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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   J$ O3 |! q8 t) V; j/ l, D
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解释的不错2 f( E! M! M& X

3 s* N# R' }0 F: x2 V7 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。- t2 H4 |' \1 Y0 V6 z( h. v  S

: h# J  a* c8 \ 关键要素' N, X9 c1 b0 w- m
1. **基线条件(Base Case)**8 w; ]2 x1 p7 E) j  O* y1 G1 O
   - 递归终止的条件,防止无限循环' W  l! `+ e. y8 N0 \$ {. p
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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2 [3 K, t# R9 a3 Y5 H7 {4 b4 S, @8 A2. **递归条件(Recursive Case)**7 U: ~6 P( {8 }9 c* i# r+ |, v
   - 将原问题分解为更小的子问题
, f( B8 P+ f8 M8 u* I3 v/ x   - 例如:n! = n × (n-1)!
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经典示例:计算阶乘8 D, |; }" Q/ w0 w
python
' R+ Y8 X4 j. B7 n' g6 y; Udef factorial(n):+ B: ]( J8 T' y! |
    if n == 0:        # 基线条件
: {% R9 p! X2 c' g8 k        return 1
& {. O1 X6 Y3 @6 [/ b5 g    else:             # 递归条件/ K7 \6 `* Y( f
        return n * factorial(n-1)
. C2 E2 c3 l" r执行过程(以计算 3! 为例):; ?2 O+ i; U! t
factorial(3)
" Z) R+ }- f# B3 * factorial(2)2 Y; U' \) k% e6 S  ^, k4 e, |
3 * (2 * factorial(1))
! }6 k3 m7 G, p/ t3 * (2 * (1 * factorial(0)))/ f- z0 A( F( d
3 * (2 * (1 * 1)) = 6
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递归思维要点- L8 q. Q: O% A! I% K2 b
1. **信任递归**:假设子问题已经解决,专注当前层逻辑
" a7 q+ @5 Z0 }1 v: p/ @2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
+ D- b4 P$ g$ B) W9 v; X0 B3. **递推过程**:不断向下分解问题(递)  }- |5 t, k2 W! \
4. **回溯过程**:组合子问题结果返回(归), O/ T" n1 ^: ~! M1 T0 C& l3 [$ u! n
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注意事项
. i8 v/ E5 Q  j4 @* G必须要有终止条件; S5 f/ R2 s/ e: @
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
" @4 y# q1 A# r某些问题用递归更直观(如树遍历),但效率可能不如迭代7 Z  S/ K( H2 i7 s
尾递归优化可以提升效率(但Python不支持)
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递归 vs 迭代
& l$ z+ ?1 g$ i* a$ O; Z! i% i|          | 递归                          | 迭代               |
1 \1 c2 E, c; S- a|----------|-----------------------------|------------------|
, A: _; x( b4 C1 q) P9 t, l| 实现方式    | 函数自调用                        | 循环结构            |/ d& ^1 G0 i9 ?  D' Y
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |1 H0 W+ J/ E2 [/ 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         |, G& K$ o; I& N8 }4 d# g( N
| 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+ S1 i$ Z2 Q5 H4 {
1. 文件系统遍历(目录树结构)
) R3 f  }& }5 y& z2. 快速排序/归并排序算法& u2 R# m* k- w  q$ C/ {
3. 汉诺塔问题$ l( c% [; s) ^& P, N, N8 B7 T' B7 W
4. 二叉树遍历(前序/中序/后序)" d) z- R' L3 s" U% L& I8 M5 \. q
5. 生成所有可能的组合(回溯算法)
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# @8 i! D* P8 \) d1 i" L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,
3 Z8 h" f" L6 K* ?( s; z我推理机的核心算法应该是二叉树遍历的变种。
+ M, E+ p' n0 a/ Z0 ~: n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 Z& f! _* ^7 f
Key Idea of Recursion. v2 f9 n. u3 m6 `

; v' o: ?- J: \( ^0 h" aA recursive function solves a problem by:0 e2 Y) o1 r2 K5 e# E0 `+ e
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    Breaking the problem into smaller instances of the same problem.6 M. ^1 _' v# N
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    Solving the smallest instance directly (base case).0 v- F6 {1 m* w  r' k- S

9 K2 ^- T; {  x8 W* S3 Y  |! ]/ D: B    Combining the results of smaller instances to solve the larger problem.
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$ M% ?% E2 F! {2 X' TComponents of a Recursive Function" G* R0 l- F( \. P' o
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    Base Case:
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9 }: f) b# l4 n$ v4 h8 w+ x        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.5 z+ o, s9 W( z

8 A& K: p( h. O* e2 ]$ V: P' `6 k. @        It acts as the stopping condition to prevent infinite recursion.+ {# b$ U1 R. i2 o- e' ^
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        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.& _: _4 ^* p1 E4 D3 o
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    Recursive Case:3 L2 f: m1 @" h8 C6 ^+ e' i

