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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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9 \, |8 |& X" c+ R, \" _2 p- @解释的不错7 G( L) A3 a4 X; q
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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。
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; Q- J: [8 l# x# V! z1 r 关键要素5 S  L8 l9 f1 L
1. **基线条件(Base Case)**. Z+ P9 b2 r* ?# b# v& l
   - 递归终止的条件,防止无限循环
& o0 K  F6 p" E# U; _1 p0 s3 z   - 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)**7 K. E- X# T. s; ~8 \$ ~
   - 将原问题分解为更小的子问题2 l- D) m9 A9 B4 t, w
   - 例如:n! = n × (n-1)!  I3 x2 f, v! @# e2 d( B

6 s1 f6 }8 E. `) ~4 P/ x1 \ 经典示例:计算阶乘
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& e' j9 Q, d) }* u& odef factorial(n):
8 }( w+ }. W9 I5 I8 f4 c3 }    if n == 0:        # 基线条件7 g( o- B7 ?/ t+ n! `" I" g% H
        return 1
  i# t$ c" [0 l/ I% {; Q# m    else:             # 递归条件+ o+ s' }& S) c
        return n * factorial(n-1)
% _, _5 {) [0 r' H执行过程(以计算 3! 为例):: q8 o4 F2 G" h
factorial(3)
6 {9 n& L. X0 f3 * factorial(2)
; D" Z. g' o: n( S3 * (2 * factorial(1))- @4 A3 }' t4 r3 }. I# a* f
3 * (2 * (1 * factorial(0)))
) ?9 ^& Y& ]' V5 e) O" E5 V  Z3 * (2 * (1 * 1)) = 6# r$ X: A8 c% ^( g4 Q; C# b

: \2 D0 j2 {: E. E! v 递归思维要点
2 g  B$ l" k4 U* M- V1. **信任递归**:假设子问题已经解决,专注当前层逻辑
5 a& Q. Q2 `4 y1 V. L2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
6 y) i( Q8 e  K: s3. **递推过程**:不断向下分解问题(递)
( Q. j' c6 ^4 P# I4. **回溯过程**:组合子问题结果返回(归)
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/ I8 d* y" G0 [& V. z 注意事项
: j5 _: T6 e* o. ]必须要有终止条件
0 M( m  g8 J' R0 |$ j2 ~6 k递归深度过大可能导致栈溢出(Python默认递归深度约1000层)/ z$ v; {- [- Z* e1 E
某些问题用递归更直观(如树遍历),但效率可能不如迭代1 U, |- W3 t. N' h& R9 ~* T
尾递归优化可以提升效率(但Python不支持); h% Z3 a5 H- x8 e/ d; j: \

% U3 u4 `3 @- a1 Z" `4 g) a 递归 vs 迭代5 j) C/ ]1 L# u& ^: F3 @6 o
|          | 递归                          | 迭代               |
0 e2 ~2 C5 z5 D% w1 P|----------|-----------------------------|------------------|7 p) k4 B( F8 b! F+ _. R
| 实现方式    | 函数自调用                        | 循环结构            |+ R- c6 d4 g0 J+ L; j
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |) N3 X  Z$ G( z  m
| 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         |8 Q/ B6 k* `# ?! o
| 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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经典递归应用场景
6 Z: w% x) R6 Z, V1. 文件系统遍历(目录树结构)# ^0 z! ~" q4 C5 S' z# b2 g1 J
2. 快速排序/归并排序算法
3 |  }; ?- G' L$ g% s* c- Z3. 汉诺塔问题8 b+ T' z# u7 k, o3 M8 J' @/ x
4. 二叉树遍历(前序/中序/后序)" A6 J, [0 ^& `5 K# i
5. 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,: E* ~2 m- B6 ?# p6 ~: i; d
我推理机的核心算法应该是二叉树遍历的变种。
3 P0 A$ k0 Z# s3 F$ b7 F! x, ?3 ^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:
0 \, @; F, v3 O0 V- I8 y7 LKey Idea of Recursion
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0 d; K; Z2 S9 J+ t9 p8 ]5 CA recursive function solves a problem by:9 l, a) `9 y9 E! O, u# U0 K0 f

- a3 T' D4 ]  M6 Q. F    Breaking the problem into smaller instances of the same problem.
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% \/ j8 I0 T3 [5 h# A    Solving the smallest instance directly (base case).
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+ p0 t8 `3 k; ]    Combining the results of smaller instances to solve the larger problem.1 g8 v3 i2 L! t9 L

' M( K1 x) B+ M+ MComponents of a Recursive Function6 x0 O/ d- }' S* A" L$ B

. o; s# J$ G5 @- K! ~8 m* Z0 I    Base Case:
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        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.! V& Y) s) N& S
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        It acts as the stopping condition to prevent infinite recursion.6 e1 S% p0 G$ H8 w, L; c

' E2 }4 A4 E' e$ @        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.9 T6 e- e, v$ ~6 R+ e; {( {

