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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
    % ?0 R" |; x" Y2 a: l+ R3 r6 r' e5 Q+ F, h! u) A* R" E5 }
    解释的不错% x. G0 S$ |) W: j0 m+ Q

    ! N- }6 g9 \3 v( u  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。$ k" i4 V. l) W4 {& b7 Z
    ( Q+ X, d) D8 X; P3 J% Y; L4 t
    关键要素
    ( M4 @* P- R  o3 z" D5 k8 l7 g  m- v; u1. **基线条件(Base Case)**2 g0 C/ {) q% J  S' I, Q2 y" @
       - 递归终止的条件,防止无限循环8 p% u; Y! S* b7 v7 U
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    / T* G1 _. ~! a8 Q: [2 j3 I# p0 [  J* z' l) Y+ W3 j, F- H$ Z' ^: z
    2. **递归条件(Recursive Case)**% U8 V% C/ a1 `0 N8 ]
       - 将原问题分解为更小的子问题
    6 J' O9 \6 T: f+ N+ b% y" J7 c8 V, u   - 例如:n! = n × (n-1)!6 z" V. K. `$ A* E4 e/ W5 `7 O. M

    3 K+ z# O& S% b9 f; l+ m 经典示例:计算阶乘
    ( h. T' G- W# tpython3 g" R, g6 [0 L9 H5 ^3 s
    def factorial(n):+ C2 Y& Y* O4 L
        if n == 0:        # 基线条件
    8 E+ g: T$ X% Y% ~( y, w1 @        return 14 R& v+ f& G  T7 a
        else:             # 递归条件9 j. P  `$ ^1 f# I  k4 u
            return n * factorial(n-1)8 e7 S! \7 R$ p: x% j- |6 F
    执行过程(以计算 3! 为例):
    0 Q0 j1 j  x# j0 t' `/ Ffactorial(3)
    , w3 x' ~( N* E9 S8 a3 * factorial(2)! D5 I2 [6 h' |( W" S! N
    3 * (2 * factorial(1))1 Y$ P/ N# `* q9 i
    3 * (2 * (1 * factorial(0)))
    % s# Q4 U8 w3 u$ R" h& z- Y3 * (2 * (1 * 1)) = 6  g, V0 Z6 b# r# r

    8 f1 U9 ~7 z- v. t3 h; [ 递归思维要点
    9 Z0 C# R  G7 t' U4 T1. **信任递归**:假设子问题已经解决,专注当前层逻辑9 O' T. A1 ]: D4 Y9 H4 z
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    4 l  m  A" M5 W4 n! Z+ t  v3 W3. **递推过程**:不断向下分解问题(递)  a+ S1 [# G" ^4 N( n5 }
    4. **回溯过程**:组合子问题结果返回(归)6 y& @8 C) b' p; ^& a1 s! d# U

    , Y& Q/ i( ?$ o. k 注意事项
    9 Z! J" x2 P* f, N5 i6 U必须要有终止条件5 S# [  d, V$ V$ s
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)& r0 A  }0 R) R' v; F- o- ?! D# h
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ; {' |% L( B  T; B6 o8 _" [尾递归优化可以提升效率(但Python不支持)  \+ e4 M/ X% V/ g3 `% R8 h# T

    ; n% k+ X6 v! x( I- w1 g 递归 vs 迭代' I  _8 T  e, F+ g+ T) ?
    |          | 递归                          | 迭代               |
    - V5 N. Q8 T3 v; H0 n2 i|----------|-----------------------------|------------------|
    + z3 A9 h) L: Z( ^& k/ F| 实现方式    | 函数自调用                        | 循环结构            |! C$ F  l0 a/ R- @% N
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    % b5 c! |7 c' 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         |
    ' c( r# a9 |; W% X| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |8 G( c) v$ M! I! O* K4 \" i( Z
    1 H3 |! |8 `2 M* [! n
    经典递归应用场景
    + M8 f4 v' m0 b7 e& v$ h7 }1. 文件系统遍历(目录树结构)0 r% v) s5 N: S) y1 \2 _0 k
    2. 快速排序/归并排序算法( t/ P( F6 A! q% p4 `6 D' h- v
    3. 汉诺塔问题8 o9 n+ T" d+ H# }( ~6 y3 @
    4. 二叉树遍历(前序/中序/后序)
    2 v% W- Q- [1 Z$ {" X/ d. E% F5. 生成所有可能的组合(回溯算法)4 a% w2 b) x8 l5 x
    2 R2 R+ w: \0 ^9 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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    昨天 00:10
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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,( @/ g3 D) w4 f+ v
    我推理机的核心算法应该是二叉树遍历的变种。/ d0 e$ [0 j- w$ T5 M) U, H
    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:
    # p2 `. Q  o- V7 `- r: o' v1 ?Key Idea of Recursion0 b) W0 x- k0 M' h6 |. e' b

