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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 5 d0 V$ {+ Z8 R" K9 Z, ]% L

; v: G. t3 `3 E; Q. U* Y- t解释的不错4 |: [* s9 Y+ e% ?3 U

' q% y8 j5 m& E8 T& [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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: X1 ^& k0 d& x' C0 i5 W$ A" i4 S 关键要素( g0 y7 V7 p( G5 L; o
1. **基线条件(Base Case)**2 `0 C2 b( c1 o7 S( L3 ^2 }
   - 递归终止的条件,防止无限循环
% X1 R' l2 A  X9 c5 l( S# p) |( x   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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: m( Y/ d) I; o2. **递归条件(Recursive Case)**: A+ C$ ]9 B% P6 c
   - 将原问题分解为更小的子问题
. i" {/ n3 s% Z   - 例如:n! = n × (n-1)!+ p" {: c8 ^0 T4 c3 B8 x# V# \

  `2 {; k! r0 n  S 经典示例:计算阶乘
; [5 o4 l( a$ R" Spython
+ S5 m3 V( ]; [* I; Ddef factorial(n):8 h4 R% S4 w3 b1 i4 r
    if n == 0:        # 基线条件& `' W+ ~6 j9 o) {0 Y5 K
        return 12 p2 p: q- T; r9 ?2 F
    else:             # 递归条件
! C* {6 Q6 U# E( ~9 c# K        return n * factorial(n-1). Z0 f$ u& L; c( q4 m
执行过程(以计算 3! 为例):
$ n0 ~' }( B* @factorial(3)' r! Z( t2 V; O7 u) B
3 * factorial(2)- n+ H% K( U. P( F7 m
3 * (2 * factorial(1))
$ }2 O3 _& w6 O3 E9 H, Z3 E8 H3 * (2 * (1 * factorial(0)))3 ~1 }  {0 o8 b, p/ K+ ~" p
3 * (2 * (1 * 1)) = 6
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2 _+ T4 G6 S. {8 A  O6 Q% J. q 递归思维要点
. G; f0 r. v3 ?; G5 w6 g1. **信任递归**:假设子问题已经解决,专注当前层逻辑( m1 A1 ^; S& t5 Y, }5 R" W
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
; }4 |5 f6 W( E  ^" X6 x8 Q3. **递推过程**:不断向下分解问题(递)
5 t* D8 v* U3 ]4. **回溯过程**:组合子问题结果返回(归)1 ^, F; w& N* m; `3 Q0 Q
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注意事项
" o* G; D, t* \/ u# @$ g必须要有终止条件! M5 i) T# n& j8 l
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
3 z9 A8 C& x% d2 [某些问题用递归更直观(如树遍历),但效率可能不如迭代$ f& h4 ?5 {3 B2 C: z, x9 g" x* s
尾递归优化可以提升效率(但Python不支持)8 M3 B( o( |3 L& w

! F7 p# Y4 |1 V+ `4 k4 V4 n 递归 vs 迭代# R9 r. Z1 @! L* Z
|          | 递归                          | 迭代               |
: p2 v6 _( \. v* q5 p6 K|----------|-----------------------------|------------------|: B/ v1 ^8 h  ~. e) H6 s' s
| 实现方式    | 函数自调用                        | 循环结构            |
9 Q2 m; M( K: B; K: p| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |5 A, M5 C- @" S6 l/ u
| 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         |
" }2 K5 z( i8 m# M3 P' L7 || 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |% N5 Y* O) R5 f; l# y. Z- |
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经典递归应用场景& ?5 t% j/ R0 [! S
1. 文件系统遍历(目录树结构)
  c  R' O7 v4 X- F; k5 d: W2. 快速排序/归并排序算法
3 ~6 c6 H* R# H  w% k4 d* r/ s3. 汉诺塔问题) z: a+ [4 u5 b! V9 o: K
4. 二叉树遍历(前序/中序/后序)
( R) ?/ P8 e- [9 `+ Q$ t# a, F! x* E# P5. 生成所有可能的组合(回溯算法)
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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,
- ~6 Q% C: |# b, m) i2 m我推理机的核心算法应该是二叉树遍历的变种。+ J; B% J5 M$ U# M, s# S, @2 n
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:& O2 L% g1 p- |
Key Idea of Recursion
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  l: p8 i+ v1 c) g8 J* BA recursive function solves a problem by:, B; y) y* m) v; V8 A

) f2 C' Z1 @8 T5 O- r' I    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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1 d; ~0 C+ Z8 Z+ p+ ?    Combining the results of smaller instances to solve the larger problem.
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( i+ J) S" ?' m+ s& A/ \Components of a Recursive Function/ F, E7 f+ @' N( I& A

