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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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9 @1 v9 X  [6 M解释的不错- O. }5 E+ L0 M

% m' o: ?9 V5 V$ D4 q) u$ f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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/ l1 A; X# O7 }) c6 y0 @ 关键要素9 F5 @% X: i( J8 H8 F6 Z; W; g
1. **基线条件(Base Case)**
! G% R- Q" f% ]: [   - 递归终止的条件,防止无限循环$ H. @6 t$ R2 z4 Z
   - 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)**
' ~: p% e  G4 H   - 将原问题分解为更小的子问题
( T/ r, Q) `9 k) J" d, M# O   - 例如:n! = n × (n-1)!3 }$ ~( |1 Q. w  c* x8 q; I
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经典示例:计算阶乘: a0 k) R% x, K6 h; y. B
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def factorial(n):7 h5 i8 b6 s( g6 ~; {# O9 I: X
    if n == 0:        # 基线条件1 V- {' o7 C, L  J
        return 1
2 N0 |4 f+ w! p  Y    else:             # 递归条件
1 l4 C; x6 H* f6 p        return n * factorial(n-1)9 D$ Y  c; i2 Q) A
执行过程(以计算 3! 为例):1 H/ D0 G! N8 C4 ]  F5 ?, o  M
factorial(3)
5 v- t' I; t7 o3 _  {% {: j! P3 * factorial(2)- Y' f6 O6 O- J
3 * (2 * factorial(1))
6 v0 K. C# W  t) {$ ]# o' {$ u3 ^3 * (2 * (1 * factorial(0)))4 A* B& V, J$ O% d4 J) Z% k# Y
3 * (2 * (1 * 1)) = 6
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& J  y6 `3 \, l3 o0 |  a 递归思维要点
! k' y* a7 x+ N2 o* [; w* [+ H1. **信任递归**:假设子问题已经解决,专注当前层逻辑; D! h" O2 D  e' h1 h
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
. p4 K2 s) I# I/ |( z0 i& v$ W3. **递推过程**:不断向下分解问题(递)- r' Z8 L" V& t: q* }
4. **回溯过程**:组合子问题结果返回(归)! _. q/ ^3 l  R% t

7 P5 H) |, J9 W- M 注意事项
  d. _) \" S! S# G/ z必须要有终止条件
0 E7 j! l' r: b  a4 D% D: `! q1 M递归深度过大可能导致栈溢出(Python默认递归深度约1000层)2 j! D  _! Q8 X. i. N6 \* G
某些问题用递归更直观(如树遍历),但效率可能不如迭代9 y) u1 Q4 I2 y6 ?( L- W9 b7 `
尾递归优化可以提升效率(但Python不支持)) \2 `/ t# w+ i0 X) K8 C

% x& X/ c& {+ z8 j, p3 c+ ^' A: q 递归 vs 迭代/ `7 g2 ~. V+ v# M: a
|          | 递归                          | 迭代               |
& m) O& D+ N. r5 F5 ~|----------|-----------------------------|------------------|
3 A5 S% E' B& c2 E* }| 实现方式    | 函数自调用                        | 循环结构            |( Z" ~# }2 }* A) ^) j1 G
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |* o' w/ h& F: }2 R8 x+ \* P: c
| 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         |
! I+ c' ^% U9 z: L/ t  {3 v| 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- l  J- ~3 J1 n
1. 文件系统遍历(目录树结构)
1 P9 t1 n! U' w$ }( Z8 F2. 快速排序/归并排序算法( Y, L& J; S; z; p0 @+ p& S, \0 k
3. 汉诺塔问题
4 \! ?6 j6 a+ [- f8 F4. 二叉树遍历(前序/中序/后序)0 q6 f! n, A4 o! @( J
5. 生成所有可能的组合(回溯算法)
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1 ?2 w* e% i; f4 b/ m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,9 h3 p1 J* j  w( g" ]; p
我推理机的核心算法应该是二叉树遍历的变种。8 j" W; d: J# z( {0 o: C; Q1 o+ k
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:7 ~- f- y$ Z& Y( ~3 C* b- K
Key Idea of Recursion; D) B' a( G$ Z7 T4 X$ ?

* f& t) z7 T  M: QA recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.
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    Solving the smallest instance directly (base case).
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# B6 z7 R8 B/ ~, V/ Z" |/ W" G, B    Combining the results of smaller instances to solve the larger problem.8 y( b4 N) X( g: F4 I0 M, u

( c- [+ o% }! X; l4 _: b0 yComponents of a Recursive Function; `. w6 T! I! S
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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.! v! B9 o9 Q, V1 t: U

, c9 G  M% s# b# K- H/ m        It acts as the stopping condition to prevent infinite recursion.
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  r3 X9 v! y* O0 \        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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5 J$ |2 F% a* D& H8 K2 Z    Recursive Case:, A) b6 K$ ^5 t+ G* e; Q7 P. m

