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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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1 l# o0 z% k( ^/ X3 a; F' ~" \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8 S4 x. j; s+ B) }
1. **基线条件(Base Case)**0 M3 e! O) m8 w5 d$ U% U
   - 递归终止的条件,防止无限循环
- m1 r3 Q' _% R+ e( }) Q   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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. `, r) o; [) A# r5 q! @2. **递归条件(Recursive Case)**
1 B5 E9 G  H( ^5 k  @1 E& ^3 P) O   - 将原问题分解为更小的子问题: u( L# J7 _" w$ ]) l
   - 例如:n! = n × (n-1)!1 R/ Z4 A$ U$ }1 y( Z2 R
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经典示例:计算阶乘& p% n9 D% S9 n: C8 t- f
python
% E$ N" p, n" kdef factorial(n):
6 u- R0 f* u: P    if n == 0:        # 基线条件5 ?" J0 ?  p! |/ T
        return 1+ o# n4 N  O# R! E, B) f% `
    else:             # 递归条件
8 e5 V7 `3 Z+ I* a5 W' g% Q        return n * factorial(n-1)
, g# I; u6 M7 C2 T% q( t; I执行过程(以计算 3! 为例):* @  S8 o/ Q& S- b3 B# f+ F
factorial(3)
( G. m5 o- v  P" T( R/ x3 * factorial(2)
0 P; O- {' n$ @3 * (2 * factorial(1))# H- g3 f' g; R) z2 s: e) a
3 * (2 * (1 * factorial(0)))5 g, m* T1 f5 t  d1 }
3 * (2 * (1 * 1)) = 6
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' m; Z( W, T; r1 c 递归思维要点0 o5 T( e9 Q/ _. d
1. **信任递归**:假设子问题已经解决,专注当前层逻辑
8 t: h7 o# o8 c! @( Y2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 p1 C  k- a) x1 p# A+ O
3. **递推过程**:不断向下分解问题(递)
; y% f! h9 f7 D. P5 P: Q2 c4. **回溯过程**:组合子问题结果返回(归)
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& S: f# }# s2 T6 H 注意事项
$ V: |0 a( S, z' ]. }  S必须要有终止条件
. \7 h, k2 H" B. l  j递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
$ i- M4 v1 D' b2 j) k& ^; y6 ]某些问题用递归更直观(如树遍历),但效率可能不如迭代. H: z( J: y. ?! K# @9 z2 h
尾递归优化可以提升效率(但Python不支持)( |% R; X" s1 l4 `

. n5 u* O/ l0 O$ Q1 \+ s 递归 vs 迭代
3 ^; X5 U- C3 K|          | 递归                          | 迭代               |2 ]) k7 O9 ]8 |; D$ X9 H  Z
|----------|-----------------------------|------------------|7 |, R2 W7 o+ G2 h0 z" r' ^2 D
| 实现方式    | 函数自调用                        | 循环结构            |) G! Y. d% i4 X' F# J2 o$ L+ l( h+ o- H
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
  t$ M1 c8 q4 B' V% Z| 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         |% b% s; a/ Z; {% b" S0 F$ 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
" J; [. i, Q( m- }/ P' T+ [1. 文件系统遍历(目录树结构)* x! d) u3 Z( c8 F( D
2. 快速排序/归并排序算法
% g/ _( l' k! I  Y3. 汉诺塔问题: j" }' q( O4 o& t. T( j' d
4. 二叉树遍历(前序/中序/后序)$ y: G; E% I) k+ ~1 f6 ]2 {
5. 生成所有可能的组合(回溯算法)
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3 H+ E+ J) u) }4 i" {$ X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,: I$ l- m% }& |) p9 ~' J: ~
我推理机的核心算法应该是二叉树遍历的变种。
/ l$ |( q$ Z8 m6 I: K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:7 E" f$ R1 @# ~: |/ V1 W! y
Key Idea of Recursion
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2 j9 n! A4 b4 Q$ `A recursive function solves a problem by:  E  Q& N0 S: x0 {4 M& k2 E
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    Breaking the problem into smaller instances of the same problem.% g! \$ a2 N0 f1 W& l( [/ P* x' |
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    Solving the smallest instance directly (base case).! [6 E" v! z% y' k# x! Z  t

8 L. b+ X) I: d    Combining the results of smaller instances to solve the larger problem.+ p# }2 L2 y' k) L3 ?

/ E: m" H, r* n; z2 cComponents of a Recursive Function
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    Base Case:7 K: ~( |/ u: d6 _+ q

