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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 6 {+ U0 O6 F! n
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解释的不错1 Q6 ^. I  u* A1 H

( K) m$ y& i6 q+ g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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$ A9 P9 I' t+ m& g! |% h3 N9 w 关键要素& s& F& x9 K/ l. _7 N2 @/ Z
1. **基线条件(Base Case)**/ Z( h6 `. r! M5 \7 Y
   - 递归终止的条件,防止无限循环
* j( W( `: D: X4 \5 Q# Z, b  e   - 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)**
4 {+ u4 m% i0 X/ X, y   - 将原问题分解为更小的子问题! ]# E3 O+ c. s) t1 F8 ]3 q& i
   - 例如:n! = n × (n-1)!
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经典示例:计算阶乘
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def factorial(n):
: {% s1 h6 F$ k" B$ e    if n == 0:        # 基线条件" S3 }1 D# E- P6 N& m
        return 1; k3 T0 G3 y$ G" n. F& C
    else:             # 递归条件. \" x; Q2 F2 ^5 `1 A- I
        return n * factorial(n-1)4 ]3 f$ t: U$ H* y9 g
执行过程(以计算 3! 为例):# H* r. W' d0 x3 T, q
factorial(3)  Y: O- s5 E3 V" f3 A
3 * factorial(2)5 t& f& i9 Z* {- q, l
3 * (2 * factorial(1))
8 y: t' I- a% i* |% s; K# H' l3 * (2 * (1 * factorial(0)))3 _: X/ |( \( `
3 * (2 * (1 * 1)) = 6
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! G8 @3 D  V( p$ |+ D 递归思维要点
+ f+ r/ D! _/ S! j/ \1. **信任递归**:假设子问题已经解决,专注当前层逻辑
2 C6 P6 s; _8 W5 e. Q2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( m  P) O6 t! o; x( z
3. **递推过程**:不断向下分解问题(递)
  n" P9 c) m, z+ g  Y1 d4. **回溯过程**:组合子问题结果返回(归)( _4 |4 i" c9 J2 O

) ]+ i  l( w$ c. I 注意事项1 n% n! i/ V, ?& Y1 V" O
必须要有终止条件
) V) M3 ?- {6 ?6 f+ 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 x3 L: k. N$ B1 C3 n某些问题用递归更直观(如树遍历),但效率可能不如迭代
+ M& `$ F" Y9 T: b尾递归优化可以提升效率(但Python不支持)8 o( ~4 W7 r+ ]* F/ U0 c/ v

  \1 \) L5 [) ?1 {: w; g5 t. G 递归 vs 迭代
9 K+ [6 d9 K5 r2 @7 j; U|          | 递归                          | 迭代               |
  e* R9 p7 V1 o; K" @|----------|-----------------------------|------------------|  X( t1 ~7 E$ [/ G& T. ~# @
| 实现方式    | 函数自调用                        | 循环结构            |# l4 v8 w& R7 M1 I
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |; @/ H  `! k5 M+ q- Z* v$ r* N3 l% i- u
| 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         |* E" k% v. p1 A
| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |) y; T$ A1 Y* a7 z" C

  h$ n( c$ h; W9 u$ R% m! Y4 }  {4 p 经典递归应用场景3 M' f2 N9 N4 o0 k" j: \1 D* M
1. 文件系统遍历(目录树结构)
- @, B4 j3 h. i0 S4 w* p( _2. 快速排序/归并排序算法
3 l9 W9 M' D; f) w: f; b+ V3. 汉诺塔问题- D6 w1 z: K( n: p4 e7 b7 [: ^
4. 二叉树遍历(前序/中序/后序)% |5 H# T6 h( j2 ^  D1 {
5. 生成所有可能的组合(回溯算法)" A) N& S+ J# b2 \! `, K6 ^

" ~8 I4 q  m5 R4 C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,( T& @  c6 t# e  \, S2 ?
我推理机的核心算法应该是二叉树遍历的变种。2 @& u; H8 G# F/ Y/ j/ E
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:
* i  ~$ b0 w$ ^8 v5 NKey Idea of Recursion
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- f; Q. k; R+ b; }: d# F9 X4 ~A recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.6 V: i- t4 T. u& i1 h! O/ O, ^1 X

5 Q) {: j4 Z6 `7 n; V    Solving the smallest instance directly (base case).5 d# e* S% A$ Z& U" B2 S( w0 {+ I

5 U$ Z3 Y' K5 ]* R+ K; @5 ]2 Z    Combining the results of smaller instances to solve the larger problem.
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# U) L. Z) E% F/ z+ rComponents of a Recursive Function! `, i" k& e* w" s8 [7 ^8 L; A% {

. f; |3 s8 a: P    Base Case:
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8 `% C& D7 x3 S" b        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.& `! S# p8 D; v( J+ S
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        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 P& l4 _0 E* P. v0 S: Y7 j3 Q    Recursive Case:
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+ u( V) c6 Q# b) [3 v        This is where the function calls itself with a smaller or simpler version of the problem.6 d0 H3 D6 W" s; E; Z
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% V' N1 p, W* }7 M
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Example: Factorial Calculation
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& P8 _/ S1 R3 Q; D2 x  ~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:' J4 r# k3 J/ Y& H! K) l

