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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 9 ^' T! A# b# X- J  m# Z( w' t1 p

    1 g/ W4 D2 |! Y" ]) j2 Y. G解释的不错3 `0 ~& ?* V4 q% n0 T; K$ Z
    ' R* f, U5 q9 I4 n
    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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    , w2 Y" J( W  @. P5 f 关键要素( {" W" \# ^) T, X
    1. **基线条件(Base Case)**( l: X1 U* g& O% F
       - 递归终止的条件,防止无限循环
    9 T( x. R+ h) C8 G0 m8 ?   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 12 ?5 g& Q2 l3 ]" ]

    6 |; U$ X/ T' P9 l2. **递归条件(Recursive Case)**9 n  z) J& j/ K3 A; E
       - 将原问题分解为更小的子问题" N# g) U4 d& ^6 V5 L$ \3 i
       - 例如:n! = n × (n-1)!) h# l) [1 |5 t& Y0 r

    : `" |& k. b+ s 经典示例:计算阶乘
    # }, G0 `2 R7 dpython
    6 M, }9 R# |9 b  d' U; i- rdef factorial(n):
    ( p* a, L$ m7 C6 K/ W/ J  m% O+ u    if n == 0:        # 基线条件3 t: N% s. y9 J1 B% j' s- f) K
            return 1
    - W, u, f, Y* V( H1 n    else:             # 递归条件
    ' |) T! k/ [$ W. o/ Y& [6 b        return n * factorial(n-1)
    ; D" {2 B* J& q- I' R* O8 @% ^执行过程(以计算 3! 为例):
    # {7 h2 j& F4 w. |  Lfactorial(3)
    ; d6 S9 {5 n! S) I; M% }3 * factorial(2)
    - S3 }+ ?0 T3 U3 * (2 * factorial(1))5 ^; @+ D1 a3 w$ O" W$ G9 j" Y
    3 * (2 * (1 * factorial(0)))8 g5 L, S' F2 q! s
    3 * (2 * (1 * 1)) = 6/ h" T/ v: B0 J: l
    * d4 g2 `  A- I  J
    递归思维要点4 O4 O" V7 d# T
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    $ ?8 Q5 c& h5 x, V: o0 d' H  o" D& T2. **栈结构**:每次调用都会创建新的栈帧(内存空间), Y# F" f5 |) I1 u+ n; ^8 O
    3. **递推过程**:不断向下分解问题(递)$ I# V0 I4 t0 j
    4. **回溯过程**:组合子问题结果返回(归)
    5 n; Z1 _% e; F) v0 T! O; Q: n+ N7 Q5 R3 g) g. L
    注意事项& g4 A8 V4 U8 y) c$ ~3 ]5 v
    必须要有终止条件& G5 U# S0 S- N9 |+ t% K: {9 {
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    1 w1 z- z  D, K* v; T: |0 D4 G2 m* d某些问题用递归更直观(如树遍历),但效率可能不如迭代! Z/ M% X2 T1 k8 t: n3 l
    尾递归优化可以提升效率(但Python不支持)
    5 [; S  X3 q- E+ j* Q( }; f6 r; d7 {0 \9 }
    递归 vs 迭代
      P! E; H4 F' n; i: U3 q. e9 b: z|          | 递归                          | 迭代               |
    ( B# T3 p9 {* |" M) {|----------|-----------------------------|------------------|( i3 Q2 Z7 @1 i9 _  T1 n- U* S/ X% \- |
    | 实现方式    | 函数自调用                        | 循环结构            |, b# M8 P- \- s$ d) h$ p
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |  y; j( o! t/ 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         |- g$ Z  s1 V- i& o% R
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |* p) D. c! `( H% n; |+ Z* N0 j* `
    3 ^% j; C$ S# c: p, `
    经典递归应用场景
      Q2 G" _  \- I, e1. 文件系统遍历(目录树结构)9 X: C& n1 g& ^6 v9 _
    2. 快速排序/归并排序算法
    + ?) M2 a% |" A3. 汉诺塔问题9 C% b+ R* e) |4 H# z# g
    4. 二叉树遍历(前序/中序/后序)
    * q* @1 x/ s9 I5. 生成所有可能的组合(回溯算法)1 U$ f! L& }% V% ~
    9 N3 m% g: H; O9 ^7 M) d/ g7 K
    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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    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,$ r) i* u1 P/ q1 j( p  \: z
    我推理机的核心算法应该是二叉树遍历的变种。5 r+ i8 M- d: Z. |
    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:
    * P8 E, w! j4 oKey Idea of Recursion
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    * `5 z8 Y: t6 S. t% PA recursive function solves a problem by:6 y2 _; Y0 U8 m* x# Y' s
    3 p$ w! o  P' i# w
        Breaking the problem into smaller instances of the same problem.7 V$ I* o& a6 q) W9 Z! U& Y
    $ m0 w) M1 _  s% c+ n0 g" S
        Solving the smallest instance directly (base case).9 o, A& I9 R6 X* n* c
    ! b* `2 m/ l3 Z) i% @- H6 V
        Combining the results of smaller instances to solve the larger problem.& m! B4 d3 u4 u$ l' F' u

