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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 i0 s8 I, U) y- y* O5 i

1 P* n! z" S2 ~, K! e5 y解释的不错
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1 |1 l+ {: ?6 C4 c6 u4 a8 R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。2 {4 f% J! y% j3 h: \7 ^4 B' F
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关键要素
, K3 O9 j2 n1 o2 E# x7 t1. **基线条件(Base Case)**
' a8 I5 g4 q' i- e" |! o   - 递归终止的条件,防止无限循环; V2 M. ~3 _  U1 g! w; r8 b
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1: w2 {! e' }$ K3 g8 d& f$ h

; D6 w% `- ?/ [2. **递归条件(Recursive Case)**: z3 n/ A$ }8 i( h4 `' H% e
   - 将原问题分解为更小的子问题
' _* f1 f, X$ t   - 例如:n! = n × (n-1)!
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经典示例:计算阶乘; p- c& m" j/ P8 e
python
& g' J9 q7 V" @8 i9 }- P6 q. Ddef factorial(n):. c% T. A3 d3 G- e* Y
    if n == 0:        # 基线条件8 ?: u0 M8 P' X. Q; n; W) ]9 J$ D& @
        return 1
8 r8 c+ m# p0 V& x# `, n; u    else:             # 递归条件
7 A9 P8 G! B8 H. x6 O        return n * factorial(n-1)
& g- r8 p+ R) }9 s9 ]执行过程(以计算 3! 为例):
2 d: Z' p! t6 {2 `, {  H! Bfactorial(3). o/ c% i6 ]1 b( t
3 * factorial(2)
1 W* n5 }/ d2 s. H& t3 * (2 * factorial(1))6 x9 ^3 \, K+ c2 ]- k+ e
3 * (2 * (1 * factorial(0)))
% e: Z* ^: g: n. U; C3 * (2 * (1 * 1)) = 62 _( _* R- Y: A: `) W/ ^9 W

# e5 R. p" ^0 d% k$ l- ]8 i! @) h2 | 递归思维要点' q* u8 u* u; Z
1. **信任递归**:假设子问题已经解决,专注当前层逻辑
: [, d. N% k# A, X: F2. **栈结构**:每次调用都会创建新的栈帧(内存空间)3 o: k. M4 L  ]  X$ L' Q3 b
3. **递推过程**:不断向下分解问题(递)
( K& K4 i& h6 q) ?  i* Z- {( D4. **回溯过程**:组合子问题结果返回(归)
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( C* }" w! n7 z" C6 x; C- M1 S 注意事项
; P1 [* e9 z9 n3 j0 I0 g必须要有终止条件: {' V3 O$ g2 j
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
. z6 R! R9 T6 i( X0 [8 _% b某些问题用递归更直观(如树遍历),但效率可能不如迭代
# _* l. D, u/ E尾递归优化可以提升效率(但Python不支持), a1 e. R/ Y5 r4 b) R$ b' L0 @
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递归 vs 迭代
8 n  h+ A1 n3 ?$ F4 b! `' T|          | 递归                          | 迭代               |: i; K2 Z# p. l. v; ?' Y* k3 w
|----------|-----------------------------|------------------|
. i) g; Q  `$ U2 _5 U% k| 实现方式    | 函数自调用                        | 循环结构            |. O% C! o2 p2 ^
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
0 c/ K" g* J  h8 X| 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         |
" y& H$ {# v, K% `/ z. N3 G) v+ [| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |3 f$ j& Q6 G/ D: L
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经典递归应用场景7 _9 W' [* e% M$ F
1. 文件系统遍历(目录树结构)! c3 E2 v  D  D  @! D# Y+ F5 M
2. 快速排序/归并排序算法
, k! W" y/ V0 J" C( O" l3. 汉诺塔问题0 y$ i$ J6 f  x" j
4. 二叉树遍历(前序/中序/后序)
( P, p/ y) e6 B5. 生成所有可能的组合(回溯算法)4 v& L7 a3 Q% c( N9 D
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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,# D' G# R, t. ^
我推理机的核心算法应该是二叉树遍历的变种。
8 p% ]' v1 h5 V( {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:
& j8 p' G" y, U8 e0 r( k. AKey Idea of Recursion
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A recursive function solves a problem by:
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8 C* j6 s4 a  ~    Breaking the problem into smaller instances of the same problem.
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3 W& n& P. P4 o1 ?: U    Solving the smallest instance directly (base case).2 ^, j" x5 |; T
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    Combining the results of smaller instances to solve the larger problem.1 j" m0 K6 M+ y& R

( C! U( ?' D6 N3 x! X* K3 S' \- yComponents of a Recursive Function( A0 A7 [( X/ ]2 C& I( j
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    Base Case:
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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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, {* Y7 ]1 H! K! S- A2 o        It acts as the stopping condition to prevent infinite recursion.5 B# W* d1 J) |9 t

/ {4 k3 G. ^: I7 Q, O( s# s        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.  b( s* t1 \4 b' T! Q0 H

