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标题: 突然想到让deepseek来解释一下递归 [打印本页]

作者: 密银    时间: 2025-1-29 14:16
标题: 突然想到让deepseek来解释一下递归
本帖最后由 密银 于 2025-1-29 14:19 编辑 " `! z0 d4 Q4 U8 j9 g! S

9 _% r( q, U4 z( F6 U解释的不错# `8 {# H3 n! V2 R0 w$ Y! c

) c( G, [8 J4 d1 W8 Y# P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。
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! Z+ _! t! o$ ~6 H! x  M  I, {6 L 关键要素
- }; V" A5 Y7 G6 E( J$ |+ P1. **基线条件(Base Case)**
5 `/ b. T, C5 t' ]/ a6 l   - 递归终止的条件,防止无限循环
/ t, I: I: f, ~  Y+ G. h" Q. v   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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* N& c) q, I# r# E& a2 w) t2. **递归条件(Recursive Case)**
2 k! a% `. x1 s- a- Z* c9 x   - 将原问题分解为更小的子问题" Z4 @/ X6 Y" b/ l# {
   - 例如:n! = n × (n-1)!
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经典示例:计算阶乘
+ Z! ~& |+ [; @) d% Rpython
1 g: ?8 R5 z$ H9 y) j" E5 y+ rdef factorial(n):
; l0 {! y! B  _- F    if n == 0:        # 基线条件
, }3 q# M( }0 q        return 1
# L7 k7 l) ?( h: C3 c    else:             # 递归条件
, v! z9 @. E8 F- p        return n * factorial(n-1)% c5 ~* e8 r0 n
执行过程(以计算 3! 为例):9 @- M2 a. s- a+ F! n/ g) l
factorial(3)
8 F8 t* z2 x- u0 Q4 f3 * factorial(2)! Q  }: X  E8 ?
3 * (2 * factorial(1)). R( g! e& H2 L9 b8 l: ~+ J8 r/ z  c9 E
3 * (2 * (1 * factorial(0)))1 ~0 @, u& k' R2 h! f
3 * (2 * (1 * 1)) = 6
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% z6 ~8 \) T- r4 d/ { 递归思维要点
" Q" i+ ]* g% k) P1 }1. **信任递归**:假设子问题已经解决,专注当前层逻辑
+ N4 z) @: x0 L7 _$ f0 O2. **栈结构**:每次调用都会创建新的栈帧(内存空间)- [+ m0 P9 O- x# t
3. **递推过程**:不断向下分解问题(递)9 p) K) Q7 ]$ F2 A0 w9 A9 Q
4. **回溯过程**:组合子问题结果返回(归)
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注意事项
: g7 g9 r" P; H% _, u必须要有终止条件
# ]" V4 T- a/ U; C8 `9 h! n递归深度过大可能导致栈溢出(Python默认递归深度约1000层)' J8 y! @- F; d: O
某些问题用递归更直观(如树遍历),但效率可能不如迭代
' y9 A/ \7 H2 |# x; |, _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7 g  h- B1 d4 F2 z% p' ^8 Y9 U7 W
|          | 递归                          | 迭代               |
, d% C- V! d5 W|----------|-----------------------------|------------------|
  a1 e' j7 a7 Y, h7 w| 实现方式    | 函数自调用                        | 循环结构            |
! G$ h( X# E6 H- [3 z| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |* |" o7 w/ d' o$ c3 W! K# ]! e
| 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         |) n5 D  M% i. k
| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
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+ i# T) k6 H, t( D  R 经典递归应用场景( g3 I1 v% v& {, I9 e7 d
1. 文件系统遍历(目录树结构)( J# U5 Y7 m0 L9 D. d: t3 c5 S
2. 快速排序/归并排序算法
; N% n7 B. }& r  T# Z3 }/ P& x- l3. 汉诺塔问题# s) l; s- l; B' P2 w, f
4. 二叉树遍历(前序/中序/后序)+ i7 s4 K2 C. r5 n2 L7 X* q+ V
5. 生成所有可能的组合(回溯算法)
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# ]0 M! |4 I0 E4 X' L5 K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。
作者: testjhy    时间: 2025-1-30 00:07
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,
9 G1 ]" p: {. Y. G我推理机的核心算法应该是二叉树遍历的变种。
3 E+ l& n7 x0 ?/ Q* F7 D9 |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。
作者: nanimarcus    时间: 2025-2-2 00:45
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:
# F  K: c$ [% p$ S9 {' A) l- pKey Idea of Recursion8 I) i. ]$ ]+ Y3 o: m! D) I; }+ t
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A recursive function solves a problem by:' N8 m: |# F4 T$ w' |/ Y
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    Breaking the problem into smaller instances of the same problem.
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2 W; t6 V; {- g    Solving the smallest instance directly (base case).
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    Combining the results of smaller instances to solve the larger problem.0 X: Y* \6 Q3 K$ F; v! b3 {  y7 X4 R

. T/ A5 N& N" ]! C. |1 P5 H$ g6 OComponents of a Recursive Function4 [3 k; x) x  F% J# D# a) q
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    Base Case:7 ^. K- \! P7 |

