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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 / v6 g: g3 i; T: V, u" w
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    解释的不错" g% K/ F& s! Y* t* ~& w# P8 b
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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。
    ( h. b- y3 E  I/ C9 L+ i+ e
    * l. v" w5 ^0 g/ Z. A! | 关键要素
    # P; b! j& j# o8 f1. **基线条件(Base Case)**
    , a2 ~+ X' g6 Z   - 递归终止的条件,防止无限循环! Q( o5 C1 r5 l+ ~8 n3 j8 u
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 10 }; K4 n6 M. N$ K2 `0 y

    " t: Z' P: ?* q1 e1 D1 p  n2. **递归条件(Recursive Case)**
    7 ^8 c3 x# N( D1 R* m2 ]0 U   - 将原问题分解为更小的子问题$ P( F4 G6 J* w  q( o9 ^
       - 例如:n! = n × (n-1)!9 q% i. R/ ?, ^6 E
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    经典示例:计算阶乘
    - X6 g* Q" @! z' Jpython  P1 V1 L2 N8 W
    def factorial(n):+ ?% c0 \  U1 L9 |$ j& ?! a4 e
        if n == 0:        # 基线条件6 @  T+ H7 N0 g
            return 14 ~/ h+ G- I7 r& x0 t. |
        else:             # 递归条件* H$ n! W  P& P; T* p1 o
            return n * factorial(n-1)5 p% y8 g' X. s0 \
    执行过程(以计算 3! 为例):  b7 }$ ]) _' e4 Q' [' ~" r2 V
    factorial(3)8 T& u( m6 W& P: n2 z; R
    3 * factorial(2)
      B( ?  j6 b+ m- X  Q- X4 w3 * (2 * factorial(1))7 [9 Y) S- m7 q" U& d) G0 W9 U
    3 * (2 * (1 * factorial(0)))
    3 O3 P& {6 n1 A1 v9 f3 * (2 * (1 * 1)) = 6% ?) ~: V: M0 y7 x+ B0 Q+ D% Q  C
    ' ^- K) c! t0 P* ^
    递归思维要点
    6 T, g6 w5 s7 B5 i+ E  i1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    / j% r+ R$ g- k+ G+ z2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( @; u6 Q8 y5 Y5 i6 k
    3. **递推过程**:不断向下分解问题(递)
    % p; V4 ~* c* m; {( K" L; }4. **回溯过程**:组合子问题结果返回(归)- D& W7 K7 I) p" d! m2 y$ a
    6 C8 @% z8 u* J2 X
    注意事项1 h7 V+ g: N+ u* X
    必须要有终止条件/ O5 u( k( U' z: y
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    + U+ D9 S$ G/ \3 Y' g, U0 n! R5 {' N某些问题用递归更直观(如树遍历),但效率可能不如迭代, B7 t: k3 r7 B6 s& K2 g& X( t% F
    尾递归优化可以提升效率(但Python不支持)5 m) M/ l/ _$ j; [+ m4 b

    7 \$ J2 C  V* x: `& z 递归 vs 迭代9 c" @" @: V  i  L' V/ R" d: X5 d
    |          | 递归                          | 迭代               |
    " T8 d2 `) V& m|----------|-----------------------------|------------------|% J; L, X5 P" M& t) n
    | 实现方式    | 函数自调用                        | 循环结构            |
    ' k1 r) U( p5 F# j! V* [7 X2 K+ y| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |1 O( y: ~: z: c, X' g4 X2 x
    | 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         |
    2 a4 [5 a/ w% S# V| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |" o% t2 Z4 g% K

    % N9 V! q; i% N 经典递归应用场景
    2 O! Q( M9 c4 D1. 文件系统遍历(目录树结构)
    5 s" o2 U/ X) q" Q/ ~5 B2. 快速排序/归并排序算法+ k) T. W! r6 ?* s$ X; K
    3. 汉诺塔问题
    % _$ k4 K7 }. m7 M: E4. 二叉树遍历(前序/中序/后序)
    + K2 M8 g* Y( x# `! L9 a3 g6 v5. 生成所有可能的组合(回溯算法), B) j% ?. s" k$ x! m
    ! V- Z0 F( a- l1 b7 s  V3 A
    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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    慵懒
    13 小时前
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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% a! N0 I% k, k
    我推理机的核心算法应该是二叉树遍历的变种。
    " e- C9 b1 J7 j: r. |- E. O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 q1 U, Q! _$ M  h2 U
    Key Idea of Recursion0 u6 H/ m3 u9 F

