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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
    1 q8 v( p4 t* K& [! U3 d' s9 P( I) ~- d+ H3 T; f$ U
    解释的不错- H- j# q3 w$ x" M5 [+ ?

    7 u  Q$ O5 ?7 ~7 J4 w3 v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。& w4 n# ^8 N# }
    ' S+ x: K  V, k' p+ @) _' d" O
    关键要素# ~9 l. q8 X/ ~0 _4 l: u' b% b
    1. **基线条件(Base Case)**
    . Y$ e* [0 s8 |7 I2 G$ Y! l0 V* T" s   - 递归终止的条件,防止无限循环* h+ ^6 y% Y. x% t1 O
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    - J7 K2 c5 x: \% z6 Q$ a3 J$ b7 C9 [  P" J# f) {6 z% x: L6 {% e  y
    2. **递归条件(Recursive Case)**
      D% c* n. ]7 ~4 T3 s4 m3 V   - 将原问题分解为更小的子问题
    ' o! @& J+ K  O& }* w  @+ t! S& ~* A   - 例如:n! = n × (n-1)!; A' t' N0 V9 p2 u4 t

    + m4 w' C) \  H* }6 o7 r( i 经典示例:计算阶乘% F: U* }9 D) x' `3 @
    python
    ; [) l( l0 w" \/ \# xdef factorial(n):2 A( w% J& \  ?: N( I: g4 y
        if n == 0:        # 基线条件
    1 G% w. |# m) I2 A* i# R        return 1
    : q4 n# f7 w0 ~. }$ b) Z+ Z: {    else:             # 递归条件& O/ R+ b- Q" i
            return n * factorial(n-1)
    * S% y  e! i5 j& F1 d2 _' z执行过程(以计算 3! 为例):, ~: A0 W( z; Z7 `
    factorial(3)
    1 J. @8 d5 Y; w- N- E3 * factorial(2)
    " `; G$ E# b8 w* ?, \3 * (2 * factorial(1))
    7 P1 m! C9 S, E: W" R3 * (2 * (1 * factorial(0)))
    ! U  C- s: M7 _; a7 w2 v' T# c3 * (2 * (1 * 1)) = 6
    ) B6 x' @& {4 z8 d3 j3 U. K
    + ^- k. P$ Q0 s4 v: B 递归思维要点1 `2 y5 N9 u2 J4 h
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑% {* N5 M8 D3 e* h! I" v" p+ a
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    2 _" N% i2 }2 W& L9 r3. **递推过程**:不断向下分解问题(递)4 m+ r" \4 |# L. }
    4. **回溯过程**:组合子问题结果返回(归). G- B/ u0 U# |0 E$ B: j- E

    ! I/ K! R1 z9 O& f" ] 注意事项' q7 D' {( u6 j/ t
    必须要有终止条件
    % H) e& A! v( G* D" @递归深度过大可能导致栈溢出(Python默认递归深度约1000层)/ \9 O0 \& e2 `& A, q, T
    某些问题用递归更直观(如树遍历),但效率可能不如迭代. z. Q! F# ?  L. Q8 A8 e9 v
    尾递归优化可以提升效率(但Python不支持)7 {$ G/ R( }1 \- f/ Z  O
    ! \/ D# P( a3 x3 O$ r3 V
    递归 vs 迭代& A% r, R# [! L( B
    |          | 递归                          | 迭代               |9 ~) |( u- q- R# _
    |----------|-----------------------------|------------------|- {& [) h9 P4 z. c5 D. E" t
    | 实现方式    | 函数自调用                        | 循环结构            |3 s/ o( y7 D4 `) c6 Z, c2 G/ ]
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |2 q) H6 V0 n1 n$ n$ Y; a
    | 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         |
    3 T, u" ~: Z3 E* l. s| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |6 l9 l/ I- h/ W, x" A: I

    7 n+ _: r2 N% v' P% @3 c+ @) r! F 经典递归应用场景9 }- u- r! _; O, v3 z
    1. 文件系统遍历(目录树结构)) C' F! M/ b3 z4 k  R3 V# S
    2. 快速排序/归并排序算法
    6 w3 ?9 D, u  f& Z" U3. 汉诺塔问题1 m- _! ~8 M" z) x; b
    4. 二叉树遍历(前序/中序/后序)$ W6 ?; q' I- S/ b- k( u$ a+ g: _; R
    5. 生成所有可能的组合(回溯算法)
    5 i1 f; D  E" A9 q6 ~0 W/ w! \# z3 ~
    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
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    3 天前
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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,9 ?( {! I* I' F/ ~& G
    我推理机的核心算法应该是二叉树遍历的变种。
    ) {0 j- x  _4 q/ ]  V+ ^; b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:- U/ e* l2 |, P( q8 H" \1 u
    Key Idea of Recursion9 r+ E7 U, j( a2 {

