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,. c6 b6 d8 S- u( b1 O" k# B
    我推理机的核心算法应该是二叉树遍历的变种。
    ; J* C) i& P4 Q3 h. Q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:
    % B/ q% `% H3 `3 _' C: OKey Idea of Recursion
    ( N9 [, m: d9 B+ R
    4 Y# M0 m9 X- ~9 L! K1 mA recursive function solves a problem by:9 U9 f$ |7 k  s+ C

    1 [8 j1 X+ G2 l- W- k    Breaking the problem into smaller instances of the same problem.# Z- w7 q9 s& ~8 t& p" ?8 f

    ! J$ M3 k( S* O    Solving the smallest instance directly (base case).. i, L) O( i2 _$ W2 ~8 J' E3 o

    / F" ~5 @( A. z    Combining the results of smaller instances to solve the larger problem.
    " {2 D8 W4 ^. ]' E% O+ I* r) T% P/ g% ~2 F) e% X- S( l
    Components of a Recursive Function* |+ Z$ p8 K# V
    9 a) f# }2 \' b8 v1 a+ v
        Base Case:
    ( l4 [* p0 B3 a  c4 s0 T! T' a) N8 @' B! R" `) m3 p$ W* e: f+ m
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; u( \1 B8 v. G9 ?2 B* D3 q0 t" K  {# b. s) ^
            It acts as the stopping condition to prevent infinite recursion.  n1 u" o# I' N3 L% t  f

    0 b2 d3 ^( Q. h; G7 Y        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    . y9 Y% l+ B1 ]0 f1 K. G+ J) e+ [+ ^& I- x/ o* ~3 w" k5 |
        Recursive Case:4 W. X' S! V& ^8 v
    ; p$ Z  X: w! G) F% A, ]3 z
            This is where the function calls itself with a smaller or simpler version of the problem.. [; c2 ?2 i5 w1 g: d6 G# @. v
    3 g7 ], ~3 c$ \+ w5 [) E2 _1 ^. L$ h
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    4 R: p6 l+ s7 O; h& {) [: J* f* y; n8 h# Y
    Example: Factorial Calculation) S9 d% m* X4 r* |

    0 o1 H2 X1 {; wThe 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:
    ) ~0 ~7 A1 p3 N0 i! @7 ?9 e8 S
    1 f' v% r. J1 B; K# K" |    Base case: 0! = 16 @9 I7 T- d  ?$ ~) Z5 |1 `' x
    * t7 A) T' \8 X* |# d& i$ _: w- i" _
        Recursive case: n! = n * (n-1)!
    # D$ \% P' B$ ?( Y$ Y7 i: T0 |- }. N; A! N4 e# g/ p# ^& V
    Here’s how it looks in code (Python):% j# E" E& x$ ]& Y: j1 b
    python8 x3 O! _5 T8 c& [+ d4 {
    & t3 n5 m2 i: y; r

    - W+ _! f5 z$ U% G; rdef factorial(n):; s% v' _: T0 M8 K3 y9 u
        # Base case
    1 K+ Y$ ^% Y2 g% ]6 T8 N+ ?    if n == 0:
    . W0 M& P  Q& ?; X$ U& r6 {0 R        return 15 w8 Z6 a0 m$ F9 w% W" b' l8 D4 l8 }
        # Recursive case
      h& W: P% h) h' z, C/ D/ H    else:; H9 _9 l; l; W1 p  `& ^
            return n * factorial(n - 1): S/ u1 i' [  j- ]' f, K6 n# _
    - x1 H  b! `- l
    # Example usage
    0 A2 K4 k% j6 k) Q0 u# Wprint(factorial(5))  # Output: 120. M( g: r! d7 k! f9 c0 A- ~. R

    / a6 E: h0 l: O3 N9 ?2 O" R7 \How Recursion Works
    # \8 P  @3 H' c. @2 O
    ) O: `  l( d% b+ l    The function keeps calling itself with smaller inputs until it reaches the base case.
    ) }% _+ N% U8 ]" ?4 `
    7 L* [# G3 W2 }* B0 E    Once the base case is reached, the function starts returning values back up the call stack.& ?! A4 b# @# W/ g  [% o4 `

    ' I# ^0 @7 r/ N+ ?    These returned values are combined to produce the final result.  p& z: U6 X* {- I

