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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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解释的不错
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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
6 ]/ ~/ x. {" Q; {( `4 p1. **基线条件(Base Case)**- A; n0 I: _) r) A$ l! T/ W
   - 递归终止的条件,防止无限循环
$ ~# g* q  H- J0 x. G$ D3 ^4 z   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 15 S$ a9 s( a3 _6 m5 t% I

0 u, K1 x* K9 j! O, E" X: _2. **递归条件(Recursive Case)**7 m! J+ N6 o  ~3 Y
   - 将原问题分解为更小的子问题
! N4 K, t3 U2 W) v8 \) q   - 例如:n! = n × (n-1)!: w' f1 [8 i% k* u$ P9 D

0 Q- i) y) j# {0 ~4 q3 b, L 经典示例:计算阶乘
% N3 Z! M& s0 t) j7 O: gpython
: N" V0 f" t& |$ Cdef factorial(n):
% p, }  I) U* J, G    if n == 0:        # 基线条件
# _1 H4 X1 H$ V& V! W6 j        return 1
. @% U4 ]6 x  \7 e    else:             # 递归条件8 o+ A% ]0 w) p$ b) }  H+ y5 B, d6 X0 ^
        return n * factorial(n-1), z# B4 n4 a' r& ]" o9 M% b5 z9 |( B$ V
执行过程(以计算 3! 为例):. a" ?/ f' I8 C; `% A: ^
factorial(3)
+ |" m* o2 r7 u# ^/ r" O3 * factorial(2)
2 N7 p* ^, k9 u8 I* D3 * (2 * factorial(1))' e& u. u0 y- }# ~
3 * (2 * (1 * factorial(0)))
! z; w" o: M- G1 N# G3 r3 * (2 * (1 * 1)) = 6
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, ?* h% i2 K! H: E 递归思维要点6 L# Y+ p% J0 F! F* Z
1. **信任递归**:假设子问题已经解决,专注当前层逻辑! J' ~- V" N+ a! a8 P6 @
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
! p# ~) \+ V8 J3. **递推过程**:不断向下分解问题(递)6 t4 [2 q  G3 Y- R3 g
4. **回溯过程**:组合子问题结果返回(归)' I2 ?5 J7 i. ~5 Z/ k

4 S: T* v* k" d' @4 q 注意事项
" Z4 }; y7 {( r) Y9 R) q必须要有终止条件
. [8 a% p$ d6 p4 ~0 M: Y( u递归深度过大可能导致栈溢出(Python默认递归深度约1000层), z7 q. i2 y0 D# ^* s
某些问题用递归更直观(如树遍历),但效率可能不如迭代
% j2 Z0 T% w% D2 m/ P8 J1 N( x0 U, u. p尾递归优化可以提升效率(但Python不支持)
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3 [, Y2 {1 k: U 递归 vs 迭代
! J! i% \2 a) W+ C* |4 E|          | 递归                          | 迭代               |
6 y2 I! ^" o0 t, J# l  M+ S. n* V|----------|-----------------------------|------------------|
# e0 N  s" l) G1 v6 r* S8 l6 A| 实现方式    | 函数自调用                        | 循环结构            |0 e$ D' I1 P6 _
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |- ^0 k# Z; r4 ]' W! }! \
| 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         |, V+ C% W* s  ?# f8 G
| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |2 ~( b  _8 i- o! c6 M+ j/ \
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经典递归应用场景; n3 |: }$ [1 X, `5 r/ K+ B
1. 文件系统遍历(目录树结构): V4 {6 t" o+ {  \/ }( p
2. 快速排序/归并排序算法; v5 e3 q. q+ v2 L' ~1 S% {
3. 汉诺塔问题% P  q6 t3 r; ^- K: l( Q+ t
4. 二叉树遍历(前序/中序/后序)
- M; \* \, @( D" Q3 b5. 生成所有可能的组合(回溯算法)$ q) ?0 f5 {& H; ]# W6 N. Y

& M' \" N/ n, ]; \% P* y, o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,5 O9 g6 W% {9 \& Z; o
我推理机的核心算法应该是二叉树遍历的变种。2 E+ y$ _1 d& e# [
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:/ A. F  G9 I+ m+ ]2 N/ R
Key Idea of Recursion
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A recursive function solves a problem by:0 B8 R5 X' L) P& ^$ R. T

6 _) N; A6 Q" J/ ]8 g+ ]7 R0 X    Breaking the problem into smaller instances of the same problem.
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    Solving the smallest instance directly (base case).8 V7 j4 W2 |+ Y2 m8 L. D. u2 ^

/ H; F0 g6 b: F, Y( d! V  B    Combining the results of smaller instances to solve the larger problem.# {% b/ y4 M1 b6 _; n/ s& X

( y  U' h$ R- _2 ]/ O: e& V* {+ qComponents of a Recursive Function
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# _) w& F: i+ b0 p. X4 O2 k, o    Base Case:6 C; _7 }  \+ h7 d( M  j8 f8 l) S

