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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 + W+ g8 v8 |' P; G$ b4 g
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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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关键要素
6 F. Z: W8 l6 f6 G$ p1. **基线条件(Base Case)**
  i- B" P6 |- M6 ^: Q   - 递归终止的条件,防止无限循环" q+ m: M9 n& F
   - 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)**. P- }' f" W- b5 D* E/ {
   - 将原问题分解为更小的子问题: k8 z. \8 ~3 S7 `
   - 例如:n! = n × (n-1)!
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1 M- l% }/ c# u" R# L 经典示例:计算阶乘5 S7 I9 ]. u( f& ^
python
  h- M  C6 j8 h9 x$ P+ _def factorial(n):
- {/ P( ^3 Y2 g9 e, r; X& X! O    if n == 0:        # 基线条件
$ c9 I" s  D3 E( {6 _& e4 U        return 12 V9 H! c  @  E. S* v1 h; Z; H
    else:             # 递归条件7 O$ c- V) S" Q8 C0 Z) h' ~
        return n * factorial(n-1)' O- Y& `  ?8 F4 ?$ J
执行过程(以计算 3! 为例):
& G+ G+ B, T: s( k3 K7 y, ~factorial(3)- x2 I% o3 }. n6 [/ N2 B
3 * factorial(2)
$ i) @/ e% G) R/ O/ O0 w+ s5 R3 * (2 * factorial(1))
5 r( l4 G# _$ |+ k3 * (2 * (1 * factorial(0)))
! A4 P, {3 y0 K0 j3 * (2 * (1 * 1)) = 6
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& D  l' _# A3 Y+ u 递归思维要点
( Z: R; D* K7 J5 J( S0 a1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 m7 U- q, w' ]+ c: Y6 o: \
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)5 E, w/ K- E& L5 t! @! w) P! l
3. **递推过程**:不断向下分解问题(递)
+ G$ w! D' W( u" f9 {( @4. **回溯过程**:组合子问题结果返回(归)
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注意事项
5 r% R3 }: w6 r0 z) ~必须要有终止条件
- D8 e' M: S3 E! _2 N# Q递归深度过大可能导致栈溢出(Python默认递归深度约1000层)% B7 A* c8 r' }9 T  C* N
某些问题用递归更直观(如树遍历),但效率可能不如迭代5 |. Z$ x9 h: G" T9 `: ^
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# r& a& P* I' ?4 p" c6 l9 R1 e
|          | 递归                          | 迭代               |9 K  U/ P5 }! o+ d' r- c4 Q0 T' U
|----------|-----------------------------|------------------|0 r% p( J8 y! C
| 实现方式    | 函数自调用                        | 循环结构            |. U% U! W6 f# W( U* ~9 B2 W  _
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |, ?& n/ S( |; t; l/ U# o
| 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 I8 T' f( @1 n3 d0 V6 I| 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
1 m9 ?5 w- {% a3 V, x$ @1. 文件系统遍历(目录树结构)
* ]2 n- j* o* `/ n2. 快速排序/归并排序算法2 e0 X0 N) d  j( b
3. 汉诺塔问题7 {* x$ a" F: ~' s" b/ C# [5 U
4. 二叉树遍历(前序/中序/后序)
8 |3 j1 p& N. g. |  s3 r5. 生成所有可能的组合(回溯算法)1 M" a0 ?, x4 A* _

3 ?. V% U7 d; L) A) j4 ]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,7 Z5 }4 J; |7 c/ y2 x* V( p
我推理机的核心算法应该是二叉树遍历的变种。
2 M/ y' `! V7 R9 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:
5 x8 B( L: }2 \* TKey Idea of Recursion
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1 h; N1 W( }' l. |7 I; u: @A recursive function solves a problem by:+ i7 X3 F1 ~! ~: ~
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    Breaking the problem into smaller instances of the same problem.3 j4 z' H8 q; I' L6 J
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    Solving the smallest instance directly (base case).7 s; a: r# \) W* l

/ L" W3 c) k; D! ~1 g. O' E/ |    Combining the results of smaller instances to solve the larger problem.. h+ B% D, u7 D5 {& V

) Q1 {& n, j$ b) s! c3 ZComponents of a Recursive Function! L( g7 `$ y: ^

, x# e* F% i! m- Z* }    Base Case:8 k2 w* ^- Q  }" W% e7 g0 [
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        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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- ~: s* d. y7 H        It acts as the stopping condition to prevent infinite recursion.
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3 B. \( z) o5 a! o6 S3 G' E1 f" a) u! j        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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# e9 q0 Q. u8 i    Recursive Case:$ |4 w: t8 O+ @$ F

- E- x& x. b# Y1 V  |/ \8 g" g# K        This is where the function calls itself with a smaller or simpler version of the problem.' X5 o  \# ?7 T+ P+ _5 J% m& n
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)." A* B9 B3 k- K+ h# f7 ]8 D