    5 @3 s6 x$ B4 M( m) v) l5 sFor factorial(5):8 h7 y& j2 \1 y( Q5 D

    ! U/ {9 T+ U4 T8 Z  E5 J
    ! E: k9 T; t; v! Tfactorial(5) = 5 * factorial(4), i0 I' C/ y- ~; O$ F, ]( P+ e
    factorial(4) = 4 * factorial(3)
    # s* g8 d! A2 ]# y( mfactorial(3) = 3 * factorial(2)- z0 y+ R9 ^( j
    factorial(2) = 2 * factorial(1); z& x/ N+ Q0 @
    factorial(1) = 1 * factorial(0)
    4 q: i( x  P+ i, Cfactorial(0) = 1  # Base case) P& t' \0 ?7 J! \" q, _+ N1 O

    * y( x# |1 J8 ?9 H& J2 K% eThen, the results are combined:/ i/ j1 x  X. r

    ; x. R6 |6 }) V7 ?0 {1 N# v4 S" ~: k$ E+ q$ J1 E
    factorial(1) = 1 * 1 = 1
    * V  X9 D( q! I% i6 G2 Kfactorial(2) = 2 * 1 = 2* h' s  f! [" F: d! R6 J
    factorial(3) = 3 * 2 = 6& t% f( z& Z0 U0 Q& [- C; e8 Q5 u
    factorial(4) = 4 * 6 = 24
    4 D* I# Q! H4 X+ q# v) Ufactorial(5) = 5 * 24 = 120
    6 `( F+ T9 c' ?( P0 f! j( Q
    5 ~* a) a0 x% }2 |  `9 x: T$ [Advantages of Recursion
    & q& O4 }+ B9 |/ J5 S2 q/ U0 F1 P9 T! G6 u+ B# _( D! F7 c5 v& G+ [
        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).
    $ _5 R7 b/ N% |8 C5 ?$ W/ n
    " Z2 H$ ]" I, C" U* m    Readability: Recursive code can be more readable and concise compared to iterative solutions.; x/ P- J% M+ @1 J5 s
    & o  A0 d6 _7 M8 c0 G5 Q2 s1 t
    Disadvantages of Recursion* a" ~! S' n0 b% t) ~* a

    3 N1 b! Y& a2 c$ }- U; S  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.
    - g% M- L" V) n
    & @2 [. h2 u/ M    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ( M- K, U+ r( B' M) W0 a$ |- Z4 o$ \) H+ y, \
    When to Use Recursion
    6 R4 M1 \! i" C5 L( C- Z- N! T9 X. [) w! M2 d3 Y& f& F
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# M$ s  J2 r1 \$ g  J6 M; W

    * ?8 o/ E; w: v$ L1 h9 [    Problems with a clear base case and recursive case.
    ; f2 O' e$ H" x* u. z1 A5 T  [* e$ j% s
    - v1 {1 B7 O! c* VExample: Fibonacci Sequence' f# o/ B4 a# P; I( J

    + ~1 B" q, o6 r/ J- W- sThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:( J- R* g7 n: L! J. c- {
    9 W; M5 \2 q+ w% Q8 @
        Base case: fib(0) = 0, fib(1) = 1
    9 q+ e; m7 x! m  B! P, b$ @3 R* n( X% o" i7 o0 W% ?4 Q& ^: J! h5 n
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ; j4 f: G" ?/ t" M! E) [% N; O
    - T1 N0 n3 u8 dpython
    8 A* n7 D9 t- u  q8 ^
    7 h+ ]. ]9 U. }" k# }$ e5 T) m  B1 ^- d: w4 Q
    def fibonacci(n):
    + d" t, }' B$ R0 w    # Base cases  a; w  s, R# v+ r9 X- Q; A; @
        if n == 0:
    1 L' G" O4 u3 l6 J        return 0
    / c! S& w0 H; e    elif n == 1:; x' m+ @9 @$ D+ y' s
            return 1
      G, p, j% W- H/ Z    # Recursive case
    4 r* J( \; Y) K' K1 I    else:
    # {5 }; }' h: k. R1 n        return fibonacci(n - 1) + fibonacci(n - 2)9 s# [2 a4 B: j% Y( G+ {
      w, i' Y1 X3 i$ u
    # Example usage1 v( M2 H* I& t- \
    print(fibonacci(6))  # Output: 80 I1 W" H! O0 Q# L3 R
    ! G+ k# a0 y/ c; f/ C% y  n; `
    Tail Recursion
    2 h& `" i) x( H
      T, y9 g/ U7 m& _( s2 ]& ZTail 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).2 ]* j8 B, ~3 H
    : Z3 t, g, O& d) F
    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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