    ) `" i3 j0 j! @* ~; [A recursive function solves a problem by:, w5 h. u* l: q
    5 h. y- w/ d9 E, `$ Y
        Breaking the problem into smaller instances of the same problem.
    ) r# O' T0 k0 g: w+ r. S+ M% a7 k6 }( O8 D+ J# R
        Solving the smallest instance directly (base case).
    , Z  W) P& F4 e- R5 F% D( F$ F) n, _5 x
        Combining the results of smaller instances to solve the larger problem.
    / ?2 T6 D$ V1 l' t
      w6 m; K* w# }) |Components of a Recursive Function
    ' T% j2 n( C/ k% `
    / ?, {$ y, H" _- R6 Z: N4 S& X) T    Base Case:: Z8 [$ r: j( U- T/ M
    - m" M# P+ x$ _* \
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    1 r8 e; z# V* s/ m6 D8 p$ Y' B3 D5 i( q
            It acts as the stopping condition to prevent infinite recursion.
    . @( d% n# S/ j* L/ a: i) A- @
    " g8 a. z0 S: ]) P3 z# U4 |        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    + e# Y/ u/ B8 V( \& y5 z
    9 x6 T3 g4 J% P, j    Recursive Case:
    7 i1 G, ^% \4 _* i  k& v3 v
    & w; p" _  C, x, J, F# y        This is where the function calls itself with a smaller or simpler version of the problem.
    . A3 w8 y3 P( ^; _( {& l7 u$ ]" ~, P7 g& ?7 S* d0 x  }9 k
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    ' l: p& @" o# R& l5 j: a  B0 O' }5 v1 j
    " v% t$ }9 l* j6 y7 ?  _Example: Factorial Calculation4 z2 f, y& F( t8 f# o* y+ }5 W

    5 ~% N0 g% T7 g% S" _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:/ [0 ]4 [! v4 S4 f  R
    7 K+ n- i* W; O( L
        Base case: 0! = 16 B% o8 d. n& _' ^; y( t/ T. E* e3 I

    : \& @( W( y( @    Recursive case: n! = n * (n-1)!9 }4 p- f6 g. h( [( h; ]+ L
    9 k1 R+ S; d: }/ V" r6 a
    Here’s how it looks in code (Python):' n- T3 ~: u* i% i2 j/ p7 j. {6 i
    python+ x( H5 R! v3 T" {9 {. z  w
    8 O: C6 E% l% w, y" F# ^& l

    5 V3 \9 H. w5 e' P8 Hdef factorial(n):5 _5 u' Q; d* J* Y7 B
        # Base case
    - C9 x% q3 F* E- R( z    if n == 0:7 [* o* @/ @. E, Y. s4 J" s
            return 1
    : d6 }- V3 `' v" Z, X    # Recursive case* C) U) \; X% }- u3 Z1 K% f/ o
        else:! G) b, l0 W) p( ^% K
            return n * factorial(n - 1)
    6 k2 l+ A" }5 B( |2 N- j9 P" ]( p; F4 Q2 F' h4 U. ]
    # Example usage; M2 _8 M" I, K6 ]+ h' h
    print(factorial(5))  # Output: 120
    5 g: c- r, ]$ A) e
    " B8 {" y2 g4 CHow Recursion Works
    " {2 z* C  }" y) t- f- u) z$ s0 p) w0 `9 f0 C
        The function keeps calling itself with smaller inputs until it reaches the base case.% k. N' J8 O; ]/ l; G2 b

    : o" j& B9 P" C    Once the base case is reached, the function starts returning values back up the call stack.) S. \8 v" H: z
    + `- l, s6 b% P. E% R0 Z1 X
        These returned values are combined to produce the final result.0 Q' V6 l: c* V' ]