5 V- M0 t1 i* A8 p8 @0 C! r& ^5 w        This is where the function calls itself with a smaller or simpler version of the problem.8 ^4 f+ ~7 {( H3 r

% L( k8 v  [: b" @! R4 l3 V# b# g1 c        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).* z5 f8 K; u7 j
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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:7 r2 Q* K) \! {: V

" _4 G& O6 t; r% T- T. K    Base case: 0! = 1
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' |1 s' y3 \/ C    Recursive case: n! = n * (n-1)!2 H% k; j/ ?$ {0 ~" a& f
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Here’s how it looks in code (Python):, l* _+ U) V# Q* Z
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def factorial(n):
% ^$ q/ j+ I; s" [    # Base case
% _$ q3 h4 q# w# p4 B    if n == 0:
2 r" z: s  s# c        return 1" N. V: C' Z  k& S4 Y& R
    # Recursive case
. u8 S" m$ h: n' L# S3 w+ V+ g7 |    else:- T4 X, r  b$ ]+ V9 K9 Z8 i
        return n * factorial(n - 1)
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# Example usage
: c% G7 }( M( rprint(factorial(5))  # Output: 120
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. i% u: R; N0 {2 OHow Recursion Works
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    The function keeps calling itself with smaller inputs until it reaches the base case.( q- l/ @! S% ~9 g( k& o# W& ^, l
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    Once the base case is reached, the function starts returning values back up the call stack.
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    These returned values are combined to produce the final result.
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# ?4 T$ l4 D4 |% {; z# XFor factorial(5):
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& t2 ]' }3 b0 U& K3 p; Efactorial(5) = 5 * factorial(4)
$ }, X" m8 r8 S' p1 Rfactorial(4) = 4 * factorial(3)' \! o8 h0 f5 M
factorial(3) = 3 * factorial(2)
# V4 r0 ~0 [& Qfactorial(2) = 2 * factorial(1)" n' U( a1 B- O! z, L
factorial(1) = 1 * factorial(0)$ C1 P. p; S) a9 L" r
factorial(0) = 1  # Base case
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Then, the results are combined:$ W3 n# |" B0 ^, y# ^
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7 w7 \0 B" h1 G9 u! n# A3 Lfactorial(1) = 1 * 1 = 18 k% z$ J7 e- n9 {1 Q7 z
factorial(2) = 2 * 1 = 2" J; \  e4 `- S  S# ^9 H
factorial(3) = 3 * 2 = 6  P1 I# a9 [$ ?2 p: X. r3 }* U  V
factorial(4) = 4 * 6 = 24
* Q) \$ j6 ]* ^5 w" e6 X, i+ J; Q" ufactorial(5) = 5 * 24 = 120
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Advantages of Recursion
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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).4 e3 o4 f) m- r$ R
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    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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/ W, T  w3 E# a+ A% SDisadvantages of Recursion; ?5 H& n  W; Z
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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.: l* ]0 B' C; p4 i1 `
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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When to Use Recursion
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    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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    Problems with a clear base case and recursive case.
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1 p7 K' l  x9 L3 kExample: Fibonacci Sequence
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0 t4 `* u1 W: l: H2 jThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 U$ ~. \) A, M: [7 a. k* z

$ N! ?  k( V* }9 y3 M' E    Base case: fib(0) = 0, fib(1) = 1" Z; M- W3 a9 n) D2 x: }
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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8 \' _- L2 ?- S% S: c' g3 O' i3 Xpython/ {8 k. I& L! {

# ^6 C$ V# o# J5 V# k" g0 @7 ~" B# m5 ?
- E7 j& E6 l% s. G/ S- m0 ?9 Cdef fibonacci(n):
3 P% ?, T6 ]5 c; r8 ?, N    # Base cases
1 n  @: D4 X/ {( @# j. r    if n == 0:
/ [, B0 [  ~% x        return 0
3 g. f5 ~. T. C! j) L    elif n == 1:
& |6 N6 D+ J/ b" V8 r3 E6 g$ Q9 M/ T        return 1- r) e+ O5 ^3 ?. L) \; H& W
    # Recursive case
* ^8 }" S3 P+ Q% X# G% h4 G    else:$ Y0 @& V7 f6 o+ z& K0 d0 h5 s
        return fibonacci(n - 1) + fibonacci(n - 2)
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& e  E/ n8 v) @1 H4 h. e# Example usage* S7 ]5 a$ N% r, `& N& e: O# f$ h
print(fibonacci(6))  # Output: 8& V0 a2 v, A. x8 j" ~

) I' j+ f% V& c8 G# W$ HTail Recursion* @) K$ `, r/ b9 N2 |) T: T

& F7 ^5 j% Y2 cTail 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).& m' [9 U* B" L, G: C. U$ M" V

  N9 L" ^$ N; }4 wIn 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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