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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
     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 * h; `9 r" x# q% X, D
    0 M4 C# d0 j8 l: }
    解释的不错
    5 R/ O3 Z& S. X& P' ^: |; g. t) W# s/ e3 O: V& |$ U* a
    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。# y% W# \/ c4 e" X

    ; {( q/ {3 K) N: x+ l4 g 关键要素
    # v0 m( a  b: o0 S; _& E1. **基线条件(Base Case)**; X: c  d$ g- k4 J: g6 M% `5 q. t
       - 递归终止的条件,防止无限循环
    : @/ N) h. q. V, a* \. d   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    8 N' j' ~7 R5 J
    / S% R2 p; d; n. I+ |2. **递归条件(Recursive Case)**
    - ^  O& P/ R7 t3 A   - 将原问题分解为更小的子问题
    $ V( d' A: k, B8 \8 A) a/ [   - 例如:n! = n × (n-1)!8 q% n% {- h% H
    3 h& E9 B; {: ~8 X4 {
    经典示例:计算阶乘
    0 X, m& E1 P1 M  P: E- V# {' npython
    3 \- I+ p! K# L* S* edef factorial(n):. l/ q9 I! ?1 s
        if n == 0:        # 基线条件
    & |# ^$ i+ W8 _) ?        return 1
    - n% h& h$ M8 O    else:             # 递归条件: h1 h# ~7 m3 V5 V! ?
            return n * factorial(n-1)
    ; J7 M2 `3 d' ~; y3 Y, D6 g执行过程(以计算 3! 为例):
      T) A* E; P$ W% afactorial(3)
    5 d8 U" M* I0 ]& y3 e: P. W3 * factorial(2)
    % t; u/ u8 z0 x: c! I3 * (2 * factorial(1))
    2 w; N) T& s4 y% D& a3 * (2 * (1 * factorial(0)))4 |; T0 N; G* ^+ N0 f) ~8 I
    3 * (2 * (1 * 1)) = 6  E- Y  I# B( i

    . Q2 p+ M7 K( s' C  f" C 递归思维要点0 ^6 b  u" E7 ]# i8 T
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    / A6 B, x+ t+ K9 `5 ^. }2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    7 p; e& e+ W2 h+ t2 L9 e3. **递推过程**:不断向下分解问题(递)1 f' l1 y' o3 R4 x. n
    4. **回溯过程**:组合子问题结果返回(归)0 E3 v7 ?" D# ~  R: s7 K& v

    , Q: r+ w0 z$ L, |4 j' y 注意事项
    : l5 Q/ R4 U' b" l4 `必须要有终止条件
    + P. ~& u' z& T  \6 n) ^5 {递归深度过大可能导致栈溢出(Python默认递归深度约1000层)3 k+ I* C: f1 [/ ]( q9 v) b( r
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    . s/ I0 X" ^2 l尾递归优化可以提升效率(但Python不支持)) Q( m% v$ Z9 z6 _' i
    / I$ H3 p% }# Q* R- B/ r3 C- F
    递归 vs 迭代7 D6 K% j( R( y: R0 @0 f5 s
    |          | 递归                          | 迭代               |
    # _% ^, U$ m8 e$ |4 G|----------|-----------------------------|------------------|
    ) A& i+ ^- p( {& @( w/ j& l| 实现方式    | 函数自调用                        | 循环结构            |7 ]- y1 d' |, F7 P5 K5 j
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    ' r  d# T. Y: I6 p& z' y1 {| 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         |: m8 e! o# b' ]! e
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |; ~1 p, }: J- s+ M& k$ o, `

    ) y; s8 ?# j4 c5 ^+ ?4 d5 c  w 经典递归应用场景
    6 F' V0 a7 R( U# f' m8 K7 d1. 文件系统遍历(目录树结构)
    3 b5 R2 O  X, D! Z7 g. ~  h4 U2. 快速排序/归并排序算法7 `) ^8 l# h4 ?6 U# j; h
    3. 汉诺塔问题# y) Q) g$ X# |/ |! K4 L. X8 x* V
    4. 二叉树遍历(前序/中序/后序). j6 j+ Q% e8 L
    5. 生成所有可能的组合(回溯算法); e# {' G7 V0 E- d. k
    ; [% T8 _. F/ w, @. x$ f' ]- e6 V
    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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    郁闷
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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,$ `* Y4 D$ t2 p0 m6 X
    我推理机的核心算法应该是二叉树遍历的变种。8 c! {6 W4 \8 l" L
    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:; h& ~, R9 J6 g; p, n0 c( Q$ n3 o
    Key Idea of Recursion' ~( r3 Q9 ]% O

