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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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5 `9 R1 |7 y5 W解释的不错( r! L9 W$ k7 a$ p- I6 j+ R6 _2 Y
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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。* a# c1 R3 X& X2 t* L7 |

) j/ q/ z, q; O" e 关键要素. j" x5 Q7 H! u1 O$ M
1. **基线条件(Base Case)**1 B3 |& X* {+ x1 Y6 }1 p/ p9 N
   - 递归终止的条件,防止无限循环2 Z5 C$ R/ O$ l5 j1 u2 ]: o
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 18 b& J: Y/ c0 f1 H0 M

: ~; a7 W' y' Q7 b+ u: ~8 t$ w2. **递归条件(Recursive Case)**. ?0 {& V( \$ q
   - 将原问题分解为更小的子问题* u; U2 Q" R2 f1 V/ \# p- ]$ r6 ^8 q
   - 例如:n! = n × (n-1)!7 S. e% }8 q0 M( H! e

* W& Z* u% D  s3 ^+ `& v3 U 经典示例:计算阶乘
4 s# Q0 c" N* Y+ g  ^4 @python
: `* s$ p. l' L7 F' sdef factorial(n):
$ F; V) ~4 ]+ W. N    if n == 0:        # 基线条件
# Q1 V4 O/ S8 j+ V% ^! ^3 ?& ?        return 1$ y% b$ }5 C- B, S
    else:             # 递归条件
* Z. x% _9 G5 p, I/ O6 l        return n * factorial(n-1)5 F4 S( ~4 C( D# g3 ]
执行过程(以计算 3! 为例):2 P' n; w' W! m$ o
factorial(3)1 b0 m4 t4 D/ \
3 * factorial(2)
0 G$ M; Z5 p) U0 _- M/ y3 * (2 * factorial(1))# r. K3 G; }0 |
3 * (2 * (1 * factorial(0)))9 _$ [) S0 _' b+ H* b* Y
3 * (2 * (1 * 1)) = 6" B' K! t9 f) X& f
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递归思维要点5 j# y& ]8 E3 r* X8 W* A) g4 ^
1. **信任递归**:假设子问题已经解决,专注当前层逻辑
8 A# G- {% l" C6 I' t. q& c2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
5 Q+ Q8 u- B3 ]1 F' B3. **递推过程**:不断向下分解问题(递)' P8 ~' T1 W/ S$ I
4. **回溯过程**:组合子问题结果返回(归)
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& z) z% F' J: ] 注意事项
! v' b$ Z4 y8 p& |3 T必须要有终止条件: |0 F* V( N5 D# W1 T/ S3 n
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
( K* t$ L/ u$ v3 r某些问题用递归更直观(如树遍历),但效率可能不如迭代
3 t: R3 Z) ~' b4 }' Z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
) }- ?( ?. S' z2 J. p|          | 递归                          | 迭代               |1 H( n! J& |6 w& q# O
|----------|-----------------------------|------------------|. S+ e* M, a, [6 L2 j+ ]" P) P' |& r
| 实现方式    | 函数自调用                        | 循环结构            |
; n# D( E) ]! S* b8 P| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
5 W; P8 F! c& L| 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         |
. j5 ?) }; ^! m% @, `' o# P0 O, 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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# s' E$ N* N: V3 ^ 经典递归应用场景
8 G, s' J8 i. Q0 k1 s0 M: X# E: H1. 文件系统遍历(目录树结构)
2 w0 z0 {: w9 r; a& w" N9 B2. 快速排序/归并排序算法& u* p. \8 Q2 N- n8 X+ F) e0 Q
3. 汉诺塔问题: A- Y0 X; j3 L
4. 二叉树遍历(前序/中序/后序)9 W& D% }& I8 Q8 ?" c# d$ V) l
5. 生成所有可能的组合(回溯算法)- S1 _+ Y4 W. }* S+ d

2 }9 l* i# G! [9 b' S2 E: I5 j2 Y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,
$ J+ Z' z' m" t: C! Z我推理机的核心算法应该是二叉树遍历的变种。
, R: U  i' H3 s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 T( o( F4 o9 Y: s1 }. c3 q) \! O
Key Idea of Recursion
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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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! ~+ o; x8 W, c2 h& @" t% C% v    Solving the smallest instance directly (base case).
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    Combining the results of smaller instances to solve the larger problem.- _4 T" s3 V, c! c5 O
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Components of a Recursive Function
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    Base Case:
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6 U8 D  X1 T* ^/ i( P# M/ h        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.
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6 y  L5 l7 ^7 N9 F' q        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.# k' Y0 W" O6 ~2 ]6 g4 B, m
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    Recursive Case:
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; K0 }& z  V6 W  G: p5 ?        This is where the function calls itself with a smaller or simpler version of the problem.+ n0 ?& _4 m7 _4 t
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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Example: Factorial Calculation
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! x1 b( V3 ^6 E$ p* S# R3 KThe 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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9 Z1 u1 n  f. e$ W# [5 p% H% V( j    Base case: 0! = 1
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    Recursive case: n! = n * (n-1)!0 g5 M3 y5 `) y' h

