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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
    4 b3 A! y, @: ?$ y3 h% m* K& H! P8 B2 U# |) V4 b
    解释的不错! u5 H8 I6 n- [
    ! m  a) V" b% c
    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。: f2 i) Z; t7 s2 D

    6 L# y# R8 X6 |7 k# E 关键要素
    + U5 F7 ]. Q$ b; ^; f$ R7 u1. **基线条件(Base Case)**# p- ^3 C$ r- r% T7 q3 V3 ^) c1 D
       - 递归终止的条件,防止无限循环) s$ a, v$ a' x! y! D& N  y+ i
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    & ^1 i% s' Z; Y0 A7 y4 [# n+ \' n9 _+ W, g4 J- H
    2. **递归条件(Recursive Case)**
    5 z  H/ b4 B4 X   - 将原问题分解为更小的子问题1 S7 x, p* @. e, ^; E- {* @: x, N. s
       - 例如:n! = n × (n-1)!6 `- [( c8 q: O$ Q# i) P( I

    1 V# S& V( |1 B' ?+ `5 ^ 经典示例:计算阶乘
    2 O) `4 I" y2 W: B' F/ p+ kpython
    - Z1 G: J! j/ K0 kdef factorial(n):
    8 {! S; B- l8 p: y5 m# W2 x    if n == 0:        # 基线条件3 K( J4 S8 v* {! v- W7 b0 z' W
            return 1
    6 h0 M4 Y6 B. X6 U) |/ ?; z# t    else:             # 递归条件; y/ V' t9 J. o% O
            return n * factorial(n-1)
    % {* m% q* M6 P# ?6 k. b- F执行过程(以计算 3! 为例):
    7 [% n6 `* |/ o9 ]' T2 ?factorial(3)3 Z5 ]1 \5 z9 f; `% N7 A
    3 * factorial(2)6 f4 q. E4 t+ ?, k* a
    3 * (2 * factorial(1))
    9 ?. k+ k/ ]$ V- C5 b5 a, M( A& W3 * (2 * (1 * factorial(0)))
    3 f$ y2 ?( W# g3 * (2 * (1 * 1)) = 6
    ) Q- n& ?9 Q0 }) R5 C7 I& J, p8 U
    * i+ i0 l, {7 e+ ~' Z; m% |1 v 递归思维要点$ X- P' d& d* F' {% c8 v
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑+ P6 z0 Y! }6 W0 n* W
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 Z4 F1 L5 J4 T5 y
    3. **递推过程**:不断向下分解问题(递)
    " U: i* x- ]. H0 L1 _4. **回溯过程**:组合子问题结果返回(归)
    4 A6 o& Y3 h; y! G) L% \
    0 z: g# P! }) J 注意事项" b# \$ |, V2 A8 M  R# r% |
    必须要有终止条件* g& E3 R7 n! O! q3 e5 I0 h3 h' A: B% S: h
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)2 @* \  P' K5 g3 _$ t6 k5 a* i
    某些问题用递归更直观(如树遍历),但效率可能不如迭代1 K& t! b5 j! @5 y4 R
    尾递归优化可以提升效率(但Python不支持)
    ; V( [1 L* C& r% p0 I' w% t2 B' {  o! N' l; R- D: `/ j
    递归 vs 迭代/ t# v6 D+ m0 i) @. j& g
    |          | 递归                          | 迭代               |6 F5 E/ U6 T6 H6 b# R4 r% Z2 I( t# l
    |----------|-----------------------------|------------------|2 }& }  f& X9 L: x7 m6 M4 j0 m
    | 实现方式    | 函数自调用                        | 循环结构            |
    ; e* V% x) B: T4 A| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    7 d# H. L- ^' J| 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         |
    . |9 q8 `. u) \" ~$ ]| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    , D2 ?. `/ k8 {( ?; ?8 ]% U1 {9 d
    " v! n3 v  w% K4 E# m) o( l 经典递归应用场景
      \' V% {1 e* Y3 i8 j1. 文件系统遍历(目录树结构)0 ^+ [+ X2 g3 ?# u9 T5 Z. O* D
    2. 快速排序/归并排序算法: O+ g, \& D0 C( a1 T: x/ Z# L
    3. 汉诺塔问题" c; s4 {. y1 y8 c# \9 k& }
    4. 二叉树遍历(前序/中序/后序)
    + B9 ]/ w$ i6 X1 k& U1 D5. 生成所有可能的组合(回溯算法)
    $ y0 A/ O6 w& f! s+ z, E& m0 B& S1 @& L# M! h' l' i7 i$ b& b
    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,( z: k9 b! g! @. o8 l
    我推理机的核心算法应该是二叉树遍历的变种。% i) e! `9 u; O% D
    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:
    4 p. T. O* W) A) R' l3 E& gKey Idea of Recursion
    . G; d( I: i9 u) t& C  _( M% M. X! L& }4 b* c
    A recursive function solves a problem by:
    / q' {; E+ J' P1 P/ Q6 [, C  h( ^! v. z' N
    7 t* e7 w/ S, H. y  P9 a2 W7 i    Breaking the problem into smaller instances of the same problem.% c/ A% i8 b$ z& x+ N9 `
    1 @) p0 R2 M9 \1 c: Q( M: \
        Solving the smallest instance directly (base case).
    * U1 S. }% Z5 B. X( z; S
    5 ^9 _0 ?+ R# e( l7 u0 b# P; `, K    Combining the results of smaller instances to solve the larger problem., |% L, j0 X; B" P8 Y. f