+ A# ]( m' h6 r( M; V    Recursive Case:
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0 x% w2 e7 l/ I2 d0 h% o        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/ y% W$ a9 z; B6 B

+ W4 s8 n* j% H* YThe 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:/ J' r; O; e6 Y# y) s

7 E7 h( _- l3 `$ K1 \) M' |5 L- z    Base case: 0! = 1
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* q4 z# n" t% e, Y: U4 a  x    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):
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def factorial(n):( s( y4 s& ^; P, t4 a! p' g) N
    # Base case6 X0 Y$ V4 K3 Z
    if n == 0:* z5 ^3 I& `% U+ z( B1 x% h% I
        return 1
; Z. h- R; J' R# c/ m( `    # Recursive case
* `3 ]0 `8 j) u) t3 H' Z7 i* v) p    else:2 z% ?+ J; ^/ q  F2 f" Y4 b
        return n * factorial(n - 1)
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& p' V+ |; z0 W6 g' f5 ?# Example usage/ a. g3 S: o& D, @" ?$ w8 z
print(factorial(5))  # Output: 120) W  C( D% v7 X! r. q# N+ q# N! r
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How Recursion Works, M5 G$ n4 j: T
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    The function keeps calling itself with smaller inputs until it reaches the base case." Y# P8 w6 A5 d2 }5 \3 Q! `

6 d0 P$ q+ O3 K/ k0 s; k    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.& @1 y$ r: N7 a* L4 p
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For factorial(5):) n1 D  v1 N# Y5 u6 m; |

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3 v. t6 k" q) {) |factorial(5) = 5 * factorial(4)0 S5 k  S1 e6 _2 A6 `$ M$ }* a
factorial(4) = 4 * factorial(3)- I5 E  E  E0 K7 D
factorial(3) = 3 * factorial(2)) Y* ]1 |; [8 V* M
factorial(2) = 2 * factorial(1). p5 Q" [6 X6 p  j* ^; J3 [2 f
factorial(1) = 1 * factorial(0)1 ~. L) A* K" m* ?& M* [7 f$ W$ p
factorial(0) = 1  # Base case3 c# E3 r, b' `, Q

/ M" N; U, R) h  S' ^Then, the results are combined:
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& x3 d' J8 C1 k" Q7 N' u5 E  T" ?# `factorial(1) = 1 * 1 = 1  n6 y9 Z1 y# C, [+ b6 P3 W3 I+ l
factorial(2) = 2 * 1 = 2
$ q/ J8 h+ ^6 t- \7 [0 f5 Jfactorial(3) = 3 * 2 = 6
$ _$ U; Q( w* m) N! D& ?5 Z, S( W4 M* h0 }factorial(4) = 4 * 6 = 24& A: b8 F2 i( Z3 Y. A* R
factorial(5) = 5 * 24 = 120
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/ Y' S, I6 b0 `  L* `Advantages of Recursion( D$ `! P" s4 r# M

( g6 u: m! B9 z( f( T6 f    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).3 N0 M, @, H' G6 D! D6 o

; _( _) u- {; T. J# g    Readability: Recursive code can be more readable and concise compared to iterative solutions.7 M$ F; g, H5 n4 b% H% v
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Disadvantages of Recursion/ A4 O1 X" h6 ]$ U

' z; k" D& }1 `( E- a7 m    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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  w% y3 A7 `9 E0 E3 `9 c. z2 cWhen to Use Recursion; V6 Z- c) T4 f; S" i% Q. H

; r; B: R* f/ k/ N- A. u    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)./ E0 C7 _0 R7 }/ g* y3 V3 [! ]

7 X% X, d" ~6 C" ^7 H) z& J/ d4 C( D    Problems with a clear base case and recursive case.9 J3 G, W2 L& M0 V
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Example: Fibonacci Sequence
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2 s2 v; c. n- b3 ~5 E8 n) NThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:: i& }! H( x' J/ b) S

7 U' ~9 W" K" F& S. r9 T/ I    Base case: fib(0) = 0, fib(1) = 1) l% ~2 |1 |+ U! h

/ S5 t; g: S8 ^4 R" W    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python
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! |0 `. a2 V, {" Xdef fibonacci(n):  r, V2 X) W2 ?: h5 u! [0 {' e. k
    # Base cases
' [  w! ^& u! T& ~    if n == 0:8 }# Q, M# B7 l- q! `3 j: P
        return 0$ H1 m( z# S% i4 I. P/ h2 |
    elif n == 1:
( T, N1 ~) Q, t2 R& s+ P  a  b        return 11 u4 n: ]3 h2 _& r- x0 l- X) L
    # Recursive case/ {! H3 T: U' b% m1 c: Y
    else:
# ~  w2 t( x" |- {4 ^        return fibonacci(n - 1) + fibonacci(n - 2)
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# Example usage3 T( u( ~  V% Y3 P& x3 h# |6 E
print(fibonacci(6))  # Output: 8  {( F- s, B' a: t( n# ~9 Z
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Tail Recursion
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  B* |2 R. M% s# r2 c3 rTail 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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/ S- e3 s+ Y- vIn 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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