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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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    2025-9-8 05:08
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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
    ! C  m0 Y6 J7 f, S. ?  B! t4 c
    ) \( l+ K$ T& t  a解释的不错
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    9 O: B$ O7 o! K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。
    1 Z; B, a) c# F2 t
    3 s4 V, S8 p5 F7 U6 o% L4 l; V/ R 关键要素
    , v  x* ?0 O3 u. X, b6 i* h1. **基线条件(Base Case)**
    2 k8 L; z  k% S' r3 X" Q/ n   - 递归终止的条件,防止无限循环, J7 c# S+ X2 p. G% Q
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ( g& `! Q2 ?/ b2 t' ^4 A6 O$ }: i4 c" f% g" s2 X: G0 k
    2. **递归条件(Recursive Case)**9 _/ ^2 O  U2 ^( l
       - 将原问题分解为更小的子问题
    5 y1 B, l8 q8 B! D8 j   - 例如:n! = n × (n-1)!
    9 t6 _/ E# r. T8 r: v) ^3 z: [$ Y! a, `% j
    经典示例:计算阶乘6 n! c# r# P7 Q6 E# G0 p
    python
    ) e! n, g' Z7 s; Q3 }$ z* Xdef factorial(n):* _6 _, B+ V# [7 W8 H* B3 R9 n
        if n == 0:        # 基线条件2 L* ~8 W! P; l5 M
            return 18 ^4 I' ~2 T7 ?2 K
        else:             # 递归条件' u( D* }' _: m$ v- L: V
            return n * factorial(n-1), H4 M0 C% |3 V
    执行过程(以计算 3! 为例):
    0 J1 A! F" c) n7 u1 q) cfactorial(3)
    " a2 B' D0 V/ t* H' k+ m/ _3 * factorial(2)6 ]& ], N5 N( ~  F1 b
    3 * (2 * factorial(1))+ p, Z8 T. n; [$ m/ A5 a/ M
    3 * (2 * (1 * factorial(0)))1 u/ s* |" o) i0 \
    3 * (2 * (1 * 1)) = 6
    0 n7 e- O; i5 f- R. g
    7 r, X4 J9 J2 y  I% N2 \; B; O8 \! S 递归思维要点! `# j* l1 x1 u
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑' D" J6 T4 H+ k* _# v) A
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    $ @+ a0 `5 Z8 M3. **递推过程**:不断向下分解问题(递)/ y- `! n2 J$ P, n2 d6 c
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项: {( a2 i) t" S6 Q1 L( d* U
    必须要有终止条件
    " _0 p4 A7 N8 s递归深度过大可能导致栈溢出(Python默认递归深度约1000层)7 C& v# o2 A4 a5 G- |
    某些问题用递归更直观(如树遍历),但效率可能不如迭代2 j1 U% ~! H. X4 C" v- l- d
    尾递归优化可以提升效率(但Python不支持)
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    递归 vs 迭代1 t: S1 }( \9 p8 _
    |          | 递归                          | 迭代               |( \1 f  a  J( F, o
    |----------|-----------------------------|------------------|
    6 d2 ^8 D: v5 _1 O| 实现方式    | 函数自调用                        | 循环结构            |
    7 K! I9 v. Q/ P# S$ e| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ' T% Z: }  S# Y) ^) S| 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         |
    ! S$ H  X$ ~1 F& r9 ]7 ?| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |# p) w" F, }4 E4 O
    5 W7 d2 V* j+ X) u
    经典递归应用场景! v9 h' D: k8 i8 d6 T% W8 o' N
    1. 文件系统遍历(目录树结构)
    ) ?" |, n3 z8 H, m7 u- l2. 快速排序/归并排序算法/ X- [/ Q( C( {9 a
    3. 汉诺塔问题
    ! \- w' ~: S! [" a+ {# u' e4. 二叉树遍历(前序/中序/后序)7 \6 P1 N2 E( s- m8 Q2 P
    5. 生成所有可能的组合(回溯算法)
    ( l0 A- T5 k$ I( c+ o* j' Q# n7 a) k0 U( K& A+ H3 m
    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,
    $ s) d/ P5 H& d8 @5 X, q我推理机的核心算法应该是二叉树遍历的变种。& P3 p& k/ x* U; C/ P
    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:
    % v* a; y+ n& C/ b6 c( F+ w# \5 fKey Idea of Recursion
    ; K0 G" M4 |  V7 n/ q% }4 m5 f3 W
    A recursive function solves a problem by:
    5 H# p0 i2 P+ o! m5 C  w9 s. x
    % G% T# ?3 m3 g2 U* {0 a: ~6 f! A0 w/ W    Breaking the problem into smaller instances of the same problem.
    ) k( i9 P# g  b- i4 w( d+ t0 g/ i+ x0 j. n3 l. m9 m: o; {. Z
        Solving the smallest instance directly (base case).% F2 ~& V: [5 x" l' B# k

