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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 ' I# r6 z  ]0 ~, T1 |
    7 D2 y! X; g6 }' D) l" [
    解释的不错
    " B0 {4 @4 I3 N% |# f/ i2 ^; L, ?8 \
    5 E$ k" \& D& J  C( T7 e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。
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    5 T9 ]* ~+ H- g6 P' e 关键要素( K. |! g. z( q6 J' W
    1. **基线条件(Base Case)**
    & ?* Y7 ^3 U" a4 H4 K   - 递归终止的条件,防止无限循环
    4 M$ L' z2 i% g: V3 k   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% |) ?7 z' p: W# N: C
    & I& l5 h7 U9 F2 k) A1 [  G
    2. **递归条件(Recursive Case)**
    # R8 Q, |3 Q6 c   - 将原问题分解为更小的子问题
    ) G; _2 Y7 A* I) X2 H9 R   - 例如:n! = n × (n-1)!/ h1 }  [/ d% I4 _% J) i
    - R* V: }# h) l1 }& j* X* l) {
    经典示例:计算阶乘
    8 M, v: g" t' w5 I& I  a) A4 D- M; o, ~python& x# q. k, f- D% S7 ~, k
    def factorial(n):
    ) }" }4 j1 |8 v% w4 a    if n == 0:        # 基线条件7 c) p8 v0 M; X" \8 D1 D+ \; o
            return 1. l* _. q4 Y6 o1 i" `
        else:             # 递归条件: n  Q6 j3 F2 p7 E) X
            return n * factorial(n-1). |$ X" d  q7 _7 d) X2 k
    执行过程(以计算 3! 为例):2 k* r8 A$ ^. i% p3 D- k
    factorial(3)
    8 C4 R; @$ `- k% J8 T3 * factorial(2)
    8 @2 n" K: s' E2 s" W1 A+ d3 * (2 * factorial(1))
    * x% f3 {% {  ^( m" j3 h$ a' j0 u! j3 * (2 * (1 * factorial(0)))
    8 U; @" \8 _/ y4 A3 * (2 * (1 * 1)) = 6
    $ h: W/ M$ _( s( O6 v$ u& h* L
    * k5 \/ {& Z! L4 g$ @ 递归思维要点
    . H' |) c. t) K0 K5 |* ?" ^$ X4 K: n. z1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    3 O5 C3 S# n  [1 O, [9 u2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    6 j5 T1 V8 I3 L2 c' F9 n3. **递推过程**:不断向下分解问题(递)4 y/ Q0 ^2 k; |# d
    4. **回溯过程**:组合子问题结果返回(归)
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    ( s4 [  Q* O/ H3 }/ v) G 注意事项. {7 S% [: n# W- J3 c% Z- y+ N
    必须要有终止条件
    $ [5 D! O4 p( [' Y7 q9 }/ A递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ; x: n( H5 u' R- o! E: v2 K某些问题用递归更直观(如树遍历),但效率可能不如迭代" E: {/ }% \$ u6 y
    尾递归优化可以提升效率(但Python不支持)
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    " x- \: u/ I- ]! o 递归 vs 迭代: U! D9 w* Y9 [3 V2 m& R) l+ |
    |          | 递归                          | 迭代               |
    7 M) u$ w$ H! }) x/ j8 g|----------|-----------------------------|------------------|
    ) o2 z7 Q5 i9 v0 U8 ~6 s| 实现方式    | 函数自调用                        | 循环结构            |3 x, ~- r0 W9 ^0 I
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    & J* Y* x! r9 f1 ~$ 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         |
    ' n& \# T, T2 u: Y- Q6 R# M| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |$ v8 R: {) w1 X
    " s* v5 h! I8 }0 w
    经典递归应用场景  C8 Q1 R( G) V
    1. 文件系统遍历(目录树结构)3 W8 M8 @9 p' |0 X! }( C4 u5 j: h6 z- m
    2. 快速排序/归并排序算法
    8 L# n* ^; [# ^; F: W: @7 c3. 汉诺塔问题
    0 h5 B& g/ |3 P$ I: [/ w5 {4. 二叉树遍历(前序/中序/后序)! t" u' a$ b1 v# }' t2 b
    5. 生成所有可能的组合(回溯算法)
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    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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  • TA的每日心情