/ |" N5 i( w  S3 i2 n, b* |    Recursive Case:- M+ e, V7 X6 U4 J8 P" ?
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        This is where the function calls itself with a smaller or simpler version of the problem." T& P* T' v$ J. N
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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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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:1 d, B+ `5 q6 I  e

$ h3 d8 X* V7 J: T    Base case: 0! = 1
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    Recursive case: n! = n * (n-1)!
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* C& g: W# e7 r' i4 E$ mHere’s how it looks in code (Python):
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* |. [! h  n" Bdef factorial(n):
5 u. s& I" ^% r/ O! q" Y, N    # Base case- l1 t" g4 o0 n" M" C9 _1 [2 r
    if n == 0:
" g% w- J* L; C        return 1
8 d! p5 R7 Z; L$ _) y    # Recursive case
5 D9 {  i, G% B    else:
7 O" Y" g$ ]6 N2 j, S6 e        return n * factorial(n - 1)- ~  T, D: n# L$ o) H2 Q: I# {

$ B: x, ], L4 n5 \5 |8 V# x# Example usage( n1 i: h3 O4 ^% w' e* a% L# g
print(factorial(5))  # Output: 1202 P/ {9 k5 [! n3 v
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How Recursion Works9 l4 a: O& q: |' k8 V# n
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    The function keeps calling itself with smaller inputs until it reaches the base case.1 v8 z  X5 T, r  e! k6 x; Y/ l2 s  H$ T

5 y3 i' U' h$ [- o    Once the base case is reached, the function starts returning values back up the call stack.! g' Z- P5 V( ]7 L) v4 z

2 z* T! c7 Z; }2 Z& U9 r5 t6 G6 G$ ~    These returned values are combined to produce the final result.4 c0 W/ V6 {  Q( T" ?" z! o) R

* L( R! F( t6 }+ _For factorial(5):
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factorial(5) = 5 * factorial(4)
: Z. ^  d# N$ g- @5 W, I4 xfactorial(4) = 4 * factorial(3)
2 h& S1 {" Z# `factorial(3) = 3 * factorial(2)4 a9 j( I+ A+ ^, s8 V' E! f
factorial(2) = 2 * factorial(1)
: a0 h+ o  a5 z, g+ L9 Afactorial(1) = 1 * factorial(0)
/ d# D) k8 {/ _  C2 {factorial(0) = 1  # Base case3 T, b" \7 R* }$ k) k
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Then, the results are combined:
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factorial(1) = 1 * 1 = 1$ i7 ~# z* l$ M% T4 S
factorial(2) = 2 * 1 = 2
- f& D8 M, C$ `# e; E, Jfactorial(3) = 3 * 2 = 6
: }9 Q2 E$ U- F" T; Tfactorial(4) = 4 * 6 = 24' f& |9 B2 g7 K. b0 A
factorial(5) = 5 * 24 = 120+ q6 t6 w9 O" z7 @; J
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Advantages 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).
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    Readability: Recursive code can be more readable and concise compared to iterative solutions." G( {$ W, M) R: u* D& Y
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Disadvantages of Recursion
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- E) b; B, y. y- P, t0 w    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)." [* V" R6 h# ^& j; l

, [* R, K: Z& g# Y+ A1 [" ]$ M3 `8 fWhen to Use Recursion% Y4 G( d/ b6 s# y/ z
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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.  q5 g* R1 X: i. Q

0 v/ f9 O/ x" r+ U( r& G6 YExample: Fibonacci Sequence/ H  A1 x+ @; y+ V8 h  V' o

' ^, `, ^7 S& YThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:; ?/ _! G) t% f4 i9 \4 `, I3 f  x
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    Base case: fib(0) = 0, fib(1) = 1  m  R% T- g( F. I5 u! B
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python# C9 Q& t$ L; t$ r8 {8 `$ U
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def fibonacci(n):( T( I9 a( D+ g; i9 E
    # Base cases1 G3 ]5 t% }% o* Q! I5 f7 y
    if n == 0:% C: L) U# l; [7 |3 P/ p) Q+ Y9 Z
        return 0
/ m" l6 L6 C( b$ p1 G) g* p' _    elif n == 1:
( e. o; p* W: V2 s% C9 m- c        return 1- f  k, `0 Q5 W3 D
    # Recursive case
/ z$ y9 S0 ?1 E! d0 `  ]4 t+ T    else:
+ p6 R2 M% b( A5 z! b+ ?        return fibonacci(n - 1) + fibonacci(n - 2)8 k' Z1 w" N" P
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# Example usage
7 E3 L7 p! T% [% ^" l! J. zprint(fibonacci(6))  # Output: 8" w" u3 {2 ^, s& u  e7 Z, P
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Tail Recursion
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2 \, I  w9 A! O7 STail 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).6 f1 [: w+ D) Y6 i6 r7 }  u
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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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