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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 ; t! x+ l8 v/ ^: S1 K% F
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    解释的不错- s  Y6 D6 C3 {9 Y+ W

    ' P" k8 I' _# Q+ i& ^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。$ v9 a: B2 T: p* Y" F& z9 A8 j

    % A) m4 M$ f3 u 关键要素
    + [% u4 ?  _( i  a" R( ^1. **基线条件(Base Case)**
    8 V( _: l$ S, x   - 递归终止的条件,防止无限循环
    / g5 j6 D9 B0 v   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    0 z$ P. o: C  U. _$ J; B% h9 j9 M7 Z; X
    2. **递归条件(Recursive Case)**
    & j  ]8 K  ~9 o( k0 J) r1 H% A' k) j   - 将原问题分解为更小的子问题
      X0 H% ?. z/ {1 H   - 例如:n! = n × (n-1)!: e# u9 t+ W1 {! g( n) c, x) n

    1 }" R( a% ?, ^- ` 经典示例:计算阶乘8 R! T/ X, z$ L3 A# B+ F
    python
    * [* g$ Z( t! o2 sdef factorial(n):
    8 u. O5 Q) W: v# g0 t' ~+ l    if n == 0:        # 基线条件) v4 f# v3 V. E) \( l# z
            return 1* L* f  X1 E- d9 j
        else:             # 递归条件
    5 B3 O+ J5 v4 {1 I2 J5 c3 K) k        return n * factorial(n-1)( t2 G* |0 T. H3 d7 e) l7 F
    执行过程(以计算 3! 为例):
    , C. s( f9 ~- D1 `( o: t( Nfactorial(3)4 q, t  P  m' w
    3 * factorial(2)
    * e$ c* Q3 h$ @0 U; S" F2 L3 * (2 * factorial(1))
    - K6 X8 ~, x7 F; Z& f' o  n3 * (2 * (1 * factorial(0)))( ]  F9 N, \+ s6 G
    3 * (2 * (1 * 1)) = 6
    5 A! L7 i; u4 Q' b/ z% d/ \& p# _2 ]4 Y* j
    递归思维要点& I+ t" W; w, u8 f3 `; N- C
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    ; y3 h0 i& _4 i2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    : R2 @5 L) t  |# u3 C3. **递推过程**:不断向下分解问题(递)( [7 J1 Q+ E- a! S+ L+ ?
    4. **回溯过程**:组合子问题结果返回(归)7 @! `  n$ X& r! w

