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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 , [( L! t) g  t' x, z: o

$ x3 p6 J+ N0 B( W% j解释的不错
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  g( x1 m, m8 a" P3 o) c- a4 l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。; N# s, y" F  o' z9 ?, X  j& G1 ]
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关键要素
: n$ E8 N" H! h' [$ o3 i1. **基线条件(Base Case)**
  _" C7 A5 P. I' D- _1 H8 L" k   - 递归终止的条件,防止无限循环
# U% s8 X1 Y% o   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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' g% r0 k5 K' r+ @8 `' }2. **递归条件(Recursive Case)**
# `9 \% b% \+ U8 i; l0 U   - 将原问题分解为更小的子问题  D' D+ F$ w% D/ A5 V5 |0 ^. r
   - 例如:n! = n × (n-1)!
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+ g+ d  z  \% E0 t8 X6 H( Y: K 经典示例:计算阶乘) m- y$ Q' z; d& x3 a' }
python
/ c/ Q% k. j2 c$ sdef factorial(n):
3 B: \5 C7 T5 _' M0 E    if n == 0:        # 基线条件+ [( k; z3 K$ q6 o
        return 1$ B$ r! `+ e* o! J! F8 h
    else:             # 递归条件
1 r: r( ?4 L; }" u& G6 P2 Y  S; O% k, J, M        return n * factorial(n-1)% }& \6 C4 i: U& I/ w: V; K
执行过程(以计算 3! 为例):& U& i# j- s: x  x
factorial(3)' O# ]7 ?* m% C8 S
3 * factorial(2)
/ f( m+ U# X  @: g% ^( h3 * (2 * factorial(1))* U# q, [: [; \+ k% ^$ W$ S
3 * (2 * (1 * factorial(0)))4 F% ?* p9 P) B. I1 R% ~* r8 g
3 * (2 * (1 * 1)) = 6
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递归思维要点
) y" Y2 U7 E0 Q5 `4 d) x: d# k1. **信任递归**:假设子问题已经解决,专注当前层逻辑, D9 y2 ~+ R, P$ U  o  X
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( j3 A2 v7 D8 f1 s9 l
3. **递推过程**:不断向下分解问题(递)& f2 a. \" X- }! r# q4 n
4. **回溯过程**:组合子问题结果返回(归)) J" n( m3 q' r2 l

0 Y4 p* w; r& B. l* I- u 注意事项3 Y5 z, m% }' E9 ~$ H/ h- @
必须要有终止条件8 h9 n. c$ y. W4 `0 X8 P% `
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
! A9 V. Z3 l7 o, P某些问题用递归更直观(如树遍历),但效率可能不如迭代3 @1 r2 w, Y( {+ [* ^% i" ^. ?
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
; r0 V( y: e( M+ D5 z|          | 递归                          | 迭代               |
* I, J/ \" H: g: ||----------|-----------------------------|------------------|- e. ~3 x4 M" a, D$ Q! |" F7 P
| 实现方式    | 函数自调用                        | 循环结构            |
. i. ?7 u$ [6 H+ Z, R/ z/ ~1 @8 e0 f( d, D| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
$ D, q% x) Z+ T4 ^8 {+ T| 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 @" T! C4 ], b; t! e  J
| 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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5 A* S% j& B  S* r% ~4 Z: ~: M% x9 d 经典递归应用场景: o; e; c& d! }/ I. ~' {
1. 文件系统遍历(目录树结构)
* W" [( e, Z9 M& J. A8 c4 p- N2. 快速排序/归并排序算法- v; B/ o- D/ r7 W, _4 f* J0 ~- b
3. 汉诺塔问题
$ w! e+ B0 [" a/ [4 t# n4. 二叉树遍历(前序/中序/后序)% D) y5 ^; E" c; w3 Z* V- \
5. 生成所有可能的组合(回溯算法)* Y0 @! |/ G& J' y2 u$ p

- n: _; S1 L! _- c. S! k" h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,& g9 m: H" x; f. Z% B+ ^8 \
我推理机的核心算法应该是二叉树遍历的变种。& [6 B) h! @# H1 J; i6 @; b8 [! f
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 A" A* y% ^* F
Key Idea of Recursion7 o) O* V5 Q! S, n

- b  }5 G* b* _  U: zA recursive function solves a problem by:
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6 \7 N( |: n, k7 V( r- @5 |    Breaking the problem into smaller instances of the same problem.
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    Solving the smallest instance directly (base case).
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    Combining the results of smaller instances to solve the larger problem." b& O2 o& p" h& }7 a
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Components of a Recursive Function
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1 M( Z0 j6 z" T2 z  Q( a& \    Base Case:/ G6 j; d; L% a8 `
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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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        It acts as the stopping condition to prevent infinite recursion.
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8 Q  T2 Z2 c" K) ]        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ p! `$ V$ t6 A( U2 }4 \
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    Recursive Case:
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' P$ K' ]+ d1 W+ w% B        This is where the function calls itself with a smaller or simpler version of the problem.- O# c# ?* G4 F

