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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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% e# w$ L) l# p解释的不错9 f* n/ q# z8 v+ d1 ^: R- H  ]7 W

4 A. D& b- |# Z8 w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/ g" }; s- F. s/ L) c3 F
1. **基线条件(Base Case)**
: m5 ~! |& u! W  \; b, j& F   - 递归终止的条件,防止无限循环
7 {& l/ H+ @4 |   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 10 k4 A, h/ H+ y  D% }
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2. **递归条件(Recursive Case)**
# `% X$ w; _( E$ A3 T+ y1 Q$ s8 s   - 将原问题分解为更小的子问题, |9 k/ ^7 m; [7 f/ N
   - 例如:n! = n × (n-1)!
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经典示例:计算阶乘
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def factorial(n):
) U# F( r$ R5 X$ @. H* Y    if n == 0:        # 基线条件
, ^8 `' W& K) P% u: G: J' k3 g        return 1
' Q& o4 C  ]& _. i6 F    else:             # 递归条件
! s9 {& P! _) e0 V: `0 C. w" _: l        return n * factorial(n-1)
7 L& N. C6 U. J6 u# f执行过程(以计算 3! 为例):
+ T! G/ l, W7 i  @" w( M; Xfactorial(3)
  g3 K* v) \7 Q$ Y* Q4 [3 * factorial(2)
% }- G  `" g/ w$ c+ x3 * (2 * factorial(1))7 U. g! R$ \: E( b* E3 g
3 * (2 * (1 * factorial(0)))
8 u7 w: t" M3 L8 e. E3 * (2 * (1 * 1)) = 6* O; w' Z7 {, a
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递归思维要点2 ^$ A0 M9 j1 s: C* D
1. **信任递归**:假设子问题已经解决,专注当前层逻辑8 L# L4 \8 r- B9 [6 B7 }
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)9 J  a% e0 B" b: c8 T8 B  {
3. **递推过程**:不断向下分解问题(递)$ l; d( B" @6 @3 I, t+ ^& n. b
4. **回溯过程**:组合子问题结果返回(归)8 c" c$ i$ T- o5 _9 _

/ ^# H1 D" ]1 G' j. M) Z5 y 注意事项
' K5 c* u; u( x6 c  [( T必须要有终止条件0 j2 ^* o% t/ |
递归深度过大可能导致栈溢出(Python默认递归深度约1000层); }  V/ `& s( X) W
某些问题用递归更直观(如树遍历),但效率可能不如迭代$ }5 W) N- N7 _% l  L1 C
尾递归优化可以提升效率(但Python不支持)5 X. p5 z9 j: e

1 O( J7 j8 C9 f) P7 t 递归 vs 迭代
* a' E* e. D) n9 I7 m4 T1 v|          | 递归                          | 迭代               |: I" b; O1 N6 d% X8 J- D4 ^
|----------|-----------------------------|------------------|8 T" P) g% P5 g4 X
| 实现方式    | 函数自调用                        | 循环结构            |0 }9 D0 P' B7 i" ]
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
% B: g2 N9 m5 u$ i/ ^" K| 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 o" v- L* s5 \( Z& C5 T
| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |0 d' h% b7 j* A: z, c

$ t0 }) y5 p3 _1 _ 经典递归应用场景) ?1 U5 g+ E" K  r1 m
1. 文件系统遍历(目录树结构)3 K# o. |$ q% ~" p
2. 快速排序/归并排序算法
5 T. _- z7 _' _3. 汉诺塔问题
/ S: z! G' I1 E8 n; T- {4. 二叉树遍历(前序/中序/后序)
7 S# {* F. f2 `" ]: j5. 生成所有可能的组合(回溯算法)
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/ U0 `6 t5 a0 n4 w! d# M! r: h1 g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,) ~2 r6 {  b+ C: e/ W* x/ s
我推理机的核心算法应该是二叉树遍历的变种。8 P9 f. x8 [$ ^( 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:$ r- O3 |0 c2 k( O0 p! s) m# e; r
Key Idea of Recursion
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A recursive function solves a problem by:
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7 o$ W( s; ]8 r" G  z3 ~    Breaking the problem into smaller instances of the same problem.0 |" x3 ^$ J( }) k# _# a. W6 R
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    Solving the smallest instance directly (base case).
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    Combining the results of smaller instances to solve the larger problem.
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Components of a Recursive Function0 V9 W, W+ s: p* X
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    Base Case:+ A2 d7 @4 C2 A0 t% ?) x; m% P- s8 B
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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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# H* h0 V6 z1 I$ p7 b  F        It acts as the stopping condition to prevent infinite recursion.1 v0 U% j$ X! c! O  d

! C& L& }$ A# O% f        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% ?; _, J% X4 G6 y% q

