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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
    2 P4 {- z2 _: V2 Y
    # r" a8 Q9 N! a$ |+ {" J" i9 N解释的不错
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    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。+ G" z% B) l2 E9 \

    4 Z6 L, u* J7 N2 R/ U 关键要素" t% R; k5 x- z1 W1 F0 ?
    1. **基线条件(Base Case)**6 a- a. w6 O- O/ C2 l8 v
       - 递归终止的条件,防止无限循环' B$ c  _8 _% u* E6 d+ i
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    & B9 N( w) L" T. o4 N, J4 I
    / B8 o' D, B! @! b# `. \1 f$ ~2. **递归条件(Recursive Case)**
    ' r# s4 T6 ~% L/ R0 w) P& ~   - 将原问题分解为更小的子问题* d8 J. s6 |9 e
       - 例如:n! = n × (n-1)!
    # ]* Q* }$ a- t4 P; }) J& e  f' p/ p0 U  F
    经典示例:计算阶乘
    % P/ r+ a' L* o: h+ a" O" f- Mpython' U2 ?. A9 A6 R4 q3 e
    def factorial(n):  y3 d9 |4 Q: j5 d& Z- j/ H4 t* J
        if n == 0:        # 基线条件
    $ U# z) ^4 j/ w: m- V: f1 E        return 1
    + F; i; a5 J. S. `    else:             # 递归条件  T4 b/ Q% ^9 L# K& E0 _+ W
            return n * factorial(n-1). `9 b' Y: j9 U9 F9 D
    执行过程(以计算 3! 为例):
    5 G, k' u) ^5 v; J" V5 Jfactorial(3): c/ j( P, n6 {' x9 A& {) R
    3 * factorial(2)/ S$ T* P, i9 u, w
    3 * (2 * factorial(1))* }- z' m4 d) i  F. U  V
    3 * (2 * (1 * factorial(0)))
    & O( _/ d. b' w0 o* z3 * (2 * (1 * 1)) = 6* v& r9 P7 G3 G+ t0 A

    # u. J  ?! i$ z+ [% O" \! g 递归思维要点
    + X- y" w% O  ~! Y1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ( }) }' D2 ]! {3 z5 F2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    2 K% l" @+ |* ^3. **递推过程**:不断向下分解问题(递)
    1 G9 M5 b/ e5 o! U$ \4. **回溯过程**:组合子问题结果返回(归)
    / o* |1 w! O( K/ j5 a- ~; |% X8 @& j8 `% Z5 m# L& q/ \' d
    注意事项& S/ {! @8 V5 c' I' X' R5 p! H
    必须要有终止条件. g8 N8 [& ?" q+ u+ |
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)1 n& E( C& w4 B6 [
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    6 L9 Y" i5 e4 v7 ?, l6 `3 n" j尾递归优化可以提升效率(但Python不支持)" O6 p* o1 [5 K! y

    / O4 W/ @; C4 c5 y1 l3 k 递归 vs 迭代) J5 c$ N; o7 O/ f0 z) f
    |          | 递归                          | 迭代               |
    + q8 a# c( y& j# W( [|----------|-----------------------------|------------------|
    ' j' s, ?6 }  t/ d3 A$ w| 实现方式    | 函数自调用                        | 循环结构            |' x4 ^0 \# ~. S4 M; f6 ~+ A
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    , ]( M9 t  _9 T- d. O| 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         |
    / [1 w" ~3 W3 p$ r| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |9 t9 `. q! r. F- Y
    ( P  ^5 E9 o7 ^; e
    经典递归应用场景# D* Y4 m) G6 [6 M6 R
    1. 文件系统遍历(目录树结构)  V2 ^( H! }4 x/ ?
    2. 快速排序/归并排序算法7 x# H6 C$ [) g
    3. 汉诺塔问题
      E+ d: R8 G9 U7 H' c7 H/ l4. 二叉树遍历(前序/中序/后序)
    * {% M  Y7 a0 ^5 d! W. a% a1 O5. 生成所有可能的组合(回溯算法)
    - h" Z, Z/ U6 O3 e+ U4 |
    , M, B9 o) q3 u+ F$ E: s+ S  J, u; J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,
    ! `0 K; r/ C& b: i& t我推理机的核心算法应该是二叉树遍历的变种。6 f( M& Z% N0 ]& G2 c- j( 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:; q$ D2 I+ r1 A4 f2 `/ m# p' Z9 P
    Key Idea of Recursion
    + H  v' g0 G: r! B' i: o/ n) J3 t. {+ `. H0 h
    A recursive function solves a problem by:! J7 [8 C6 z$ n  Z3 Q- n
    8 c: n+ T) Y& O% U' j2 W
        Breaking the problem into smaller instances of the same problem.: |# g: W% W7 ]% n0 K5 @

