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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
      z/ [! E# r1 z. m
    / A' X$ r- P+ a; U6 U7 v% I解释的不错4 |& F: ~( F; Y+ x' ]1 [
    - b4 w" g2 u3 v8 _
    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。# n7 U9 ]! {/ i

    8 V# o3 E; N' e- ` 关键要素
    * T) y2 R( D' r! g' d1. **基线条件(Base Case)**6 M3 G( i+ r2 r, w4 G& l
       - 递归终止的条件,防止无限循环
    ( u3 W% f- Q; k9 @  h   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: d7 K) p5 p* i- p5 d8 k

    - k6 Z8 \( G7 z4 ?9 u2. **递归条件(Recursive Case)**/ Y. q: V$ U" p
       - 将原问题分解为更小的子问题
    4 u1 }+ ~7 h0 h$ y# Q   - 例如:n! = n × (n-1)!
    " }6 u2 E5 J: n, ?. p- J8 q3 x
    ' s2 r4 c& u7 @; ?8 K% O) ^ 经典示例:计算阶乘9 |4 Q( j7 O. B
    python
    6 T, ^5 m5 W' L$ ~* i0 y1 xdef factorial(n):: H- d& P6 E5 R; \2 @
        if n == 0:        # 基线条件9 b& x, v! i1 F' P
            return 1
    & \6 w" Z1 _5 u    else:             # 递归条件+ [' y' S' e) i% s
            return n * factorial(n-1)+ {, d; A( @4 K1 R" Z# a4 C2 \' M
    执行过程(以计算 3! 为例):9 v5 @1 k; d9 U4 Y7 j2 q8 ^0 q1 c
    factorial(3)
    7 j! H/ Z' H" T) }; r3 * factorial(2)
    0 |5 Q- X8 l# d4 z6 X/ ?+ a3 * (2 * factorial(1))4 X" r6 y3 X; l' ?4 Z% q! S
    3 * (2 * (1 * factorial(0)))0 A3 O# ]* c5 v; s/ S/ @" y. u
    3 * (2 * (1 * 1)) = 6% J: u, }% T" T. _7 ^

    9 ?2 o" B5 {  u% K" ~ 递归思维要点
    1 q( `- M+ i. s1. **信任递归**:假设子问题已经解决,专注当前层逻辑; ]8 H: x7 W) W  R
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)7 ~9 f2 f: N. B- Z' A
    3. **递推过程**:不断向下分解问题(递)
    4 A* N6 u/ }+ s; ^0 c/ c) W4. **回溯过程**:组合子问题结果返回(归), S3 K  Q/ g' m& F: ?# v. m

    / R, u! U/ G3 L. x6 W$ c5 W 注意事项$ n. l( a& S( {, D' ?6 ^
    必须要有终止条件- L1 S, R3 D7 M# \9 m5 |+ s& S( v
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)' ^1 ?4 O0 ^) L+ O  u3 N
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ! k% t% j/ H5 W( d/ v3 h6 \+ m尾递归优化可以提升效率(但Python不支持)6 d$ D( p9 s7 U4 t
    : F' R: p# ?5 b. M
    递归 vs 迭代# v" ^% p4 \8 ?- o# p# u. I
    |          | 递归                          | 迭代               |
    " ?9 B% g9 }, M2 x. \|----------|-----------------------------|------------------|
    ' H3 g& k5 J% i0 G- m| 实现方式    | 函数自调用                        | 循环结构            |; y( j* i, y) d! m# J( Q
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |  d1 t+ J6 B& w  r2 d
    | 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         |4 z, G" B+ U- R2 M. j
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ; \- P7 {6 s- {% w% e; s$ q+ H4 v. q* N# G  t$ h
    经典递归应用场景
    7 g7 l) E5 J# T* i5 Z* `1. 文件系统遍历(目录树结构)7 S* C& E& E6 K
    2. 快速排序/归并排序算法
    # o6 e, o; z) O: m; H5 ]* y3. 汉诺塔问题
    ! X) H" y) Y1 w( X$ {  n; j4. 二叉树遍历(前序/中序/后序)
      i* \4 ?7 k" [/ x8 c5. 生成所有可能的组合(回溯算法)
    1 k+ C! i: @: M' b* Z# h' @6 V% {! {/ {( n  Q2 Y2 n5 r. Q; X: T
    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,
    " q( @2 x# Y5 ?! W: F, P我推理机的核心算法应该是二叉树遍历的变种。
    8 ~* T2 j& z! P. X' O8 O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:; k4 p* _( C  G. M. s1 S
    Key Idea of Recursion
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    * X6 v' L/ ?- \2 ~" G: `A recursive function solves a problem by:# B( ^/ q3 G4 d3 ]  A