/ q2 p' t7 A: N) b, p, {Here’s how it looks in code (Python):
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2 T1 L4 q, ]& b) e6 xdef factorial(n):$ B; {$ i0 k" P. u0 e# `! \
    # Base case
2 E; \( q; R% L5 }    if n == 0:
- m# D8 I0 J1 H        return 1
6 ]9 Q1 t1 Z8 Q9 H+ r+ m/ u+ h    # Recursive case! v5 D5 {9 X4 _, G8 b& y
    else:/ A& w- [/ r; d7 D
        return n * factorial(n - 1)* i- r" D; }6 q3 \1 ^; n( P5 i

+ W  O( c2 l. b% p: }/ F/ I5 B8 `# Example usage6 N( ]- ?0 j4 P2 j: [
print(factorial(5))  # Output: 1201 c2 W8 [# X( ^* \

4 n0 Q& N- A! E8 O. VHow Recursion Works9 M" @  s0 C. @" b

) E( ?9 w. z. s; t- ]    The function keeps calling itself with smaller inputs until it reaches the base case.
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9 w, p8 M: h* \- ?0 N0 [1 F3 |    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.2 N# E& u! ]4 K% ~: k
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For factorial(5):7 N2 a" A+ ^! _9 J3 o. A% V6 v
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factorial(5) = 5 * factorial(4)/ q2 ?3 G7 O, v; v( x
factorial(4) = 4 * factorial(3)
6 N  }/ _# i* K- c6 _factorial(3) = 3 * factorial(2)9 k8 P: m& u; o
factorial(2) = 2 * factorial(1)
3 E1 T, c+ Z' W& n! Yfactorial(1) = 1 * factorial(0)
7 s* i$ N8 X; ?3 c) efactorial(0) = 1  # Base case
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, W' Q" h9 p- |* I9 k% BThen, the results are combined:- x% K& k0 I% {3 u" B

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( o' W" V3 M4 J' @8 Nfactorial(1) = 1 * 1 = 1
: ^5 D/ W; P. F5 ~5 gfactorial(2) = 2 * 1 = 2
# A5 a; ]* M6 Q& Mfactorial(3) = 3 * 2 = 65 x+ Z. p, q! C+ m& p; R
factorial(4) = 4 * 6 = 24
3 w$ ~0 {) q( u) z! ?factorial(5) = 5 * 24 = 120
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+ v  o) t( Z9 X2 a( \, O% jAdvantages 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).* W. e, E; e/ F# x
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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Disadvantages of Recursion' q6 a8 X( H7 ~& p

: g8 l$ V  D; k+ v# E, _# u" ]    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).% @( i' C2 t  f0 V/ O3 [; U
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When to Use Recursion( Y4 b% ^( [6 e  T& b" P
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)., i5 E9 |% s7 d6 h) g1 [

7 q+ A' t: N+ H( m; e9 L0 W: Z    Problems with a clear base case and recursive case.
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+ T) B  g/ B' r$ Q& kExample: 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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    Base case: fib(0) = 0, fib(1) = 1/ u, ]8 R; y* `
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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3 g/ I" K( @4 X: Y* \( J' vpython
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/ v% l- ^9 z. a; ^def fibonacci(n):
# d& M: m6 ^& |5 v% V    # Base cases2 K" C5 D& C  T5 |  l
    if n == 0:
) _% y( C" F5 H, o' k% }        return 0% v, A) r5 M0 P, x0 a
    elif n == 1:" c* B$ E* V$ }1 F! J  q
        return 1& O* N  G) l6 ]+ t3 D
    # Recursive case
8 V( D0 n6 a3 s% Q" A% M3 D5 l    else:- C3 c, z2 s1 k& {% X5 F
        return fibonacci(n - 1) + fibonacci(n - 2)" Q% |; u1 G7 u4 T

, f0 u. ~; L& m5 n& m# Example usage" z3 J( P* u* e+ Y. z
print(fibonacci(6))  # Output: 8
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) l4 Q, V  k: q- aTail Recursion
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3 K6 T) u4 k8 Q" p) l" ]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).+ p9 \9 `  k: J1 W
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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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