    ) N. J9 w% W4 J1 g) X5 O- {3 s 注意事项3 B5 |( c6 \5 C; y' Y
    必须要有终止条件8 f6 \: O0 k* U2 z! V1 Y
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( Z, {" ?  V" _& i: ~% O
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    3 p) A1 @) n/ p: \" ?4 H4 H尾递归优化可以提升效率(但Python不支持)
    5 ~5 \; E8 `- e
    + u. q& a; T) R) V3 @9 j 递归 vs 迭代
    / J6 F( J$ S8 X  v# {|          | 递归                          | 迭代               |4 l* ]( j7 Y# P  T5 `
    |----------|-----------------------------|------------------|8 D; Q* \& {& A! s  [
    | 实现方式    | 函数自调用                        | 循环结构            |/ V" h% N# X" M' \
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |6 v5 Y% p& Z2 ]: B; F% w
    | 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* v' I' W8 D  L7 Z, ^| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    1 e' s/ `1 O# }# E) v; _: U! V2 c7 g' p
    经典递归应用场景
    - K, u8 q* h+ [; D8 C1. 文件系统遍历(目录树结构)
    # T! b/ q7 G+ \8 t' @2. 快速排序/归并排序算法
    . ~% F- W9 y) ]" r9 }- v" i: z" e3. 汉诺塔问题5 L9 u& w, D0 H0 f
    4. 二叉树遍历(前序/中序/后序)
    ( x& Z: H" N6 {* Y' s& p5. 生成所有可能的组合(回溯算法)
    # A4 A' K! J; o1 s- H& g$ X9 W, E: U8 e8 ]  D6 F
    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,+ L# H% y. h' h6 W5 }7 w
    我推理机的核心算法应该是二叉树遍历的变种。
    + f' r8 D. m; r$ _" b- D) S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:3 R2 s% P' U; y( Y5 t1 P& R
    Key Idea of Recursion
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    A recursive function solves a problem by:
    0 ?4 }5 {$ e* D7 o9 ?) e' v0 E; m8 _, @* y; X" u. A4 n
        Breaking the problem into smaller instances of the same problem.
    4 i8 L8 i2 k% Q5 a4 F0 p  U3 {9 {" g
        Solving the smallest instance directly (base case)./ y! N! C; q- o! I' [7 N
    / v( X9 c7 a' R6 y2 q; A
        Combining the results of smaller instances to solve the larger problem.
    ! R. K( n, G( c1 O
    5 p7 X. f" n, OComponents of a Recursive Function
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        Base Case:  f/ F  a9 ^2 L& n
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    6 \) |+ u" `4 g& u# G7 e3 [8 n/ h* J$ D3 |
            It acts as the stopping condition to prevent infinite recursion.
    - h( `/ R$ v; Y
    6 o2 X3 q4 [; O2 g% y1 k        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.- Y5 f2 c  h8 d2 f
    * ^) S! A: ^, G9 g$ ^' @
        Recursive Case:
    5 i* B. j( A+ l$ X& g) g* j( p3 h5 |9 h2 v5 Z2 ?! S
            This is where the function calls itself with a smaller or simpler version of the problem.
    6 q" O' t- _! n. f. q! U  ~. G; Z# ^' g6 N3 {+ p! }% r" v1 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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    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:; M$ m- _6 ]! w6 l! |
    8 J) U" V9 h( ]$ O; E
        Base case: 0! = 1
    7 p- a1 _/ ~( K4 W
    5 ~- C* p$ c5 F" M    Recursive case: n! = n * (n-1)!
    4 C( w2 F5 ?+ R! a  A# z
    " ^1 O" T; g0 n$ j; I0 a% uHere’s how it looks in code (Python):2 h' `. v- a+ N1 ^! w+ R
    python
      _4 i9 p! X: j2 M: \1 U* Q% w1 [* D

    # G5 [( B' d/ C$ G5 j( X5 _def factorial(n):9 U1 x) t# Q( D% p1 a  l" X+ Y" b
        # Base case
    2 S& D# T0 f& t* B; @6 D    if n == 0:
    7 B1 y8 V( x! W/ z! s' i( z( V        return 1
    7 r+ f! n& |8 D, Z    # Recursive case
    $ _/ [0 h2 C' e, K- |5 O7 x    else:4 F- ]8 b& i* ~( M* |2 a0 m
            return n * factorial(n - 1)4 A1 t. K; N# ^5 G7 u7 r- o
    0 K6 y$ r. b& s* h& V
    # Example usage
    ; g- o1 e/ i+ l! g+ H. c! @& Iprint(factorial(5))  # Output: 120' M3 X- V( Z5 ^; K2 m
    , a0 Z" N) n. {+ p) _% \
    How Recursion Works7 g  V) s( g( R' v5 k

    ) i* ]0 z, V8 _$ |  r    The function keeps calling itself with smaller inputs until it reaches the base case.6 W0 @2 W' e. f4 f# d8 V( F6 }+ z

