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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 d# L& s6 R+ {解释的不错! S% A6 \* Y; ~- x5 X  [( M3 W

1 K+ q  D( K4 s8 [% y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
4 a  h; r. o1 _& s! a5 L1. **基线条件(Base Case)**
* P' T! H: o4 W9 }- @3 g   - 递归终止的条件,防止无限循环5 O7 A( \7 v: O4 c! j
   - 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)**, u1 m% f7 X" S& m* t
   - 将原问题分解为更小的子问题6 t) B- N0 ^% b  V) F
   - 例如:n! = n × (n-1)!
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2 n# r9 `* {$ O9 O  Q 经典示例:计算阶乘2 u( G6 \% d4 E( z
python
  f$ M, o5 p) X( m" i2 Wdef factorial(n):
9 m1 I+ [( f! F) [- y4 p    if n == 0:        # 基线条件
: s7 F' U  J( f# z! _6 h# _$ `7 k        return 1
/ ?4 y( O7 i  V* w& O+ L    else:             # 递归条件
6 {: L! |, Y( E* w0 d$ k        return n * factorial(n-1)
  D- M* t* Q; @- y执行过程(以计算 3! 为例):: M6 R3 j. T/ l/ q* H
factorial(3)3 }6 `* _3 |) G  e
3 * factorial(2)
* V+ k+ k. K; t- P( Q. {8 t! f3 * (2 * factorial(1))% R( ^! N7 g, \' [3 z! J9 A1 S3 ^
3 * (2 * (1 * factorial(0)))
! }! v) Z. J5 e9 L, S! r3 * (2 * (1 * 1)) = 6$ T5 o, c8 d3 o! Z- s" V' f6 A
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递归思维要点
; u% {1 G/ _) F$ o1 S1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 W* }/ G+ A" f8 Z) x7 \4 u0 E
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)( F; E9 }1 x6 u
3. **递推过程**:不断向下分解问题(递)3 P- W- z4 n: O3 ]( `7 [) m/ a
4. **回溯过程**:组合子问题结果返回(归)3 y6 g, v. Y; o) c, c

( f7 O" ]" \1 m8 F/ h 注意事项9 \% M( i  m0 K5 _: @
必须要有终止条件
+ `6 g% W7 O- m3 L& E# _递归深度过大可能导致栈溢出(Python默认递归深度约1000层). m* X* y4 x3 q0 ]0 m
某些问题用递归更直观(如树遍历),但效率可能不如迭代
. b* @( v8 z+ C7 \/ D尾递归优化可以提升效率(但Python不支持)! h2 ~1 c+ {" P5 W

. u- m3 Z* x+ G! m- ? 递归 vs 迭代- |) }) I& i6 Q5 P
|          | 递归                          | 迭代               |' n' |1 t7 |% L3 c& f# r
|----------|-----------------------------|------------------|
/ n$ K2 ^+ U$ N' D, F5 H| 实现方式    | 函数自调用                        | 循环结构            |: }; W& G. Q, J, S
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |! U: t& n* |9 d3 B& 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         |
8 X. u3 V; {5 A: i- b" r0 A| 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
3 M4 c, ]3 Z) N8 Z; R! g0 ^) P1. 文件系统遍历(目录树结构)
& ~! N1 t) z2 i. a2. 快速排序/归并排序算法
( p+ M! S( S* u. g$ g8 i3. 汉诺塔问题
' r$ f( w! [+ `4 B4. 二叉树遍历(前序/中序/后序)9 P. G+ {. r/ x& h1 G
5. 生成所有可能的组合(回溯算法)" q" g. |8 e1 E- S% h! \% G0 u

$ n1 c. s' o" D9 K( z% t$ i* F, \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: q- p- J& {; W! V, [8 t8 s9 C我推理机的核心算法应该是二叉树遍历的变种。3 w: H6 N% \) c$ q
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:: [+ I' k9 f: a
Key Idea of Recursion
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4 M+ d( T2 b8 j" n, V* a+ UA recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.
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/ I7 ^8 j# N2 Q% T! X    Solving the smallest instance directly (base case).& E7 M. l' ?8 @/ d
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    Combining the results of smaller instances to solve the larger problem.
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Components of a Recursive Function- y9 `. }9 }1 x. {, A5 Y) D
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    Base Case:' h$ _9 G: b# [/ s  v4 v- m9 ~
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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 u7 S! A  e9 p) g/ J6 j

& k9 u. e4 L2 S  j. Z5 u        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; R+ k8 U. G$ y, C  s

7 K! e" {& ^6 ^    Recursive Case:  [, @# U/ W5 Z& J

* Y5 f- j# P% P: O; \. F        This is where the function calls itself with a smaller or simpler version of the problem.( e9 @( e, `' o$ J3 E
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).; C- B! f) n' s

