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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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解释的不错7 H6 o; r" K* n

( p7 V0 i. |  n; T  g4 P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。1 h' u$ G- M  T" e, @

7 x) g' {1 ^' o 关键要素
+ s' z" K2 D2 k* R+ x3 l1. **基线条件(Base Case)**4 {3 ?, k' x# E. A
   - 递归终止的条件,防止无限循环3 G  ?, _! @& h9 @: V( Y
   - 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)**
. D- E) i* E/ Y   - 将原问题分解为更小的子问题5 m  e* b" N2 a: [$ J1 B
   - 例如:n! = n × (n-1)!
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经典示例:计算阶乘
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9 H& y9 t! Q% m# ^9 Xdef factorial(n):
0 i- D, a2 G1 T4 y) E8 t    if n == 0:        # 基线条件
! V' g% m- b. ?9 a        return 1: k2 P, Q, g$ l' ~
    else:             # 递归条件. d  ]8 L& C# r7 g: L% R3 C0 U7 W
        return n * factorial(n-1)
$ Q4 _$ _- z6 [0 j9 A$ \3 u! F1 F执行过程(以计算 3! 为例):0 W, p: B9 [* h) Q' [
factorial(3)
7 ~4 j7 q0 @" k+ H7 \5 C# D/ W3 * factorial(2)
# M" ?4 r/ j' K: v4 i5 G3 * (2 * factorial(1))
, O" }' E3 J6 _; @9 ^3 * (2 * (1 * factorial(0))), n0 v! P! {) Y# K" M8 s) N5 j2 S2 R
3 * (2 * (1 * 1)) = 6
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递归思维要点
3 D' N( y' A" R$ e% r8 L8 a4 l! m1. **信任递归**:假设子问题已经解决,专注当前层逻辑* I: H  k' C7 U4 Z
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)* a7 h9 Z# v' F- K; D% y9 q
3. **递推过程**:不断向下分解问题(递)
& _" k: j& h. R& D4. **回溯过程**:组合子问题结果返回(归)' ?9 s" M9 j, @9 Z% }
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注意事项* y/ x, P4 ^; u( {9 t7 A
必须要有终止条件- b: h3 p  G( M  B" B
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)" p8 H  L+ f! H' S7 @6 O9 h( Y
某些问题用递归更直观(如树遍历),但效率可能不如迭代# _( |0 T5 l0 j/ n: {
尾递归优化可以提升效率(但Python不支持)
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递归 vs 迭代
  N7 [" I* E, r; m* ~1 w|          | 递归                          | 迭代               |
  m; z6 j) Q; @. n9 F4 D|----------|-----------------------------|------------------|
- t) C8 n7 [2 {: j% T| 实现方式    | 函数自调用                        | 循环结构            |
  g. I+ ~" e- Q2 B: `' D7 G| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
4 h! K; ~. r9 c2 j, s9 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         |0 ^$ `7 m5 |& W( b) M" R' D! k. ?
| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |  t9 T: r$ S) d% d! {3 J; h# V

6 p1 N" m8 ~0 N( h) c 经典递归应用场景
0 u9 B6 I6 c# G& T+ Q% ~0 R1. 文件系统遍历(目录树结构)1 g- P) `* `# P9 C8 M2 w
2. 快速排序/归并排序算法
2 r8 l/ I9 q: M8 e3. 汉诺塔问题% X% ]3 V$ ]( I1 D9 I$ q
4. 二叉树遍历(前序/中序/后序)
" `6 F/ e; {5 l& E5 f5. 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,
$ R* {& S9 d# d* ?3 v( f我推理机的核心算法应该是二叉树遍历的变种。' g( O( k8 N1 _' R
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:
2 w2 r. ~( m, _Key Idea of Recursion1 a5 N! q0 g7 g; n

: q- J6 J1 t% [( i5 gA recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.' S, v% H4 E$ B+ h
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    Solving the smallest instance directly (base case)." m' k& X4 ^3 M+ S: {6 ^
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    Combining the results of smaller instances to solve the larger problem.
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0 k1 p# n5 J, h! `5 m4 H  D( UComponents of a Recursive Function7 i; A8 H; i% j, C6 _; E
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    Base Case:
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+ {+ R" G8 k( w        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.+ t4 W/ U* u6 `8 @
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        It acts as the stopping condition to prevent infinite recursion.
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9 H/ [1 Z: h3 P0 K* ~7 F( l, t        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    Recursive Case:( a6 K2 y& e9 B  ]! X

4 z. c9 c# t! b& S        This is where the function calls itself with a smaller or simpler version of the problem.2 k# w0 T/ L' ?0 W1 _) o1 t. x" C
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).& i9 f1 {7 b' p8 B8 Q, z

3 ]- e: z7 {& G( Q9 S/ pExample: Factorial Calculation* a1 o% R& w- ^
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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:0 b- Q* R7 D1 `( `* U. V

