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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 # N7 P- v) g' h6 b; s
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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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4 t' p) s8 w- o8 @% o 关键要素# ?" a2 s: i- D3 J  l" C) _2 l* ?
1. **基线条件(Base Case)**4 K2 j: O+ @# C- S
   - 递归终止的条件,防止无限循环( H: h+ _" x: p# W/ D
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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2. **递归条件(Recursive Case)**
; f1 w( S! U4 l; Y% t   - 将原问题分解为更小的子问题- G/ A& Z, p, a7 w3 V
   - 例如:n! = n × (n-1)!, t% [( E, N& t' J2 F5 R  h% i

/ f" l5 d3 }! s* p6 [; N: F 经典示例:计算阶乘: X8 Z" a( g4 ]) s' O: t& a( X- `
python" B! @' z2 N  |7 E4 d
def factorial(n):# `, K! X6 y: b9 }8 K/ P
    if n == 0:        # 基线条件
" G% d& [; @  u9 w        return 1
* x2 u. V" L9 F. s9 [9 Z    else:             # 递归条件
0 X# a* B: l) V; g# _        return n * factorial(n-1)
0 g4 Z$ U) B- C$ q  X. }执行过程(以计算 3! 为例):
) ?* X  x% H; P3 j- sfactorial(3), D6 C6 w- l/ H7 ~( F
3 * factorial(2)$ C/ Z5 Y/ ]- i" G
3 * (2 * factorial(1))
3 j. ]* u) [; G  o4 q; E% D' t  B3 * (2 * (1 * factorial(0)))/ I! \! z, A; v& P$ r; G5 D! T9 l+ @
3 * (2 * (1 * 1)) = 6
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* z5 G6 g3 }9 C2 f% h& \4 N0 t6 c 递归思维要点
) x4 A/ e& r% K5 }! z1. **信任递归**:假设子问题已经解决,专注当前层逻辑
, z3 J6 r; N/ F) A7 z1 c6 W9 @2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 a% U1 R, a3 x0 v3 t
3. **递推过程**:不断向下分解问题(递)
5 M! O. `* A: y- Y9 Z* O1 l8 h4. **回溯过程**:组合子问题结果返回(归)3 R" ~6 o& R; `- o/ J8 t1 Z
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注意事项7 r& P$ J/ w: n5 Y7 B9 y" f
必须要有终止条件
" M4 G: y2 b# A# {: H3 w4 S4 J8 O6 p递归深度过大可能导致栈溢出(Python默认递归深度约1000层)5 x2 }; e4 T5 `, l1 E, T2 g1 S
某些问题用递归更直观(如树遍历),但效率可能不如迭代
2 A7 {, I+ [: v+ Q  K尾递归优化可以提升效率(但Python不支持)5 y: w; T# {+ {  V& \: P# T
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递归 vs 迭代' E5 B0 \( R3 H2 Q; V0 |
|          | 递归                          | 迭代               |0 g- {. c* B% L" S
|----------|-----------------------------|------------------|9 j+ ]7 q- w0 c: {5 s& B1 \9 ]: t
| 实现方式    | 函数自调用                        | 循环结构            |  l0 X! {$ K2 H- ~/ \0 `" m1 ^
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |2 n3 t/ v) z; B* M( M
| 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         |
; r, g$ c# _; ^/ b5 G& D) j| 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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经典递归应用场景
3 y, n- p3 L; }$ d! w  m: b) b1. 文件系统遍历(目录树结构)
0 j  a8 y9 i7 z7 k2 y2. 快速排序/归并排序算法
1 e( f' c" l1 Y3. 汉诺塔问题& t8 |; o3 h* e: l! l3 U, i
4. 二叉树遍历(前序/中序/后序)( w+ }+ e3 D+ y( ?6 r
5. 生成所有可能的组合(回溯算法)
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+ p5 H1 m, G# P( ?$ [! T0 a! Q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,
8 R- N/ `& {; D% C) @! _( U- [我推理机的核心算法应该是二叉树遍历的变种。
3 p# p- e1 T% b0 s/ U) N$ m  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。
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:3 k2 @' v) c4 ]# k6 x$ o
Key Idea of Recursion
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. _4 x6 z; Y2 F3 x) OA recursive function solves a problem by:8 J; W! R3 w- F+ |$ M7 A

. E" s( n# I- a! T( }( ?    Breaking the problem into smaller instances of the same problem.: c$ |. e7 ^0 ?* r8 M$ _8 ?