    * X% U+ e9 e" C; z1 u    Breaking the problem into smaller instances of the same problem.
    % v; u6 Y0 e* B) w1 N
    - }, ?% J% }7 H7 Y3 C" @    Solving the smallest instance directly (base case).
    6 D6 l# k8 [! l" H; _
    / y4 n8 |) g; q, I0 a5 }    Combining the results of smaller instances to solve the larger problem.; h  B( A/ E% Z7 I3 Z) P; f
    ) q+ L+ l( w% s: g% W
    Components of a Recursive Function( F; r6 {& m' \7 K/ I7 n
    # W  C) ^" R* p5 T* I3 m
        Base Case:( Y, ?8 W( ?+ F* c# t6 V3 f1 l

      v; D+ ~0 T8 V        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.9 U- o- H( b) d% l) v
    , f+ A" D9 G4 M
            It acts as the stopping condition to prevent infinite recursion.
    6 Q* R) r2 D- s& I1 B5 N  R1 f4 ~) @% s- z/ O0 l2 D; d8 \
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    0 q: x4 U' i! u5 P; @- I6 r2 T) W8 ?1 d
        Recursive Case:  n( m; _$ \# Y

    1 `7 k( D/ X1 w+ S( \        This is where the function calls itself with a smaller or simpler version of the problem.9 Q  O; Y  }& |) @" N
    $ K& {7 w9 b3 t9 N
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).4 O. h; |9 t. T

      |1 [  E! S7 F3 v  WExample: Factorial Calculation& J, H" ?+ D: L0 _/ R% b* X; S5 y; k2 U$ L

    - K% l( w" _7 W) p2 t- a  E8 c( {5 yThe 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:+ p7 ~2 M) j; i9 i, G! `: V2 D

    " Q+ w& b( _: j& P2 E( l, X3 |    Base case: 0! = 1  e& ^% v! @! U) A2 v

    9 r8 C% i/ f& S* p, N    Recursive case: n! = n * (n-1)!
    ' t. g" M) Z8 G1 s# g- J' Y# M1 V$ b/ t+ D% Y
    Here’s how it looks in code (Python):
    6 o" o% n4 V: Q6 F% U0 Rpython+ b5 T, e0 t2 }4 g2 k/ S6 t

    $ r# o- [" b; u) X* L( E& g' E% H% R6 U. B( n; U4 b
    def factorial(n):8 p6 P$ m4 Y1 E
        # Base case- \4 t3 S; y5 s# [  k% P. |8 K
        if n == 0:. N' X$ V4 N4 H5 P, p1 }1 ]
            return 1
    1 J2 n9 ~% L! |5 ~7 G' G+ K    # Recursive case
    3 ~6 F$ b0 p( P  ~9 Q# x2 m) f    else:
    ' h$ c5 q6 h( ^. X+ x. x        return n * factorial(n - 1)
    4 P0 m% y( g- ^7 T' a1 D
    , X+ Z$ R* a& F# Example usage
    & |1 ]# s7 m5 ^* v; sprint(factorial(5))  # Output: 120
    3 P: ~  Z( a3 p' Y  m+ [! K% V
    , m6 Y7 b  {0 Q0 e1 b# I9 CHow Recursion Works9 M1 o" N# e0 A4 p6 k

    % J" }3 D$ v2 O7 Y- L1 A! Y7 i9 M    The function keeps calling itself with smaller inputs until it reaches the base case.4 @; C( E" t. Q' v0 V/ N
    " Z- C1 [; `% \. r1 A% j
        Once the base case is reached, the function starts returning values back up the call stack.7 E4 l2 O! p% K' ?# s/ |" X/ ]/ Y
    * d5 y( d1 x' n/ p; k0 F
        These returned values are combined to produce the final result.0 e3 c  k0 O9 }& K6 N1 H) [4 h: w