    9 K7 u$ y! v  L4 T% y3 MComponents of a Recursive Function
    7 \% C: [8 Z2 C! _
    0 \, i* I3 Y9 O* F    Base Case:( C4 `! q9 t# X' f9 s
    4 @4 A& U. c' ?# z. d" E
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.# r: u  W8 b* b

    # r0 K7 {* J3 y9 H: U        It acts as the stopping condition to prevent infinite recursion.
    ' K2 `9 w! k& L: {3 P- W+ [. S  {5 m+ J2 G; n* t
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.4 {9 Y5 p6 u& a1 G5 W6 M. A

    + {$ w) d1 v9 c- e8 m& Y; g    Recursive Case:
    ! e9 h! V7 Z. l: w
    2 [' B, Z' j0 R) g/ q5 j3 K        This is where the function calls itself with a smaller or simpler version of the problem.% l# F( Q. W' `0 ]
    . S6 T% Y9 g1 Q: W& ]0 {) x% y0 [
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).. V) k5 ~' Q- J( S+ U
    6 a) C9 F" ~% g# G8 T% ]3 [
    Example: Factorial Calculation+ t& d! \/ w( y! j
    " b" A6 F4 S! ]( g
    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:
    ( F6 O* A& t6 w! `. S' r" W2 \
    * p* [" X7 U/ q: ^, E    Base case: 0! = 1. R' L( G% ?$ l7 h6 B
    + A2 S, Q2 r5 L2 U
        Recursive case: n! = n * (n-1)!6 @( _" t3 }: U0 i

    7 Y# i2 V2 J' Z# B) B: @4 SHere’s how it looks in code (Python):/ G; n. G9 J8 ~' q3 T. N
    python
    , N1 J+ W, N9 j5 b' s: _$ C- |$ G0 r+ N' l% h" O

    ) ^/ J% E/ R$ @5 F& ~4 xdef factorial(n):  Z" D7 l% L; l- v  X9 e: _# {
        # Base case* P3 w* |+ ]4 V" k# {3 ~
        if n == 0:
    / y3 T! x% e0 c* B+ c. q        return 1
    4 K- ?  H- i% s; q; ~    # Recursive case. Q( C" K' `2 X% {( k2 a9 F: N
        else:% O( q2 R( \6 e0 k9 {
            return n * factorial(n - 1)
    : G9 A7 Y* i' ~0 O8 o( I5 ^* d5 A$ u+ j
    # Example usage# _6 f1 e) ~% v5 c& Y, v( h1 H
    print(factorial(5))  # Output: 1204 t( ]; A# _  C$ U
    6 W7 t( M  |- Y2 D. z
    How Recursion Works! d" L4 ^4 Y" C- e
    ' R: H1 f- A4 b1 ]
        The function keeps calling itself with smaller inputs until it reaches the base case.
    & A9 ?: i5 @; A, ?6 ?5 c( n4 _0 V" Y) Y% W1 V' B% r; G
        Once the base case is reached, the function starts returning values back up the call stack.
    # Q. D* q1 }1 A+ a: e6 w. f" o7 K4 c( n1 Q0 a) P1 C. b# `
        These returned values are combined to produce the final result.
    ( d" P9 i, r# Q7 r4 O- U$ w% _# g# U; w2 @  x  q
    For factorial(5):& y& o: l& k; R4 q  q
    % v5 x, t; m; @8 V0 W* r' s/ |
    / k; C# T* y+ J- h0 y% |
    factorial(5) = 5 * factorial(4)
    : p2 ^9 v1 Y! y2 \, B# Ffactorial(4) = 4 * factorial(3)
    0 W& U, o3 D( N" D3 Ofactorial(3) = 3 * factorial(2)
    7 E9 _$ b+ E% L8 [7 ffactorial(2) = 2 * factorial(1)0 }& n; L; ?4 O* y
    factorial(1) = 1 * factorial(0)" @$ m" U4 b& G3 G: b
    factorial(0) = 1  # Base case8 m  O& \' X: D1 H
    0 K0 K; g6 S+ \+ [
    Then, the results are combined:, G1 @0 k0 s5 ~% _
    3 e5 L/ E; W: Q" Q7 `4 B7 A