    3 D% @7 L3 \9 A1 H7 I9 t# v    Combining the results of smaller instances to solve the larger problem.: L, w( p3 B' G$ g* W
    4 u% t9 r5 K5 O4 _; t8 c) _
    Components of a Recursive Function2 f& G3 \' J' A! O5 T. r
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        Base Case:+ D& }. d4 s. Z& M. X0 J" M

    . v+ \3 Z  V# H: ]        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.1 p' i# v! v; y( X/ n5 }

    * g: N/ F3 q$ n$ q/ P        It acts as the stopping condition to prevent infinite recursion.
    ; L5 z, H$ E: Y, `5 R4 d% B
    & U. a1 x9 ?: t1 ]3 V3 F- v4 h        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.' U0 b/ E5 |! G1 p

    , U/ p, l. T) D  J% g, ~, ?8 N    Recursive Case:( l" D6 Y+ B& x* y2 {0 W0 i
    3 L1 p' `- U# s' o5 {2 q" T
            This is where the function calls itself with a smaller or simpler version of the problem.
    * W; ~+ B/ t! b, t9 y5 u* q1 I8 g+ v2 j; h
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    1 T' S$ [6 R+ {7 {1 Q% r& W' s0 t1 V) x+ @. h4 x3 w* {, q
    Example: Factorial Calculation9 Y3 K9 q8 h. e/ k6 u# T5 p) k
    ! Q) [4 q( t- I! m. B, [
    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 O8 e! m" F& ~" V5 c* R: ?

    ; ~9 B3 V- S. T4 ~" M6 G    Base case: 0! = 1: O: @; ^2 U) o4 g8 t% v$ i
    # n( k5 }' n6 E$ Y, |1 g! {
        Recursive case: n! = n * (n-1)!3 O( m- m2 \' _$ J5 m- D( H* m
    1 Z/ p( ?8 c$ f" W: ^* E; P
    Here’s how it looks in code (Python):; W$ Y. u, i3 I
    python( d" u6 y5 H1 R0 u" F4 a
    % s$ V/ I8 A  X# N3 b1 W1 e
    & z& }9 R* W3 o& {' P: s/ s& x/ t) H
    def factorial(n):$ `- p$ x1 p* @( N* L1 }& b
        # Base case% z* Y* C' s/ B: i1 r2 v2 R
        if n == 0:
    , L- ?* ^  o0 x1 f# C        return 1
    " R/ b( t6 R5 w    # Recursive case: \9 l/ Z: O7 C$ F* H) e8 i4 l. T& G
        else:
    + u6 `# k* n; R$ \5 s  D/ y5 S$ h        return n * factorial(n - 1)0 v/ T) o7 B  ?/ @: C- `

    : M9 E" o. }+ {& F. P8 q/ d# Example usage
    * k+ E1 Y  r. l$ Nprint(factorial(5))  # Output: 120
    ' h% V3 Q0 ^! s0 k5 T. j8 g' b& ^1 r
    How Recursion Works# _+ O* @7 |5 t' ]. J" ~

    9 _3 s6 f( y" y7 j$ b    The function keeps calling itself with smaller inputs until it reaches the base case.
    1 p) h& ~6 X+ ?
    ! w2 [9 h- \" c& C# k: ]" c  H    Once the base case is reached, the function starts returning values back up the call stack.
    4 ~, ~& {( O" ?
    ( R1 f3 w' ?5 F5 ?3 }    These returned values are combined to produce the final result.* I' W; g- T: J, Q+ r8 V, t
    3 N- G* [- T  c
    For factorial(5):" V, C- [  A" Y- l8 t0 u