    + z7 ?9 d. Z) }A recursive function solves a problem by:3 E  W0 V5 Q' |; O3 I: N; D

    ( b3 m- R+ [- |, Y7 k5 a    Breaking the problem into smaller instances of the same problem.6 ^! d. W- o4 X8 o
    0 w) A  v& z! H4 @* U3 y
        Solving the smallest instance directly (base case).
    6 o' R% a- E% g$ k
    4 u0 q7 t; `( H5 y+ {: ^    Combining the results of smaller instances to solve the larger problem.
    7 h* ?  Y" |8 o7 n. `+ ^* P/ G
    6 w- Q! N+ q: h- P: WComponents of a Recursive Function7 ^( B8 G* `2 {8 R! ?9 r

    8 E+ A. R0 X: |( r. J+ ~    Base Case:6 P, Z2 ~% W* q: J, k' z

    ; f- j. h! n: d6 g! B* T4 b: G6 `        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    3 q# w) g8 [3 N0 w4 t: e
    0 A7 @- w/ e6 I$ V7 u4 z        It acts as the stopping condition to prevent infinite recursion.& O1 r& H8 \9 N" h. A( g4 G

    ( @: h  Z; Q0 j/ h        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    # q, G1 P: [! y" t5 ]3 l2 }  D4 Q& i' n3 @/ t
        Recursive Case:' Q- [; W, F# m, f) l% ]1 E

    2 t& l) Q/ ?* U4 I( f        This is where the function calls itself with a smaller or simpler version of the problem.
    5 b3 N2 U( z( p+ N& C$ F- R
    + s, ]  [: @  Q        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., ]) L* t- B( Q1 `- f/ U
    - [" ?! m8 ^/ c) l& x# P
    Example: Factorial Calculation
      e& w* [3 S, e4 v3 g7 S$ _) p# H) t
    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:
    # B* \$ a% v. R! H" N/ w2 T4 t+ z; f' j/ v, K! X: H
        Base case: 0! = 1% r, q; D2 m/ B7 h! k  C2 u6 r6 e
    1 J6 Y9 L0 K3 A+ ~5 ^& Q5 H0 c
        Recursive case: n! = n * (n-1)!$ V8 x3 [% m' z1 U* x

    0 H. Z+ Q+ n3 g' L6 t5 NHere’s how it looks in code (Python):' w6 B) o- ?$ L& D$ r
    python' {5 s" b* W  {) B, A! v, x, J

    8 X: {; P8 L9 H% K! p5 S! h3 Z% d# {( a- E, I
    def factorial(n):
    4 G( u+ u* j. o4 O3 _  c2 j' W    # Base case
    ( B) P$ O0 T2 F$ L: j, h7 O8 P    if n == 0:
    , K4 e+ O* I1 E5 V9 S$ g+ X        return 1
    $ p. f6 H3 X$ ~$ u$ D    # Recursive case9 ~3 a0 y0 w+ |) Y/ T2 p: H* r
        else:5 W1 e" Q8 E* \1 ~% k1 Q$ [
            return n * factorial(n - 1): l0 J5 W; \' e8 ^

    ! Z+ }1 ]: f: O0 f6 l' |# Example usage9 J, }, }6 I% T3 Z9 h9 x
    print(factorial(5))  # Output: 120/ U: W; q7 H% X7 n2 y
    8 i: w5 M1 w+ ~# J4 }- i8 ?6 ]- P
    How Recursion Works+ a, V2 a% s' Y. q& f

    1 V! a2 x1 p5 L  K- W% ~    The function keeps calling itself with smaller inputs until it reaches the base case.4 j3 w0 I' A$ a8 K

    9 I" b9 a% P; R& f2 Z% v# i    Once the base case is reached, the function starts returning values back up the call stack.
    * Y- _3 W: [$ }% _7 m# L8 p& J  S
    1 {: J* K: o! k2 W    These returned values are combined to produce the final result.
    ) H3 v4 ~/ v% x+ d) ^1 J: E9 p' ?$ R+ N* z
    For factorial(5):. N1 R! A2 Y" f7 z

