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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8 E! G3 ~/ a1 t0 t& v) z, l

9 W( p) k2 u/ l- o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。: a2 s& r% ^& k# R8 s

: W/ E6 ~! i1 s( T$ { 关键要素
$ F& ?# j  q# `' w1. **基线条件(Base Case)**8 a. A/ ~) ]8 q& B  c6 ?. O. r
   - 递归终止的条件,防止无限循环! Q1 K9 d9 G# i5 F
   - 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)**( N, Y& G4 C3 r9 ^( d0 F
   - 将原问题分解为更小的子问题
! h' |2 f# |+ s' [6 F   - 例如:n! = n × (n-1)!
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0 A+ i& c1 H/ v  P5 U3 L 经典示例:计算阶乘
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) Y3 d4 `  E8 P- x2 n5 Z- ldef factorial(n):
: B0 H& e2 k& D) ]- C* \' P' H4 R7 `    if n == 0:        # 基线条件$ P; X3 K. o* m# i% ^% ]4 G7 B* a
        return 1
2 K* P( R7 z  `! d, Y    else:             # 递归条件
5 j' L5 s( v" y        return n * factorial(n-1); b3 E, c/ W; J4 @! M: S3 G* a
执行过程(以计算 3! 为例):
' R; M% ?1 n; C" p* f9 @- N, Ifactorial(3): G) i) Q+ w, N1 b  @, G  D5 k
3 * factorial(2)1 q7 O- |8 a- m. \" k' y3 z8 f
3 * (2 * factorial(1))% v. j2 ?+ i5 n* b/ \0 O
3 * (2 * (1 * factorial(0)))
/ N9 D6 Z! B4 ]2 \2 c3 * (2 * (1 * 1)) = 6
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递归思维要点
8 Q. l* q0 t2 V8 p, @1. **信任递归**:假设子问题已经解决,专注当前层逻辑
  {+ W3 Q$ X6 h4 k& L7 ?2. **栈结构**:每次调用都会创建新的栈帧(内存空间)3 p! d: |. s1 {  M5 C
3. **递推过程**:不断向下分解问题(递)8 }/ k' T/ H+ t3 o3 }% J
4. **回溯过程**:组合子问题结果返回(归)$ j# T# D" o- L
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注意事项8 ?, j) O+ F, |3 @0 _, s
必须要有终止条件2 q6 u; e* C0 x8 M
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
) h4 f3 \6 Z, f, n  o某些问题用递归更直观(如树遍历),但效率可能不如迭代
& \$ ~! J; v# g9 _# S) |尾递归优化可以提升效率(但Python不支持)
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9 l6 L5 v3 O2 _  V. a4 a# p$ I 递归 vs 迭代; j$ D; f) |2 z
|          | 递归                          | 迭代               |% o, l( Y8 A3 D9 X( R6 r) I. N2 ?
|----------|-----------------------------|------------------|
* A5 M- h" v7 W* j# E6 j: C| 实现方式    | 函数自调用                        | 循环结构            |# r8 u/ G# E1 Z; [. u
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
' h& {+ A, `* F& B3 n: L| 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         |
5 w* w. x$ v9 J+ }1 f  I| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |( k7 g% v/ D: h$ V; _

7 _/ C0 ^0 G0 l8 n3 S: k 经典递归应用场景) q# g  z, E1 r, Y
1. 文件系统遍历(目录树结构)+ b+ {- K4 h! |8 r
2. 快速排序/归并排序算法5 @5 V$ v- Q4 q. i) {4 I) `
3. 汉诺塔问题0 c7 f( F+ G7 h: A# G
4. 二叉树遍历(前序/中序/后序)
# d  b# }8 o+ K& l& u. r; p- F' {5. 生成所有可能的组合(回溯算法)5 R' y5 i# O( ]- f8 P; A

, _7 X/ p) {, J5 x2 P: s. C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,
5 f' t" X8 f2 m2 M5 }我推理机的核心算法应该是二叉树遍历的变种。& |$ I/ I3 s" M
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:1 _9 U& o  E: G
Key Idea of Recursion
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6 g; ?0 ?. s1 O$ E8 d: bA recursive function solves a problem by:
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8 h1 r& O* j  ^% B- v" r% D    Breaking the problem into smaller instances of the same problem.0 Z8 ^9 \& q3 E# B

9 e) y0 K' X% I" E    Solving the smallest instance directly (base case).+ L5 R: L- K0 N; ?1 |% E/ a5 \
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    Combining the results of smaller instances to solve the larger problem.
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Components of a Recursive Function
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    Base Case:; G! o; P7 B( @* A

& p. Z4 ~. Z" |, V' N        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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) l7 O0 C. Z8 ?" w) ~* h3 g        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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" Y2 }" N+ }- G6 _- k' v9 m5 ~    Recursive Case:9 U& n" [$ a6 M  w9 _

