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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
     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 7 m, D8 M$ c6 [4 Y- T, w
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    解释的不错/ {" M" r6 R- Y+ [- O

    * R' E5 z% |4 F( 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。
    8 I: z' x+ m/ W. e; S: P# F, X; h6 A- w  v7 |1 `
    关键要素) w- \& G. t+ U' S  o6 I" k" @
    1. **基线条件(Base Case)**
    1 _6 P' S+ ]9 x% F* W   - 递归终止的条件,防止无限循环3 z" N. S% D3 ?5 A& d
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 11 u- G' R) s8 A7 G% x
    1 d; }7 m, r- j' P' S" X4 X
    2. **递归条件(Recursive Case)**  A# Q2 \, U0 k7 n  j; Q6 x. l3 U
       - 将原问题分解为更小的子问题# L2 e5 v8 r+ R* Y
       - 例如:n! = n × (n-1)!
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    经典示例:计算阶乘3 j) m* Y; k! H( z  M
    python
    6 ^8 Z6 x8 X8 ]% m& K9 n8 Q( V4 \0 t( ldef factorial(n):
    ( G1 P1 }! r8 |, p; [8 x' {    if n == 0:        # 基线条件
    ' h0 }" I/ b  P( s6 {  P        return 1" b. b# J, ]$ Z" r8 U
        else:             # 递归条件1 H. S) F+ ?- V$ K
            return n * factorial(n-1)2 @6 H2 a2 r2 W' P/ X2 U5 [
    执行过程(以计算 3! 为例):
    2 V0 ^/ C7 |% G' K1 ]factorial(3)
    3 Y% `! D6 u8 P' U  b% e, ~3 * factorial(2)
    6 f; e9 q0 {2 f+ L5 c3 * (2 * factorial(1))
    ( s, z. y3 m* Y* F2 L' M% ^3 * (2 * (1 * factorial(0)))' S$ A% H% a9 J2 X2 z8 g/ r; P
    3 * (2 * (1 * 1)) = 6
    $ x0 q+ h" @1 \$ C+ E/ V& G4 v% M6 `- Q1 Y# j" Y) j
    递归思维要点, {* J/ }' ?/ l- N9 ^, ]
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑+ u/ R2 f: h. x# o( }6 [
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)/ H- c6 ~% i9 A; |4 Q6 w& t
    3. **递推过程**:不断向下分解问题(递)2 h1 n. o7 T! d& W: [
    4. **回溯过程**:组合子问题结果返回(归)
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      l5 E4 C7 i) o; F  Q3 r8 ]0 J6 I4 M( I 注意事项! l; Z+ I0 ~7 n4 _3 R/ O
    必须要有终止条件' a" T* b' o! H7 c) j: l4 g
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)' b' L& u, N7 U- J0 |
    某些问题用递归更直观(如树遍历),但效率可能不如迭代- I& X% j( n! V- Y2 \
    尾递归优化可以提升效率(但Python不支持). Z+ `; o" g8 p* k! \* r/ Q
    & O3 j8 \- B8 y: {/ b
    递归 vs 迭代: p, q  D+ I% l8 G. ^
    |          | 递归                          | 迭代               |
      Z' p# E% ?/ z! X% ]7 ?, h1 h|----------|-----------------------------|------------------|  {# Z' O0 N/ O5 k: _
    | 实现方式    | 函数自调用                        | 循环结构            |9 ~4 e, ~" o- W) {
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    2 X6 V$ C9 _  `; S: j7 L| 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         |+ ~. M: u2 }# v( o. b* i
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ! x$ c3 a9 b+ b7 o) s. j9 F  U; E0 }! V) R
    经典递归应用场景2 ]4 F$ B4 M: ~- V
    1. 文件系统遍历(目录树结构)! R0 R6 Z) G. J; q
    2. 快速排序/归并排序算法* C2 M; e4 t2 x5 t6 y4 C0 j- A8 k
    3. 汉诺塔问题
    3 M7 a$ q) F5 j. j6 Z4. 二叉树遍历(前序/中序/后序)  F1 z7 p, }4 a
    5. 生成所有可能的组合(回溯算法)
    7 [$ l: W/ y% v( v3 q# s. k0 l- ?# s
    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 07:17
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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,
    8 D! L+ B( |7 P) M我推理机的核心算法应该是二叉树遍历的变种。7 ^  V% Z: V& N
    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:
    . u8 q$ h& h& n% m0 m' V3 y3 AKey Idea of Recursion
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      c' [% q- [2 `. I+ SA recursive function solves a problem by:4 o( {# l$ e/ P: |8 Y! o( ~
    * U* y4 T! u2 @3 i9 J/ q
        Breaking the problem into smaller instances of the same problem.
    4 z  ~% F& }% I- i/ p2 i8 ?* e
    ' [. c1 i* v% \& K& r. w; S/ A4 ^( N    Solving the smallest instance directly (base case).
    * K; N7 m0 u- j3 y% Z% E7 \. I+ ^8 [( W. d" v; H
        Combining the results of smaller instances to solve the larger problem.6 W+ |8 V0 E  i8 ?, E  ?
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    Components of a Recursive Function
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    $ n  B! x8 F( z6 U2 N    Base Case:
    0 f9 _# L, y8 o4 b3 j8 M& E/ \/ x- W- Y& R! U, m: u
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion." M) w' X% J/ u) K. a
    9 s5 D9 {( j7 z6 `$ b
            It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.2 D8 R4 H$ G3 Q/ `6 Q4 D
    4 A* \$ m, Z/ p" P! r
        Recursive Case:
    + t: k( X4 S1 Z
    1 I% E0 K  W' {3 \, y* b        This is where the function calls itself with a smaller or simpler version of the problem.& U1 C5 n2 m: l+ z  D4 J" m7 ~