    0 }3 X; }5 v( P# y/ a, i' y! {- a+ B4 I2 c9 V
    factorial(5) = 5 * factorial(4)$ T: e# b7 t7 ~. u0 a' W* F0 }
    factorial(4) = 4 * factorial(3); l, y1 @, ~' {9 ?# ~
    factorial(3) = 3 * factorial(2)
    + e0 V3 N' V8 g  m9 Q8 yfactorial(2) = 2 * factorial(1)
    9 P+ r5 F+ k( j: u4 y$ b, efactorial(1) = 1 * factorial(0)6 R! W! [$ y- j# x7 Q( n
    factorial(0) = 1  # Base case
    : A5 m5 i2 m. a6 l; u
    $ s0 X) j; B+ u1 BThen, the results are combined:
    / U  Q8 J6 k0 b, d  C7 a
    2 W+ m- G4 d. I# A5 V; T# {
    # Y) U- Q* f# I' `6 l! H. G% wfactorial(1) = 1 * 1 = 1
    / j, N! e8 q* D0 x# v6 `factorial(2) = 2 * 1 = 2
    6 g& \0 O# O. j7 W, b; Efactorial(3) = 3 * 2 = 6$ I7 \+ `* e. _1 j
    factorial(4) = 4 * 6 = 24  Y7 |3 S4 e* N, ^8 i, w2 c* N/ D
    factorial(5) = 5 * 24 = 120
    ( q) v9 Q" Y/ ^7 \' w3 o% j
    ( X2 n; L  t6 V$ J& \$ m& eAdvantages of Recursion2 m; |2 j8 u$ z2 e0 ^9 `- S8 M* o
    * X6 `  a, H+ l
        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).
    * f! j# T! m5 Y" t- q; H2 |$ x2 @6 Z0 [! }% o
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    " \0 h6 T( E; Z/ U, c# [" I9 s
    / q4 W+ x4 l, q" y0 R  W: m/ G" L; VDisadvantages of Recursion4 ?3 l- W. }' n% o6 A
    # H9 P4 B4 ]" }3 L9 G: v' Z- m
        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.5 w. v4 ^( g  w( u
    ( |3 m1 h, z8 y; j. W9 B' L2 S% p
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).5 T  R' h2 P( x$ h" w

    " S' J3 h1 v. i- }* q- T" XWhen to Use Recursion
    ) ?$ v1 I, L0 k3 C) i4 Y" {4 t8 j1 d, i
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' g/ g7 Q0 L/ h, T, x$ c/ I

    $ p( U9 S) M; o; K5 s    Problems with a clear base case and recursive case.. K5 _  N+ ?1 d( v

    0 X: H+ j! w$ g# LExample: Fibonacci Sequence2 o* C) o$ b; R# Q6 A- k* I
    7 T% l' f1 @& Y& t- i; l
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    $ \& k. l% n) [/ ?
    ! q& Z0 e/ i3 R$ [( |, |+ K: Q+ [    Base case: fib(0) = 0, fib(1) = 15 c* w3 a- a5 h4 r" a7 V) s
    3 S) B, t) e! M
        Recursive case: fib(n) = fib(n-1) + fib(n-2). n9 i) `  @0 Y5 l+ Q2 X
    0 t7 `9 P, N+ j+ c6 i
    python* C( e8 c4 M) Z0 u2 P& P% A9 K

      z) _! k9 Y7 t, J
    7 e$ p" l6 ]/ R9 j! F" Ddef fibonacci(n):& Q' F0 O/ t5 `+ s# U
        # Base cases
    ) q4 {5 x2 C3 e- i# n& d6 `8 D    if n == 0:
    . i1 z6 \3 @" s# X4 l: K- X        return 0" A5 Q# [% L3 B5 G: P
        elif n == 1:
    ! ]  t5 S  d( x  F0 S/ e3 R        return 1
    0 H1 i* @, f4 q& M0 ^: T1 i% }8 |    # Recursive case0 N1 c8 L% \5 O8 l- n  J
        else:9 W7 N* y1 a$ z+ a" h! B+ i
            return fibonacci(n - 1) + fibonacci(n - 2)
    0 _! }* g. z2 ]5 D: f* h5 {2 V( J; }. E( [, l; O
    # Example usage
    2 m4 J; g0 a( B' V4 b* ^2 D; R4 Yprint(fibonacci(6))  # Output: 8
    # J% _2 I4 D4 U# T9 Y% a! p4 e: W# R. x* i9 {
    Tail Recursion
    - I. L. q9 V/ I# Q3 O0 P/ @' G& X. Z
    6 g3 z/ e$ [6 }; `; b. 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).
    % Z/ W4 w- P  Y- ?  m/ h: O9 [, g/ [/ [6 b
    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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