    * A) D/ N, ^, ^/ Y$ `A recursive function solves a problem by:( Y5 [# i; r" B* }9 m) H

    * J& j0 ]* |9 y1 S( `5 _" Q, i: J    Breaking the problem into smaller instances of the same problem.: X; K9 ~9 y: R& t4 O2 T
    2 `6 g; r# R$ M$ w/ D) Z
        Solving the smallest instance directly (base case).
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    ' u) ]2 ~* O) n8 A7 q' i( g  M    Combining the results of smaller instances to solve the larger problem.; {) Y/ L; X* M8 l" h) p3 s

    $ X, m9 S  o) `) lComponents of a Recursive Function
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        Base Case:( k# C  ^8 P3 \# ~

    # @  e* D* ]* r) r" J) m& X        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.2 f% b% P" q3 a% w# q5 _1 T, W
    ! V  [( Z0 S' O7 ~  y  x3 j
            It acts as the stopping condition to prevent infinite recursion.  h, ]' G/ m! Q$ E. f

    & d6 h) N1 g, a! U        Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ Y7 c' b) Y. V& Z  E

    ; E: O- N5 Q; i& n3 y4 T( N    Recursive Case:* u: Q2 D* J) D4 p* X6 \4 X' e" L' B

    % Y: K  R; L, E1 q) E        This is where the function calls itself with a smaller or simpler version of the problem.
    6 K* ?" L5 s0 I. C( b$ p1 g; B
    9 v  x- K( H" y( V. N3 e        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).! ^1 F# T# m( n# [( L

    5 s& t+ R) W; y/ b& E  HExample: Factorial Calculation
    2 Z5 c, J0 I, M: W0 W* X0 k9 G/ m
    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 f1 p" G. m: H" K5 m8 A) b

    # O- ~# e' P7 o" `- G$ A    Base case: 0! = 1
    0 \3 v0 ^& B/ W3 N/ c7 d% l4 r9 r% b# l3 N% c3 j
        Recursive case: n! = n * (n-1)!
    3 b. s7 D: B' v' h  I0 ?! g
    . B. s/ b( ?2 d& z- ^1 zHere’s how it looks in code (Python):
    ) V: z( h: J* V; G9 H9 [+ }# s0 @python
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    4 Y$ W; _+ A9 ~, n& C
    def factorial(n):3 a$ p5 z8 Z% S6 c4 k
        # Base case
    0 {$ F) w- Y5 d    if n == 0:
      D% r2 I; W) F3 p% O2 x8 U        return 1
    5 S& A# D% ^/ [    # Recursive case1 q) H0 }/ m/ m- A4 G8 h$ A
        else:
    ( Y. F  I, I5 s% S) q9 r1 m4 @        return n * factorial(n - 1)) p* ]  l' i. E4 E2 j
    ' {3 T% U, h2 a4 [
    # Example usage% C9 @1 ^3 }+ ~8 X0 I
    print(factorial(5))  # Output: 120
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    How Recursion Works, c7 {. P2 [/ r* X/ |, Z5 K
    7 ~* c  u4 l2 n9 v
        The function keeps calling itself with smaller inputs until it reaches the base case.1 M  |. c0 u# l  [1 F- X
    2 V, ?) j% U! X1 z
        Once the base case is reached, the function starts returning values back up the call stack.) _8 c! Y: L+ g& Y7 b6 Y7 K3 ^
    4 m7 b/ v+ N( V5 n4 @
        These returned values are combined to produce the final result.. n1 [$ c1 E) {) H3 p
    9 n6 W% ^% s8 k. o" f
    For factorial(5):$ T4 E- ?6 {2 B3 u. N
    * L3 T, `: o5 W7 b
    : T8 [8 g' J$ ^3 f) t
    factorial(5) = 5 * factorial(4)2 T4 d# `% K- _, i
    factorial(4) = 4 * factorial(3)
    $ M9 K7 u6 Z- Q* I/ s7 cfactorial(3) = 3 * factorial(2)
    ! G0 T2 _  S8 X6 G0 d1 J1 Jfactorial(2) = 2 * factorial(1)% c7 K  Z# c5 |6 O/ i
    factorial(1) = 1 * factorial(0)" G8 v0 |" p# B2 W
    factorial(0) = 1  # Base case
    4 a# a  n6 N) W. L+ w. L1 [% S$ o$ R6 w
    Then, the results are combined:% W- R! S. r7 N8 U2 |% `
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    , a) r, \' D& N5 t! R
    factorial(1) = 1 * 1 = 1
    * D9 H, q" [; H6 ]factorial(2) = 2 * 1 = 2
    * s( w' [/ v! ]3 {) {8 R( ^$ qfactorial(3) = 3 * 2 = 6
    5 Z2 ^% P- P/ U) ?; Jfactorial(4) = 4 * 6 = 24
    8 F/ D  a; s% P4 N; z8 z# c' Gfactorial(5) = 5 * 24 = 120
    * J. Z' t+ @* E0 u8 N+ ]; }+ d* j4 h' V4 }) L4 c1 L
    Advantages of Recursion
    + M" D# ?3 F2 I7 p& M, U: ^: b( M9 f# ^% l; {) e
        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).* P; |/ E4 ~! Z