    0 M7 }( c  \7 \0 ~1 M! nComponents of a Recursive Function
    ( A- O' `1 G) D' [
    ! B; K7 W/ l. t    Base Case:
    ! A5 V+ e6 G' B  X, U) R0 u* a% g4 Q% c1 w3 r* m/ K" f' A7 m
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.1 M# ?# k  b7 m7 H7 \- Z
    % \, E" \; K2 Y% r" D( q' {
            It acts as the stopping condition to prevent infinite recursion.' i( f2 _; X. ^$ e: j

    - Y. p9 H: u( l0 G( U4 ?% E3 x  @        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ' u# J& I3 K% ?4 I1 \8 P2 e$ h& @0 M5 G9 ^  R  Z7 i, j
        Recursive Case:
    ' I8 T+ e3 f' S! [# v, G2 A9 b% M  N: E! F) _
            This is where the function calls itself with a smaller or simpler version of the problem.
    6 M* X: u, a1 J) c7 b" g7 U
    - ^  G# s) v) b- p        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).9 T' f6 r. S; A- g1 T; h8 D4 k( Y

    - q) O) ~# Y$ ~) LExample: Factorial Calculation+ d" O- c& O. F2 Q
    0 c, @: H8 p5 p0 i
    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:2 {1 T) x& p# c; N
    - |2 E& y! }, b# C6 M
        Base case: 0! = 1
    2 \# L/ P  j  C* D' [' @1 k, c: U. v: v* A$ M$ j
        Recursive case: n! = n * (n-1)!
    6 ]5 |' Z9 s1 D7 v6 D
    # E6 C  y; ~7 Q6 d& V, PHere’s how it looks in code (Python):( U" d& l& |0 [8 k6 ~; ^
    python
    - P- ^/ _9 F3 O& v$ M3 w
    ! e* q& \2 G# T/ B! t! U+ ~* U4 A# E* G' g; Y. |, ~: c0 f
    def factorial(n):6 f3 L0 `3 H0 d1 _4 E/ ~% ~1 p
        # Base case
    8 o% }: C! S" c0 F8 j    if n == 0:
    5 ?7 j; n* l& ^+ Q& G% O        return 1
    & t4 ?- Z: W( j6 S    # Recursive case8 N/ n0 }) @- [( a' W' l0 ^
        else:, k( v2 @1 }- T+ ?, z
            return n * factorial(n - 1)0 Z  r3 v4 ~' }( S  i9 p( A

    ! v! _- f+ W( h/ I6 R# Example usage
    5 N; b7 c# D) \4 Z# aprint(factorial(5))  # Output: 1209 s: @( c7 ?# k" e4 i7 H

    9 x: t. Y$ V& U& J  x) L  eHow Recursion Works
    5 M6 s2 w6 Y6 x; r- J& y
    ) h9 z" e$ o& ?7 X6 S2 I" y    The function keeps calling itself with smaller inputs until it reaches the base case.
    * v* J: T6 U3 b4 @$ V+ Q1 A$ |+ i. e; X$ @' x" m
        Once the base case is reached, the function starts returning values back up the call stack.- g, a) P% I$ ~4 q5 a6 c
    & D) H' M9 x1 J1 M- m
        These returned values are combined to produce the final result." [  ]1 ?' V6 H  T3 u1 a
    ) w* U' f5 E2 z' |. v
    For factorial(5):" W" Q& ~, |: u* E& i( p
    ' g0 E6 v. j' ]- E/ }
    % f& v- ^+ y# w) e# {0 Z% z# \
    factorial(5) = 5 * factorial(4)
    ( e  N. V" {  m$ G. `: X$ `factorial(4) = 4 * factorial(3)/ ?. o2 Y# `/ B8 V# N& J# N$ y& V7 k
    factorial(3) = 3 * factorial(2)1 R4 S* i' V$ G2 z' J
    factorial(2) = 2 * factorial(1)0 K6 B+ z6 V5 o. d5 v5 e: M
    factorial(1) = 1 * factorial(0)3 `6 s9 {1 X* ?
    factorial(0) = 1  # Base case
    7 {/ @% }$ \  j, L  T( q- n# J$ E2 O* ^! a6 \1 t4 J+ s% q5 x9 V% Q
    Then, the results are combined:4 b( c8 z# e8 C. h- t, o

