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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
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/ F% e. K& n6 a解释的不错
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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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关键要素6 [- w, w5 A9 ^' j$ g7 Y% u% `/ w
1. **基线条件(Base Case)**
3 y8 G1 \& K7 a9 l) U+ N; s   - 递归终止的条件,防止无限循环
1 h' w0 f$ E! M. |   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# [  [5 k. b, y3 _) K; e
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2. **递归条件(Recursive Case)**, V1 `; c8 [* U4 E1 c( t. Y
   - 将原问题分解为更小的子问题
' n0 [. Z% B6 H" ]# _   - 例如:n! = n × (n-1)!  G7 |7 L3 k/ @1 P( ~
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经典示例:计算阶乘
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def factorial(n):  w5 x# k) J7 H; \7 s5 I. \# t
    if n == 0:        # 基线条件/ e; u; o% J$ {) r
        return 1
4 @% P6 Y9 l1 G+ p. E+ t    else:             # 递归条件5 S  i  q' T* G' c1 \4 s
        return n * factorial(n-1)/ x7 c- B- L. X- `5 s
执行过程(以计算 3! 为例):( M* L! t( ~  T, |: z
factorial(3)9 @8 @8 k9 F8 I. Z
3 * factorial(2)
' ^7 h# E; \8 k) b8 w3 * (2 * factorial(1))
3 u/ [: }# q8 f6 M3 * (2 * (1 * factorial(0)))0 `: s9 t' f6 H- E2 T9 }0 x
3 * (2 * (1 * 1)) = 69 n% p! j$ q0 g+ t; E% n: k4 F! E4 \
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递归思维要点% e  ]. N( C- _: q& f  v
1. **信任递归**:假设子问题已经解决,专注当前层逻辑% T- s  I0 m4 L; t# ~& F5 g& B
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
2 i- S3 Z1 v$ ~' R# u+ M- ~3. **递推过程**:不断向下分解问题(递); F3 p+ C6 X9 R7 H' l
4. **回溯过程**:组合子问题结果返回(归)7 U/ D  _. ~( f( ]% i

; F4 l, M2 X' H7 F6 B! H- F 注意事项
; G$ M  ^. f# G$ Z4 O# n6 d必须要有终止条件
1 g  M8 U- M% U" S/ T递归深度过大可能导致栈溢出(Python默认递归深度约1000层)8 l7 W  m/ x: x
某些问题用递归更直观(如树遍历),但效率可能不如迭代
+ F) Y& W. v/ x: u5 [& @# c尾递归优化可以提升效率(但Python不支持)
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9 ~" B( U. }6 ~4 }, C 递归 vs 迭代! Z  S; w  g1 C! H" O
|          | 递归                          | 迭代               |) C) x+ S9 {. h, b7 n/ B4 Z
|----------|-----------------------------|------------------|% E0 L7 y4 ]& R9 I) }  N& B& g
| 实现方式    | 函数自调用                        | 循环结构            |
& y% n$ v: L% K; Q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |. a1 d1 ]* P: P1 W3 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 E" m( n$ `5 z' T8 r
| 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, V$ Z8 N/ \& O' t9 S0 x 经典递归应用场景5 z# U0 F8 A% r8 t4 {) }6 d
1. 文件系统遍历(目录树结构)
6 Q5 }# u# |- O; V2. 快速排序/归并排序算法
& R" @! j+ w; f: k4 e3. 汉诺塔问题
3 N3 w; K2 m( m4 x* n& m4. 二叉树遍历(前序/中序/后序)
0 |$ F/ U8 P7 F, @5. 生成所有可能的组合(回溯算法)
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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,
6 p" [" y) L7 c我推理机的核心算法应该是二叉树遍历的变种。! m* I# g, R3 N* C! k
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:6 m: H7 P* n% b5 U$ S/ Q1 C1 ?
Key Idea of Recursion
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A recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.' t3 }2 i! f6 Z: r% ?4 G$ b0 ^  W
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    Solving the smallest instance directly (base case).
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! T- G% K: A5 ]: C    Combining the results of smaller instances to solve the larger problem.0 S" S5 m0 W( E; X, t* n% K

4 z  P7 R$ I$ mComponents of a Recursive Function0 q! M6 l1 e+ {0 Z
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    Base Case:
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! J! s. ?9 j; W! N! b        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.% R9 r& d% }, |6 f" w' R

; y1 O; z7 h: h        It acts as the stopping condition to prevent infinite recursion.
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        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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( Z* B7 e& ]/ O, M    Recursive Case:
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        This is where the function calls itself with a smaller or simpler version of the problem.# _. Y$ ]. _7 I. U! d
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 m9 _, M! @/ {9 m- |: K8 N8 m8 |. z

