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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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4 a/ M/ m4 Y9 h6 n' `" y* \! I( M解释的不错* U% N& Y- F" k

* h$ E) _) w6 o, Q" H& a$ y' P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
+ j5 }( Q$ b+ @  x& x1 F1 v. W- q1. **基线条件(Base Case)**: V  M) _2 {4 i  k+ l; d/ ^
   - 递归终止的条件,防止无限循环# o( p0 ^  D! g( n5 L
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1& e5 i) O9 e) ?5 O& |9 e$ C) G

# h+ |/ h0 n5 C2. **递归条件(Recursive Case)**
4 I% m  G1 w) g& Z5 u   - 将原问题分解为更小的子问题& a! x- c9 I+ o" k$ y9 w& T
   - 例如:n! = n × (n-1)!0 c  f' n8 D* `4 Z
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经典示例:计算阶乘
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def factorial(n):
7 r6 f  d& \7 t! g+ J9 y    if n == 0:        # 基线条件
7 n0 c, W+ _: D        return 1& T7 J2 W* W& \$ D; ~: m1 _1 g
    else:             # 递归条件
. \. V  L# g9 H% X% P+ L: L5 x        return n * factorial(n-1)0 Y0 {$ Q; f, C, \
执行过程(以计算 3! 为例):7 K* Z& @' o% ^% e# m! k( H
factorial(3)7 A$ K, e9 N0 B$ h8 U# r
3 * factorial(2)5 v# P, G+ V# B% M% s$ d
3 * (2 * factorial(1))
  o! O' U' ~+ L, P3 k5 n, V3 y3 * (2 * (1 * factorial(0)))
$ @% b  r4 O% |) F6 z- S3 Y3 * (2 * (1 * 1)) = 6- H7 h8 q- Y9 O% g

: n2 x* k$ I/ N% t3 L/ I  D 递归思维要点) A  N9 W/ I* U2 R$ I
1. **信任递归**:假设子问题已经解决,专注当前层逻辑9 ^4 y3 h5 s( q9 k: {( `8 T9 J
2. **栈结构**:每次调用都会创建新的栈帧(内存空间), ]  X7 X) z8 Z. b( O' Y
3. **递推过程**:不断向下分解问题(递)/ y( p/ G$ k. y0 g' F1 J
4. **回溯过程**:组合子问题结果返回(归)# k! ^0 G1 a! M6 }" X% @  J1 S
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注意事项# w, B  D# H# [( z) O0 W( y+ w  B/ ^
必须要有终止条件" j: M. O$ v& z) A
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)! B6 }! T7 T% z! D/ i
某些问题用递归更直观(如树遍历),但效率可能不如迭代
3 C( B1 f7 V; d" @; L8 U1 f尾递归优化可以提升效率(但Python不支持)5 k; n$ {6 e9 H" ^7 Z# H3 l3 X

, ~/ \& l: A( p+ b) X; o' V  x 递归 vs 迭代
+ Q6 `) g( N5 E, m6 I$ C|          | 递归                          | 迭代               |- B; |) z+ X9 `- f, Q7 a" I4 p
|----------|-----------------------------|------------------|
: J  Q: q8 S7 n  Q, U| 实现方式    | 函数自调用                        | 循环结构            |, Z) I) L# o" ]2 h. P8 N+ k
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
# j6 b5 U  K$ g0 t4 D: m| 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         |4 Z8 k1 U; h# D* z/ w/ d6 y; 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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) P, f' B. ^. W; H 经典递归应用场景
% a+ ^8 ~" y7 Y1. 文件系统遍历(目录树结构)& S( a6 o4 U. K; L
2. 快速排序/归并排序算法
' G" l. q1 ]! h& Q( l# J# i3. 汉诺塔问题; {' k* a# C6 v9 m
4. 二叉树遍历(前序/中序/后序)
6 [7 g. a% N$ j5. 生成所有可能的组合(回溯算法)7 \0 D+ A: m+ }7 S
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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,
; I6 o$ G4 d5 g我推理机的核心算法应该是二叉树遍历的变种。/ ]6 s0 S. W  |  F
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 c9 M; V5 _& G8 C- J) j+ K
Key Idea of Recursion& X5 Y) x, U% W- S- F( ?! v) @
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A recursive function solves a problem by:
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% H9 b6 q3 k1 L% [! E- J; Q    Breaking the problem into smaller instances of the same problem.
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7 d0 t1 D" ?' e& I# g& s8 W, t    Solving the smallest instance directly (base case).  |3 s  R* I$ G, o, J; Y  I

% K: x: \* ?0 C5 U0 t6 A6 O    Combining the results of smaller instances to solve the larger problem.
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Components of a Recursive Function! y$ F1 m# i  C

