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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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解释的不错
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8 C& [2 A2 b& k) T1 e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。9 B/ j, g% h* {
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关键要素5 L# J- k* D& E( M
1. **基线条件(Base Case)**
( W4 {: K+ w/ V# F  x   - 递归终止的条件,防止无限循环
( `+ `5 I: Z4 t. a  a   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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% S4 O( r% D4 }/ y) A2. **递归条件(Recursive Case)**6 b7 r. b3 ]% B  r/ N
   - 将原问题分解为更小的子问题  J/ M. F: K' N9 W
   - 例如:n! = n × (n-1)!8 N6 L2 ^5 M0 X7 D: W
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经典示例:计算阶乘
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/ j2 v# \" V% ]def factorial(n):! u) ]7 x5 f9 F( H
    if n == 0:        # 基线条件# O/ O% P$ b2 j* c5 S
        return 1
' @9 \) r/ e- |0 @    else:             # 递归条件
% `) P* F; r4 g& d5 A& j        return n * factorial(n-1)
) }. F' }+ |. H3 B# v/ g$ J9 _执行过程(以计算 3! 为例):
, D* i$ t, h$ }7 T7 Gfactorial(3)9 f6 X7 g; b/ p) k
3 * factorial(2)  m& j% w$ f& b1 V7 x: c- k3 ^
3 * (2 * factorial(1))+ d  B! [& n& k4 F; H0 a
3 * (2 * (1 * factorial(0)))
$ L! y+ D; {! J3 * (2 * (1 * 1)) = 6
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; l4 R! Z) a6 v2 A, @- `. \8 k5 Y 递归思维要点
" r5 z% o6 U! ~( B! ?- y1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 i* d# U8 N- A$ c7 g% ~
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)$ Q' k7 d; E: Q- R0 ?+ [
3. **递推过程**:不断向下分解问题(递)
  s5 J0 X6 b4 d4 P, L! H- E# G4. **回溯过程**:组合子问题结果返回(归)
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注意事项
4 m1 ^* d5 t- e7 |) U) w% L必须要有终止条件5 @. K  O( r3 S9 w* X. ]4 r2 B
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)4 w( ], M, D! q" B5 ?1 _9 Y
某些问题用递归更直观(如树遍历),但效率可能不如迭代
% y4 j2 Q& V7 J+ ]  b( h7 u$ z5 O5 i尾递归优化可以提升效率(但Python不支持)3 t8 q$ O5 `' }! c8 _  C, i

: a% f. l; o: d 递归 vs 迭代
* F6 e. |" C9 n3 Z8 s|          | 递归                          | 迭代               |6 H+ `' n; X  U9 d8 u5 J
|----------|-----------------------------|------------------|
9 |, B4 b& {# U# |% p2 i& K: B| 实现方式    | 函数自调用                        | 循环结构            |! z) z% b1 b; B2 z6 U2 \# u0 i' z
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |. X% Q0 a) E& V8 J/ a, \$ @" B
| 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         |
, W9 w0 {4 s, {/ _- F  s| 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 L9 v" ~1 L8 V: J, |
1. 文件系统遍历(目录树结构)) L" c8 }% I! v% _! z* |, f7 g
2. 快速排序/归并排序算法7 }; \! ~1 x3 M, s
3. 汉诺塔问题
& O( ?% T; [0 y5 g2 l; {. d4. 二叉树遍历(前序/中序/后序)* T" B0 e: x8 X8 _! N* e6 L
5. 生成所有可能的组合(回溯算法)
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! Q% [$ D7 H% Z) x) w2 D4 ~! R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,
% P- R) g6 Z. Q* `( `2 t我推理机的核心算法应该是二叉树遍历的变种。1 N$ `! \. k* Y2 L' |( [, T1 q
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:" V4 z, ~" ^. ^
Key Idea of Recursion! K& g! @: _8 i
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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.
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    Solving the smallest instance directly (base case).2 M7 E9 ]4 |& N# W( C  U7 u0 r

  o" `. Y9 y. M    Combining the results of smaller instances to solve the larger problem.  X3 x3 L2 y3 u0 ^6 n' f# Q0 g
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Components of a Recursive Function
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, ^7 v2 S0 j4 [4 m$ T8 r    Base Case:! d2 W% g) Y, j$ }: @
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        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.3 f  I. D  K+ p5 b" M0 N