    * H  c- e9 C+ `% b: I2 `        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    % v/ e- n: k; ~  s+ [! D0 N) ^1 @; g
    Example: Factorial Calculation
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    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:$ r, U$ ]* A8 M

    3 t) \1 P; B7 s* O$ W0 i! Z. M3 j    Base case: 0! = 14 D7 p9 V5 W6 X
    2 v8 p: P2 l) P
        Recursive case: n! = n * (n-1)!* D, @; l" O- E: L  F

    * k# B4 S  R1 z/ [5 j" CHere’s how it looks in code (Python):
      w: ^7 r9 |7 L* Q' L6 w, j6 ]& @python
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    1 A3 \2 U, R( w$ A2 A% T; ?
    ! `; G( L! p( e6 \8 E" }' I0 g# }" {def factorial(n):  _3 y5 o7 L' ]4 R" B
        # Base case( E3 o4 @( d2 M7 u  N
        if n == 0:3 A' b2 k2 A* s) M' N, f5 f5 Z3 `
            return 1
    , P% Y- \7 B; h, m! @% _* H" }. `) T/ V    # Recursive case; m& H( ?. ], g& c& p8 m2 q
        else:6 ~( \9 C( H; S5 X
            return n * factorial(n - 1)
    / g) ]$ e$ E( s1 c
    ' R" B# k" L4 W# Example usage/ \+ E1 S  n  U7 k+ z# |' [9 ]
    print(factorial(5))  # Output: 120$ V& ^) w5 q  E5 \1 L6 U- ~