" \6 E8 H, x4 O" M    Base Case:" P0 e8 U; |" A6 z. n. |. R1 z; t

8 g; e! V& t' l( X4 q; J! N% L        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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% R( ~* W" \# K- m: [4 N        It acts as the stopping condition to prevent infinite recursion.
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! H. M9 l/ [5 E7 n        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.! [. u- t  p/ l$ M) f8 B9 X% e" K
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    Recursive Case:
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: d7 e9 a! b, z! G& T7 r        This is where the function calls itself with a smaller or simpler version of the problem.
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( E7 P0 x' ?. Q9 U: S        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).& N! _+ [5 \+ i4 K/ h- l
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Example: Factorial Calculation
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, `; S. s% @- H! c/ i: v, RThe 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:, U1 g" m) k# N0 V$ p$ m

9 [* A4 b: b0 k  o, F0 l    Base case: 0! = 1
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    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):9 t8 l; T, v# |  J, [# I0 E* o
python
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def factorial(n):: o2 J4 }9 Z. W: [' C
    # Base case
  j! ?1 Y! o1 b( S7 W% Z    if n == 0:7 b% X; W7 |+ `8 C' x
        return 1
  d/ c! ~) f  K0 R4 i8 [* Q    # Recursive case
8 q! a6 R8 z5 U    else:4 U" g0 M4 S2 E9 n% Z+ ^" y
        return n * factorial(n - 1)4 v! o' O0 ^& r: E/ G% M
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# Example usage# j& f+ d/ }1 K5 W8 J: f/ U! R
print(factorial(5))  # Output: 120# J$ ^, Q5 u2 o' m4 h0 z7 N% M

2 j! T# B& L! n+ V1 j/ e* ]7 |How Recursion Works" m2 q+ b9 f5 a# c% A5 E

# j9 F* P, @9 f) X" L3 `8 ~    The function keeps calling itself with smaller inputs until it reaches the base case.- }) J4 Y$ G+ Q8 N/ v, O& p

1 W5 `0 {* S/ D' e    Once the base case is reached, the function starts returning values back up the call stack.5 o6 W0 @) ]: H" K7 @/ g
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    These returned values are combined to produce the final result.
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5 _7 p; \/ ?: U5 S1 t4 v  HFor factorial(5):
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factorial(5) = 5 * factorial(4)
2 ]* _* L7 r' u  i- F1 W6 kfactorial(4) = 4 * factorial(3)5 y2 }" s! }! b
factorial(3) = 3 * factorial(2)
. C8 Z& l8 ]: c# W( e; \" [4 tfactorial(2) = 2 * factorial(1)* v+ j( e5 Z% f6 h# O/ f8 A
factorial(1) = 1 * factorial(0)
3 r0 h' m( k4 d+ R+ T1 q1 Yfactorial(0) = 1  # Base case
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Then, the results are combined:
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# o: H$ c( c8 efactorial(1) = 1 * 1 = 19 G7 a: |% d7 d0 i
factorial(2) = 2 * 1 = 23 S5 P+ P' Z3 j8 F6 i
factorial(3) = 3 * 2 = 6
: I' p3 h& }; I+ I' p1 {! Wfactorial(4) = 4 * 6 = 24# m7 q4 X* R# E: Y2 r
factorial(5) = 5 * 24 = 120
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; x& t1 B5 q" T  F6 iAdvantages of Recursion7 z; k1 V* ~  {  p' e
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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).2 g, u: I4 z/ T7 C
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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Disadvantages 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.- o8 g$ W% h" I$ }3 T* `
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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4 E0 G4 h- x! g/ F& rWhen to Use Recursion4 L9 L6 m# K' Q6 ^5 N
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).3 U: A7 w* m% X" D2 \3 I$ k

& C- V0 U3 u' V4 n( l8 Q" a2 J' H    Problems with a clear base case and recursive case.( |  U; e* ?/ Q) E; R1 ~3 v

) l6 l6 w# X5 f! DExample: Fibonacci Sequence: ?9 f4 o- O( @. h/ |  o$ 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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( N& Z, j% N+ Y. [9 J    Base case: fib(0) = 0, fib(1) = 1
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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python" l+ N1 S! N% G3 N' M+ c
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def fibonacci(n):
- [9 c4 X0 t% R1 t& ^) s/ d    # Base cases
8 z. G2 q% d$ A% S. d, D    if n == 0:
; N, D" P- o1 Q. E. w        return 0
1 y9 \" R1 w" b& ^4 e: S    elif n == 1:. Q- ^+ \# S! F( [/ ?% _0 x
        return 1. @, z8 p: `1 Z8 t8 S  d& Y5 d
    # Recursive case! W5 N0 M7 O' L# |) h& |
    else:
# n" e/ {! M% R2 @        return fibonacci(n - 1) + fibonacci(n - 2)# J: y6 n  ^4 w

5 f3 G1 |# [1 ]! _8 H# Example usage
: r4 z6 M2 b: R) y4 G0 ?* sprint(fibonacci(6))  # Output: 8* V" a" s% T8 F7 N, c; `

+ s4 K* z- m1 v3 y# ITail Recursion2 \! o3 D8 l# u

3 k- ]$ U9 y  Z6 UTail 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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) z0 j+ ?1 \1 m5 e: JIn 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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