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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 D2 p, E4 d$ b- U, o

  F8 u& H: U- e/ d8 Y5 I2 V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。
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关键要素5 C4 P; e# n# \" ]% k, S2 X) W0 M: m
1. **基线条件(Base Case)**
% q8 A* h' P/ ]! P. _( a6 u   - 递归终止的条件,防止无限循环9 X4 b: _% i) [  B& j: S, r! k/ y
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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' D: D& Y1 f3 ]- v% z7 m! K) X* A2. **递归条件(Recursive Case)**5 n/ ]) H9 f. U0 k
   - 将原问题分解为更小的子问题
  j) j) {0 L; L3 ~- e   - 例如:n! = n × (n-1)!2 H7 i' {2 o% z$ P# l5 a

7 n! _5 x9 B/ ~; M, r3 C6 n 经典示例:计算阶乘2 W: }0 U: T* r8 X$ f- R7 X
python; Q1 j  I4 w) i. ^0 V. G9 b1 N) r* ]
def factorial(n):$ p; ~9 }/ D% g5 U2 V
    if n == 0:        # 基线条件, O5 P9 K* J& g
        return 1
" E- k' H/ R0 q5 p! {1 r    else:             # 递归条件
* G3 g# {( T( w5 U; X  Z1 e* `        return n * factorial(n-1)% t8 U& ]6 c6 D
执行过程(以计算 3! 为例):% ^$ n8 ]1 M  ]( C  W; d- L
factorial(3)+ e  x: E7 g2 d) ^: Q) A3 I
3 * factorial(2)) U- _: j$ w" e$ s% I6 h
3 * (2 * factorial(1))
7 _1 w- @  o# a5 h; T+ ?3 * (2 * (1 * factorial(0))), U. |8 H$ r: c5 H, p2 B7 R: I
3 * (2 * (1 * 1)) = 6
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0 n& t. S# r" Q1 k 递归思维要点
  o" ]8 o. w6 M1. **信任递归**:假设子问题已经解决,专注当前层逻辑
2 `* c# {' p) h, c6 ~( X3 |2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
# A- V+ h( \4 f$ x* b0 L, p3. **递推过程**:不断向下分解问题(递); P  n# B/ B0 |$ G  j, }2 {  p
4. **回溯过程**:组合子问题结果返回(归)
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7 \; B, F* W& K: q- I( `" [ 注意事项6 B' i1 `: J) \
必须要有终止条件" p8 `- g$ `; m/ ]- b( W0 ~
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)% d4 i8 `, c9 G" K2 [, m- R1 k
某些问题用递归更直观(如树遍历),但效率可能不如迭代
/ c* T5 |% E' m尾递归优化可以提升效率(但Python不支持)  K8 c' q5 ^  {& [5 R  Q
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递归 vs 迭代* m; _+ ^4 A2 n+ x+ q+ e9 Z# X% l
|          | 递归                          | 迭代               |* h% ~* I5 A; ], |2 C0 a
|----------|-----------------------------|------------------|% K) \1 L: L) C; u/ P+ P" _+ k( n
| 实现方式    | 函数自调用                        | 循环结构            |, {+ _! R$ c  \8 I8 V8 H
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |( f6 N0 N, T) I6 i! W3 K# t
| 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         |
! p% F2 w# }$ P  {8 x. [- p1 e| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |  c" \# }8 F5 ^% p2 D; w
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经典递归应用场景
0 l6 }# W9 w( o" i$ r1 g1. 文件系统遍历(目录树结构)7 e/ U: W1 ]5 H( y  {
2. 快速排序/归并排序算法6 [3 p9 P$ t, S  @1 g7 [
3. 汉诺塔问题
9 W. b. f2 t7 Q6 g4. 二叉树遍历(前序/中序/后序)
! u2 x  O$ h9 [  e5. 生成所有可能的组合(回溯算法)
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+ J$ }& T7 g* c  m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. e! e* H. ?我推理机的核心算法应该是二叉树遍历的变种。
' ?; ?4 b7 A' T5 I) ?, G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 K( [! ^. t$ E+ N/ e+ w( Y% D
Key Idea of Recursion
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2 ~4 [4 c# @; h$ ~A recursive function solves a problem by:
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) E9 t' i; A9 X: M) k2 c9 n+ s    Breaking the problem into smaller instances of the same problem.; k3 o* v% s/ E2 ~7 G

: ]! f$ `" u- }5 n    Solving the smallest instance directly (base case).. I: l/ f& f/ @. A
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    Combining the results of smaller instances to solve the larger problem.
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Components of a Recursive Function* I+ s- M0 d  ?. G6 z
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    Base Case:) ~9 i- `# W7 d- L
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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.
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        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    Recursive Case:
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9 @( s( H6 d2 d# O4 S        This is where the function calls itself with a smaller or simpler version of the problem.- y: X2 [! A* L. n; M
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).: O7 o, H* X/ m* V