    + u# G' V  ~% t( ^! `7 q% x; U$ Q- |2 H+ E. T* m
    factorial(1) = 1 * 1 = 1
    7 }; ~6 ~, z5 E9 W. a: Q! yfactorial(2) = 2 * 1 = 25 a3 R9 r  A9 [& m' d! w
    factorial(3) = 3 * 2 = 6$ M5 h) R" \& M7 c" ]
    factorial(4) = 4 * 6 = 24
    ' F- R0 d; t; d% R, R5 D( zfactorial(5) = 5 * 24 = 120
    $ V% d' U6 A/ {* \. q" B; N  Y! `- K$ z
    Advantages of Recursion
    ' i: N; u& u+ [* R  B: u( M9 o
    8 o( s; u2 Y; \, D- c  g1 a) V  Z/ ^0 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).
    + H. s; @* B0 O- t6 u  |% F" O% d4 R7 }) @+ I1 l! |# t
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ' _4 n& M, I, P& m* d' z) S7 ~( w" |; x- q" f8 \
    Disadvantages of Recursion1 `# c# k. G- `8 j6 ^

    # t: L% ^; g- a% @8 \) d' l6 l' r2 R6 F    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.5 k! i% [# y' }( j; Z" L

    - _( B  j# V% @3 c    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).; Q# y3 ^, ^8 d$ t9 f8 c* {, o! p
    4 b  ^0 g. n! l! ~( L0 k* V
    When to Use Recursion
    ! \+ N; ~' \, [4 o7 ]( v/ C$ f. @5 S/ A
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& ^1 }9 H0 i5 G- e

    4 d# v5 q- v& k$ E8 W    Problems with a clear base case and recursive case.
    3 v* ?& `+ q& U3 t0 `  J) q1 J) C: X" c; V6 [# K( @
    Example: Fibonacci Sequence# |2 C/ F% x- L5 d9 e5 m+ A! y8 t
    ' n) @+ X& i% _" _* V; u
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:. Y, Z3 t( U% f+ A5 Z) h
    / }0 O' M( `# d. j/ m; v  T
        Base case: fib(0) = 0, fib(1) = 1
    ; h+ W/ |! t( }7 c% y7 o
    8 I; v1 A! f1 H: g+ ]2 f9 v) u7 [4 M3 t    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    9 j& x/ S# u1 |( ~
    ) x  m- X( ?" ^0 Kpython+ ?' z5 h" z% y+ ~
    $ C2 {/ K/ l3 Z5 P
    " b: R) F( @- b1 L2 b, Y+ C. d# X
    def fibonacci(n):) u) B6 I3 x$ x, |
        # Base cases6 E: f& D2 L- S2 l
        if n == 0:
    1 h+ l# m! Z1 r$ x/ x% ^# c5 T" X2 x        return 0) R/ L( i) {8 L7 d1 V6 Q8 K; I
        elif n == 1:9 V& O9 O8 W$ V& D$ w! E9 U3 u! A
            return 1% L# H, S( s: x& F+ Q- F4 l! `4 p5 n
        # Recursive case' d- ~% s" F+ P3 y+ T! i! t# s& q
        else:+ h) h% Q& r) T$ L4 p
            return fibonacci(n - 1) + fibonacci(n - 2)
    # w1 I* M/ T3 J/ {, U6 _% E, J& o
    % b$ @0 |* t3 s, ^% \5 G& u# Example usage: X7 K9 C! O& C! |" ~- }
    print(fibonacci(6))  # Output: 8
    2 z$ U5 L) d7 z% ]& h9 ?- I* ^. |8 ?& t
    Tail Recursion) d: m9 A4 I" Z! Z: _0 k
    % ^; Z9 Z, y( A, j' l4 a3 j: _) Z
    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).
    4 Z% b8 z7 E  L! v- d" [* g6 l0 ]6 G  w/ Q
    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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