    . \3 W  ^7 M: y" i, Z2 yHow Recursion Works. M1 X' M2 X3 @9 c1 ^0 z! `* E

    / C2 L4 ^. j# D- \    The function keeps calling itself with smaller inputs until it reaches the base case.
    0 {9 ?: b) X7 S1 T- s/ I4 C( Q1 ~: o. {4 Q: D% o1 O/ p0 [
        Once the base case is reached, the function starts returning values back up the call stack.
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    7 c5 g& C: ~4 g    These returned values are combined to produce the final result.
    % m8 b1 D* Z; m  a/ O: C
    ; _7 y3 J5 n! cFor factorial(5):
    * @' {! b( \/ ?+ Y9 `1 K7 `- E% K
    1 a! n# b8 A# Y; U: s0 g
    6 [5 o' w3 Z3 ^. I( Nfactorial(5) = 5 * factorial(4)
    : h7 ]' O$ b$ D' ^factorial(4) = 4 * factorial(3)8 ?( O& r5 ]' C, M$ ?  c
    factorial(3) = 3 * factorial(2)6 h! ]" w( g' x& F
    factorial(2) = 2 * factorial(1)
    ( U: z' J: B2 d& x( {, x/ y$ T* qfactorial(1) = 1 * factorial(0)
    # W- X( ?/ x1 [7 `factorial(0) = 1  # Base case
    8 F3 ^" t; z1 q% U' {! }" O/ [6 I8 ~. l
    ! W8 Z, ~; u4 P+ uThen, the results are combined:
    ( Z6 Z5 j) {, r, n% E
    & W- d" R% U3 t2 w$ P( `% G2 o* ^' y+ ^9 {, d. X
    factorial(1) = 1 * 1 = 13 X+ ^* B+ G% X1 M0 w3 l
    factorial(2) = 2 * 1 = 2& L. t$ O% {# ~3 V* y3 I/ l
    factorial(3) = 3 * 2 = 6& E' E1 N# }9 t$ m, N9 i
    factorial(4) = 4 * 6 = 24  q: s2 h0 m4 O9 c( ~( d1 d
    factorial(5) = 5 * 24 = 120
    + A+ {5 o9 y/ Z. F7 K7 ?. V5 f
    6 C4 f) w+ d7 P6 xAdvantages of Recursion7 M2 h3 H6 [# x/ P# U4 W1 g- T
    # _- Q7 h. A' l# [, Y2 \; ?; b+ @3 O
        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)./ ~# k2 K' a. L: ?& f0 i5 `
    ' E8 ^4 I1 ]* U% F; ~- o7 n
        Readability: Recursive code can be more readable and concise compared to iterative solutions.' P: b" x/ d8 e4 P; h
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    Disadvantages of Recursion
    3 d4 I. d; E( ?6 v  c0 n# n
    2 X5 i- O! r) s2 p    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.
    ! _& E7 X! s- [( r7 W; e" H  _8 i: d2 V4 r* h7 M1 [/ I
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    & Q% P, q8 L* d, E1 E; L1 d# E
    * x: m) Q+ W5 q% ^+ N, T* nWhen to Use Recursion4 |+ d* q& z* h" ~- q8 _7 [' N

    ' ]/ D9 v1 M; D5 O5 @5 d    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).8 e% k: _3 E: ?# x! G+ D
    / F6 z: M& j* h/ @
        Problems with a clear base case and recursive case.
    ( o$ B" N1 z: {
    4 G" L- g$ B' z# MExample: Fibonacci Sequence
    $ C7 Z- V8 s6 Q; k. E! |/ S; _- u4 W) g* _8 w% _" N4 d
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:4 A. d6 }4 B3 V- z

    # q5 n: j7 _0 \$ g    Base case: fib(0) = 0, fib(1) = 1  E( b8 o% ]  I6 |  a
    - i. n7 |* M7 v' B5 _
        Recursive case: fib(n) = fib(n-1) + fib(n-2)4 H9 R+ ~# G! S8 \

    9 O( c( A! {3 x  x3 Npython
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    ' v0 V+ q5 q+ G6 A4 Z$ l' B' k- t8 J1 j- q3 ~/ }
    def fibonacci(n):% x& g4 W3 p7 F1 W* F! K
        # Base cases
    6 t+ n3 P3 @" I  ^    if n == 0:
    # d" ?, z( [5 P1 D' z3 l        return 0
    - s1 Q5 x* o7 I  S( x; W: |    elif n == 1:
    # D! P# Q7 n' c! X( _' \' a        return 11 K! |! D& W1 ?$ b
        # Recursive case& o9 t6 e) w+ C$ C+ B7 B9 F& W; w4 z6 Z
        else:
    . }# Z+ M: q- I1 l$ [# u        return fibonacci(n - 1) + fibonacci(n - 2)/ l1 o5 R* Z5 K3 _6 i+ W$ l

    3 w' j7 O; d* t, X# Example usage
    . H6 ^8 B( p  M  p8 Y6 J+ R( P& uprint(fibonacci(6))  # Output: 8$ v1 ~' d+ z' P, S! Q% d1 U
    % R& U. j  d( \& ^
    Tail Recursion
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    2 e* U( h( |4 v. z" N2 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).3 O$ E6 A- R5 f: I" V. b& P3 a
      ^/ k6 O+ x$ V
    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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