( Y/ E  Y. L1 \* @; xExample: Factorial Calculation2 M+ O' J9 ~( N6 o2 Q
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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:8 Y- [% W0 ]2 V  q7 W# C
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    Base case: 0! = 1/ D; P1 k4 S! b" H

! N4 V4 ~. Z5 L- v& c0 a    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):
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1 m3 S* A/ c( O; P! D9 e& }0 adef factorial(n):  A5 M$ ~$ e- \, ]% a- r! [
    # Base case
% O# Y/ j3 @1 F- H. @9 c* \& n    if n == 0:5 e; ]/ j+ T  |4 g* N- R
        return 1
5 c- X6 x$ I" M$ E9 p. j( _; M    # Recursive case
6 u  a! K; ?2 s# f0 a) C. E    else:/ Q- h7 Z' g6 E' E- E1 M5 ?0 C
        return n * factorial(n - 1)( w6 |6 y6 N- D8 @7 U- E" i

3 z) Z! r. B5 q3 J# Example usage
% j: Z$ m! S9 r5 s/ pprint(factorial(5))  # Output: 120
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How Recursion Works9 U) ^2 X4 b9 E( D
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    The function keeps calling itself with smaller inputs until it reaches the base case.
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- O! M' f2 P2 P! N" @. k2 e5 M3 b    Once the base case is reached, the function starts returning values back up the call stack.  x/ t0 V- X  t- u* h9 V3 L
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    These returned values are combined to produce the final result.
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8 h" S6 `& D8 B9 VFor factorial(5):
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5 H- H3 W: t6 @) u2 A0 u. ^* s: ]  Jfactorial(5) = 5 * factorial(4)+ z* u* U9 g0 I3 d3 D
factorial(4) = 4 * factorial(3)8 Q2 L4 E, z+ h' C' y' g
factorial(3) = 3 * factorial(2)  p( f7 a9 X. L% ~4 _  j# u+ J
factorial(2) = 2 * factorial(1)
! Q" }3 b! {6 t3 X) Xfactorial(1) = 1 * factorial(0)1 T. w/ q1 l( j
factorial(0) = 1  # Base case
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Then, the results are combined:( J: i3 ~  x3 }8 f" c

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2 z" E) v5 v/ I  M7 r. ^4 s9 t: j! Afactorial(1) = 1 * 1 = 1& Z; ^( `% i% w( K
factorial(2) = 2 * 1 = 2
& a7 Y3 ~; u0 u" O0 v+ K* Zfactorial(3) = 3 * 2 = 6
" ^' h: q9 i2 h! gfactorial(4) = 4 * 6 = 24% h2 x7 q/ U0 `, J; o" W
factorial(5) = 5 * 24 = 120# F8 Y, C2 ]% ^8 s1 h% e
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Advantages of Recursion$ N, f( U' w- g7 [6 T
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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).* \* M2 _0 Z* n# }7 e

' m5 R1 u9 g: a, W) I    Readability: Recursive code can be more readable and concise compared to iterative solutions.; P$ P7 D! C1 k$ X1 @3 p

9 _! z/ V7 ?; n2 Q. ?& c, WDisadvantages 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.# g8 E1 M, `! F- J$ S
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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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).
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2 d( z4 z: A" A2 x    Problems with a clear base case and recursive case.
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Example: Fibonacci Sequence1 G2 I8 u) ^/ ^' ^# M

# d. E, R9 i/ `5 O+ G- wThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
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' E5 q: |. i4 J. R! m8 X! ~    Base case: fib(0) = 0, fib(1) = 1
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; G3 U' }9 m9 \/ I# `) L9 g! _    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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. E8 N/ l. Q% apython
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  T( l6 T/ i7 n" g! u7 ydef fibonacci(n):
8 D: C* p- \! e; r& w    # Base cases
4 E+ G2 F; E# z& l$ B- U7 I% W: I    if n == 0:
  X' n4 l/ ?6 z" _        return 0( @$ K9 w( m) Q/ A0 `
    elif n == 1:9 ~# k$ O2 |' Y
        return 1
. V" P" f3 `. u, [- f8 }" h    # Recursive case3 F  E! b/ R2 x6 v& ^* j3 o
    else:
+ E$ F# U, [6 ]        return fibonacci(n - 1) + fibonacci(n - 2)
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# Example usage
/ A6 Z6 Y$ {6 T2 c9 o. U6 `print(fibonacci(6))  # Output: 8
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8 ]) m6 m  e6 }" _( JTail Recursion3 v: ~. [9 X7 C+ J9 H  V
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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).
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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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