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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 7 R4 X1 j, Y( ^. |1 R3 a

# W" r6 i3 I; x7 U解释的不错$ d" H, l0 y- }. ^5 ?: X7 `% u

' R" U* z8 W+ d8 z" V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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  a' }( F+ j2 ^% r; S3 h 关键要素
; u7 q4 y, \4 p( f0 L6 [1 p5 h1. **基线条件(Base Case)**# Y. X8 _9 c4 v  w! ]
   - 递归终止的条件,防止无限循环
$ h' m# ?% L/ ^1 s1 p  A1 B' o# v& y   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1' r$ e& m* N+ f  d

; K# D5 s' Q: @' b2. **递归条件(Recursive Case)**
( a5 S& b# W# I, k) q   - 将原问题分解为更小的子问题
5 B  V/ `0 u* r5 I% @3 f5 [   - 例如:n! = n × (n-1)!: x- h3 {( x* y- C3 i9 m* O

7 v' L" n7 {, {) r 经典示例:计算阶乘
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def factorial(n):
$ b' ]# r7 a% Z: G3 U, L    if n == 0:        # 基线条件
) _2 a- t7 G6 }) d: P! |: i        return 1
' s5 C2 x' {. e+ ^8 R, X    else:             # 递归条件7 e7 `: u2 r3 _" J% v
        return n * factorial(n-1)
; O8 K6 D+ D# C0 V% z' T执行过程(以计算 3! 为例):4 s- Q7 G' u0 R* F" g
factorial(3)
8 q* m8 c$ ~/ s! f: U8 b0 l, G% X3 * factorial(2)5 X5 Z( s  ]1 i. t9 k
3 * (2 * factorial(1))
9 C% D8 m; Z( B5 b, q" h3 * (2 * (1 * factorial(0)))0 C  y. S- n* D! \1 j6 Y2 q; T
3 * (2 * (1 * 1)) = 6: N; p! Q. ]' r  V

3 k. H; w! s. w. [! J& v. g 递归思维要点" ^  a3 y4 f- B
1. **信任递归**:假设子问题已经解决,专注当前层逻辑! c7 W" d3 d' L& e8 d( w
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
+ i, }( D6 y0 s1 C3. **递推过程**:不断向下分解问题(递)! X& i, Q- R4 e# [
4. **回溯过程**:组合子问题结果返回(归)* F4 k- M0 X5 O2 u$ ^7 t
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注意事项
/ s; M' Y7 k7 f0 N5 @- k- Z$ g  x必须要有终止条件
# e( I+ N. c7 Q0 z递归深度过大可能导致栈溢出(Python默认递归深度约1000层)8 ]1 E1 J! O* J' Y' {! I2 a3 U9 J
某些问题用递归更直观(如树遍历),但效率可能不如迭代8 a9 G+ W) ^: w4 ?
尾递归优化可以提升效率(但Python不支持)$ p3 y6 `) O; x' j3 h7 m* ?

5 @5 L& Y7 j& {5 y  q& Y9 Y 递归 vs 迭代' `5 v4 B  e& ^4 w4 f, M8 K
|          | 递归                          | 迭代               |
, X0 Y5 v; c9 f3 G8 e8 f; C|----------|-----------------------------|------------------|
* m3 h7 K1 j$ H( ]8 q| 实现方式    | 函数自调用                        | 循环结构            |
" H6 c$ l& \! `6 Y$ \| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
" ]6 c4 h* T& i/ x2 o  A' C9 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         |9 x+ M' J) [5 f# h
| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |$ s8 e# h( S$ Z' |) U

3 `3 h0 U! W, j% n0 U9 u! _  ] 经典递归应用场景
  Z1 f: Z( j( S0 V1. 文件系统遍历(目录树结构)
0 {8 e0 z5 v8 |' l1 r3 O- u6 i2. 快速排序/归并排序算法/ v% @  }( r' |8 t
3. 汉诺塔问题
* g# b5 }0 ^; M" e  U; V* M% Z4. 二叉树遍历(前序/中序/后序)
! S% u4 F6 I- L; b% D0 n5. 生成所有可能的组合(回溯算法)1 a5 m3 ~, [0 B. d6 V

1 |' \& Q) d, b  W8 s" z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,
! Q+ K+ q/ ?9 X3 ~我推理机的核心算法应该是二叉树遍历的变种。* G9 L$ o6 v+ P& C! X; E8 m
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:- e" e: B0 o$ N& R( c) }& I2 [' W
Key Idea of Recursion3 O* I5 Q# ]6 R- b1 s4 y

7 B8 S+ L6 s" lA recursive function solves a problem by:
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- s9 U# X4 O/ P$ |    Breaking the problem into smaller instances of the same problem.
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; |% E) S0 d1 H3 D+ [    Solving the smallest instance directly (base case).; M7 k% G0 j8 O5 o4 ^3 J