    ! E& F- O9 i$ U1 f! w6 ^& D/ |9 X3 W) z2 }* ~( l
    factorial(5) = 5 * factorial(4)+ z% }6 H, W0 e+ K( z9 r" J
    factorial(4) = 4 * factorial(3)
    2 }# L5 T3 I. g& c) E1 zfactorial(3) = 3 * factorial(2)
    # {; s6 m% z7 m" m0 ]2 |factorial(2) = 2 * factorial(1)
    + L4 R$ c& l. p8 n9 qfactorial(1) = 1 * factorial(0)6 m* M+ J: M. }
    factorial(0) = 1  # Base case. Z9 Z  J) B7 z2 k- g' a$ M

    7 d8 l( W1 B3 x+ G( M$ bThen, the results are combined:
    ! D% s+ [9 X: L( u7 e9 b/ f9 @6 P; d
    - t: q2 d0 b8 F  [9 n) h& T8 Y
    factorial(1) = 1 * 1 = 1% V( g- @2 g, J+ G
    factorial(2) = 2 * 1 = 2/ l! I! V$ N4 ?' _8 D2 b" ]
    factorial(3) = 3 * 2 = 6
    1 l# e9 x7 y' Bfactorial(4) = 4 * 6 = 24+ h1 Y/ p* ^$ K- x! i2 }- w' \- A
    factorial(5) = 5 * 24 = 120
    ) ]* a) `/ Z# J$ b8 p; e# b- ]# K* b( l& }4 o+ W# Q
    Advantages of Recursion: h) X" I7 ^7 |. |3 `8 W
    8 l" K4 d! s/ P5 z8 z
        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).
    " B8 x* R* J, H
    ( a/ F  b  r/ f    Readability: Recursive code can be more readable and concise compared to iterative solutions.. d) p. Y: ~; D

    0 H" f$ x: [& ]2 R, qDisadvantages of Recursion6 X+ u1 r( N* _! ^. ^4 B3 u' T
      D1 {+ R1 f# e% G( |
        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.4 z2 ]$ O, B/ ^* A3 O- t: h- L9 p! O; K
    : n5 ^) N) C: m5 \2 `
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    8 \' D# L0 C, O6 r4 S! Z) U- U0 t4 I
    When to Use Recursion
    : o4 g( A) G& k4 E2 @6 }8 X# z% A, ~- N9 O: X/ E: d
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).. y6 q/ Z! x- r2 G- q% n

    0 B2 K8 h1 l9 H5 ]8 A    Problems with a clear base case and recursive case.
    6 p8 ]- k# s, ^, _' z1 ^' P$ {7 \% U* V* W  @
    Example: Fibonacci Sequence2 E" T1 A/ S0 C( \2 i; F

    $ W% i4 B! U1 Q# }7 a3 eThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:. ^. Y- y& R8 `
    7 p) X) {5 a1 Z) p5 n  v
        Base case: fib(0) = 0, fib(1) = 1
    ! ~3 B# O6 \& G9 C3 S
    3 M: j/ }3 C. a  u% q0 V    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    2 s3 Z; j7 B! f3 M" p4 t9 a+ ~; M$ F; A' r1 @, w# G9 h
    python
    - x. _/ w5 |& _  W- `( i# s9 I) y- U5 Z1 `& D

      c, d) c7 h$ i7 C7 qdef fibonacci(n):. P+ W; d) W0 f5 \/ _
        # Base cases6 \' T9 C$ [' Q8 R
        if n == 0:
    ! d4 {  D- Z% y* r, R: E  T: E7 m        return 0$ ?  r% T1 g  E6 W) O! u
        elif n == 1:$ B( I: v4 d  G+ R0 E
            return 1
    ; \9 C- z) s, B- x! p. l    # Recursive case+ a/ Q$ [) t& @0 y7 v, _$ U
        else:( i9 h  C4 k' Q& m( z# d
            return fibonacci(n - 1) + fibonacci(n - 2)
    6 z6 v% O0 w  |+ y
    & N# ^6 d6 a8 o# f8 Q& \# Example usage! D: i3 E6 e' a8 \8 a
    print(fibonacci(6))  # Output: 83 X' b1 r/ O, b5 V) ]) O; j+ Y

    : \; p( a8 l2 f( |Tail Recursion
    - ~; |) ]% U% C4 n+ `0 I0 B- c- t8 N
    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).; z  V! p; l/ ~
    : h* @5 P! h: k1 Q* v0 A/ X) j
    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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