    ) h4 @4 o2 h* O    Solving the smallest instance directly (base case).! C+ E; @' ]- H% c1 P# N/ j
    8 W9 _$ {: V  L  U5 J; W1 m
        Combining the results of smaller instances to solve the larger problem.
    " h! R3 t! v1 b) x6 x% a4 W' k/ z; a  h$ S$ {4 ?7 K8 q7 }4 E
    Components of a Recursive Function
    4 R- g- v5 k# `. g. R
    & B+ t$ k% b- M6 W" h9 E# ?5 m0 Y    Base Case:8 A' C% E8 e  C& V5 d+ p

    3 x; F' g0 B4 @( u2 d4 P        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    4 M$ H+ W( z; A0 l9 f' `* Y, @/ [3 H$ g
            It acts as the stopping condition to prevent infinite recursion.
    6 y  ~/ D9 T" E2 a; i
    % `1 W9 Q2 u7 y8 h* G9 i: P        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    3 S7 l" e" F9 H: N2 {4 y1 V; ]4 b" ]! k: h8 B0 F
        Recursive Case:
    % g3 L) U2 ^, U) E9 ]5 C
    . I. L$ a& q0 c  E. K        This is where the function calls itself with a smaller or simpler version of the problem.
    - b/ I1 \) L$ v+ i0 N7 Y+ F# d9 O+ `; W1 L+ V4 w! o
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    3 [4 q- d% [; h1 L8 o
    7 w" O( j- E1 N: [5 E( B; iExample: Factorial Calculation
    : P' J2 w$ @+ U7 Z9 R( m
    ! [; l; I% _) v. o" X2 AThe 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:
    9 d, o) Q+ B6 Y. ?8 P
    . W0 s4 B$ l0 T3 Q# e* q9 v    Base case: 0! = 15 z5 v9 R, @9 J: y; d- a0 v

    1 m* K4 g+ r: u3 x8 {    Recursive case: n! = n * (n-1)!( C6 r* A, T, D1 r1 J& G

    9 t, g# x4 X: RHere’s how it looks in code (Python):1 F& ?* x9 [$ r
    python  L+ B% \  I" v! @6 U: c/ O

    4 }4 i5 b6 V: Z; C7 n, h5 z6 E6 E! {( h) I! N5 r7 z
    def factorial(n):, n8 L. a& h7 c' r8 B6 ]0 l& @
        # Base case: ?% M, z9 h* A. [1 @( \
        if n == 0:8 @8 V4 z' j( R6 ^+ A( a
            return 1' f, [/ j+ }2 R4 M. M7 v: Z, b
        # Recursive case- w$ b/ X7 _; H3 C% s% T
        else:
    ; G9 a. u- m) D. I        return n * factorial(n - 1)
    . o3 E) a! I, ^7 p- Z! f4 ^( Z5 `* ^( S9 ~
    # Example usage
    5 Z, n7 \  T# Z4 v4 ]print(factorial(5))  # Output: 120
    5 i8 f2 O, a9 J7 R  q1 o" _. }3 a/ g$ P/ g, K) c8 y( p: |
    How Recursion Works+ V; v- k0 ?. y

    + X2 l1 k  Y2 [# V9 m, V/ A    The function keeps calling itself with smaller inputs until it reaches the base case.
    3 }3 \1 k& {5 b& Z1 S1 ?
    1 K# D+ k5 }4 G/ Q    Once the base case is reached, the function starts returning values back up the call stack.% a) F" M+ u, G7 d/ F
    " {1 }1 T7 u# q8 N) D3 ]& P3 O
        These returned values are combined to produce the final result.
    0 U; I! `. f# ~: a# c1 b
    / j/ z5 E" Z* |7 j" a1 {; H2 v" K1 P8 ZFor factorial(5):
    6 v. a7 K; G  o. t: Z0 z; ^
    9 q" p9 ?$ ~/ T2 ?4 g4 f! V$ r+ O* g9 m: X7 J; V
    factorial(5) = 5 * factorial(4)
    8 Z0 b1 c; V! M& C. Yfactorial(4) = 4 * factorial(3)
    " Y# G: l3 F6 e- e4 ~factorial(3) = 3 * factorial(2)4 p7 @! \2 c0 }) ~, \
    factorial(2) = 2 * factorial(1)$ |; r1 s8 O( y4 ?- Q
    factorial(1) = 1 * factorial(0)
    1 n7 {: _& g' s) ?  h, Ofactorial(0) = 1  # Base case1 G: y/ N1 V" `$ F; d% j3 b7 I

