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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 ' M2 C0 y, ^( B* q, b4 ]1 T' J

+ ~  H- S4 F' B% I- E5 f8 E& O解释的不错
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7 z; y  e! \9 D& p! g0 M; B) {; B! X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。) G) w4 B4 d' ~4 a/ t# J, e* J

9 W! d( {5 P. Y" |1 j; ? 关键要素2 p/ v9 p% m3 T
1. **基线条件(Base Case)**
' o" Y7 C, M. }8 ?1 @0 F; ^   - 递归终止的条件,防止无限循环
" q, o8 ^/ M1 q5 \; S4 n2 I9 C   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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5 H# |  R$ R, i( B2. **递归条件(Recursive Case)**
- u6 N: D1 Z& @4 X  f3 z   - 将原问题分解为更小的子问题. ]& r; a, V9 h; x. V: {
   - 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):
/ `# w7 `( s2 u5 j    if n == 0:        # 基线条件0 |1 z% s- b: U% e) Q
        return 1
; D0 y/ O+ O7 _' t% n2 h    else:             # 递归条件! O5 d' X( L& B1 Y8 Y9 z
        return n * factorial(n-1)
8 B3 O7 N# U( K! Z! E; r6 B执行过程(以计算 3! 为例):
; w' i- W7 K. ^$ ufactorial(3)6 v+ n) Z3 v; g9 ~$ `6 w0 d
3 * factorial(2)1 [* d- J' r* h7 H- t" u
3 * (2 * factorial(1))9 @2 T! }+ h9 R; f/ q4 f
3 * (2 * (1 * factorial(0)))
3 d- p5 ^! J* N6 r: A* u) ]  s$ H3 * (2 * (1 * 1)) = 68 I; a, i; I8 L' U2 K, Z3 Y: J

' `" q7 _! P- ~' \8 e 递归思维要点
" e2 [0 \+ |' K! ]6 B5 r5 L1. **信任递归**:假设子问题已经解决,专注当前层逻辑
$ }$ K; Z9 i- w2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 G2 @- k1 T" _+ Z
3. **递推过程**:不断向下分解问题(递)
, M% F0 g! m3 s3 D2 z7 N4. **回溯过程**:组合子问题结果返回(归)6 |  e, X; h( W: p5 g5 V7 u

! I6 ~0 S  _( g2 |. @ 注意事项
2 c. u) e) C1 n# A. H! U! b必须要有终止条件
; t) r3 t0 R/ ^( W递归深度过大可能导致栈溢出(Python默认递归深度约1000层)9 v7 Z+ a0 A# q% P% c4 B% l1 a: _" B
某些问题用递归更直观(如树遍历),但效率可能不如迭代8 _0 Q) j* w  N/ I' {1 A
尾递归优化可以提升效率(但Python不支持)( v) O5 |. R- b# f

% n) t$ `( ^* V% Q* a 递归 vs 迭代
$ I& Q/ N& q( O8 y$ T% E# t|          | 递归                          | 迭代               |
$ x2 n1 k0 K9 u" B* z2 n* [+ p- X% A|----------|-----------------------------|------------------|3 Z$ H0 }( L& n( S: P3 w. ^
| 实现方式    | 函数自调用                        | 循环结构            |
; Q3 |3 @# k" v8 G| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |( r/ Q, @% H% Z6 G' ~% A
| 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         |
6 z; k4 ~! I6 E4 T$ j* ~( \  V| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |1 b# A. i8 F) k: Q( P( p% E, C/ S
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经典递归应用场景
# Z% Y5 m0 p' x1. 文件系统遍历(目录树结构)* ^% X- o) f+ ^5 p/ s4 M
2. 快速排序/归并排序算法
) A% L. Z; m: V3. 汉诺塔问题
3 v% x$ Y5 p$ y# S' |& c4. 二叉树遍历(前序/中序/后序)
: u) z. L! [: F5. 生成所有可能的组合(回溯算法)
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1 R0 H; t" W; i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 W: A. Q  @3 n( M* D
我推理机的核心算法应该是二叉树遍历的变种。
7 R: g( r3 V" O- |8 i7 V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:
/ ]6 p! i) R0 c. F  o- EKey Idea of Recursion% k$ l8 {) q$ C3 k* d: X8 v0 j) R0 X" J, G

% J0 x; J) u+ e5 L$ ~8 VA recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.9 v- s  B9 h1 L/ Y4 O& [2 w

' t! M$ q. {: ^: U! u( {    Solving the smallest instance directly (base case).
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8 m! w. r2 _- ]+ p    Combining the results of smaller instances to solve the larger problem.  a* ^* J. g. W& V7 {+ i, p
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Components of a Recursive Function' ?1 R& }5 {' L
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    Base Case:
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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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- h% \9 ?# T" c2 ^5 S        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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9 Q, p+ `( l: l" H3 P0 b2 d6 }3 N    Recursive Case:
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' v* W- m7 ~# ~! W0 }7 i! y3 B        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).  q, `2 R. N1 N' Y" G! h' S