2 `/ Q7 V% m: u- G4 c8 ]    Base case: 0! = 1
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% y9 O- {- D( x8 H6 I    Recursive case: n! = n * (n-1)!" @7 U5 f. c" ^' I
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Here’s how it looks in code (Python):  R  d; n, A$ T+ e9 @4 C
python
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& b9 t# q) n/ F0 @def factorial(n):1 ~0 S  P6 E/ h0 G# o
    # Base case
% H5 }& l4 ~+ d) M" ~    if n == 0:6 q4 C0 G- C( J4 Y9 y" S+ v5 L2 T
        return 1/ D' Y! h! [3 C( l* V( I- y* y
    # Recursive case
9 R, P  Z) a' y    else:
( `( j% r, d* Q' P' R        return n * factorial(n - 1)& c+ f: O+ ^/ V% u$ |2 v8 U/ F* J) a

4 r2 ~( e3 R: T8 X# Example usage( }% L) \& L! H
print(factorial(5))  # Output: 120
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8 q- X  y5 q6 E% _' a6 FHow Recursion Works& @( S2 ?: Q1 R7 ~( v! Y  X$ @$ l

; a/ Z0 f6 ?- t" o2 e* p    The function keeps calling itself with smaller inputs until it reaches the base case.& t8 O+ i# A! q: Y: @+ b
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    Once the base case is reached, the function starts returning values back up the call stack.
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& [! t- S! d) E, v% o' M( t7 C) h5 d    These returned values are combined to produce the final result./ H  m4 C2 T8 M/ J, e! d0 l

8 C% L* m9 g* P. l1 K2 Z3 G) ZFor factorial(5):* G. I# ~' N5 @9 b
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2 t# W+ o" R% G' ~( D+ x7 jfactorial(5) = 5 * factorial(4)
1 @% }! @' O- v9 n3 {factorial(4) = 4 * factorial(3)8 i" V3 `, x7 {# ^$ f
factorial(3) = 3 * factorial(2)' s7 }2 e! n9 h' x" }1 b
factorial(2) = 2 * factorial(1)5 m9 ?* K& E% a( i% t3 y' M8 g
factorial(1) = 1 * factorial(0)
2 ^" ^4 t) K, j" U/ f2 ?factorial(0) = 1  # Base case
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6 y8 B, _8 I  D! d2 SThen, the results are combined:
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9 K9 G) U2 r% Z& j1 c9 N/ S  U% Ifactorial(1) = 1 * 1 = 1+ b/ k( k8 a2 q! o  m. F& l
factorial(2) = 2 * 1 = 2) p/ V# B  Z* }2 f1 l
factorial(3) = 3 * 2 = 6- N4 x( o& b9 m  ~# ^
factorial(4) = 4 * 6 = 242 {8 l4 X4 v; r0 Y' B" \
factorial(5) = 5 * 24 = 120
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8 `. B9 D2 [0 u# KAdvantages of Recursion: q+ P5 |; B# }8 w2 h
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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.- i0 V7 Z# i& u3 ~
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Disadvantages of Recursion
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    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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% o% ]* L- a8 y& M    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).7 F5 u- o* p5 ^/ Z( m( D0 J

) ?% j0 S+ e0 f! xWhen to Use Recursion
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+ W/ }+ M+ g/ j# Y) W7 D5 i; [    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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$ K; R0 l& n7 F5 T0 e9 z. |. I    Problems with a clear base case and recursive case.
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3 M* k2 C! @5 P& g8 ]% X) N8 h# [' [Example: Fibonacci Sequence
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/ Z( _* I/ s2 \4 |) QThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:3 x( u# X6 C, w) g  \6 q3 N$ j
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    Base case: fib(0) = 0, fib(1) = 1  T. M! I) I6 }9 |
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python! ^9 e6 Z4 H% c+ g& w9 Q( x$ A5 M

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def fibonacci(n):3 c" Q! H4 @, o2 ?3 w* b
    # Base cases# z4 r) r7 C4 Y: k/ U4 `- ]0 F
    if n == 0:
2 ?* `- \% U' y        return 03 P4 J3 S) H. H, o& G
    elif n == 1:
! }' K; j& N7 I2 x: W4 y        return 1' e* b9 s: i/ z6 I/ p6 x. c
    # Recursive case
1 Y% l$ W! o! P9 q0 Q; g$ o    else:  Q( J  {) W$ j5 B4 v1 h* L
        return fibonacci(n - 1) + fibonacci(n - 2)
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# w4 U9 c1 U; A* R. a/ k# Example usage3 T+ j& }# f$ q
print(fibonacci(6))  # Output: 8
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Tail Recursion
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9 T' y' B1 h$ FTail 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).: d" |# K" |! V) b. a) O; e

9 ?5 s. I' s7 u* b. HIn 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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