    - y( ?: e& X  G0 y* D# J, l4 V7 \    Once the base case is reached, the function starts returning values back up the call stack.$ |7 c/ y+ q$ g% ^- |
      N# u) B% h* p& [  o( W
        These returned values are combined to produce the final result., L- Q6 c/ {5 {- W0 [1 L4 ^+ j
    0 r0 I3 u1 m& _4 L
    For factorial(5):
    $ A+ a: S7 C! b6 _6 U$ W$ C
    / h7 R2 @$ @; H& w: r# S0 F8 r) k5 y9 x% u1 i- Z+ P0 S4 j  C
    factorial(5) = 5 * factorial(4)
    3 ?3 p, W) o. L. Q& }factorial(4) = 4 * factorial(3)  j, M4 t) l3 [) d
    factorial(3) = 3 * factorial(2)
    : ^+ h! }8 ^, [  J7 Zfactorial(2) = 2 * factorial(1)
    , m- t9 t1 y" J; @. Jfactorial(1) = 1 * factorial(0)
    * O8 Y; Q4 Z3 U1 }4 ^2 @9 ~factorial(0) = 1  # Base case6 I$ s( V, _. ~
    5 {6 @+ q# L; |) R/ [9 H4 ^
    Then, the results are combined:
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    9 ?! V  L8 q/ [+ z
    factorial(1) = 1 * 1 = 10 Y7 c, p* @* ?1 H
    factorial(2) = 2 * 1 = 2) i+ K# X( z8 ^/ w) [- L
    factorial(3) = 3 * 2 = 60 z$ Y4 x" o' _2 k# m( \
    factorial(4) = 4 * 6 = 24
      P# n! P. o: i9 kfactorial(5) = 5 * 24 = 120* M" L. Z! t- |$ O

    ) K& T" k" n: C/ o3 vAdvantages of Recursion
    2 k. S, N+ `- Z0 M+ ]- [) a6 z: p2 @) K( q1 t5 J
        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).! w5 D: m; P/ }. o

    ) J2 b  n8 N+ X. h: A4 a" V    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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    Disadvantages of Recursion
    ( Q4 A5 {6 D3 p  p( v; j1 x0 S1 S! f
    , j0 _+ h5 Q; a: b: `    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.
    1 I% Z+ K: S! I4 X
    . G( f, O2 `$ E  d8 }8 y    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).. m# h8 h( I+ C2 |  l

    8 v: w6 T8 s4 p6 Y3 t, DWhen to Use Recursion
    * T3 U4 D. E* [* D. U
    - u' l2 ]$ O' I! i    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).' E6 Q) e' z1 ?5 m9 X

    ; v  U5 G$ Y0 C7 Y" v    Problems with a clear base case and recursive case.4 M" i9 [! C! G% U# Y

    : O, o" Q6 @9 m) D+ PExample: Fibonacci Sequence9 h1 V0 w. W3 c2 \
    " L) F8 r( b+ I% u9 c
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:; x5 l4 b" ~6 p+ k7 X
    # m/ Y; ]! T7 c3 n: w- I2 r9 k
        Base case: fib(0) = 0, fib(1) = 1
    % A* v1 @; [- R# D1 c% \4 w; k- o$ \4 @4 M8 U
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # h0 D2 W3 k. L) x; A: o% U  Z
      E. s7 C" `2 Kpython
    2 @  Y. A* \+ |! \- h$ ^& H! {( \5 D3 S6 T5 z  `8 W
    6 ?3 {) {$ M8 a  N3 e- r
    def fibonacci(n):
      R* d% `: t8 v0 \# w  X    # Base cases" I$ X' A" Y# O: E! O3 C% {0 ]5 j- `( w
        if n == 0:' d) e3 p( m. X: H
            return 0
    3 ]5 p& v3 s7 \* C4 P1 n; C    elif n == 1:$ |  A/ t% K8 P5 z# F
            return 15 {$ F6 C& w4 z# z$ M% Y; t
        # Recursive case) Q8 H5 d* y; I# A
        else:
    # {# s- `9 J/ M; f6 t        return fibonacci(n - 1) + fibonacci(n - 2)) |7 {  o. C& U% K/ p" [. t- |
    " X0 T! X0 f5 }8 p' b: p! r" v
    # Example usage
    ' t8 ?. m& C! @! Gprint(fibonacci(6))  # Output: 8' @. Q# B* I" c
    5 ?# k, \0 s) X! w1 q- F0 u
    Tail Recursion4 C1 G2 J$ T) R' E, h
    " ?' ~3 Z+ F( K# L' `
    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).
    0 d/ O  e) d0 t; A' R; h' k! e, ~0 D8 P4 i! _9 F1 `1 V9 V2 A
    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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