' J* ^/ _% U  D    Combining the results of smaller instances to solve the larger problem.( g, C4 {9 Z$ w$ Q3 d
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Components of a Recursive Function
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    Base Case:
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% M# g( c- f+ J/ n8 C- ], S        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.* i: S" w5 j% h) K$ S& l

8 e7 r/ @) _+ T7 S9 k8 U' i        It acts as the stopping condition to prevent infinite recursion.( N* g0 N: w8 `9 n$ i& Y) v

; I4 g, f7 c3 o3 |6 k+ q- j        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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1 G) w* N" r9 R' x& |: B% H    Recursive Case:% [+ o' \! c, J% C" o! d& I0 a/ X( J/ @1 a
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        This is where the function calls itself with a smaller or simpler version of the problem.9 c: M# A" N* f2 p; g2 p1 A9 G4 J7 A

( ^. \2 n9 L, C6 U2 X2 S! |        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).+ p& H7 Y+ K1 Y/ c

: Z$ B+ }! [- ^9 J0 j1 J6 RExample: 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:' Y# H  d4 H, U0 P2 s7 B- e

$ z  e# F2 O  M    Base case: 0! = 17 H; m* S: w8 H, ]4 R+ R

. f% D/ j' q3 F/ E! g    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):
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def factorial(n):. U+ I8 ?# m6 ~+ h- j( J- c" X
    # Base case
+ V: g7 Z+ Z! j    if n == 0:6 h# y2 X6 g8 ^2 y4 u7 E- L
        return 10 i( ?0 t, c" l% s- e) B' U
    # Recursive case3 f  y1 j( Z& ~- H1 Y9 X
    else:
8 T5 E' J9 W5 h+ |3 u        return n * factorial(n - 1)
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! Y3 J$ i9 }9 M- g  \0 B# Example usage5 u: [- A# N& z# E: D+ [# E- l
print(factorial(5))  # Output: 120  c0 p& t1 _0 s' J4 X/ R
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How Recursion Works
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8 ^3 [9 {( n: f& m( |. {9 _* S4 ^- f    The function keeps calling itself with smaller inputs until it reaches the base case.; L1 s0 B1 n# r

; z8 a& c1 ]" N    Once the base case is reached, the function starts returning values back up the call stack.: Q% r; z& n9 \" |
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    These returned values are combined to produce the final result.% i1 e$ I+ h$ Y4 {/ r$ J
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For factorial(5):; H6 W+ p% E/ J# |& d! y9 C

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factorial(5) = 5 * factorial(4)( E/ \4 S+ c) a$ \" q
factorial(4) = 4 * factorial(3)9 q0 k' m; \- p
factorial(3) = 3 * factorial(2)9 Q$ e2 L. S/ j' S& G
factorial(2) = 2 * factorial(1)
0 w: S' E+ z6 W8 g" Tfactorial(1) = 1 * factorial(0)- U; ~$ ~( ]* U9 b
factorial(0) = 1  # Base case
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- f: f7 i5 A$ DThen, the results are combined:3 u8 f4 k4 \. t# B

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6 P6 E/ t1 V' l8 F/ ?. w: `; {7 Zfactorial(1) = 1 * 1 = 1: j& P2 q8 T% C! j& L6 N+ X
factorial(2) = 2 * 1 = 2
9 K. D2 Y# }+ j/ Ifactorial(3) = 3 * 2 = 63 D/ o. |# ^7 ?7 U. a* k' _
factorial(4) = 4 * 6 = 24
2 `- O) c- Q. R" T7 w5 Q' dfactorial(5) = 5 * 24 = 120
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Advantages of Recursion  w  \0 _, q3 |" _+ n7 V" V5 S: O
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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).2 P. H2 e- `5 f+ Y. v

4 J3 q) ]+ q. q) o' D  ]    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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Disadvantages of Recursion
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0 n8 X% e; @4 S5 @0 Z    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).
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When to Use Recursion
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    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 Sequence0 u# z4 J9 Q1 k, _& K+ O

# |  N. H: n. t7 ?& SThe 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
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)* ~0 L0 \( n. s7 l
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python
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def fibonacci(n):2 c& m$ c  S" K
    # Base cases! B  e; @% R0 P4 L% \- b5 B
    if n == 0:
3 E. f% j4 M, v, ]( I        return 0
  w" y- J, @9 b" N) A    elif n == 1:. m  a/ ~' N$ u- L
        return 1' u( H( m0 Z  D5 q: X
    # Recursive case
# n( J4 V" V4 b  O- d* o- s$ @    else:
$ ~  M' ^: H0 E7 ^        return fibonacci(n - 1) + fibonacci(n - 2)
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( Y" b7 e$ O# a! T* X- i# Example usage* q. u3 G+ t5 W  h
print(fibonacci(6))  # Output: 8) F; B. W- L% |( O" X

: W: W, C; T2 GTail Recursion/ c) B7 n" W6 |9 A6 L' L5 [

' T6 I; |6 r" s% cTail 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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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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