5 y+ i; P2 R, |  A) A    Base case: 0! = 1  O2 R" I7 t6 k6 r6 _' F
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    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):% W3 l- t) r! s$ z! B
python
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& j/ L( R. n9 d! s2 V7 ]7 ~4 o6 K! Sdef factorial(n):  Y* k. s  F% W
    # Base case3 e0 r- N2 E: _0 Y: A0 b+ j
    if n == 0:
) f" J, F, o9 h' \        return 1
/ B5 W8 a0 z" `9 K  E0 N    # Recursive case6 F$ A1 R# t* n3 T4 d6 |0 D3 V
    else:
% _- [0 v) \( Q; e0 Y0 s        return n * factorial(n - 1)# R5 X$ Q, R: |& x/ K( S' |

( K* N9 P/ _: L, l1 T5 N# Example usage3 w2 ~* |3 y/ V% P) U; W' k
print(factorial(5))  # Output: 120
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4 k2 q. ~& S: BHow Recursion Works
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: [6 {- }% a& E/ L9 _0 x' K    The function keeps calling itself with smaller inputs until it reaches the base case.* q% _5 w  }  o
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    Once the base case is reached, the function starts returning values back up the call stack.  J4 Z9 s; x% e! u- M- i* X3 z" I
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    These returned values are combined to produce the final result.
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$ A& O: D# S: n( X1 }( \7 nFor factorial(5):, o1 L2 G8 v& h  B

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; u6 a6 M5 p+ L! N0 {factorial(5) = 5 * factorial(4); A7 r; F' ^% x- ~
factorial(4) = 4 * factorial(3)* s* l9 Z  |9 {+ |2 @5 z/ t, E% _
factorial(3) = 3 * factorial(2)7 U8 o% I( \5 x& `- ^
factorial(2) = 2 * factorial(1)
# X- z6 p0 C3 I  v- O# j- Gfactorial(1) = 1 * factorial(0)  `- T- U# g* I: L$ b3 l
factorial(0) = 1  # Base case8 E" {1 [% S  J! R3 ~( w" K/ c

$ R, y+ C. k, Q# t; mThen, the results are combined:8 X1 ~; D& ~8 m

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# R& j5 p$ S+ _; x: _# Ifactorial(1) = 1 * 1 = 1; S+ ]' }& L) P0 Q& J( o2 A( A8 y/ b
factorial(2) = 2 * 1 = 2( ~/ C# c+ _$ L7 K# J
factorial(3) = 3 * 2 = 6& e3 G% r# {- N0 a% H
factorial(4) = 4 * 6 = 24
: f4 u1 L. |9 l0 y, k9 Q0 ?2 Tfactorial(5) = 5 * 24 = 120( a- A5 E  B: b; u% \+ q
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Advantages of Recursion$ y6 |" E6 u" @- ^% `, y, l+ z% }
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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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+ l' s) O+ A9 o+ z& n    Readability: Recursive code can be more readable and concise compared to iterative solutions.1 z$ y1 y; S2 P  h: G1 D

. B+ G! m# N+ R. pDisadvantages of Recursion
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    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.! b) ?0 D5 T; ~. O

* x: C: r0 N5 |    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 b3 N8 V6 U4 J8 K' R* r
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When to Use Recursion
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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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; J4 t4 u8 [5 ^: z    Problems with a clear base case and recursive case.0 p: r: N7 S5 I  x" V

) h$ G- f9 N" `4 w0 c+ w5 p2 BExample: Fibonacci Sequence' X6 @9 o+ B/ C7 @
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:1 W0 ]3 ]0 o, n3 w4 L2 p
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    Base case: fib(0) = 0, fib(1) = 1
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. v5 f; R/ D, E6 E- o: L( t    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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* M/ z. k- Q& D; Z% s, Kpython$ O! D0 n' z) n4 m
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def fibonacci(n):' V9 u6 K6 p! \3 E9 P! x
    # Base cases
1 F# `  J% k; R& b6 S    if n == 0:
% s) c) Q% k7 N( t8 a7 h        return 0
% o! m) N* Z7 _6 D    elif n == 1:
% h1 W3 G% v" l5 h0 O- x4 o' D        return 1
+ U/ f* V5 h3 I/ Z/ g" n) C    # Recursive case
5 G1 ~3 i4 l3 y: g    else:
- |/ U+ _3 l% d2 i1 @( N7 e7 D' Z+ G( e        return fibonacci(n - 1) + fibonacci(n - 2)
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# Example usage
1 M) W/ _4 J$ f  Nprint(fibonacci(6))  # Output: 8
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Tail Recursion
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9 A2 q+ {8 a; d) Z; ?5 F2 iTail 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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9 m4 C2 p' ~/ S4 |# pIn 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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