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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 & k, \; V5 U: y' M. Y
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解释的不错
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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。8 \5 w3 m4 Q: U
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关键要素
0 h6 i4 T/ ~* t, f1. **基线条件(Base Case)**
4 q- I; Y5 P7 d   - 递归终止的条件,防止无限循环+ n5 B% t& W; R6 h  V/ f: A* s
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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2. **递归条件(Recursive Case)**$ q; z5 P' ^" F+ v
   - 将原问题分解为更小的子问题8 @% ~8 I8 M! K% ~
   - 例如:n! = n × (n-1)!
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! l, Q2 S! ]" g 经典示例:计算阶乘& h1 ^8 O& ^$ X' ]& I
python+ s9 j, S' ]5 c
def factorial(n):1 U3 I% d  C- F3 ^
    if n == 0:        # 基线条件- K5 {& d; A% [. t8 |
        return 1$ V: ~! b" Q- I
    else:             # 递归条件- x4 \9 F# M! V
        return n * factorial(n-1)8 @, d1 l% q' {1 N8 `
执行过程(以计算 3! 为例):7 y! x4 M& a+ R
factorial(3)4 M$ S! Q  o9 S# B! T0 l: s4 ^
3 * factorial(2)9 M( B! D( E- o' |' e
3 * (2 * factorial(1))7 \1 I8 t+ }: _0 J4 N4 s
3 * (2 * (1 * factorial(0)))
6 W- c: e+ W/ p# o) d* B: k3 * (2 * (1 * 1)) = 6
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/ r  A7 n3 M" y9 Q- p4 { 递归思维要点
; D# A* d) b4 E) m1. **信任递归**:假设子问题已经解决,专注当前层逻辑- l& H' ?  {$ a1 i1 R% Q$ j3 k
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)2 W, C0 \  r$ [5 B/ s
3. **递推过程**:不断向下分解问题(递)
3 G# W) g* \- U3 ~; B( w9 z/ V- j4. **回溯过程**:组合子问题结果返回(归)) k+ N3 i8 w' S4 R2 B8 i& k
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注意事项
" t' J  V- d+ K必须要有终止条件
; I9 m% a# @/ s: x* k! i: ~递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
) N; c( @" M) j/ k" J某些问题用递归更直观(如树遍历),但效率可能不如迭代* W. t; V8 I- |
尾递归优化可以提升效率(但Python不支持)/ ]: Z2 j7 I% o: X4 S
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递归 vs 迭代
2 }+ q5 j% z  t2 U6 v) l1 T|          | 递归                          | 迭代               |. p! R% J: {" M
|----------|-----------------------------|------------------|
# c9 l7 T) B0 H  K/ U# o- a: }| 实现方式    | 函数自调用                        | 循环结构            |) C) N/ ~; a+ r7 z& k
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |" ?- b: S' t) W: X
| 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 H8 o0 s1 {( j/ P6 m| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |4 t' z/ ^. R3 `$ V. h8 Q

4 e4 a0 D2 a; I. \" A' T/ P5 t; k 经典递归应用场景
$ E4 ]; z* H. Z" h: A: `3 o7 G1. 文件系统遍历(目录树结构)
- K6 ^3 b  U) T8 W6 X: L# O! {3 o5 C2. 快速排序/归并排序算法% ]/ f/ ^) s" e' Q. C5 e$ j; X+ R0 f
3. 汉诺塔问题
+ X  T- j: p. d6 r4. 二叉树遍历(前序/中序/后序)
7 t+ ^* y6 W3 {4 `7 Q5. 生成所有可能的组合(回溯算法)$ M6 S& B" j$ q% H9 Z+ E# Q0 \# s

) l) }  I) {: I6 e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,
/ u. h2 h5 U% \( X7 y( l我推理机的核心算法应该是二叉树遍历的变种。
* A5 N. b" y; j0 u& U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:3 n5 G# v6 u) M3 n2 `. m
Key Idea of Recursion; `4 t. D) d4 Z: ~
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A recursive function solves a problem by:9 L: j3 w7 B) v2 y. W, M! z. X+ z
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    Breaking the problem into smaller instances of the same problem." _7 j$ D4 o+ |7 p1 K4 K

( e' T  r- `+ l' U2 M" E    Solving the smallest instance directly (base case).
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4 Q- E7 A: m1 A% J- U7 _* V    Combining the results of smaller instances to solve the larger problem.3 Z0 _7 X: C, P- F- @! r
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Components of a Recursive Function
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$ e# W( |5 N% R- t: T; m& J    Base Case:
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0 ?- Y9 h5 K8 @$ N" x        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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        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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- e" e5 J/ U7 H+ L    Recursive Case:4 h5 ^" J  C6 l& }) V* p5 B
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        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).8 g# R4 @1 a! _# g0 C