    ! i9 |9 M* A- ^3 b. `A recursive function solves a problem by:; l/ x! {" g( n# Y0 Q' L: a2 K3 |: r' b

    8 K8 G6 S8 M$ f# C1 x- y/ m6 V" L; N    Breaking the problem into smaller instances of the same problem.$ B$ _' o8 R7 U3 _( T2 W0 E: s+ v

    ' F( Z$ e/ G& l, C( C" |/ f+ f    Solving the smallest instance directly (base case).
    3 x- d, X/ E' m% w5 A$ F# {) D( k- s/ Z) U' M
        Combining the results of smaller instances to solve the larger problem.
    3 _* _4 c; b4 |5 u- H, q( j  q$ `7 v" v& P
    Components of a Recursive Function5 O' h& u0 L! O9 p/ z2 A: ^; M
    # j2 T2 _# J' n1 t
        Base Case:2 f$ a/ u4 |% I6 _( ^; |; ^2 Q

    5 O+ U' l3 ]% ~        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.# l* f5 t/ E$ e  s2 I  [
    $ I8 R% _* c1 _  Q  r% q) s; D, F
            It acts as the stopping condition to prevent infinite recursion.
    ; s6 [+ U3 G% m/ ^
    3 H6 O. D- {8 F# }6 j  b        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.2 X& H/ u$ U! @# m

    ! v7 i5 H$ q' @  }    Recursive Case:
    0 s, Y1 |& A3 h7 m
    ) A; E. k7 H: K' i2 V' G& K+ [: r1 z        This is where the function calls itself with a smaller or simpler version of the problem.
    . F! w) w  j% b+ Z+ _. _7 Q" t6 y+ R/ A4 H0 R- E, m
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).9 G5 j/ }. Z" {
    4 m' F8 Y/ g9 a! `- C$ u$ ?. v
    Example: Factorial Calculation
    5 b  [4 j1 c; d7 g) o; W! Y; q7 L. {/ i0 ^1 |8 ^! Z* o' D
    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:
    . G$ i0 @. A$ e2 M% s( O; e8 _) Z( F
        Base case: 0! = 1
    7 \! X7 c! D, h3 m8 F. U% z7 e9 ~- S% t2 X0 P: Q9 Z
        Recursive case: n! = n * (n-1)!& {! v5 I8 P6 }7 |$ s
    ) c2 K& \2 N- U$ Y2 |
    Here’s how it looks in code (Python):
    ) x4 _" ^& @0 K$ ipython
    ) v3 `! r: `' u/ M$ S! U/ C% H: `; M8 b  k+ o' b
    ' X- P) V6 F2 n* x
    def factorial(n):
    " p; Y7 _8 Q* w1 Q0 n4 e    # Base case9 i" E$ v+ o0 u! K
        if n == 0:
    & V* S9 L& P. D+ i+ |+ r( b8 }        return 1
    6 F1 `" Q  M. t3 Y6 \; `: p    # Recursive case0 L4 t5 c) s3 v( w. F# Y
        else:2 ~' V2 B: Q9 b/ E
            return n * factorial(n - 1)+ v1 {' x8 {. [3 o  }! L8 L

    0 o% E: k! ]8 R6 k# Example usage
    2 b+ k# p" E5 o, s9 w" t5 i( Fprint(factorial(5))  # Output: 1202 W! }8 M9 i1 ?5 A2 u; _) Y) J$ t5 K
    & I/ n- C$ d6 F: G& u& n
    How Recursion Works
    ) s8 R, H( c9 D: }* v7 \( L" k# K* z& z9 g# a; b
        The function keeps calling itself with smaller inputs until it reaches the base case.) t) j. O8 {! H& q

    , u9 V# {9 H( e7 O    Once the base case is reached, the function starts returning values back up the call stack.% e: H7 i  i" ?# c. M* W

