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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 + Y8 E. x! _* m

8 F: [" }( H6 ?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
/ g' n. h! ~: U6 m1 H! h7 p3 r1. **基线条件(Base Case)**" C7 ^8 p$ N! X$ d, H$ W9 i
   - 递归终止的条件,防止无限循环7 j) R, j0 z8 M4 E8 ^
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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# L) O7 q/ u9 A6 t+ }/ q2. **递归条件(Recursive Case)**) B1 G, B) \' u# w7 d. ^$ C
   - 将原问题分解为更小的子问题
2 u2 h, j* o3 S! ]* l5 w   - 例如:n! = n × (n-1)!
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经典示例:计算阶乘$ T! i* H) g; ^6 \
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def factorial(n):1 L: W; U6 O) o
    if n == 0:        # 基线条件
# k* M4 z- P% ^8 f& t) w3 L/ p& f2 U        return 13 e7 y! k2 y! f! m7 x( w* u$ C+ d: w
    else:             # 递归条件
2 U5 m5 b0 m' B        return n * factorial(n-1)
! b( ^* u( D! M+ C6 d2 d) f- B执行过程(以计算 3! 为例):
& u  W: f$ t4 @% X9 Y9 Zfactorial(3)
% V3 O3 j2 Z. [4 H7 ^( `& h! K2 j7 C3 * factorial(2)
/ R8 y+ L1 L; i7 J  ^3 * (2 * factorial(1))
4 W1 f4 ]) H9 A% \, q- n% L9 a3 * (2 * (1 * factorial(0)))
+ e5 n4 u0 d4 ?5 D3 * (2 * (1 * 1)) = 6
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, {+ |$ O# x$ P. t' N2 u 递归思维要点5 q+ T( p$ P. [" V
1. **信任递归**:假设子问题已经解决,专注当前层逻辑
7 I6 V5 |9 j( e% S5 M/ h- q2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 ?6 F' N( v: v+ L
3. **递推过程**:不断向下分解问题(递)% {8 j5 i( ~- V$ \
4. **回溯过程**:组合子问题结果返回(归)( l# B4 @7 @# u& Z7 N; M# d0 [5 O4 a& g

! q) `( C2 o  C! ~* L0 H& \ 注意事项
3 `8 E/ S0 V3 p; X必须要有终止条件
2 n$ k& i5 J- ]7 Z: I" |$ s) q$ C9 b递归深度过大可能导致栈溢出(Python默认递归深度约1000层): I% [4 v$ L- O2 ?8 T3 J& F, ~
某些问题用递归更直观(如树遍历),但效率可能不如迭代
2 \0 p6 Z; R4 r7 W" S( Y6 z6 |尾递归优化可以提升效率(但Python不支持)7 P) t# Z4 s9 ~, r2 J

; G# I. k- F' C! {3 h 递归 vs 迭代, H7 n+ N, u+ z5 U
|          | 递归                          | 迭代               |* V5 Y" u# \( _( B- F. D4 G8 _
|----------|-----------------------------|------------------|9 l/ p. Y! c& y8 q0 v
| 实现方式    | 函数自调用                        | 循环结构            |4 [4 `7 ~1 j: X$ K
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
9 x# h$ a& l, c( Y) Z| 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         |
& j  O, l! C8 ]1 T( L7 H- f| 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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7 {0 _0 _# Q7 b% a6 q2 O5 M 经典递归应用场景1 V- E& @* T8 q/ j8 a3 ~
1. 文件系统遍历(目录树结构)1 t2 R+ R: q& ]/ B
2. 快速排序/归并排序算法
* i! h4 J- l+ Z3. 汉诺塔问题
8 u% y) I- k' ]( m9 ]5 J& ]4. 二叉树遍历(前序/中序/后序): T) l: S/ V3 Y$ w4 p
5. 生成所有可能的组合(回溯算法)8 w6 f  y* Z* {6 }. u3 h6 c# L

5 |; y0 |: Q- z3 }& F3 e* V/ r* j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,
; [9 v  I+ s) Y/ w5 g: }1 S& @我推理机的核心算法应该是二叉树遍历的变种。
) u0 L, ?3 o3 Q& H0 W' U1 M' 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:7 Z0 G  f5 y- k
Key Idea of Recursion
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A recursive function solves a problem by:
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4 m6 g# n& Q2 E3 c6 ^- d    Breaking the problem into smaller instances of the same problem.
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, O5 {  A' _. v6 @2 k! j    Solving the smallest instance directly (base case).
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    Combining the results of smaller instances to solve the larger problem.
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( ~# ?' h1 Q) {4 X7 LComponents of a Recursive Function
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/ R+ j( Z  h' h1 J6 b- t5 f    Base Case:
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, U( g8 M' x8 O. \        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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        It acts as the stopping condition to prevent infinite recursion.* [0 B) ~( l" Y% h