; G* ?, ?. r% ?! pExample: Factorial Calculation2 ^' T8 X( [% G: L4 N( U

1 m1 q5 H# H9 R% {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:- ^% |3 {, h: x7 \3 R  h$ {3 |

0 b% |8 w" o! }: J" j, K, h* D    Base case: 0! = 1
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, G8 m) i( G# m1 ?    Recursive case: n! = n * (n-1)!; U6 I' ?1 p% e$ n2 A; ~( e* K9 d
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Here’s how it looks in code (Python):
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def factorial(n):* @" a$ e$ Q/ _
    # Base case' U0 c! T! k& J; I) S% n- `1 ^
    if n == 0:
; T# o, E; h7 |2 U* A        return 1
9 s  l$ Q/ I& W    # Recursive case
  w$ e8 e6 A. S4 a' {: ~    else:8 B# b8 r+ x  R; z1 B8 T) T
        return n * factorial(n - 1)# n$ d, H$ n0 e# H% t  {2 S3 O
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# Example usage" t9 i1 c% p6 F6 F
print(factorial(5))  # Output: 120" E1 Y& i9 r/ @# C

- |2 v! x0 X4 r8 I& s) SHow Recursion Works5 ~9 I. [9 j3 X

! n$ G$ S0 u3 F: b  J    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.) g! p; d' P- q7 h% E( h3 ~
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For factorial(5):
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factorial(5) = 5 * factorial(4)
3 r; x2 l; Z6 R! rfactorial(4) = 4 * factorial(3)
" p! p. ^2 P# j4 P+ u" {: P! rfactorial(3) = 3 * factorial(2)
- n  J' i- r2 jfactorial(2) = 2 * factorial(1)
0 f- a1 z$ x: W2 ?9 d. pfactorial(1) = 1 * factorial(0)' z8 ^6 y% w8 Z3 d
factorial(0) = 1  # Base case
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Then, the results are combined:
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factorial(1) = 1 * 1 = 1
0 j0 B+ h% r) @2 X$ Y9 wfactorial(2) = 2 * 1 = 2
4 L/ p- V# V1 k7 [  m/ L4 efactorial(3) = 3 * 2 = 6
+ x8 e) b4 L7 S+ R& S. P1 `4 |2 Dfactorial(4) = 4 * 6 = 24
$ Y+ v! k, d! N- I7 h* G8 ?factorial(5) = 5 * 24 = 120" v! D5 ^8 x% r8 h9 i
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Advantages of Recursion
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* F7 O0 o# E* Y8 G- |0 F; n    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).( _+ r& x* Q& v& W2 ~
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.2 }) a+ X( u+ C+ [5 r

, ~* F$ {/ s# Z9 p* RDisadvantages of Recursion& V5 _+ `9 u; A9 ]0 O! k( [

( e' X# U; J8 W! C8 b    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.1 e5 U4 q. h. q0 Z6 R7 k1 `1 C
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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  d* {: W/ \  [9 Q$ {) ]  ^1 a% U( A6 AWhen to Use Recursion. \4 s! C9 a# s4 C

% a0 C% o! J/ |% D+ D) q, J2 j2 [    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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( W# f5 M4 ]/ ^1 x, @2 p    Problems with a clear base case and recursive case.
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" @: c; R& L+ e& }# sExample: Fibonacci Sequence
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& R  a6 n9 W* _) Y4 q. QThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:0 G# o. p! o& b9 f2 Z3 @

) W! e% Y6 ]# r( \8 Q5 Z5 l    Base case: fib(0) = 0, fib(1) = 1- j8 L$ I* }  _& P1 i
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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7 X3 i8 R9 l! m; m# hpython$ y  |6 |. `. u- D5 U
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' v2 `0 Z( V8 Q) n: Ndef fibonacci(n):
/ ^3 z3 K/ T- r: ~( E    # Base cases
- y: a2 N6 X: ], s' g6 z    if n == 0:- g1 m/ f2 L7 b) n# c: i
        return 0
. P4 e$ z2 C+ _# t    elif n == 1:$ |& `- U! v$ d  H" o7 o
        return 1
0 V; {8 G- {9 k: s; m! @0 z    # Recursive case
5 h1 N1 `4 n/ |& b* @$ e7 u% X' [    else:
0 E/ E. _* T/ e4 |% W        return fibonacci(n - 1) + fibonacci(n - 2)) q3 n; o9 q) t' q) m. q

) @, ^# ?& X# G7 p7 \) c+ @# Example usage
! a/ C2 r' |1 g. z3 w5 n) xprint(fibonacci(6))  # Output: 8- l; v% E3 e( I1 u4 i4 k
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Tail Recursion
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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).4 J/ D9 Q$ P/ e( ^# v

& ~7 x* ]6 x9 \$ MIn 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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