    昨天 07:21
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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,1 ^* c2 Q# X, o7 m
    我推理机的核心算法应该是二叉树遍历的变种。. Z! P  y/ ^7 g* R% h: d
    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:
    7 }: B2 R: T) uKey Idea of Recursion
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    ( @* a. E0 O, I# o* T6 gA recursive function solves a problem by:. i! S+ c- `" C! A  m0 N! e6 }! v# u
    5 D2 D: W! Y$ z# `2 g
        Breaking the problem into smaller instances of the same problem.6 g& [% o% M& J+ |) W4 N& C

    ; g# ]- q! t) }6 o' C  ?# v    Solving the smallest instance directly (base case).
    ! L9 }8 D) }7 M
    ; `/ G5 ]! H2 @# y7 O    Combining the results of smaller instances to solve the larger problem.
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    Components of a Recursive Function
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        Base Case:! _2 d; R# e  k$ a' U4 z2 s- _

    / t6 b# W% Z) c: H8 ^7 I2 f        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.& u, q  V6 ?; q" S
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            It acts as the stopping condition to prevent infinite recursion.
    5 }. G  R. a& O+ b: F! J. ^" R4 d4 G+ c
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:& Y) u+ Y  D% S

    6 S6 a8 z$ o) C( ~9 n; a: f2 i        This is where the function calls itself with a smaller or simpler version of the problem.
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    & q+ T$ Q& ^% K( W        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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    Example: Factorial Calculation
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    $ q* C4 t: m2 X& A$ Y& dThe 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 T  `7 f+ c- u2 H0 z
    5 F4 v, ?  q  @) [  _    Base case: 0! = 1
    % j* ]/ n/ h, z3 e
    6 D- D9 M$ Q9 M' J, G    Recursive case: n! = n * (n-1)!
    - ~( q2 C: \) M0 K2 b) [$ D& r+ X5 ?* o" v$ I! X
    Here’s how it looks in code (Python):5 A- M7 R* g! m" m' d3 u# ]
    python
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    7 r" v- w  B$ H" g% O9 d: n$ C0 j, E2 K
    def factorial(n):) u/ A8 \1 H# v  h) @0 o
        # Base case, P+ O* Y8 _$ {# s' f) Y  w% h
        if n == 0:6 K: j; _) E- O6 p; R
            return 1
    + }- ~4 i! [) S* C' n, a    # Recursive case
    $ u& V$ Y7 m9 A+ `$ ~    else:. _8 j7 e6 D: W; i* l) n
            return n * factorial(n - 1)# {0 u0 A- T9 v& s0 `2 ]' c1 h8 i$ ~1 d
    " g$ q0 c( m  D7 \, C6 P7 {
    # Example usage' j5 w5 \3 m* s$ J/ ^
    print(factorial(5))  # Output: 120
    + o# {9 d  g6 y3 O# e* ^
    * Y! Q8 i/ }& _# l5 g7 vHow Recursion Works2 R1 K: o( j1 f- ^5 }6 \1 H1 q% X
    + A0 R" {; z7 v! R7 d
        The function keeps calling itself with smaller inputs until it reaches the base case.
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    ) Y% D# ~/ L' G0 B6 Z    Once the base case is reached, the function starts returning values back up the call stack.7 g: |1 C4 b9 f

    * A* P! O' {3 e5 c    These returned values are combined to produce the final result./ o: x$ C( s, P6 l" R

    9 g$ P1 Q) g" g, U& {6 xFor factorial(5):( T$ K" C7 E+ ]  Y. ^, y4 f: D
    2 z; n) i6 H. D3 v* }