- K! w) M" E* ?% U4 k9 BExample: Factorial Calculation
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2 S, m/ d8 U& d6 \- \  ]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:& M0 G( r8 X  p; ?$ r7 Q0 ^0 e+ s
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    Base case: 0! = 1
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0 j; q) l# [$ d2 C; y, r3 V: d    Recursive case: n! = n * (n-1)!1 j7 y9 a* B7 Q$ T; p, \
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Here’s how it looks in code (Python):
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def factorial(n):* ?4 ^: o* H2 Y
    # Base case! `) `2 L- q8 [. L8 d
    if n == 0:& Q. `& ?' @0 l/ d7 K: z# M
        return 1
$ J& n( G! t: p7 b% t: u) O5 x$ _    # Recursive case
( N2 e; K* _& W* A- _1 b    else:
! i- T; a3 K) K        return n * factorial(n - 1)- Z! U0 J5 X1 o' k6 }9 X* R1 V$ J1 U

8 T* A! H/ g3 ^- q( Z# Example usage
3 N( n6 ]! n" h( c, F5 Q) Eprint(factorial(5))  # Output: 120' K6 u, Z4 L, X/ n3 t& s: R" D
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How Recursion Works2 ~& [  W% }! }3 i* [4 ?& M
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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.
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    These returned values are combined to produce the final result.
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For factorial(5):
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$ v* h, ~0 h$ y" G+ P& b% Jfactorial(5) = 5 * factorial(4)
1 n) P$ D- m+ D) e; T( w" ~" X; W: Jfactorial(4) = 4 * factorial(3)3 A/ l0 @* d# f  w
factorial(3) = 3 * factorial(2)
5 J. {' D' f3 }- `7 Nfactorial(2) = 2 * factorial(1)3 L6 T8 @5 T) ]  u1 }
factorial(1) = 1 * factorial(0)* ~# U: q4 h- L* z. ]% Q+ L  J: W
factorial(0) = 1  # Base case0 |( ?' _1 r) a2 A1 o( A
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Then, the results are combined:+ T/ T" |9 S1 t' {! z
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factorial(1) = 1 * 1 = 10 H3 f+ U7 \9 `$ {& V" F  c3 v
factorial(2) = 2 * 1 = 2) L1 [! x6 Q0 q# J
factorial(3) = 3 * 2 = 6: A6 P  W+ ^- p4 {" W0 p: B
factorial(4) = 4 * 6 = 24
: T1 ^; C  G0 Y- \8 [5 W, j+ [factorial(5) = 5 * 24 = 120; g4 I) D; N/ C- E
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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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    Readability: Recursive code can be more readable and concise compared to iterative solutions.- ]% s. U# c2 Y% X

" V3 ?0 `3 q  ?6 H- E6 w. EDisadvantages of Recursion1 \/ E) O* T- U/ S( R- x+ d

: x. u% B3 r: ~0 ^# M    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.5 P7 k3 O# }9 \. Y2 c1 `
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)., @! S; K0 R. c: z( f

6 Q1 a6 ?' \4 jWhen to Use Recursion8 M9 c# m; l3 d
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# |* ?1 U6 r* a7 i

+ n" i( _3 j) ^. V    Problems with a clear base case and recursive case.
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& G- u, G4 K0 {Example: Fibonacci Sequence! ^7 z1 q  W9 V% c

7 C3 B* w1 @5 n3 a0 j5 n. eThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 @: V, n3 ^2 |1 n. w7 ]
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    Base case: fib(0) = 0, fib(1) = 1/ [/ p& X1 q; `8 D

0 [  O- N$ E( |; _, W  n' B    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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" J) s, r# p  {python4 F/ r/ D3 T/ p) G& @! E

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0 |' h8 O7 A5 v. q) udef fibonacci(n):
+ ~: ^4 |, I: d2 s    # Base cases
2 l7 ]+ W/ A( N+ d5 J/ X    if n == 0:
/ J+ }  }9 D: p8 [8 ^        return 06 J4 s0 Q' ~/ g; }
    elif n == 1:
0 n8 J0 h( [6 Z        return 1+ y& P5 K* J2 U2 A: m" C  o
    # Recursive case
! u7 M2 c5 j# G4 k* M    else:* Z  D( S/ k# o8 T* w' l% @* X
        return fibonacci(n - 1) + fibonacci(n - 2)
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# Example usage; f) R: z7 N( W" j- `/ U
print(fibonacci(6))  # Output: 8
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Tail Recursion+ s4 U4 @; @3 c+ R9 j. s
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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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; W& T3 j8 _6 k6 D6 ?: \3 wIn 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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