! f5 x6 O+ j. f9 `3 }) n        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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        It acts as the stopping condition to prevent infinite recursion.
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1 t8 `/ V( n2 ]# S& y: \4 z- i        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.3 d) ]* J% O8 i9 I) x+ B
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    Recursive Case:6 q; j; m( _) m3 g' l4 ^8 I

8 h$ y3 y0 B; |; O# b. ?& }        This is where the function calls itself with a smaller or simpler version of the problem.
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- V1 p+ u0 a6 K( m6 c( l        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).: _! T8 c' x# a- H" t
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Example: Factorial Calculation. }' `6 c- W- I# K. b

# k' t% _5 `+ S5 N# hThe 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:* X( I/ q9 L. T
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    Base case: 0! = 1
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    Recursive case: n! = n * (n-1)!1 j- S9 }9 I( T- r

% U$ p* C! K+ uHere’s how it looks in code (Python):" i; s% O5 J4 p9 W# N  o) L
python+ [% y3 r) i& h2 y" ?

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5 ?0 e- l- x: _0 [def factorial(n):1 c: p' ~: W  b/ J4 I
    # Base case
9 l7 Z( H$ h: z    if n == 0:. I( n( s6 y$ s0 u
        return 1
6 c- E3 i8 U: n- d* `0 R    # Recursive case
0 P* v  C& s) A) D( N    else:# b! |2 \6 a% t" W- E( i
        return n * factorial(n - 1)
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# v* K! I. a9 G% S8 S# Example usage
5 d: R5 T* H' c9 jprint(factorial(5))  # Output: 1209 f: \) a) W, R: ]8 s& L+ x2 C- j

9 Y. i+ D) z) ]+ R5 CHow Recursion Works
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& u, {  ]$ B3 u* Y    The function keeps calling itself with smaller inputs until it reaches the base case.
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7 _0 ?6 X, \7 `, j$ b3 e    Once the base case is reached, the function starts returning values back up the call stack.
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    These returned values are combined to produce the final result.5 ?# E; a2 \  q
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For factorial(5):
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& Y2 u. j9 `2 l  [3 L4 ifactorial(5) = 5 * factorial(4)
) D+ Y. |7 `6 ]8 @5 ~8 s) hfactorial(4) = 4 * factorial(3)8 o" g- F  g' T1 ]4 ^8 v2 j7 U& {6 E
factorial(3) = 3 * factorial(2)
7 s# H+ \& l- u7 A2 m% Cfactorial(2) = 2 * factorial(1)
6 {, T( J# l; ?7 P8 E2 K; Pfactorial(1) = 1 * factorial(0)$ z, ?6 T6 m# m
factorial(0) = 1  # Base case
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3 r1 K4 B( r2 e1 ^# ~Then, the results are combined:& N* _) V: m, z
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$ I4 L( B3 A" \6 x0 M: e  z' o$ L: Kfactorial(1) = 1 * 1 = 1
( c# f% b' O6 R& f# V7 a5 {" @# p2 Hfactorial(2) = 2 * 1 = 2! S0 a6 D$ e: x. H; B+ d# X* ~* A
factorial(3) = 3 * 2 = 6
* I" v9 Q& C+ I% Z$ R- d) @factorial(4) = 4 * 6 = 24; `; W8 k: [" U- U3 U, j' |6 d  U  u
factorial(5) = 5 * 24 = 120
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$ h$ J2 ?  f# O, t& HAdvantages of Recursion0 Q) A% V; {* \
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    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).
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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7 ]* ]% r, S' g1 J7 `Disadvantages of Recursion3 r4 ~- N3 ?$ C" @/ w

# }" i5 [+ T, d7 E    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.
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# T( S: T7 U* g/ J    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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+ P. b- T" ^) nWhen to Use Recursion# t( U, @& j0 W: m

! ?0 ]' F/ w* i) K8 X    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).  ^4 H% e& B7 M$ ~& l/ e' D# y
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    Problems with a clear base case and recursive case.
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Example: Fibonacci Sequence
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
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    Base case: fib(0) = 0, fib(1) = 1
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7 a# m  j+ ~  j' m    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python
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def fibonacci(n):
* _, U+ j+ I$ X4 O! o) n# w    # Base cases4 i6 Y0 z) I8 s6 \& O, B$ V
    if n == 0:& w' z+ O7 E/ \2 I9 ~! ^
        return 0, b* k0 E4 Q' u( K5 K
    elif n == 1:
0 v; G) h, v5 ]        return 1
& v( d; k& A( }+ |7 U    # Recursive case
" G6 F0 I1 U* _" S- v& z1 e    else:- C8 _* Y" s$ T# c
        return fibonacci(n - 1) + fibonacci(n - 2): U$ V- @; J5 R3 i' ~9 {$ _

2 H& a$ O; z, w1 `' w0 y- N6 F# Example usage
0 O% Y- l) k- Eprint(fibonacci(6))  # Output: 8! X* b/ r- M8 U0 B" r' ?' K. Y
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Tail Recursion
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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).- B( r8 G& C) Z! S% T" o0 S2 Q7 O

5 b7 z" G, `8 R5 Z) k9 FIn 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.
作者: nanimarcus    时间: 2025-2-2 00:47
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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