) u# n2 j' w7 X% |% l% N" W        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)., N2 Q) K" U5 y, f
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Example: Factorial Calculation
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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:* |" w. {6 W! s! I. t  m. R$ m" S

0 m; ^/ `8 |) ?) ^    Base case: 0! = 1
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    Recursive case: n! = n * (n-1)!% j# C/ F0 C3 C5 K- K% ?3 B# a5 `

- l5 f. Z1 U* X3 n* b3 j; ]2 Z. E# SHere’s how it looks in code (Python):
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4 a5 f* {' M. Ddef factorial(n):* ]  P9 V7 G8 }3 g+ ^2 N+ u
    # Base case
8 @! E  m  |& `1 g1 p    if n == 0:9 a; X# {, z1 R: [3 m
        return 1
0 c# m' T: K' r    # Recursive case$ z; f1 o  o" k
    else:
; t  m) Q1 ?) ]8 t% c6 H        return n * factorial(n - 1)$ F" K( A( y- m

4 j% f2 F, c7 g, G# y# Example usage
& K0 r! a; }- a8 }& {+ jprint(factorial(5))  # Output: 120; h5 @- q! Y8 U2 O

/ w$ k+ Q" h. s+ B. LHow Recursion Works
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    The function keeps calling itself with smaller inputs until it reaches the base case.  E$ e4 f+ U& E7 V' J

7 S6 d4 T' o% x6 S. t    Once the base case is reached, the function starts returning values back up the call stack.( ^2 E* z* e! `1 X( F0 e& {2 n6 A
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    These returned values are combined to produce the final result.$ X$ L( z: n" W$ O/ U" E

4 |& f4 C+ p1 Z3 ]# MFor factorial(5):
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# E1 i# J1 c% f- Ofactorial(5) = 5 * factorial(4)
4 J( B5 s; K- t  t" Z3 Vfactorial(4) = 4 * factorial(3)
/ }9 C- A+ [. X# i- kfactorial(3) = 3 * factorial(2)1 z; l. R5 E, z5 l9 T3 M. b& B
factorial(2) = 2 * factorial(1)
8 a  J5 {% n! \$ c6 Rfactorial(1) = 1 * factorial(0)6 T1 T9 _% c, K
factorial(0) = 1  # Base case0 i3 Q" t3 V( v2 ]

  Q8 g. j) M9 p- vThen, the results are combined:- T0 c1 `) n- J" c; e8 e

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- j# m' U3 R/ P& C: jfactorial(1) = 1 * 1 = 1
1 T9 N8 Z$ a: y3 }. ]1 P8 sfactorial(2) = 2 * 1 = 2
" O* E5 {) c# V7 x+ ]factorial(3) = 3 * 2 = 67 `- s! r/ @0 ~: t$ B. I1 p5 p6 D+ D
factorial(4) = 4 * 6 = 24
1 A( O4 s2 n% X' a/ Rfactorial(5) = 5 * 24 = 120
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Advantages of Recursion* i) q2 ~4 x5 \( [

; b) X  m. g  A" ]/ o    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.6 W3 j/ l+ O6 M: f

$ Z2 ?! O4 |: q- VDisadvantages of Recursion9 X: ?1 @8 D# z$ S0 w
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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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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).7 [' d4 A+ @4 m" v  w: E% S
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When to Use Recursion  _- V, `2 s3 t0 `" N2 @* Z4 q
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).  N, N1 Z9 q; l, J/ G/ I7 Y
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    Problems with a clear base case and recursive case.
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( v. {8 q: s( ]Example: Fibonacci Sequence2 V9 q: r* G0 v. e: [2 T5 P
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:: D8 r4 G$ B3 \) n" P0 N) N* z
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    Base case: fib(0) = 0, fib(1) = 1
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, k; r3 I# m" X# I6 f    Recursive case: fib(n) = fib(n-1) + fib(n-2)8 U" R4 T- S5 |2 ]# N

" B( G: j3 x  c; @python
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4 _: S7 c/ t9 G8 \8 H8 bdef fibonacci(n):9 g* B3 i' Q" n1 P
    # Base cases
! M5 {/ Y; H+ o1 K1 M6 z+ z    if n == 0:% d/ I3 A# f; R& J, @: G2 q
        return 0' e2 \5 d, G0 ]( r6 |# ]) R* l
    elif n == 1:" n2 y' y% ^- A$ J
        return 1! a2 ]2 p! Q) b  X" f9 D
    # Recursive case
, C) U7 H% z" p$ U    else:1 f. B0 n5 q3 G8 _/ U; H" w
        return fibonacci(n - 1) + fibonacci(n - 2)
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# Example usage- y: b  _: n8 Q" n  g2 r/ `
print(fibonacci(6))  # Output: 8
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( r) `2 K$ i, _1 ^Tail Recursion
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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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3 g- {+ m9 i7 V: M7 n, l7 L4 PIn 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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