! W- O6 v! l5 y# @: B* H        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.5 M3 D/ w2 \6 ^1 u- T' z2 Y/ e( z6 \- d

$ R& L" i" ]* R5 V    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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' X0 m+ }* z' Y- t% \) v        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).1 P$ L: Q% `2 I7 M

/ x& w' v2 L7 m  ZExample: Factorial Calculation
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9 ]7 S) c( s9 r- T0 Q% RThe 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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, t3 e2 K% ]3 k    Base case: 0! = 1) B3 U% E4 g( ~. I& k

5 b; u8 p) u7 t    Recursive case: n! = n * (n-1)!, |! a/ Q8 W$ B$ Y9 C1 H. {! W& X- b" R

. S* ~5 w, `& q3 j  FHere’s how it looks in code (Python):  q% m4 o* a, A% i% }8 c- u
python
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1 `! t0 o% D' S: r  sdef factorial(n):
( G% V3 X6 R, L" F. E    # Base case
; j* ?$ J1 G5 L& d, S7 t    if n == 0:2 o% b5 k- q; W, A/ k
        return 1* }; K, w/ _5 p) K1 c3 J+ |- x# {
    # Recursive case& A) A9 Y; ^/ [8 ]: @
    else:
" W! K& W" D4 A. ~* m0 ?  f& i4 [        return n * factorial(n - 1)2 G8 A9 Y, v5 \  m  h. M' }" [+ c

( V6 E' D0 Z3 ?' |: \* j! X5 L# Example usage
7 D$ J6 {9 h' ^% Eprint(factorial(5))  # Output: 120
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9 `' k" s; n! j( LHow Recursion Works6 {0 ?4 @- R1 N$ g+ `% R5 }6 M: `

( b3 |: i* ^% E3 \) M, x3 L8 B  R. K    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.- l4 L. T& i& ^) ]- a3 x' s' ^
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    These returned values are combined to produce the final result.7 }, }8 h; l" ^( D% o" L
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For factorial(5):
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4 r6 x: F% C7 m, kfactorial(5) = 5 * factorial(4)+ k: ], u5 a) X# \- x- k
factorial(4) = 4 * factorial(3)5 _3 h# I! P/ ^2 E' s
factorial(3) = 3 * factorial(2): U# o( N( k$ {! S  D4 z
factorial(2) = 2 * factorial(1)2 G9 X, ~" ^- |, J) u- C& T" h2 ^( d3 S
factorial(1) = 1 * factorial(0)8 |3 c7 O# l/ ?" T7 @1 v% _! K& j
factorial(0) = 1  # Base case" D" `5 b# n" F
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Then, the results are combined:
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factorial(1) = 1 * 1 = 1
3 x$ e' J$ I# Dfactorial(2) = 2 * 1 = 2
7 {  A+ `' f, K; ?2 w5 lfactorial(3) = 3 * 2 = 6. n) A3 a* d- v5 n% d/ G
factorial(4) = 4 * 6 = 24
4 R' B" R6 M" \. Q. B4 y+ _* H! X* cfactorial(5) = 5 * 24 = 120: ~* P: F# _# r$ v5 J" m
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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).
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0 E. ?* ]$ A2 A& D    Readability: Recursive code can be more readable and concise compared to iterative solutions.5 ~' ]* u, i+ e& ]! D8 [

) H7 P, b8 J2 ^! \0 [& ]: S% kDisadvantages of Recursion" j( ^8 t& r. e! G& }* Y0 F
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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.
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).- N) [6 a1 _  Z* H" d
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When to Use Recursion
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2 z, q# Z0 v1 |1 J) X- O9 j2 g3 A. ^1 r    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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, x5 b# E- z  Q, }) D) q    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:/ |  ?" f1 C6 @+ q8 C
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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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python' K3 Q" I( y  q" u* f

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def fibonacci(n):3 N' b, W, H- t  A% e) i
    # Base cases) i) h3 e: l: ]6 z
    if n == 0:* \7 [& |, t1 q3 [  Q& W1 N
        return 0
' Y* z6 H# j, [0 w: b. R    elif n == 1:
; b- V5 ^# Q; [2 t" `$ c# F' w        return 1
& B/ g" S. {3 S: H1 w- ?' f+ G    # Recursive case6 j( j8 K& L7 ^3 r( R: q
    else:
( q9 W7 C) o/ O! {8 Q4 y+ \        return fibonacci(n - 1) + fibonacci(n - 2)
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# Example usage
( j' c, i7 B' y1 J- Q- Fprint(fibonacci(6))  # Output: 8
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Tail Recursion
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9 Q1 a+ P7 y3 [9 c3 n/ y7 _4 `Tail 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 E2 U. y1 I$ L& b, TIn 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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