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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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# H( c# y1 w& v: ~' h2 Y. U3 m解释的不错6 W9 I4 h! {$ f. F  y- U

  o3 Y  ?7 l, S6 b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。) ], x5 m# b1 v, Z: u0 F$ g

7 L- _- O) u' m" o9 I/ a 关键要素/ T- F& `5 a: E( K
1. **基线条件(Base Case)**
0 g3 b5 t2 @, {* B* G, V   - 递归终止的条件,防止无限循环8 c% P  {4 [( F
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 10 v% C% m: ~( }- \4 y$ u( d
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2. **递归条件(Recursive Case)**4 N& t" o' V- o8 Y/ p: T( a
   - 将原问题分解为更小的子问题, A3 ?. B5 n3 Q- [
   - 例如:n! = n × (n-1)!, V* g" `2 P! ]3 g

9 x& t5 N5 T0 v+ |5 u: a$ h 经典示例:计算阶乘
  D& z; l# T3 J6 w! kpython
1 r. B# O: o. j; e5 f! c, U: xdef factorial(n):6 c- S0 x9 b6 O5 N) r* C4 U
    if n == 0:        # 基线条件& B; o6 B- q5 l/ f# b( a! }& k
        return 1
8 c4 p9 s5 y6 ^' I2 t% l3 k    else:             # 递归条件0 i( S' D4 S. X/ ?3 A4 ?
        return n * factorial(n-1)
( S1 {) a$ J3 ?执行过程(以计算 3! 为例):
' G+ `/ I& W: G1 pfactorial(3)
3 k) T$ f3 \5 p3 * factorial(2)0 c4 L1 u2 O, ~# S. B. _6 H' S  t6 O
3 * (2 * factorial(1))
9 g/ k& c% D: L" x" |3 * (2 * (1 * factorial(0)))
* K8 T5 |, V7 }4 `3 * (2 * (1 * 1)) = 61 T9 z& Z8 X6 R. t# k% L' [, u
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递归思维要点
# ~% i& \) `- |7 _; q1. **信任递归**:假设子问题已经解决,专注当前层逻辑% ?; U+ P7 o8 A5 W
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
# K. v6 _4 m/ Y3. **递推过程**:不断向下分解问题(递)8 \1 M# ~' t5 f' V5 F1 [2 L6 ^8 X' I. B
4. **回溯过程**:组合子问题结果返回(归)- P8 O5 o9 h5 B' [" J3 l

+ _) c8 `' F3 [0 F! c 注意事项* ]% P3 d# W6 {* n7 R2 x0 C' a! c
必须要有终止条件% Y3 d. F+ _+ Y( Z; n0 L# U
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
; w. h3 q5 W8 c, L) U某些问题用递归更直观(如树遍历),但效率可能不如迭代
: g$ C  X% V5 c9 d9 ^8 Z. j尾递归优化可以提升效率(但Python不支持)
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9 @+ R+ z6 R" A! w, u 递归 vs 迭代3 F$ s% `9 r1 L7 x1 k. a4 ?. D: U
|          | 递归                          | 迭代               |
* ^: S- {0 u" T: O|----------|-----------------------------|------------------|
; Q* x  {+ F0 c0 K1 c| 实现方式    | 函数自调用                        | 循环结构            |1 ?' k" d2 N: D* ?! n8 A2 E8 [$ e# P
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
/ v! D" R  ]  h" e" b" N| 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         |
$ r1 I  i3 g& t- e| 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 O9 M/ U, E1. 文件系统遍历(目录树结构)
" Z- Q! Y" n0 ?8 B; T! M1 ]8 h) [2. 快速排序/归并排序算法
$ m# p- ~) v1 j+ J7 m3. 汉诺塔问题
: F1 N9 y: w3 T% ]& l( l6 c4. 二叉树遍历(前序/中序/后序)  `( m( q  ^; o1 W
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,! r1 r, ~# a( f. ]' ]
我推理机的核心算法应该是二叉树遍历的变种。
2 R/ B) h2 X" i; v( \" A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:4 }& u* n( G# b
Key Idea of Recursion
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3 Y$ l# ?* T; |  ~) M4 E2 s6 m. mA recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.( J( ]7 k) P! _6 I+ ]3 ^: Q
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    Solving the smallest instance directly (base case).8 D9 t! z! x  F0 {# P9 E
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    Combining the results of smaller instances to solve the larger problem.2 t; o7 Y' k  _1 o
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Components of a Recursive Function
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# H( z/ N( v3 o! k+ u( d; ~: R% e    Base Case:
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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.4 f  B/ t$ o7 F& J

( o. L% m6 q4 A7 X/ v        Example: In calculating the factorial of a number, the base case is factorial(0) = 1./ v" G3 b: c9 O
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    Recursive Case:6 A; `2 R3 D% V, Z
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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).6 I+ e# n9 B+ V
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Example: Factorial Calculation
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! d! O- w; W6 L2 a  i$ |5 J: JThe 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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& s/ d( H1 f+ n# C- L1 R  Y# x    Base case: 0! = 1
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; b; D3 S9 W* t. Y* [' {    Recursive case: n! = n * (n-1)!8 E" `, D7 _; G6 ?' `