7 U3 N8 k1 _1 R3 D4 {% R: `! ]        It acts as the stopping condition to prevent infinite recursion.
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- ]; W7 q2 r* S- S! e        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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2 G* D( Z% e  W& |" j+ |    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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; N2 A3 q4 G. n  Z        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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. v9 G" J- @0 D2 _  M' jExample: Factorial Calculation9 E0 g$ B/ J0 N% S

% v7 Z" {, W3 D3 w* B) pThe 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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. V2 g- h/ g4 U. v/ ~: N- w! n    Base case: 0! = 1) I7 g( q$ I$ w, e: H+ J" f

* T% R" W8 x1 p' r, }. A2 d0 M% g    Recursive case: n! = n * (n-1)!9 V1 V9 U& S, K1 m$ C
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Here’s how it looks in code (Python):
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def factorial(n):
( A- Y1 Y* s! O6 H8 O3 p% b+ j    # Base case
0 [9 }. ~; h, d    if n == 0:
6 N1 `) J& N/ Q( a$ t* B        return 1' `8 R6 X1 r5 h2 T+ l) _) ^
    # Recursive case/ j1 h+ u1 O  M2 {# G  s- S0 ]7 w
    else:
& C* J% x/ I1 M# z4 t3 J        return n * factorial(n - 1)
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8 O& a. b- v1 X* k4 z! m# Example usage* c% D8 t" v5 Z& w3 b
print(factorial(5))  # Output: 120& L5 P. y/ l+ t6 |1 h
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How Recursion Works4 e* r/ n& C6 U/ A6 X/ B, ~
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    The function keeps calling itself with smaller inputs until it reaches the base case.4 [( I/ E* K* R1 H( t- e. ]

9 ?1 Q( j, U) Y, 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.
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+ D4 W/ n/ N6 X# `For factorial(5):
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factorial(5) = 5 * factorial(4)
/ l! [' D4 K+ W) U  B  u0 T' {& afactorial(4) = 4 * factorial(3)
8 h8 p: X! _8 r6 i* Afactorial(3) = 3 * factorial(2)5 O0 j+ W" n+ Y$ S2 m* K
factorial(2) = 2 * factorial(1)
0 `; m! o  \# o; ?* W7 yfactorial(1) = 1 * factorial(0)
9 }" X* [$ H/ u* }1 sfactorial(0) = 1  # Base case
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$ G* @6 ], O# b  wThen, the results are combined:
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factorial(1) = 1 * 1 = 12 n; Y7 x& v# q- |, {
factorial(2) = 2 * 1 = 21 v  G' \8 I# s
factorial(3) = 3 * 2 = 6& E) U/ F2 W: K. o* m
factorial(4) = 4 * 6 = 240 n1 c1 {7 L& S/ F# ?  p
factorial(5) = 5 * 24 = 120
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Advantages of Recursion
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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).  Q3 {* K7 A1 A+ y
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.; `1 k0 v- c& y2 B

5 w% F0 }9 y- S9 Z4 a/ @! F! G( E6 pDisadvantages of Recursion
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$ K" W, H6 J1 G( @8 {    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.# H# V& \7 ^) [
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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* M& Q7 h8 w/ S$ vWhen to Use Recursion! l. f7 }7 a- d7 C" `

- |) }; ~5 x' E: W' ?* }    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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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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7 p9 |8 q: v7 D: x1 ~    Base case: fib(0) = 0, fib(1) = 1; L2 N. ^& G/ F* _
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python
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. O( O5 c2 ?( @2 ~( b* Fdef fibonacci(n):
1 T: ?& `9 b# O5 e    # Base cases
) L' z: R$ P0 g! ]    if n == 0:
% Y- |$ w  s! j$ h        return 0
2 M$ d0 A7 d+ G    elif n == 1:
1 ]5 A" s! D( A7 T' p        return 1$ p$ x9 k, O- s  D1 ~+ O9 p; v1 m( `4 {
    # Recursive case
2 R7 i4 W+ c# O. Y    else:. S$ r. Y1 s7 I+ V
        return fibonacci(n - 1) + fibonacci(n - 2). v. t  z+ B- V1 |

. Y6 {8 X1 n  N7 }# Example usage
2 V5 d5 ~1 T& H; q* X3 K* sprint(fibonacci(6))  # Output: 8
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( f+ D% z/ V7 n; |Tail Recursion) a8 ~& N* R; L2 a' V

" L4 D# r( A  D' N% ZTail 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).; \% u$ @( n: }. v

/ ~) A6 {* m3 D% p% uIn 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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