' j8 l# x2 H# H9 k  x" F5 ^; kExample: Factorial Calculation
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5 X3 L: k* S. L' R. j2 G. wThe 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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! t! d! a# X! U& Z6 k    Base case: 0! = 1+ `! e/ A4 i. n

/ e5 _$ P5 O7 g! Z/ p    Recursive case: n! = n * (n-1)!$ E$ b) Q9 {7 U9 L* r% m; l

: {, F9 J3 T: i. i9 n$ PHere’s how it looks in code (Python):
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def factorial(n):
' c% d; v! p* Z8 k% M8 \) m    # Base case
' `5 }' g  q; P9 \5 e    if n == 0:
. c6 m; i1 s$ G        return 1+ f& R/ Y- M! {% d' d0 ]' }( T
    # Recursive case( ?, X: p% y8 `$ w5 {
    else:
0 ^" P$ l4 i# [, S% J        return n * factorial(n - 1)
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# Example usage
& ~8 u- c  J& e2 p# `8 Z+ z. wprint(factorial(5))  # Output: 120) m, R4 `% _2 R$ r' M
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How Recursion Works
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% D7 p1 i/ S. [7 s" h, U    The function keeps calling itself with smaller inputs until it reaches the base case.
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    Once the base case is reached, the function starts returning values back up the call stack.
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    These returned values are combined to produce the final result.3 Z4 N5 f2 @) i( D

& B8 X5 ^& l" j% r4 h8 ^' [: LFor factorial(5):
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3 [2 X7 I8 J3 P4 R4 E/ o! Vfactorial(5) = 5 * factorial(4)$ E2 S2 c# b) r+ p$ D. P
factorial(4) = 4 * factorial(3)- Q9 g3 ~8 c9 I8 o0 P7 J: Q8 c
factorial(3) = 3 * factorial(2)0 B( i$ ^3 v1 B  _
factorial(2) = 2 * factorial(1)
/ I: G6 m) m6 ~0 r; i( Ifactorial(1) = 1 * factorial(0)
' J/ _! E7 U, r; m: L7 z8 ]factorial(0) = 1  # Base case+ J0 U# s0 p/ A, d% w* R7 H

0 u  u, y3 W/ ]& U9 j# D& IThen, the results are combined:
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factorial(1) = 1 * 1 = 1
$ ?$ x1 _9 ]% J6 o9 l3 `: Vfactorial(2) = 2 * 1 = 28 q0 {! r+ v/ G9 N% |# B7 o8 E
factorial(3) = 3 * 2 = 6
$ U9 [* N& D: k" u* dfactorial(4) = 4 * 6 = 24
& P( m1 r7 D6 v6 U' @8 efactorial(5) = 5 * 24 = 1206 O7 {; F  M+ H, m

- @. Y1 Y5 D7 C+ J" C! A2 ^( MAdvantages of Recursion- E6 J+ L. _9 o, C; m" 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.' i. [, g, R7 ^2 M" M& i

' f. Z6 ]& M2 N* P7 {6 MDisadvantages of Recursion* P- u. x1 I1 W* j  @
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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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2 E/ y( G  N0 F+ s; N& {    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)./ C( l3 ^! X8 S' R: v8 d4 s+ H  `& F
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When to Use Recursion
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; b! p; B) J8 L    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).- h+ T/ B- o; o, a1 _: t5 @1 Q  a
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    Problems with a clear base case and recursive case.+ D2 h; M  G' E  f$ ?) i0 x  e

. D  X( [6 h1 a1 g. ^- GExample: Fibonacci Sequence
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4 {0 M2 `* E; _7 WThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
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, h: N/ ~& @( m    Base case: fib(0) = 0, fib(1) = 16 y" ~0 z+ K7 n' C
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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+ \+ ~& ]4 k# |0 bpython
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def fibonacci(n):" J" d# w) g# y7 }# i' _
    # Base cases; g! ?/ u  Q2 D' T3 x& Z/ n( O9 h. a
    if n == 0:7 ?4 v2 I4 ^4 c# G
        return 0' _& P) D8 x5 S: y7 j. P* v, j
    elif n == 1:
) A) }' ^& F8 O2 G        return 1: s  x% M. H/ j
    # Recursive case0 ~2 V4 x% U9 p  N6 p
    else:
. O( B7 q  T# a& G        return fibonacci(n - 1) + fibonacci(n - 2)" W1 H+ f/ R1 s. y/ Z) C

6 n: o; _, u# N! c# Example usage9 ^9 L+ H/ X7 l: y! H
print(fibonacci(6))  # Output: 8/ i5 O* L% }# Q, x3 m
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Tail Recursion6 t& q" h/ l0 |/ B3 U
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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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6 t0 ]+ E+ P3 Y7 {+ f  X& \2 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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