) C, T$ c+ Q5 n$ z' J    Base Case:
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/ c; B- g' W; K9 l1 O$ A        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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* J, B' d# O) Q        It acts as the stopping condition to prevent infinite recursion.
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0 Z+ `( f+ u* n! w* h1 A5 j        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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3 Z9 N6 v2 j( I" B! ~/ j7 I8 F$ G    Recursive Case:1 ]3 k  V$ M) @! O- u) F; r, \
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        This is where the function calls itself with a smaller or simpler version of the problem., M0 m* C. D7 t! q0 `5 _

- B7 B7 o; `# T% p5 A        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
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    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):" x; Y* A; P& N( P/ I1 D
python
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def factorial(n):. u5 i" T& {$ p0 X
    # Base case
$ Z: N0 H# E' p% k$ l; W    if n == 0:
4 x7 S. e+ V1 E9 t        return 1+ i6 j9 w+ g& h1 K$ }/ C
    # Recursive case
: A8 a5 l/ o  G    else:
* A* H' v5 l2 a4 D* P" ?8 p        return n * factorial(n - 1)) W& _0 `) p$ E/ J

  ^% c- g7 d$ }. ]5 T% o# Example usage
3 v5 G, x0 |6 z& J9 j' N; Zprint(factorial(5))  # Output: 120
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. v3 M% u: {  A8 j8 d( NHow Recursion Works
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    The function keeps calling itself with smaller inputs until it reaches the base case.2 A  D, ?6 M# D6 l

! G# [" H1 I5 \$ O+ H4 X$ T    Once the base case is reached, the function starts returning values back up the call stack.
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9 d# E* w# p/ J+ d4 c    These returned values are combined to produce the final result., K5 p+ G6 K/ l' M$ q# y1 E
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For factorial(5):
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6 ?6 k1 R* F+ o  X, Pfactorial(5) = 5 * factorial(4)( s9 Y% v" j/ g; H* J, s7 T5 k7 n0 w
factorial(4) = 4 * factorial(3)6 F% Z" V; O. N  q
factorial(3) = 3 * factorial(2)
7 \1 V1 }+ I- Bfactorial(2) = 2 * factorial(1)" W- H; ]1 n1 D+ d" X( Y# d. _
factorial(1) = 1 * factorial(0)
0 D$ C, d$ S. @" R6 Ufactorial(0) = 1  # Base case, X; ]2 J7 b7 b( H

* Q. I7 I* M* L$ ?. f! ?" @Then, the results are combined:
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2 |9 g- `/ q5 Mfactorial(1) = 1 * 1 = 13 X- ~( }1 \( k
factorial(2) = 2 * 1 = 2
2 N( w- H4 F; D! kfactorial(3) = 3 * 2 = 6  C/ e7 |- l% I% `6 ]
factorial(4) = 4 * 6 = 24
: M6 t* k7 i7 g2 Xfactorial(5) = 5 * 24 = 120
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Advantages of Recursion
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3 y0 l6 {4 ]4 Y( a- R    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).5 h7 N  _5 I8 R% F4 ?

( F# d# _( Y6 d# c    Readability: Recursive code can be more readable and concise compared to iterative solutions.! Z5 B5 a, m: @9 ^2 J

6 G) C& X3 R( i+ rDisadvantages of Recursion. o# k- T8 Z" f' J( |* F$ F( p; V

8 h3 x: Y/ {0 O- ]' v    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.. z/ S+ z, c  Y
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).9 Q; }: y1 _" U7 u
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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).! D3 x: |- K) ~: ]: H; v7 c

% g) X0 f# z8 L6 c6 K    Problems with a clear base case and recursive case.
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4 \1 `" J% L0 Z# ^! UExample: 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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8 _2 l5 x& X- u6 N" n    Base case: fib(0) = 0, fib(1) = 12 @. @0 @. j9 c, ~" g7 `$ B1 S0 F

5 B1 s8 h0 C8 L# Y! c% g' G/ S    Recursive case: fib(n) = fib(n-1) + fib(n-2)- Y: M. ]5 L7 h6 }' O$ S% [

& B9 x/ V- c- L* h* wpython
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def fibonacci(n):% l# |7 O) r0 v2 @5 l" |5 Z5 x
    # Base cases0 T8 z! h- D7 n; P6 X8 X
    if n == 0:2 L7 q8 r3 O7 Q% v/ I
        return 0
3 [9 c1 n. ]& m% K    elif n == 1:) w0 `3 f8 O5 F/ d6 ?% g" S
        return 17 W3 B  S/ v; U4 k8 V/ l! ]+ [
    # Recursive case
* }) T- ^4 k# h  T0 l  |    else:, a' e2 J9 w$ L; H! ]5 f: w
        return fibonacci(n - 1) + fibonacci(n - 2)' K  u5 Z4 u1 g; ^' ^# t, b
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# Example usage9 [$ g; a* \# R8 ^+ \
print(fibonacci(6))  # Output: 8
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Tail Recursion. q4 s/ s: Q. q+ D
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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).! L' B& I' F8 n  W  O
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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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