1 X- p3 E: \, w    Solving the smallest instance directly (base case).
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    Combining the results of smaller instances to solve the larger problem.& H# \- N$ b- _+ A3 N

, L' O2 H3 w3 v( @Components of a Recursive Function. R) m% T2 {' J) o
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    Base Case:" B% a$ T- _. O' U7 v( H; f1 p5 E

8 O( b! x9 M$ E. b        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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9 t& M0 `3 A, {2 N5 l& C        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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8 O# a* o0 u1 ^    Recursive Case:
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        This is where the function calls itself with a smaller or simpler version of the problem.
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).2 _* v* T  @% }( E' K
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Example: Factorial Calculation
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- G9 T5 x; |. N' g- aThe 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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* t9 a* m! ?) r    Base case: 0! = 11 `9 J4 n) C. ^& D+ r, E

$ [; Q9 X( {8 M1 _0 x' _    Recursive case: n! = n * (n-1)!2 D9 ?: g" c, z2 n

/ m0 r/ i; O  _! w# Y+ OHere’s how it looks in code (Python):
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7 {- [0 v& G, i# rdef factorial(n):$ C7 r! _  g0 U) e5 F1 j
    # Base case
9 @1 f% X( Z# }: F    if n == 0:; \: l4 K* _% n% T/ I/ k3 [
        return 1
2 }% a( b* L7 M1 D5 `  [1 A, m    # Recursive case
6 R, W" h# z% B( x    else:& k& H+ k5 G, D2 y2 @$ V
        return n * factorial(n - 1), N) A6 L8 G/ l

- b& [0 J1 V/ }7 |7 @; `' @# Example usage7 M) t2 l' h! i" Y
print(factorial(5))  # Output: 120
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( B+ W. b% d7 C2 `( xHow Recursion Works
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% F3 I+ l! l; \& L3 J$ d8 Z+ n    The function keeps calling itself with smaller inputs until it reaches the base case.
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* c$ r- m  v; s( j% u: G    Once the base case is reached, the function starts returning values back up the call stack.. s; F  T8 ~+ h, @8 Z2 U

3 G* d. Y( D. o, b9 {  q    These returned values are combined to produce the final result.
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For factorial(5):" g% U1 e1 |9 c- ?1 [# f6 K

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4 Q; G& s3 D; gfactorial(5) = 5 * factorial(4)
: f2 i. u! l& S/ u7 mfactorial(4) = 4 * factorial(3)
5 |. G' U. ^( u6 dfactorial(3) = 3 * factorial(2)
+ H( i& N+ k8 _+ B6 e2 nfactorial(2) = 2 * factorial(1)
' q) a+ M6 Y5 J# C4 G/ [factorial(1) = 1 * factorial(0)* u% U/ r) l3 A
factorial(0) = 1  # Base case
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# n8 ]  `+ S, S' a% x/ r! XThen, the results are combined:
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4 _# O& k" s/ e; Zfactorial(1) = 1 * 1 = 1( |1 M) T, a6 y5 D* f* ^
factorial(2) = 2 * 1 = 2
( Q3 f( a+ j: Gfactorial(3) = 3 * 2 = 6
4 ]& ^2 @: c( p4 W: E6 r! Kfactorial(4) = 4 * 6 = 24
! S3 h; h& _8 _8 I  M" [. f0 zfactorial(5) = 5 * 24 = 120$ y# [) D8 \6 G' x

0 L+ V4 F" t7 c- z7 O& cAdvantages of Recursion. w, R/ h- f; O+ x) j2 x; g8 `2 u9 R
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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).1 V  u( |# p" }
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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Disadvantages of Recursion
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4 n2 l+ {* C0 v2 J9 _8 C4 q    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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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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When to Use Recursion9 P/ W5 j4 M7 W# L. q
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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  r7 G  f6 `/ T) B8 C9 M$ @" D$ Q7 Y4 q    Problems with a clear base case and recursive case.
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Example: Fibonacci Sequence9 n8 k  v$ J) o% D: L& F0 ~
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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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. m9 }, H( I% [* b! B0 Q3 N( u    Base case: fib(0) = 0, fib(1) = 1; H' {8 M2 }, P5 Q  g

. T/ u# D6 c8 g7 ^9 C# m& v% i- w    Recursive case: fib(n) = fib(n-1) + fib(n-2): p1 g: ]. x) ?! ?) l' o3 O1 Q! P
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python
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def fibonacci(n):
: e  s' K: f/ m. ^    # Base cases
( |; ?( R* i# S% a2 u7 W    if n == 0:( f: C) s; \1 `* N5 {
        return 0
% a( \) i6 t5 k+ \* |& K/ T    elif n == 1:
7 Y6 W9 I3 L. C& i1 l- [# }        return 12 K2 K$ G& U: |" e8 j: U0 {
    # Recursive case
  l  |3 Q1 x; u' v3 U    else:# _( p8 w# G# l, S) }+ q
        return fibonacci(n - 1) + fibonacci(n - 2)
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# Example usage
3 F* `3 Z) `/ e1 z7 B2 ~print(fibonacci(6))  # Output: 8% I( f6 s) c& L7 c+ ]8 Q* `  T
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Tail Recursion9 b2 ~. E1 N9 O" _7 g- U

8 w) T: m" G! e! b6 \7 H2 P9 C$ pTail 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).8 {4 v  U9 z& |

1 b, p- `. X8 JIn 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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