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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7 Z, S) Q+ q; Z' F1 ~1 w

5 e9 _$ V5 f$ a) G- [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。) K. ?4 j, p. u5 V$ W+ w, [4 t

! b+ Y' K6 S# d( K8 C8 A" p 关键要素
* y& P* ?; y- ?5 Q0 s0 B2 M  ?1. **基线条件(Base Case)**& h- W+ x# n+ r  F
   - 递归终止的条件,防止无限循环
- n' z  ~) e; R. F- f   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 11 ?0 b, @, ~! ]) b

; F) v( X5 w6 @% P2. **递归条件(Recursive Case)**
6 M% ?/ E6 l) h9 e   - 将原问题分解为更小的子问题3 z/ y$ P# d" _# f/ P  c) q
   - 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):# N- U. F" P( o) V* F" N
    if n == 0:        # 基线条件
( x( o3 t  w' S8 G        return 1
: I- I" J3 l, r# Q; }. V) l3 y    else:             # 递归条件) O! J' m7 b& D  y, R0 p  n; r
        return n * factorial(n-1)! s1 l8 T  R% m. b4 i2 V5 E3 U
执行过程(以计算 3! 为例):7 D8 l! N1 |4 G0 i: O& \6 a! E
factorial(3)& b! E+ Q5 K8 Q, v
3 * factorial(2)
2 P+ N2 R& c  U2 `3 * (2 * factorial(1))
' @4 V' Z. J  w6 C4 w9 V5 q3 * (2 * (1 * factorial(0)))
" P8 S( |  \2 K1 D0 a" r, ]! c3 * (2 * (1 * 1)) = 6
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递归思维要点
3 s  R  J9 s, D$ |1. **信任递归**:假设子问题已经解决,专注当前层逻辑
6 b; T. I( R3 ~2. **栈结构**:每次调用都会创建新的栈帧(内存空间)# S$ E9 x% n9 S; t  Y
3. **递推过程**:不断向下分解问题(递)# C$ U, W! Y7 x- A+ D
4. **回溯过程**:组合子问题结果返回(归)
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注意事项
; `9 ~3 P' D, x9 o0 W% M必须要有终止条件
9 ?+ J' N3 _( h+ S% T* G/ n递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
# z( S+ O( n3 @9 o7 ?( k某些问题用递归更直观(如树遍历),但效率可能不如迭代# U( Z3 _6 j) P4 W) n# ?2 l" M' V
尾递归优化可以提升效率(但Python不支持)$ Q4 j- F1 q! E4 l

1 T8 k) _: C! Q. I1 l: c& L 递归 vs 迭代
1 t  f) a! V- |3 m' P0 m|          | 递归                          | 迭代               |) {1 J( ^$ |% R7 k. Y/ E
|----------|-----------------------------|------------------|
9 G4 V  c" y' D5 v$ ^; U3 k| 实现方式    | 函数自调用                        | 循环结构            |
5 @3 P1 d' O! o" O| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |$ C/ `1 N1 k, v
| 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         |0 v* |! W, m! W9 H/ }; \
| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |( M" A" j1 u% I2 z" D
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经典递归应用场景
8 L3 X4 f" t$ W8 m6 b1 Z1. 文件系统遍历(目录树结构)
/ L7 V- h! o  r2. 快速排序/归并排序算法
+ B% h( c" Z2 K8 {8 R$ V* f3. 汉诺塔问题, z. p% A+ i* g) l/ M0 o
4. 二叉树遍历(前序/中序/后序)& W( b* X. b$ U6 D; _$ K6 ~
5. 生成所有可能的组合(回溯算法)
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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,; \9 I, o; F- g, n
我推理机的核心算法应该是二叉树遍历的变种。
' n" c1 |# t+ o5 H  B8 J6 s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:4 d" B' q" H$ j9 e! B" d
Key Idea of Recursion
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+ k) V# L$ [6 R8 k- m) G" ]A recursive function solves a problem by:
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! g, W' l) Z$ }4 ^  h    Breaking the problem into smaller instances of the same problem.
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& J* \% \. g- S3 ]* m8 R% \    Solving the smallest instance directly (base case).
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    Combining the results of smaller instances to solve the larger problem.2 z2 e9 b, c' z( V3 c3 f: \
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Components of a Recursive Function  U/ ]8 ^: q. ?/ @% G
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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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        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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& J* u. A4 U4 J    Recursive Case:
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        This is where the function calls itself with a smaller or simpler version of the problem.3 ^- {, q- r" |
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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$ O9 r$ E- _* lExample: Factorial Calculation& c: e. o, b7 |% J- E! d; p
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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:& ?" N/ _0 {' @4 q9 _# ~

  c7 C2 C5 ~2 m1 ]/ D( @) i# H; z& U0 s    Base case: 0! = 10 v& ?$ P6 M/ `4 o7 x, p$ {