2 L3 c9 e+ R. m3 G$ G    Recursive Case:! h" A+ p, H5 [
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        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 Calculation5 F5 Q3 P4 M  _- b. e& a
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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:# P0 |7 z6 M( a; j( h
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    Base case: 0! = 1
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9 ?8 c& [, T% N0 C    Recursive case: n! = n * (n-1)!& U9 c8 E" D3 L& ^% F

' M/ L) N$ v1 b5 B' yHere’s how it looks in code (Python):
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def factorial(n):
! b% j" e+ R1 q# j  ~* d    # Base case
7 j( P2 @0 U. r. \# u. V! x/ O    if n == 0:
) D0 X9 m* Y) {- H; _7 M        return 10 G! N" U9 R! Q6 U; T& N, D9 f
    # Recursive case
8 s7 U+ l# ]& _9 F3 R    else:
! G* s5 c" z8 }        return n * factorial(n - 1)! y2 G1 t8 |0 h6 g5 ~  H8 J) D% y
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# Example usage# z5 e: B7 ~3 @, {* `% o* `7 f
print(factorial(5))  # Output: 1209 S- T6 t' z- Y6 ]) U5 }6 R4 P: ]4 d
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How Recursion Works/ n4 ~6 O8 D) @( Q/ ]. D
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    The function keeps calling itself with smaller inputs until it reaches the base case.
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5 f) o2 }/ S/ }0 j    Once the base case is reached, the function starts returning values back up the call stack.
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3 R+ D% u6 W- r$ x4 d; |- J8 w    These returned values are combined to produce the final result.
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For factorial(5):* C9 N8 {1 s2 Q5 s; |% ?' w5 A6 L
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factorial(5) = 5 * factorial(4)
7 J, m: o, I5 r# K  Y* }: jfactorial(4) = 4 * factorial(3)8 b$ H% w1 b6 j; U) E4 d6 l# S
factorial(3) = 3 * factorial(2), x  @) X9 l3 P! a, Q$ J. K1 @( G
factorial(2) = 2 * factorial(1)
$ \8 L- ]8 a- l/ A% X  c6 ?factorial(1) = 1 * factorial(0)! F4 S' U# P6 W2 L
factorial(0) = 1  # Base case7 d0 }* [  o. d( _/ A

$ ?9 d" j" H4 e% z; r3 s# O7 gThen, the results are combined:' n+ R2 h% m9 V6 o! n
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factorial(1) = 1 * 1 = 1
! }$ O% E; p+ I+ l- n* ?( wfactorial(2) = 2 * 1 = 2
/ {0 v8 Y, C& ^" t# T4 X! t5 dfactorial(3) = 3 * 2 = 69 i9 v& b2 N( f0 I! n9 H  b+ g' z
factorial(4) = 4 * 6 = 24/ Z; u+ b( V% `1 j  a; N
factorial(5) = 5 * 24 = 120
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6 q2 z! I! c( h' WAdvantages of Recursion, u4 W0 J7 P# D" Y) t9 _4 ^
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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).5 P. m# \% P6 d3 U
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.( a7 i1 j( l, F0 o& J+ f

- m# Z, q3 o* Y% {6 R5 V# n0 WDisadvantages of Recursion! r# m+ D. g; U/ g% n2 N

+ i# B9 g) G! k    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.9 y( z/ z  {/ _
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).0 \' W7 U! S! u- D0 a7 P  D
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When to Use Recursion5 R5 ~# H; w- E+ W) c

6 t* @/ _- K: I! u5 ?0 X    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# n% B! N2 U+ b( x% _! x0 P
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    Problems with a clear base case and recursive case.* U! I( I& u! L8 I$ n; |) \
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Example: Fibonacci Sequence
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/ h: R" _0 J! D7 u  K6 {( sThe 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/ s, C% Y) H6 b8 m8 S
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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def fibonacci(n):8 S2 ^3 u6 l) I- H- N9 [2 o* T
    # Base cases
7 y$ L4 A/ L6 y, K" Z3 L0 z+ x    if n == 0:
6 n( P0 i6 E. D* Q, x! i        return 0
* S, X0 c1 d+ x4 }6 V7 H4 e    elif n == 1:# ?% B1 J+ w* A4 S5 U( G# O
        return 1) E6 P/ d% G0 o3 K0 W
    # Recursive case
  t" u' w) C* U, W: p' L    else:
1 U5 C5 T9 u( z6 l6 P        return fibonacci(n - 1) + fibonacci(n - 2)0 }3 ?8 S& c3 f" e. t' J% Q7 ]
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# Example usage7 \  N6 \8 Q: ?  \- I
print(fibonacci(6))  # Output: 8/ J4 |  |6 a3 M% a9 X

* V3 z2 Z; `& _( y4 pTail Recursion  ?7 k9 h5 \7 G! B( A# s/ I2 Z8 L1 O

' `+ ?- c' e8 h3 X. yTail 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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