    : K: r( W; g9 m+ d4 p! Pfactorial(5) = 5 * factorial(4)2 T- G6 y% i  g" {# I4 |, [
    factorial(4) = 4 * factorial(3)
    + P2 ?$ y3 Z. F2 rfactorial(3) = 3 * factorial(2)  ^1 {; A- s1 Q) y* V5 P! u
    factorial(2) = 2 * factorial(1)
    4 F. }: j7 J$ y& G  ?7 X0 ]factorial(1) = 1 * factorial(0)
    9 [  G; B5 K/ l: |8 B" e; J2 d3 Gfactorial(0) = 1  # Base case1 B( z! k$ u% x, }
    : \. p5 A2 ^4 m
    Then, the results are combined:  |2 G  G, B$ A. B* u0 b( [

    1 s; t* w  ?- M/ b6 |/ D3 r$ {' d- @; X1 u6 x3 ?, U
    factorial(1) = 1 * 1 = 1% ~( W5 L; I) N
    factorial(2) = 2 * 1 = 2" Y; g4 |1 c# [5 q: a9 ], m
    factorial(3) = 3 * 2 = 6
    5 E; U; \/ j# |2 s- ]factorial(4) = 4 * 6 = 241 `3 [1 b% z- o3 Z4 T
    factorial(5) = 5 * 24 = 120
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    Advantages of Recursion
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        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).) t4 d) Q  U  h, P1 E

    6 h1 N* Q* g5 y8 F& n& h. |    Readability: Recursive code can be more readable and concise compared to iterative solutions.$ e! H9 ]8 s, Y  C5 G; v. h

    # w6 ]6 |: J9 w* y. [4 YDisadvantages of Recursion
    5 d+ }0 w0 K- j0 W1 ?! a3 _: ]1 O
        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.
    " Q/ U1 b6 u2 \* Y0 E9 @) [( O0 ~9 @) m  G6 h4 V1 x
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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    " {3 M* {. n$ r" d3 i+ |& c2 C# Z% aWhen to Use Recursion
    1 n" e5 Y2 k' V1 }; ^" c" x- w- u7 y4 \0 A; f* V0 E7 H3 k
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    + B9 c2 `1 E- ]5 Z4 H; }7 r# {3 l* W: w8 ~! b' A5 {6 ?) h! I- Q
        Problems with a clear base case and recursive case.
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    6 {5 E& S9 {6 V; Y* N9 MExample: Fibonacci Sequence
    $ w4 o% V0 \5 o- E0 H
    3 G5 e" E  Y8 Y3 _2 i! pThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    ( c7 c  ]$ q5 I$ j- y: d$ W
    4 B; U( f) M9 g& ~    Base case: fib(0) = 0, fib(1) = 1
    % {, H. P2 ]; x1 u8 b! \! a8 o( a& O) f5 [% h/ i
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
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    % _# N' r( f8 m0 Z3 a; Kpython
    ) j% t% M9 U5 c9 L- e. Y* I/ G- _1 a! [- n3 q2 l' B+ V. f

    " w, D0 j+ m+ cdef fibonacci(n):
    % h" E# H" B/ {/ I  U# o* }    # Base cases! o! |" G; H; h& x% R' Z2 i
        if n == 0:& {# B1 p& H) J- K( r2 N9 x) P
            return 0
    : I: G: ~4 M5 E; h    elif n == 1:6 M6 ]: g$ S" B6 c
            return 1
    + P% R% J5 t4 T    # Recursive case
    0 D$ w5 a  A. V9 _  F$ Q& H    else:
    4 m& p0 q% n; W8 S" X, H( y( j        return fibonacci(n - 1) + fibonacci(n - 2)
    3 |+ q3 j: p3 x( I/ g, K. A) _& x- p/ t' r+ {2 L5 A; o5 x) z
    # Example usage
    & R" H7 W- v4 C* B% Iprint(fibonacci(6))  # Output: 8
    ' }9 f! f3 F+ X( B: R. Y
    9 I! O/ e6 |  dTail Recursion
    ' a  l# T, H1 {: {# K! p& R2 A
    * M) O# q- J( {. r* ]# wTail 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).
    " A! x6 c0 |$ ^8 D6 H+ T
    . A! g, E) J3 p" U2 z& ^1 ]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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