    9 D( x9 q# l. Qfactorial(1) = 1 * 1 = 11 [* M( l. h  H) h& ^2 i! M/ I
    factorial(2) = 2 * 1 = 2/ a! u: V% f) b) p+ x, k# S
    factorial(3) = 3 * 2 = 6
    $ y1 X  q- N  H  W* q0 S7 p% hfactorial(4) = 4 * 6 = 24
    2 r* f0 s1 Z* N- n) z- t4 Pfactorial(5) = 5 * 24 = 120- y+ F9 _2 E7 @. S
    / c' J' D$ Z2 l8 ^$ Y- Y
    Advantages of Recursion. N6 E: a$ s; A' I( B) W
    . u; V+ Y- Z, ~! D
        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)." K. V$ v' N3 N

    ) u/ c# S/ f& O- f$ K/ F    Readability: Recursive code can be more readable and concise compared to iterative solutions.7 T  N1 _, h+ e# z( p% \$ H9 D0 E" N) x0 Q
    - ]( ?6 [, u6 n
    Disadvantages of Recursion
    + y$ l3 V$ d# W$ M: T4 Q. r
    ( ?9 @0 B) b) a: v8 W    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., [& y2 R8 k6 @" \/ R9 A9 s3 Y

    - p) X, g, Y! o0 o    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    7 `* c7 K$ K5 f% T1 w3 q* j, Z) J5 o4 C# `+ [8 X$ a, @( }" }% y! {
    When to Use Recursion7 B4 R7 B. s# t/ G" N: B

    / u! |% n  L4 n! A- V    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).5 K. q. }2 M0 r) O0 N
    ! t$ V1 T1 h. x2 p
        Problems with a clear base case and recursive case.; F- P$ [+ o  @) l; a+ a; {" r
    8 q+ e1 X1 t3 \4 h5 X
    Example: Fibonacci Sequence
      W' o; l. s2 O( F9 D8 p% n8 u; U6 r3 c- \7 j
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:% I% e# b( V5 u' t7 s. g3 |3 B
    - ^8 j: q! X& t: _  F: n
        Base case: fib(0) = 0, fib(1) = 1: V: H& [4 B" o, J  ]: D
    3 m9 Z3 U9 b& h0 x% b4 i' b
        Recursive case: fib(n) = fib(n-1) + fib(n-2)% t. z- t# |5 q/ H( y

    - r& Y: q, L# X. |4 ]$ z4 Apython
    / T$ e* X* i8 s, M" z) [
    3 k3 X$ a0 S6 b5 W2 y( L  a
    9 M( e  W) R; j" m  X& Ddef fibonacci(n):* T, \8 ^7 }" x6 F8 G
        # Base cases/ N; I9 c+ P6 X+ O4 L* z- k1 p
        if n == 0:) t: ]7 r+ P- Q1 {! j5 ~  i$ g* r
            return 0
    . c+ c$ N: i+ x. M3 u' J    elif n == 1:
    ' _$ R$ ?, e% B7 X) M- l        return 1
    % M) j. K- o3 ~8 F8 M    # Recursive case2 p9 m# l3 [& m, S
        else:
    7 Q1 a+ u9 g# Z. g% k        return fibonacci(n - 1) + fibonacci(n - 2)" M* [' [" y  r, o: x, m" {: c
    ! m/ {" {. N- S7 Y
    # Example usage
    ( o8 F2 B" s0 e6 }print(fibonacci(6))  # Output: 8: P, n! J, O7 d; J( J$ a

    8 D  o8 [2 t- O% cTail Recursion
    - S, C2 N2 z0 u1 j/ l5 z; P( e* }" @
    Tail 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).
    . H# U- _; L2 E6 @, X- z& ~% ~. c( o0 j; N6 {* Y
    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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