, z  J6 y% g/ ?4 S5 g/ o7 {7 YHere’s how it looks in code (Python):5 U) [$ F+ p2 {/ j% g9 g
python
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def factorial(n):; C/ f1 k* X, l$ z3 o
    # Base case
# o6 ?# W4 Z$ ]( s    if n == 0:
! G7 ~7 s/ q& Q1 P" x- _3 _3 P        return 1: ]7 \9 y- O3 ]( Y' L; i9 e' J
    # Recursive case5 l! B8 Q% E$ t6 }( H3 \( ^
    else:( N) B# d$ e7 A& M% L
        return n * factorial(n - 1): B; K# B* g" l8 q' k' K

) Q8 I0 g: }4 ^6 K# W+ ]' u/ J# Example usage
5 X( K8 x7 ], R8 }% B1 jprint(factorial(5))  # Output: 1200 v; E& [# @" I* R, u8 F( n

4 H9 T+ s* b- v4 e2 iHow Recursion Works
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    The function keeps calling itself with smaller inputs until it reaches the base case.3 G/ h4 ]1 H, Y& F* M

' ^: L* I& f+ G9 S" I% i& ^, ?    Once the base case is reached, the function starts returning values back up the call stack.3 C9 ?6 l$ e$ N5 r; n6 A3 m9 T5 ?
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    These returned values are combined to produce the final result.
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8 d# T- ?0 e" p% bFor factorial(5):
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factorial(5) = 5 * factorial(4)
% l7 c* C5 _' [" M5 o  [factorial(4) = 4 * factorial(3)
4 Z7 v. }5 N8 O" r5 b9 i: J, Ufactorial(3) = 3 * factorial(2)
. |$ {8 J) z1 S4 b  Q* [# Rfactorial(2) = 2 * factorial(1)
4 E; H: U1 q  ]+ d6 Ffactorial(1) = 1 * factorial(0)
1 Z& d1 r% l2 z0 m9 U4 H1 v6 D+ yfactorial(0) = 1  # Base case
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. j9 G& L) X3 D$ J# H8 A* A/ [Then, the results are combined:
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factorial(1) = 1 * 1 = 1, v0 i* ]/ W6 w; j9 M, f1 U$ G& ?2 [
factorial(2) = 2 * 1 = 2% l; a" X4 L# S9 Z+ A8 J8 ]
factorial(3) = 3 * 2 = 63 J5 E9 N  u& m  O9 S% _! L
factorial(4) = 4 * 6 = 242 e0 ]8 x! Z% T) i5 r9 x
factorial(5) = 5 * 24 = 1203 U1 ]: b8 j; _; \) A

8 x9 ~8 x4 x: O1 _1 b8 TAdvantages of Recursion* G3 _6 j) U' ?8 B. t6 |
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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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4 O& ~# A$ L" }# v( J" Z3 o6 `    Readability: Recursive code can be more readable and concise compared to iterative solutions.& z1 M3 f9 q  Z5 e8 |3 o

2 u  X4 M! v1 S7 k0 eDisadvantages of Recursion
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5 A! m0 j9 O  ~) y    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.- u/ F# I% F& j% d* K, T* P

  H0 v, o8 }" W. ]8 v' @* b    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).* S3 G" v2 D) E- j' j, P  ?

) }3 h( b. w  c$ [, V! GWhen to Use Recursion
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$ m8 U4 U- U: A/ D' V1 B    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).4 a) \) ]4 J$ \; l# f1 _  I6 w

3 ?" D5 ^2 ?  z. a/ N    Problems with a clear base case and recursive case.
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Example: 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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    Base case: fib(0) = 0, fib(1) = 1' s" W) r* l; R( \

  X" w: R/ `1 y0 R! \/ A    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python
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0 E$ h3 y" m; L3 Odef fibonacci(n):
/ M2 {! R. Q+ X* x/ o2 p    # Base cases
: H) b4 v2 n7 I) w    if n == 0:
  w3 Q( Q  m& f0 L+ Z        return 0( g: A4 ]; A' P- S
    elif n == 1:& G* Q+ ^/ V9 c. C( D0 i
        return 1
0 H( S8 L% ~9 x, j+ b7 I    # Recursive case
- U6 L4 ~; K) ]2 }    else:
! i0 o) V  l1 O9 \        return fibonacci(n - 1) + fibonacci(n - 2)
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! ^7 v- a/ Y5 |, b# Example usage
) ^, l% _$ C6 u# hprint(fibonacci(6))  # Output: 8$ F. d: |  x( s; b1 C8 M; T0 q3 u
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Tail Recursion1 k% m/ o+ Q! X; E8 q: H

( |# S5 U+ E# t. TTail 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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1 R! L# f( c5 U( eIn 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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