8 g# c+ n1 s! C$ P( S& d        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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Example: Factorial Calculation
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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:
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    Base case: 0! = 1; U; Q3 I- `. q% I; f* f/ m
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    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):, F$ V6 @9 P3 V& g1 O
python
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def factorial(n):$ ~; L6 _& t) y0 u
    # Base case
2 |8 C' y( r+ {2 @# M    if n == 0:1 a/ w9 w- [/ f$ ]6 X
        return 1
: ^  T  A# b) {( t! R- L- e2 B    # Recursive case- m  b" w( }, I
    else:; U0 x" x' S" ^( t2 v
        return n * factorial(n - 1): U/ r6 ~  C$ h& u4 F
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# Example usage" h: k. X4 h2 o+ a, F3 D, T7 C
print(factorial(5))  # Output: 120) Y: K5 C' j* T& j* Z
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How Recursion Works$ }  ?9 j% }6 E
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    The function keeps calling itself with smaller inputs until it reaches the base case.7 ~0 `: e5 H: j0 @! K4 L; J$ I
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    Once the base case is reached, the function starts returning values back up the call stack.
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: k9 x; ]& P. u2 j    These returned values are combined to produce the final result.
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5 F, B3 O* j) G9 L" f) x) m4 yFor factorial(5):' H& p" Z* k: A1 j/ z

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6 E1 j9 L3 v5 W5 U% W" x! }% A5 lfactorial(5) = 5 * factorial(4)7 c3 W+ X  {3 t0 i7 u( ?9 m
factorial(4) = 4 * factorial(3)  i7 @1 @" u8 u3 d7 n" T# w& O2 b
factorial(3) = 3 * factorial(2)
" ]! ?0 m% f& d/ mfactorial(2) = 2 * factorial(1)
2 A2 o0 R2 d# }% Gfactorial(1) = 1 * factorial(0)" F: D: A  x6 R7 X6 q) r( x
factorial(0) = 1  # Base case
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Then, the results are combined:
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factorial(1) = 1 * 1 = 1
9 l  [( r  l) U& o. Ofactorial(2) = 2 * 1 = 2  n) E; p# Z5 z' N: ^/ [
factorial(3) = 3 * 2 = 6
0 C. |" ^) [2 b7 Jfactorial(4) = 4 * 6 = 240 v1 ^7 E- _( {$ E& T" B
factorial(5) = 5 * 24 = 120
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Advantages of Recursion
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4 w8 u2 l6 e( x" K1 F/ a    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)./ a# r" k' }1 n+ s. k$ n, f: k
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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3 m& N/ _7 }- YDisadvantages of Recursion
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& }5 N" G( [6 X0 O. a, R    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.% E( e  G, ?5 N& Z

6 L( V) [5 v4 |& r, g5 p6 {. g* L    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).8 j! Z+ N; g4 p! w6 L) H6 r
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When to Use Recursion3 I  [8 a# `: e8 H% y3 f/ l
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).- b* [6 U0 L% s- o9 o* G

" X5 A# a: }/ _    Problems with a clear base case and recursive case.7 k8 g' u7 B/ E6 k5 q" u

# B$ R% C  B' x" k* KExample: Fibonacci Sequence! h3 Q8 p* P/ F8 A

4 _; p7 Y& G3 O3 \1 }8 uThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:0 R& \: V- c# H7 G9 D- L2 x
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    Base case: fib(0) = 0, fib(1) = 1* @8 u* N5 w5 D. h+ r

- Q" H& w, T1 l9 g  e! F, B% d    Recursive case: fib(n) = fib(n-1) + fib(n-2), F' f+ u! A# _0 {2 d$ t
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python  r1 N, K9 d% l. V
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8 [4 V$ v+ g4 K3 s/ H: t; B6 \# ~# Z% Idef fibonacci(n):. T6 Y2 I9 w2 F# B' @7 ~' h$ E
    # Base cases
6 j4 r8 C! o- ^, r& @    if n == 0:* E. H0 s' {' v  [3 f' m
        return 0. I: t! M6 g4 z
    elif n == 1:
; m; R) [: \/ h2 g        return 1
( Q$ x3 [1 u8 Y& t) M8 _+ W    # Recursive case
/ N+ D. E* Y- U7 ^* j8 N    else:5 L) \8 I+ x! |) q
        return fibonacci(n - 1) + fibonacci(n - 2)3 w6 w* j& }. \! R3 E

! @1 O. i3 {, U2 O5 E" G; \/ t. n' h# Example usage- s# \* |) N( Z1 K' l5 m5 f5 D
print(fibonacci(6))  # Output: 8' v  O3 f# A" K3 Q. y( U
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Tail Recursion9 d; A3 G3 v5 x" 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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; J5 E; \0 [! L( b" R0 SIn 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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