      P! @) j) S+ u+ RThen, the results are combined:3 h6 g  o. q2 t0 L+ v( c
    # N& ~; P: ?$ E6 I. p
    9 U* W5 y) Z2 `# F  Q
    factorial(1) = 1 * 1 = 1
    4 c. q1 _/ Z& f$ |' p: pfactorial(2) = 2 * 1 = 2
    6 K! h/ D6 L  u2 R- @& z, Mfactorial(3) = 3 * 2 = 6  I, C; |: ~$ Y; j7 Y$ D& E5 P
    factorial(4) = 4 * 6 = 24
    2 U' X( M' ~6 \  T7 v' Xfactorial(5) = 5 * 24 = 120
    5 I: T+ x/ L2 I8 g; i1 Q5 X, m  J; Z: R; a
    Advantages of Recursion
      F% v6 g, k; q% `+ G% v  s# w  L" Y$ S8 `
        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).; o' F% P# z# s: a/ a! s% m' M3 u

    . v/ U( s1 L+ ~- p    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    2 T" w3 ]) @; {9 v4 n' I2 w" }  U+ c
    ) z) [. d% {$ K4 ~Disadvantages of Recursion! [+ r8 V/ W# _$ `5 S( S9 M

    1 c! |, H8 t, v! Q    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.
    , Q2 k+ I! l6 ?) H4 f% I: m1 t: R; D) r. p
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    3 G2 i1 U$ v7 w* d, s" ?+ _1 ^. n% @7 o" G* n$ ^$ m
    When to Use Recursion% Z7 Z" l6 X- g6 D# W9 D1 T

    ( u0 h. f4 b% m' a; e. t/ X    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    6 `5 U3 D( G1 S3 D% k5 p7 \2 l) ~! {& b' S3 ]9 n
        Problems with a clear base case and recursive case./ S% G; z7 N4 W% X  j4 b
    0 m, K! J# D8 u& ]& ?, a/ x4 F9 {
    Example: Fibonacci Sequence+ q- Q3 B& Y$ D

    + |2 _4 h+ i. `3 gThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:" p4 \% l: q5 o! U
    " B% D  a2 _$ L1 C4 h
        Base case: fib(0) = 0, fib(1) = 1
    9 j/ z7 s7 S- H; w) J1 h" W. j7 M# L
        Recursive case: fib(n) = fib(n-1) + fib(n-2)8 o% `' B% D# C" T

    , {( G+ {" ]: e, [3 s0 o" y' bpython
    . b5 d) q2 R- h- C( G
    " L$ G, M6 w  M6 A; E; a  v4 w% Y- C0 g- Z! v. b
    def fibonacci(n):
    7 P! M3 z6 G# g9 R6 K    # Base cases4 w0 l: Y; G  `
        if n == 0:& {7 g9 s6 G0 S8 U
            return 0
    $ e3 R4 b' H+ m( T9 y$ [" g    elif n == 1:9 x+ ]$ C: ?+ T7 _% |+ t
            return 1
    ( F; Q- n% Y3 _    # Recursive case
    # J' _" T: l! J: ^7 }5 G* M    else:9 m5 _2 ^# d2 q: L
            return fibonacci(n - 1) + fibonacci(n - 2)) N5 @2 G- a, S5 D$ d4 d& {) X7 K

    ' C* e% ^& ^# X8 T0 u7 W& `* F. F# Example usage
    " {+ @' t! g) x5 ~" k; @print(fibonacci(6))  # Output: 8
    ) I, f" e8 M! |. o' b
    6 L( H5 W5 L, a( T  {Tail Recursion* J4 l. x- R3 ]) f* L

      Q! A* W' O* R; @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).+ H8 E3 A/ P# @& j
    6 p$ z6 m7 ]! W1 b4 Q1 n  `5 k
    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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