    ' h( v" r6 s, i! q0 F" z/ nFor factorial(5):
    - c5 ^! P. V# I+ M5 X# `) p) R* [! `, n' \* Q( l1 _' M7 J
    2 o4 x' a3 n" Y) j1 `+ U
    factorial(5) = 5 * factorial(4)( N6 e9 |" ^/ r; m2 Z& V. C. t
    factorial(4) = 4 * factorial(3)
    % _% [# V5 o3 U# ufactorial(3) = 3 * factorial(2)
    ' H) e4 Q4 P8 o3 H  A, Vfactorial(2) = 2 * factorial(1)% f% P5 Y. c/ n. H1 U
    factorial(1) = 1 * factorial(0)- B6 s1 B9 g0 X* K) Z. p
    factorial(0) = 1  # Base case
    # D! z# J3 Z: P
    ) l7 S1 z9 c2 u5 J7 jThen, the results are combined:
    , w1 l; O3 F6 ~3 f) `* m! o* @6 c% t* g: T) j9 {: j7 W; L
    / s3 q) B9 ]# Y. x% I  i6 Q
    factorial(1) = 1 * 1 = 1
    ( U9 t: [: {+ ?- k) {factorial(2) = 2 * 1 = 2
    ( X  \$ s+ U: t  W  K  C) Hfactorial(3) = 3 * 2 = 69 o: ]) O) j% d" k
    factorial(4) = 4 * 6 = 24
    / k( w0 C8 @' o3 U! L7 Rfactorial(5) = 5 * 24 = 120, [6 n- Q5 G8 C/ \
    ( o/ z' \6 `- H% X# b/ l) U) ~' x
    Advantages of Recursion" b  n5 f: m  ^( [+ |

    ' p$ b* I. h8 F9 T8 c1 [    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).% x8 i6 j( ]7 W- K8 e

    ' f5 d. k/ q2 ^* S, ]. l    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    $ o3 v- Q$ _6 @8 z  U
    7 S2 {  [* H- ]4 TDisadvantages of Recursion3 p, u, K/ v* N: e2 Q6 Q  X( s2 ^
    4 }4 ~9 x  s) U# ]
        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.* `* z3 T2 g8 r1 V
    " t0 N2 B7 M$ C" Q" E7 F9 S
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    7 A) [# i8 g4 k3 `0 {% \$ \) }; |1 ^! Q5 J8 t% L2 X
    When to Use Recursion" C; T) P( C9 [, `1 @
    $ s4 n  u' U" g# u+ `/ V; e
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    9 p% a1 w/ C" Q% V1 s5 L/ w2 y$ A5 N  \+ P( g: D* m) ~& o
        Problems with a clear base case and recursive case., \8 ^& t0 R( Z5 I& n1 s+ S% L
    6 e- o" A# _9 `  O! y/ N3 u7 n
    Example: Fibonacci Sequence. I) j( |4 y% P' j# J

    & F4 K6 D+ [/ c- gThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    8 n0 N0 q" J7 V$ k* w- h' ]: M! O% n( ?& ?* A& N' ]
        Base case: fib(0) = 0, fib(1) = 1  L+ i2 o+ A0 H, q% Q
    ) q+ [6 l1 F4 X# [( e! B4 D1 x" Z
        Recursive case: fib(n) = fib(n-1) + fib(n-2)' y) \4 D* _  V# n' M# c. Y+ G9 s- n
    # x; O/ j# L9 J- X
    python/ F  h) d4 r+ c

    - d" P. |0 V7 {. J& A: m9 D. u# }' g6 `% C( v8 l9 p# P
    def fibonacci(n):6 w% ^3 ^! Y7 k% \; P
        # Base cases
    & P/ ]. Z2 i8 I5 |    if n == 0:" u3 M* X3 A9 ^
            return 0$ F/ R1 ^$ E% L) b2 O8 O2 S# E
        elif n == 1:
    ; j& e" j' V9 v) f7 S        return 1
    7 U( R4 t. A/ n0 l. t0 K    # Recursive case
    - o( k8 H/ ~& W4 V; Q4 V# [    else:
    2 H& F' h$ ?: E: }9 X9 O4 y/ Z# O        return fibonacci(n - 1) + fibonacci(n - 2)5 S5 |5 b" ^+ @9 o

    + v2 t1 t3 c+ C# Example usage6 d: U; H8 C$ h4 V/ Z
    print(fibonacci(6))  # Output: 8# O0 w! c3 V3 r& f2 u
    . T  Z) i* i8 J- \2 p' @2 B9 }4 R
    Tail Recursion
    6 x7 W5 u. M7 k2 h% D7 u8 G( y8 ~
    , w  P  E5 `. g6 eTail 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).: R6 x* `  [) M
    : V7 m5 e5 o7 L9 _; A' V6 I
    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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