8 x# C) ?7 V; x1 b& D! m( z; {        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.4 R! W0 }* t& Y) \( z1 J# p

; K: n4 A" d! }! b        It acts as the stopping condition to prevent infinite recursion.# H/ }$ a  o5 O0 _, l* A  B
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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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, X  J; T' x+ O% z( j/ {* x+ e& t        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).
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Example: Factorial Calculation" @8 w7 e5 k6 }. s! c3 j5 E
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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:, j5 d1 x+ w; y* k# T+ z$ x

, b+ l# e* j) m    Base case: 0! = 1! H* [/ K6 E& i( ~6 d

- `( u$ z' ?9 q    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):/ T0 {; z* h- ?7 L! u. `- Q
python
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* a' W6 `4 i, ?+ C2 vdef factorial(n):$ p7 R' Z- }/ f4 o
    # Base case
' b8 B9 E  W: [2 x- J3 z/ I    if n == 0:
  u, w8 q! I1 i; J        return 1
% ^# ~* ]0 B$ M  H' N    # Recursive case& l6 \$ D" t( q. u: G6 M( N' N
    else:" R0 }. X! |5 }
        return n * factorial(n - 1)
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2 M  H/ w# C+ u# Example usage4 x7 O; _5 V: l% ?
print(factorial(5))  # Output: 120
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) U3 S, H8 W; j5 `( WHow Recursion Works" v" N4 t) ?7 D8 h! s; M
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    The function keeps calling itself with smaller inputs until it reaches the base case.
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0 r4 ~) {/ V4 h- C    Once the base case is reached, the function starts returning values back up the call stack.: }% g1 ?0 A: X$ m; p: S

! Y" U2 `9 O6 [# s2 ?0 D- j    These returned values are combined to produce the final result.
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For factorial(5):7 e7 I8 k8 V+ `* r& m* e

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factorial(5) = 5 * factorial(4)
. V; Q  @; J. P: zfactorial(4) = 4 * factorial(3)9 l; Z! T6 Y; c1 u3 J
factorial(3) = 3 * factorial(2)' q1 g, Q9 S% W3 k) |$ ~
factorial(2) = 2 * factorial(1)
6 n7 j4 [8 h' Ifactorial(1) = 1 * factorial(0)* N+ _, i! u6 `9 v7 p& N6 d
factorial(0) = 1  # Base case
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Then, the results are combined:- C8 V6 U, r* ]5 d! {4 S
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factorial(1) = 1 * 1 = 1
, \& B6 l3 R9 }0 }factorial(2) = 2 * 1 = 2
0 t- ?6 `* [# h+ A0 B5 k4 I9 Ofactorial(3) = 3 * 2 = 62 v" }' H" f. j* m1 I
factorial(4) = 4 * 6 = 24
) {( B+ v4 @. J  u. B3 B" jfactorial(5) = 5 * 24 = 120
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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.
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5 u) g2 e* N7 T. o9 U2 HDisadvantages 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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* J' r7 @% L' f0 v  |    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).6 H  G  r, M/ ^+ w2 g5 ?' k
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When to Use Recursion# `2 `! r  V. r% n! T, k% k3 _
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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.
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Example: Fibonacci Sequence$ G! W# \8 t* e6 ?+ S2 ^

5 R+ `- c4 D$ z- T9 h* g8 g& C( B/ ^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
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, I# Y- T# T5 I    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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def fibonacci(n):
3 a+ E2 j! p0 P    # Base cases
0 [, d4 I8 P) \$ v    if n == 0:  ?5 R6 j' n" b, N
        return 0! u: b4 ?: p  L% y
    elif n == 1:/ q# F* C& Y4 S1 I4 D
        return 1' {! `- d) x" ~" c* P; P4 A
    # Recursive case
8 o. [) M2 a5 c# K- Z5 @  a! E" t    else:
! |) F( `# g. G/ d8 _' u        return fibonacci(n - 1) + fibonacci(n - 2)* Q) o5 `5 R8 W

, {( i# ~) r& L5 i# u: l# Example usage# x7 G  G) n/ z" W
print(fibonacci(6))  # Output: 8
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! }  k6 d2 E! w& W! k; STail Recursion/ ~, r; e; P, l' r/ \1 |

0 I2 ?7 Z( `- jTail 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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