    $ j3 b/ J, i7 Y2 `+ [    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    $ Z) P2 s: I% }5 H6 m9 R  `$ [" ?( m
    Disadvantages of Recursion1 W- ?( x  E0 ?$ k

    * i5 G* }* p9 g6 g3 K4 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.
    6 K4 N& b4 C4 i2 V% G# ^( x* j" p) }; z# p6 G' c
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ; j' |0 p' h' D/ x+ z0 `/ c& I# t+ T; u" B
    When to Use Recursion
    . B+ I  [! J( r& i' \7 L# t/ `) }* ]( r0 p+ i
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).% }3 Q6 \* T" R! l( B( P

    5 L8 e, R- Y' B    Problems with a clear base case and recursive case.
    9 ~+ P3 Q& ^3 `1 U$ J- D8 u2 W* \% j* i; B/ P: n
    Example: Fibonacci Sequence: i; M$ O: f. X$ c. I" V! X' e
    0 Q7 D/ N7 t' N! Q6 I
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 G( u) r7 Y8 _3 E5 |5 U3 _) h# F

    3 M# v! h- Y) ~' T2 J    Base case: fib(0) = 0, fib(1) = 1* _8 ^3 |4 z5 s) \! a
    0 P  Y7 D" x" l0 i1 m
        Recursive case: fib(n) = fib(n-1) + fib(n-2); S% J* S( c/ ~; O. U. P. t
    : \( D/ A; |, s% H# J
    python% b$ c  F) m! w4 C
    ; Y8 O0 W' y. X% d3 r. x3 R3 l

    3 T& E3 a; |. @/ c4 pdef fibonacci(n):6 o1 p& o* a2 R5 Y. i* ^) Y# K
        # Base cases
    ) B/ T* [, U5 G; _. }) L6 R& Z    if n == 0:3 H+ Z5 g) d+ g
            return 0
    5 z4 k1 n( X+ j4 _0 z$ l+ y    elif n == 1:
    ' S) }9 u) e' e6 c) |2 }& L        return 13 r9 j/ k( v- i- t" g% u
        # Recursive case
    8 y) s1 S; L0 v$ H* \    else:
    : u4 V* {; F7 r5 k        return fibonacci(n - 1) + fibonacci(n - 2)
    - h; J$ Z: }5 x  E- w( a& D6 i8 [4 I8 W
    # Example usage
    5 _* x  T3 S& w# d; o& oprint(fibonacci(6))  # Output: 8. W8 h6 D4 w1 ]+ \0 ^

    ) s6 f0 v7 h; f( R" K, v# JTail Recursion
    1 D1 {' T2 f8 i9 X) h
    & Z4 `5 O2 y( }4 z  }8 DTail 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 [) V' r, w+ T% k5 o, P
    7 Q! B$ X/ C" Q' cIn 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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