    ; A5 e" g8 V2 F# k: P9 s    These returned values are combined to produce the final result.
    0 f* M# n& W7 D2 s& R; W. u/ i5 R: I5 ^6 {' ~" P! `
    For factorial(5):
    ( k1 L/ M8 C, k
    * s5 Z- G) Q2 D' y& ~6 D+ v) h. m& H$ P! v! g
    factorial(5) = 5 * factorial(4)$ g1 i3 _5 R% b0 O& E
    factorial(4) = 4 * factorial(3)! T3 U4 C( \# G, e+ i: b$ L8 x8 V+ ^8 l
    factorial(3) = 3 * factorial(2)
    3 O( V9 q# n, z. H) [' f7 nfactorial(2) = 2 * factorial(1)
    ! T4 _% l( Y# ?& H, ~& Pfactorial(1) = 1 * factorial(0)# g( ^1 s3 \$ z
    factorial(0) = 1  # Base case  Z) t8 S( H/ u
    1 t5 d( i4 X% R2 F
    Then, the results are combined:4 ^& x- ?: f2 L% M
    ( s) r$ N" c( ~2 P# `

    2 C! m9 z, V8 E( ]/ s: C% Ifactorial(1) = 1 * 1 = 1
    * j0 I' D* C1 O! j" vfactorial(2) = 2 * 1 = 2
    - L8 r; v( K+ H. k$ ~factorial(3) = 3 * 2 = 6# H8 y0 U  D" j4 w6 L" g6 Q: u
    factorial(4) = 4 * 6 = 24
    4 A  Y' ^% o: l5 [1 nfactorial(5) = 5 * 24 = 120
    , G" m0 ?, U( s+ z+ n# y: j9 F* A
    ( b0 ^% V( Q( ~* E0 k' ?Advantages of Recursion  k6 p, O1 m5 S2 R: H* b7 ?

    ! c0 Y7 F/ i& {    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).: z! X7 U2 [1 C
    8 \5 \6 ?7 e! f, I: ]
        Readability: Recursive code can be more readable and concise compared to iterative solutions., s, Y2 F; C6 E: M: R+ h& g# m4 ?
    & J/ m3 u1 W3 _4 {2 z. S9 B# E% P5 I% x
    Disadvantages of Recursion, a# h% |2 v( \  R7 t. @; w

    5 ^0 C$ M  P  f. T  i- u    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.
    8 [8 \2 F' ?6 n% ^& s! n* h
    3 M8 H. ~7 g, V) f3 ?. q    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    / }( a) L2 s  z5 [5 J" R! b9 z/ @2 ?) I8 U9 J* t
    When to Use Recursion9 l3 c- f7 j+ R- f' U$ {+ G' b3 O

    6 C( |/ U9 N5 D! a( O/ j% D) R    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).+ W5 M' @5 Z% e; w
    3 B" t8 w# _( w( @6 |
        Problems with a clear base case and recursive case./ o- u' a" D7 R' s  i& j# p) B

    & W3 I; K7 C# j0 e# B# _* WExample: Fibonacci Sequence, S; b) Q* v2 a; d3 B

    : w  j& Z& _. v2 p$ l0 U4 dThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:- G6 E$ Z3 j* j7 [1 X

    4 L/ G) j) `9 ^) ?5 n    Base case: fib(0) = 0, fib(1) = 1$ \! F6 H4 K, F8 }
    7 z2 H( @' v6 O: O+ w% Y
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    # w' R  I9 f! |5 N& O' u. R, |$ g2 v) h
    python
    - }5 c4 k% t1 Z- I
    4 U) V  j, F' W# ^2 B: U" m4 {
    2 y& Q# G2 m( g1 ]2 p' |8 j$ Xdef fibonacci(n):# d0 |9 A: q* ^3 P; W
        # Base cases
    0 s0 ?2 {" I* D9 `    if n == 0:( {: X2 }7 K) j" O
            return 03 L5 j! f7 B3 G  D5 L/ {- m
        elif n == 1:
    # {0 R, D0 G: @        return 1
    - w) `! c: v+ L+ j( Q0 ^    # Recursive case8 S  [- A- a- o- x3 @
        else:
    4 ]. K( @$ c% m& T        return fibonacci(n - 1) + fibonacci(n - 2)
    5 d4 {! {" U9 m6 ~1 P5 K$ f, k7 m
    # Example usage
    8 N: b( r% ?& [print(fibonacci(6))  # Output: 8; X" t6 F! k  d: ]
    / I- A4 T) I, n7 ?  [
    Tail Recursion; v$ s2 V, [9 P) O6 l" e

    $ |: N" r  u8 U( m+ V4 p; c: _. UTail 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).
    " n6 y9 k6 b  k2 D; k1 A! {1 O. D! v- J2 d6 v% x
    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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