    " V) d, y* K+ I. SFor factorial(5):
    / u- [- {3 d1 g, U) v% _% W9 A* T+ |) I" }$ h
    3 t7 P# ~& k# K/ B3 X
    factorial(5) = 5 * factorial(4)
    ! R  }: w+ J: X  T  Kfactorial(4) = 4 * factorial(3)
    9 F2 Z1 }5 @: w5 T9 j/ t/ Gfactorial(3) = 3 * factorial(2)& C' k; u, ]- H" c5 c# C
    factorial(2) = 2 * factorial(1)2 H, }- _2 s* v  f  H1 ^, B3 A% ?
    factorial(1) = 1 * factorial(0)
    7 V; o9 G9 s6 g7 [, @  {* z; qfactorial(0) = 1  # Base case
    . S) E0 N, X2 `2 G: Z
    * e# v- H" i" J! [! X) QThen, the results are combined:3 a% K8 X: n4 ?; g: |+ M, ]
    7 j' P5 u2 q  ^6 U
    3 z0 p$ x- c  R/ u% r; g' j
    factorial(1) = 1 * 1 = 1+ R) K" y; n  e# C
    factorial(2) = 2 * 1 = 2" C8 }2 o! i- w9 j
    factorial(3) = 3 * 2 = 6
    & m) }% M' C8 xfactorial(4) = 4 * 6 = 242 i4 o. L& I/ q  H9 V. _4 k# i' t! g
    factorial(5) = 5 * 24 = 120
    : J: q% ^: {* v2 ~8 T% B$ h' B- s9 T/ P5 y
    Advantages of Recursion
    ( g5 a/ @3 N9 Q3 E$ w' p/ T; m8 F# \9 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).
      u. o6 m+ v3 A7 d& ~! ]$ W; [# u5 F6 B$ H  T$ C: Q! o! b- R
        Readability: Recursive code can be more readable and concise compared to iterative solutions./ H- ]+ S& O( K) P7 A$ f/ J  Q
    % c- n( I' g: q- L3 \9 H0 n! s& N
    Disadvantages of Recursion  i+ c; z' s( J7 B7 P7 G3 u. y  W

    ) Y( `/ B5 p1 u0 m# I. [    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.- M) c/ _5 n$ [; G8 b! @" f

    # i. q( c9 l8 ^$ E2 N  D5 D    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 B) V! O+ W, F' f8 B9 I, i3 ^0 c8 H$ g) U7 Z" M  ~
    When to Use Recursion$ S8 y- P5 y( R) z6 u! o
    ) Y; ~  K) Z1 y
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    % c0 q- z( a6 d5 D& e% e* c
    3 E; b" f% x2 u" H: y# m    Problems with a clear base case and recursive case.
    ; v1 p1 L' @8 x# Y3 r
    6 o* ~( i2 v/ `: O0 j2 lExample: Fibonacci Sequence
    % ?( G+ g0 `/ w7 T; J7 J; I( W4 G1 h: P" X: Z0 M
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:$ m" L% k& w# x1 E. }0 ?) R
    $ u9 F+ b. ^# S
        Base case: fib(0) = 0, fib(1) = 1) K" j$ ?2 G1 e: G$ `

    % W4 p  B& [# i; b    Recursive case: fib(n) = fib(n-1) + fib(n-2)2 n, d2 E; {) d! g. n
    ' K5 q9 e/ E! Y: _. B) l9 C
    python
    4 U0 y& N* ]2 m! Y. `' t1 \/ s6 S( d( X, y2 V8 o8 d

    . m! V8 w' e$ V+ L7 z2 H2 \def fibonacci(n):
    - |$ U" W0 y8 A+ y( d    # Base cases
    2 I2 E# V+ ]1 `. i& |7 ?' Q    if n == 0:
    ; m9 V2 E4 q# i1 ~- u3 i/ j        return 0
    3 y3 a& b# }0 A, K    elif n == 1:5 X6 \; y+ S9 S; U* I* T
            return 1
    ; |( O) ]3 K% D0 D5 I3 P4 M$ \2 o    # Recursive case
    + i' |& J3 E6 z; `4 j8 q    else:
    2 B# p  Y+ b3 X( A2 P8 V        return fibonacci(n - 1) + fibonacci(n - 2)
    7 J: [. T7 F! m8 J2 I* ?3 K+ c* Q$ G/ ?/ a' Z% O# @
    # Example usage
    : W9 ^7 f* R1 ]7 y, Q. U. l, p, s$ }print(fibonacci(6))  # Output: 81 t0 h. u- S  B. ^& E) ^" A
    ) i- ]3 g  @" Z3 n2 g% a1 P
    Tail Recursion8 s1 X( q6 `- y* f: r8 L' O7 n8 _* S

    ; S# L' C  V" |1 ?7 `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).+ L; t) f7 M4 r  U
      P0 d/ f% s! [; J" A2 {
    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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