4 ]$ }8 p( V1 x7 ?" ]& S( V0 c    Recursive case: n! = n * (n-1)!
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3 V. h0 S1 z1 A2 q3 y2 ^Here’s how it looks in code (Python):& N) @' `3 c/ U; x9 i/ E, y
python" }5 B! ?* t+ z' k
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def factorial(n):* s* I2 \3 L3 y, q
    # Base case
+ k6 _. @  ^6 V    if n == 0:* J8 H/ b2 Q) C2 {
        return 1! _2 ^+ k9 N/ `0 j5 J, f# |
    # Recursive case8 T1 a- a: I1 K7 o6 @+ `
    else:2 X' n5 h/ U% G8 M
        return n * factorial(n - 1)8 d; f% S8 ~$ v
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# Example usage- v3 j. H% V* l/ c) \3 S
print(factorial(5))  # Output: 120: J' Y# ~5 n3 Y. a- u

; f* `; g) X- _) d2 GHow Recursion Works! O: N: L! T+ N$ d/ a9 V, d
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    The function keeps calling itself with smaller inputs until it reaches the base case.9 P+ ?0 U5 u+ i; n& A) W

2 K+ i$ ]7 T2 t2 h7 D0 M7 j: I    Once the base case is reached, the function starts returning values back up the call stack.1 u+ \* t5 y% b
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    These returned values are combined to produce the final result.
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- u1 E5 N+ w* VFor factorial(5):9 C4 @& H! Y0 j+ ~: r; a8 D

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# h5 X5 U9 C- Y/ ^- U7 xfactorial(5) = 5 * factorial(4)
. P$ @( ~/ [( {5 h- D! Efactorial(4) = 4 * factorial(3)
' R* q9 v+ D4 U6 ^! Hfactorial(3) = 3 * factorial(2)
, Q) i4 \! m# c4 {$ O6 f7 zfactorial(2) = 2 * factorial(1)- t' k6 @) r3 B$ H
factorial(1) = 1 * factorial(0)
1 m9 p% G' J$ n* \9 w! Wfactorial(0) = 1  # Base case1 P7 ~# r- ~- o4 ^' l# o

! `& W" E- i) D- H0 r" z& s* I! gThen, the results are combined:
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* g; L  u+ O* e0 z# l  |) xfactorial(1) = 1 * 1 = 1) S2 M% k5 j+ d- H- M
factorial(2) = 2 * 1 = 2
: T" M$ ]2 ?! @' kfactorial(3) = 3 * 2 = 6) u6 ^) O, H) T) F$ ~0 a7 j
factorial(4) = 4 * 6 = 249 s+ o, s# |) {3 c1 y3 b( Y3 s4 ^
factorial(5) = 5 * 24 = 120
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8 e! d# E$ F2 YAdvantages of Recursion
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6 \& A# D1 m# u0 W    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.9 {# X' }# A! c0 Z
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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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3 d2 S* J  I2 I5 L9 V2 h) A4 }, J; r; w    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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When to Use Recursion3 {7 J0 T) B. }- y8 E

6 t6 C! M9 G# Z3 i    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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! K. m* l+ d' }9 q- W( V6 W  r    Problems with a clear base case and recursive case.
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Example: Fibonacci Sequence0 ]% L, m3 s6 J; E5 u, a0 Y

) v0 A9 u! o6 }) Y9 yThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:4 E% b2 ^0 I" G* v* t6 C" E: w4 n

0 a8 e6 u7 I: ]" }    Base case: fib(0) = 0, fib(1) = 10 x$ }$ l5 e) E3 ^
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    Recursive case: fib(n) = fib(n-1) + fib(n-2), x) ^" D" k( G4 A

: G: @- W+ j1 E2 p% Xpython) H  U$ V* U) v8 i6 a) e

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def fibonacci(n):/ J- Q/ ?: X' {9 E! {
    # Base cases
+ s) s2 p* `, J9 P    if n == 0:) E) F9 C* ~" L% R
        return 0
6 I( s, i2 l* a5 `    elif n == 1:) L  T# L, h! M! k2 l5 P$ ~
        return 1
" W/ X. M. w: K5 ?0 C    # Recursive case
& I% F/ X% G) X: U1 }    else:2 J( D" a3 X% D
        return fibonacci(n - 1) + fibonacci(n - 2)
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1 u) }3 S- K+ m' t( w; r' L# ]# Example usage7 ^1 D; I! w* K2 U3 j' t  Q9 ~
print(fibonacci(6))  # Output: 8( E; F5 H) U- g( d  g% g' X

. V4 k, N8 T' c2 O6 X1 A) JTail 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).6 a# L2 y1 O4 G# m# B1 R" ?
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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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