/ X- D7 \6 N$ x. ?3 @* ~; g% f        This is where the function calls itself with a smaller or simpler version of the problem.
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0 e1 C4 _5 O! ~; b8 ?$ ?        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).) z2 X% p8 A9 A5 t" `/ b) b$ p5 G
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Example: Factorial Calculation  j) d% ~4 Q/ ^" F  K
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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:
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" Q+ o  m% _: i1 b& o7 p    Base case: 0! = 1
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& s" R0 T% w% F8 G2 j8 i9 F/ O) t    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):  Y9 J8 S. k! \# T6 g3 Q
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def factorial(n):. n* `/ V4 {; R- t8 l- B" |
    # Base case
: X) @# a& z  O: P& ?/ }* O    if n == 0:  [) @3 b+ q# D) ^/ [
        return 1" N3 t* U4 o2 E" s& @
    # Recursive case
; d. @. G7 T/ e* V    else:: J: C# C4 S$ a8 Y- W) h* N; |
        return n * factorial(n - 1)% {' |9 v- n" `3 N
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# Example usage
. H9 G* y4 Z/ o. e9 z. [4 [0 X) Z5 qprint(factorial(5))  # Output: 120
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How Recursion Works0 Y: [: I# k; j6 B
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    The function keeps calling itself with smaller inputs until it reaches the base case.
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! C3 K0 I( Z  Y$ e! T( Y    Once the base case is reached, the function starts returning values back up the call stack.. ~3 I' G5 m* S$ l: @
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    These returned values are combined to produce the final result.
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For factorial(5):
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+ E0 Q  N* }4 l# @0 H5 h4 X2 Lfactorial(5) = 5 * factorial(4)
! W4 X9 v6 N- m$ H3 U0 ?0 d. x# T% yfactorial(4) = 4 * factorial(3)
* l5 H3 \( ^) lfactorial(3) = 3 * factorial(2)
, q6 n( o2 y# Bfactorial(2) = 2 * factorial(1)
' ~+ Z8 x! c( Q, G$ f) k% d3 l! Ifactorial(1) = 1 * factorial(0)1 m5 d: W* P  {, [5 H* G. d0 n- \
factorial(0) = 1  # Base case
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Then, the results are combined:
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% N) k( n- P7 B2 `1 Ofactorial(1) = 1 * 1 = 1
5 a/ J* P: `2 A2 jfactorial(2) = 2 * 1 = 2
% T1 J9 ?+ @/ E) @factorial(3) = 3 * 2 = 6" g! |5 m7 T- Y2 p
factorial(4) = 4 * 6 = 24
! U; c+ t, F. P# \8 t/ [( rfactorial(5) = 5 * 24 = 120% ?" ]7 V+ P4 ?

6 f' S8 c4 f6 x" n: @$ c  BAdvantages 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)." n# R3 a4 y2 r1 s
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.0 V7 r. u- ?$ y1 h2 s
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Disadvantages of Recursion% f% v' q% u5 z: ?

1 C  ?% M( S  j+ q( R" m3 F, k- L    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).
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When to Use Recursion
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4 V- F  t0 U. M5 X    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).- l" ?1 [* ^  l; c

! P) e% P2 [. K$ o% |( A( R    Problems with a clear base case and recursive case.
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2 b' u: T* O6 J$ v9 t/ hExample: Fibonacci Sequence# n3 g- N% S) @  C8 J6 ^) ^+ u
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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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& V, M) e4 @9 f, R# x    Base case: fib(0) = 0, fib(1) = 1+ b& M  `2 {) {9 Y: v; ?

5 Z" Q7 U4 @; V    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python
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7 L4 d- q( P0 \2 @. b7 @def fibonacci(n):$ I( ~7 b3 i  ?; k3 ~* \4 P! L" L$ H
    # Base cases
- t  Q" \( c5 Y7 o% D    if n == 0:
! \' }/ e  X' K. R- M- T* ~        return 0
9 f! i# ?+ V! t  e* f    elif n == 1:
6 T" U; h# X3 n3 O7 d2 Z/ }        return 1
7 l' Z# i7 Y; g2 p4 H    # Recursive case
. [' e' D! N5 V7 X& a; ?    else:
; `, s2 k! l/ l$ \. D/ n/ ^2 `# E        return fibonacci(n - 1) + fibonacci(n - 2)
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# Example usage
4 j1 W3 f5 N9 R# }print(fibonacci(6))  # Output: 8
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Tail Recursion. j3 L, e7 {$ v- d! |# [
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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).# V9 \7 ]) q" _/ G. Y" q% e

( a, p. |8 D2 ZIn 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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