" ]) T0 Y: O- R8 {. pExample: Factorial Calculation# I8 w, ?. y6 X

+ E0 G4 R. i5 K0 h( |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:
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    Base case: 0! = 1
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    Recursive case: n! = n * (n-1)!: @# m& Y5 u5 t* Y

5 e! F9 |4 K  K7 w. W- q8 jHere’s how it looks in code (Python):
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def factorial(n):$ ]! Y5 D: a0 f' N: z$ [( m
    # Base case
. ]! A2 D) G5 v+ }    if n == 0:& k& o  B4 h3 W: D$ o8 Z6 M
        return 1
* H5 ]" v* G7 Z6 l  o% \! w    # Recursive case
+ ?  [( _9 F8 v/ ?1 U0 u" J    else:  l2 T: o: \6 z, j0 g$ b8 z
        return n * factorial(n - 1)
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2 [2 Z% }$ `" T( V7 D# J# Example usage
/ F+ n2 u* ^6 f% Jprint(factorial(5))  # Output: 120# F; d  s. y+ z9 m9 l$ a# z

& s8 f4 n  B  n* W$ G/ WHow Recursion Works& B% t- e3 b' a! |$ F6 u

! L5 M$ i& a9 E5 R0 M    The function keeps calling itself with smaller inputs until it reaches the base case.
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6 z/ d9 h+ w$ t6 C/ _; l    Once the base case is reached, the function starts returning values back up the call stack.
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3 l2 k, N7 J3 c. h0 o, ]  S    These returned values are combined to produce the final result.
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For factorial(5):" |' X# m. R- \% }0 h, ~

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! U& Q& D% K9 b% e1 w2 Q% Cfactorial(5) = 5 * factorial(4)
: X  M8 A) I7 O5 {3 W9 gfactorial(4) = 4 * factorial(3)# U7 i! M( {$ R! m' @$ r- W
factorial(3) = 3 * factorial(2): M: w1 _+ ~. T
factorial(2) = 2 * factorial(1)
: }( k- J, s- y2 r; I+ z6 nfactorial(1) = 1 * factorial(0)2 }% }6 }3 ]/ D( S+ \  [- x
factorial(0) = 1  # Base case4 Y# i' {+ x8 w+ \
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Then, the results are combined:0 R9 {, F' ^5 z3 q/ H
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factorial(1) = 1 * 1 = 13 _2 w; S& j! a- a7 ^. ~2 ^' H* x
factorial(2) = 2 * 1 = 2" L: J8 Y" l* O. C, \6 r9 A
factorial(3) = 3 * 2 = 6' m7 C$ a# E6 B2 W
factorial(4) = 4 * 6 = 245 B2 H  A2 d, V5 r7 u
factorial(5) = 5 * 24 = 120
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! p! k& b  Z( JAdvantages of Recursion
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$ v" ~/ ?9 s: J: K! 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).( ?7 D2 L/ @- Z; ~
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.4 @$ S4 G6 l+ x/ k* N) q

; |3 c5 l: x" e, u" u5 U! z/ ADisadvantages of Recursion' ]) B& z3 ~5 l3 g# o2 Z

7 z1 s3 g+ I1 n8 }! Y+ B2 e  |( G9 A    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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! l, a1 Y/ S; Q$ t    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# Y( t" Q2 T. l4 [' u+ o- `
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When to Use Recursion
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. p. @0 z- f  E# ?. Q5 D6 L    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).* U+ D) j+ w8 S) N& j
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    Problems with a clear base case and recursive case.
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Example: Fibonacci Sequence
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" f/ V" ?5 m$ r# d, u2 BThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:' Z  ^' c, ]) Q# }& v$ u9 z- B6 K
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    Base case: fib(0) = 0, fib(1) = 17 ?3 T1 f, h2 _
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python
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+ X) z- o; m+ e1 Y) [def fibonacci(n):
5 E3 [' a) X- e/ |# m    # Base cases
7 S( x1 n) N9 g9 [3 s) G# [    if n == 0:8 j+ {/ I% `3 W8 Z
        return 0, L# t  q1 _+ R5 b0 r2 H: q2 B. o
    elif n == 1:) B1 ^3 i7 A1 ^, G& z
        return 1
8 {& B0 v6 O* L! h    # Recursive case
' b0 C  i' [( ]! Q    else:( G! K3 c& t& d% R1 q3 u( {
        return fibonacci(n - 1) + fibonacci(n - 2)+ K# \/ U  `$ r4 H* ~, F, M
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# Example usage
0 a7 ]+ n) s" N4 h( i2 M2 R% lprint(fibonacci(6))  # Output: 82 @8 D2 P9 {9 Z
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Tail Recursion
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4 |6 Z  M; E, D, STail 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).
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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.
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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