- j, S" k8 m' I: T( H        This is the simplest, smallest instance of the problem that can be solved directly without further recursion./ d9 d4 k( q# u; G4 S. Y

: }+ a' s- [; V        It acts as the stopping condition to prevent infinite recursion.
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        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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    Recursive Case:
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$ {* p) s( b+ r2 P& `! {5 `9 k        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)." A. |3 P# p9 A3 @# U( u

9 j2 ^- O" r' u6 |Example: Factorial Calculation( h8 _5 I( H. v! z8 `+ ^' [
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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:
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    Base case: 0! = 1
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5 ]( n) H/ t8 |    Recursive case: n! = n * (n-1)!8 ~* u- `( S# J: r

3 n9 `0 ?- F. a1 z" O9 SHere’s how it looks in code (Python):
% P3 |6 {5 I* ppython
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: k' B, s2 R" |* Wdef factorial(n):9 `, n/ c, l) g& z" x6 f
    # Base case
% c0 u: l8 `6 D- U4 z    if n == 0:# J8 s: z- R5 d. {1 W
        return 1% j, Y7 O1 F/ K6 J% T2 p/ }/ z% L6 T+ e
    # Recursive case
3 N0 V% @% q5 Q% P    else:( f' z6 B0 E% a2 L
        return n * factorial(n - 1), o" F8 o0 ~; C) g$ o: E# m
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# Example usage/ o: l7 R9 B/ ^0 m5 h
print(factorial(5))  # Output: 120
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How Recursion Works
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. j; {  O7 e$ z) g- S8 c8 Y3 g    The function keeps calling itself with smaller inputs until it reaches the base case.0 G% @$ V, x# m6 ?, E

5 ~5 j: I/ U) _    Once the base case is reached, the function starts returning values back up the call stack.
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/ y& T- S! u- ~* G' P    These returned values are combined to produce the final result.
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- g  G2 N" z! X' L  a! @For factorial(5):6 m  l2 z! q5 i  V

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! G, F8 o$ D- Z9 n1 xfactorial(5) = 5 * factorial(4)
( w3 W) ?4 D) B4 ?% @' i5 l6 |factorial(4) = 4 * factorial(3)
  a2 r+ B( Y& d/ F$ z7 X3 X: Yfactorial(3) = 3 * factorial(2)
4 G$ M0 O8 C7 }2 W# y/ z! Ufactorial(2) = 2 * factorial(1)
' j- [, D( i0 ^# |0 z, Tfactorial(1) = 1 * factorial(0)4 n# C! x! n* f9 O7 A
factorial(0) = 1  # Base case
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Then, the results are combined:
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factorial(1) = 1 * 1 = 1
" N  y5 P6 U; m- `% cfactorial(2) = 2 * 1 = 2
1 a' Y7 H5 l+ E: C9 h/ yfactorial(3) = 3 * 2 = 64 ~6 d3 X7 s6 c7 E* ?# Z
factorial(4) = 4 * 6 = 24
0 _' m6 {' l3 _# t5 I# F7 afactorial(5) = 5 * 24 = 1200 d' `% w, _$ U6 @8 r
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Advantages of Recursion: R- ]8 N2 q  g8 O9 s

, ?  p" g3 P* k* J: S- @9 z0 K    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).. Q+ b& B- c% t) z
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.1 {8 y9 X: C) s
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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.
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5 e3 Q" f2 Z* r* ?; ]    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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9 V/ ^  O: |6 j6 PWhen to Use Recursion
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6 ]- q5 \& @+ ~3 M9 U6 x9 C' M7 d    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).( l# R- ]& w* C' H1 I! T
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    Problems with a clear base case and recursive case.
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Example: Fibonacci Sequence
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/ @7 B7 W' w, m- p  pThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:5 [4 i; k# L9 |; \0 [1 ?  ?

( N: S& w5 }+ ~: k  E2 ^: t. S    Base case: fib(0) = 0, fib(1) = 1
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( y' v+ z$ g- \0 M& r3 o    Recursive case: fib(n) = fib(n-1) + fib(n-2)* s6 K- y8 }' M7 L+ [* E) M& t

: D, y- f# P% n# }python
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def fibonacci(n):/ T4 g9 d1 {/ \* s* r4 M) g0 ^
    # Base cases
0 W+ t6 v- W: s9 B    if n == 0:2 D0 X4 M: K# ~7 F
        return 07 R: j9 o" I2 p" a7 `8 ]
    elif n == 1:" l3 J3 m9 O$ Z" m+ K+ ~8 b) z
        return 1
0 T& J  b# r2 U9 z    # Recursive case+ \; u3 C& }& I5 E! @. d4 [$ s+ v
    else:3 `" z  W6 ~4 B% N* l1 Z+ f
        return fibonacci(n - 1) + fibonacci(n - 2)
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# C* d1 h: U- w; s2 c2 [1 J) D# Example usage$ |6 T- t" g! ~% R
print(fibonacci(6))  # Output: 8% u$ r, {) r) A/ T% @
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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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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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