4 S" m8 O4 }- |# H* E. wExample: Factorial Calculation
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+ j4 \* o! Y- y! b0 zThe 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:+ k( u; A- \, @9 o8 D$ p) z

+ `2 Y1 _, F& g3 z    Base case: 0! = 1
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    Recursive case: n! = n * (n-1)!
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! @4 z+ w- b- i7 \Here’s how it looks in code (Python):
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def factorial(n):
% c5 i2 e2 V- O    # Base case! S3 U& k7 K( j; }) D$ S
    if n == 0:
+ k/ X# q( Z' \/ z. _        return 1$ D2 H4 Q6 q) c. g" f$ F
    # Recursive case+ b5 j5 a* @. K; |8 N) l# b4 D
    else:1 z2 U8 H% O8 F! I6 H
        return n * factorial(n - 1)
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; v8 g/ X+ j9 v  [2 B3 ^# Example usage: O' s, t/ ^( c' p* `9 ~0 P
print(factorial(5))  # Output: 120! H7 A( x2 u: c9 q, E

  z  Z0 u8 u: f8 E7 ?. VHow Recursion Works
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    The function keeps calling itself with smaller inputs until it reaches the base case.
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    Once the base case is reached, the function starts returning values back up the call stack.$ z3 z5 H" L, V+ k& N/ d

: J* d5 ]* `  i8 r5 k+ h- i  g0 O    These returned values are combined to produce the final result." g. S$ y, m2 A: F. ^
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For factorial(5):- O% A: ?0 |* Q' [
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factorial(5) = 5 * factorial(4)
& S+ k& r8 ?& l4 l5 C/ }factorial(4) = 4 * factorial(3)
" A( \1 k# X- L: {7 [3 F: M# dfactorial(3) = 3 * factorial(2)
" a+ D, S+ ~3 D9 _$ ?+ `' H  cfactorial(2) = 2 * factorial(1)4 Z7 u. c6 q0 w) h
factorial(1) = 1 * factorial(0)" f; k. W( v5 n) i6 D
factorial(0) = 1  # Base case' S$ q( a0 {: e4 _" m& k, x

* n, U; p9 N1 DThen, the results are combined:
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factorial(1) = 1 * 1 = 1
) F# D4 ?0 p$ S* U* o7 lfactorial(2) = 2 * 1 = 2
( \3 X3 g, a2 F, n  R% H9 H& U- Rfactorial(3) = 3 * 2 = 6; Q7 u+ T7 v% r: R& A
factorial(4) = 4 * 6 = 24
- Z7 z+ U# W7 g. }2 [factorial(5) = 5 * 24 = 120
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: R1 Y% l3 W  H# t, iAdvantages of Recursion
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) s' U6 o# Z% J! Y6 |    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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0 b8 m6 z' s8 e* K& F4 `! {( m    Readability: Recursive code can be more readable and concise compared to iterative solutions.: l$ H" u9 S; b

1 `2 T8 X8 X" F  S& h+ B8 M$ pDisadvantages of Recursion
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. I/ |1 l% f9 ~% D2 O. Z% S4 i    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).. U5 E7 W* q2 p

7 |# v, s6 d/ X; [' C# EWhen to Use Recursion1 t% G1 W/ ^: B3 a9 n& B

# l5 B) W& @1 M% q- P: F    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)., c" i7 \% H% [: N3 z$ o" H
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    Problems with a clear base case and recursive case.. |8 j$ c' Q8 F

& Z9 o. N: L  |, hExample: Fibonacci Sequence9 S! J; y- ^) R6 _" p

4 D) e7 o" Q- I; ?. x* oThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 W; {& k3 d4 s2 u7 r
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    Base case: fib(0) = 0, fib(1) = 1
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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1 u" N2 q. d6 ?. p4 jpython
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* U# W6 Q! E  wdef fibonacci(n):4 L. c7 n- V1 n7 S, `$ Y( }4 S6 o
    # Base cases
4 e1 k, P- e) D' H    if n == 0:/ m& u  }3 F* ]: y3 e
        return 0* i* L* r% H! a
    elif n == 1:
" q2 f; a1 y' ~/ y; |3 _  _* A4 H% F        return 1. u& ]. u! P( l8 ~; X
    # Recursive case
" [9 q$ }5 F& |( e: w' U7 {7 |) ?    else:% f( r" F+ p/ |
        return fibonacci(n - 1) + fibonacci(n - 2)% M0 Q: F6 g  c) I
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# Example usage( I8 h" L& Z0 F: k
print(fibonacci(6))  # Output: 8" f3 g2 \) V/ R: ^
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Tail Recursion
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- x$ {8 D! o$ }+ m0 mTail 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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