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 + {+ e5 | T) k' m2 c
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解释的不错; M# b- P' f( [. P
! x+ q8 o/ | o! O, ]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。 6 D: P/ K% ^0 z& u; h9 M$ B" K5 k0 m w' I( Z! D: m. k
关键要素7 V0 I7 @! ]' D: \
1. **基线条件(Base Case)** 9 M6 y+ w" Q, E& ^- i- U# f8 e - 递归终止的条件,防止无限循环 * w6 q' S/ v0 a) ~ - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1) |3 [! z9 R! Q) k" b6 T8 j$ X
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2. **递归条件(Recursive Case)**( D5 {0 |5 ?- Y7 J- f3 b
- 将原问题分解为更小的子问题' d6 V9 [3 _" L( l0 {
- 例如:n! = n × (n-1)! 4 J: H3 q' P. Z' c8 X: ~( |9 v& v8 s
经典示例:计算阶乘* s( z! g9 K! n
python ' P( i& `* U, x8 g9 w& o" Idef factorial(n):8 t4 \, w% ~* H J( ~
if n == 0: # 基线条件7 h+ X; T6 e# Y! K
return 1 / i' ], K- X! Z else: # 递归条件 H; v0 o! N' v& E/ G
return n * factorial(n-1) $ D c( U. g |, \- H0 v执行过程(以计算 3! 为例): ( y8 g1 t3 \; T7 ~5 \, e9 Yfactorial(3) 6 L4 m- G: D$ o3 l! l: b: u3 * factorial(2)3 z8 Q9 x: X/ y8 Y. X0 a' m
3 * (2 * factorial(1)) n( W& Z& ?" q3 * (2 * (1 * factorial(0))) 4 }$ Y- O) D& d5 @5 l3 * (2 * (1 * 1)) = 6 ! R) R! E& U4 e a& U : ]6 q: M( Q9 h/ T4 B 递归思维要点 # m1 d) m5 t* m1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 k% A6 E4 D' _) s% j
2. **栈结构**:每次调用都会创建新的栈帧(内存空间): p6 J& d; R- d7 x& J) c5 p4 S Q
3. **递推过程**:不断向下分解问题(递) 3 g2 B% G; \& E' a, z4. **回溯过程**:组合子问题结果返回(归)3 q$ j m0 V- P1 N! \5 ?4 A
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注意事项+ I" ^; p, U# I, |
必须要有终止条件; U6 u+ x3 b' K1 H2 G6 f u, S) ]' f
递归深度过大可能导致栈溢出(Python默认递归深度约1000层) ' g4 N4 u" U: o, t& E& H% T某些问题用递归更直观(如树遍历),但效率可能不如迭代8 \ G2 I$ o' s' ^
尾递归优化可以提升效率(但Python不支持) 4 u, g. q, ~1 u* M3 ^ ' x$ z; i( Z* i7 ^ 递归 vs 迭代 P. i6 G' R. T$ l) ]| | 递归 | 迭代 |2 {4 x% B+ j& g/ [, N& F/ z& M ~
|----------|-----------------------------|------------------| 9 K. r" S* T! |0 U! B" T' F( v| 实现方式 | 函数自调用 | 循环结构 |5 _! f& p( }* F& {& y
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |4 t/ Z- m H0 l( D# }( T3 z% 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 |0 n! p: G( ~) k+ q) k
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | 6 ]( F) b8 \- l% A3 w7 ^ E 5 ~' U# x/ e$ J. K- F, P 经典递归应用场景 7 E/ t# s8 r& P; U, }' A6 }1. 文件系统遍历(目录树结构) ( b. x2 D) |7 t& y2. 快速排序/归并排序算法' v* E0 N5 B8 w0 }& O
3. 汉诺塔问题 ) I* [: ~0 v2 F5 h g' i0 V& X4. 二叉树遍历(前序/中序/后序) 5 E: o: E7 j7 n& W3 E$ _5. 生成所有可能的组合(回溯算法)7 A$ i% t; ]4 f/ l q7 r5 I
% h' y6 ?9 O8 H. n- B+ G" Z8 Z7 z7 b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,1 r S- r9 o1 x6 b8 J& ^: h6 m
我推理机的核心算法应该是二叉树遍历的变种。 , }5 N6 }& K* u; d' l. 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: . e9 w4 i. |7 v$ I' JKey Idea of Recursion0 \) @' E! g, J) U9 A3 ?2 m; }' R5 d
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A recursive function solves a problem by: 2 U) d) {. f9 @5 h( B3 R5 U' M" ]5 `2 n8 {
Breaking the problem into smaller instances of the same problem. \; j: X6 @- t( @
* O( M( s) k/ s) [* r8 w Solving the smallest instance directly (base case).2 a' Z' _# ?; u# d+ K. w t
/ I4 v8 ^/ k3 [ Combining the results of smaller instances to solve the larger problem.% g0 {) x8 Q' Q& B, r- L' E
. ^* D3 f6 M2 D5 W; XComponents of a Recursive Function : J) i' ]0 h7 V& P) L. `* S3 M# J, J8 [7 n! \: e* |4 J! _! l4 O& c
Base Case: 5 p; u4 E( @2 ^ " D6 I1 b8 B) } This is the simplest, smallest instance of the problem that can be solved directly without further recursion. / |$ K) T/ h: B- o% \& }) }/ a+ ~# r9 K: V. Z2 h4 K0 C
It acts as the stopping condition to prevent infinite recursion. ! C7 h4 w6 q2 {5 p' l1 v; J8 |! o; K9 Z" K
Example: In calculating the factorial of a number, the base case is factorial(0) = 1. ' r- `. x: b' D1 \( I2 P , G7 `9 }7 C2 g3 Q8 @2 j3 [: V Recursive Case: , _$ g% ~" ]$ ?4 H0 e 1 F3 }4 }1 Y+ K5 t: K) J( F This is where the function calls itself with a smaller or simpler version of the problem. ; |0 q4 O+ _% T! r) }3 T 3 G J% {* r& C/ l5 X2 l Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1)., h! r. n9 k1 g
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Example: Factorial Calculation . ?) A6 {) S; V5 o6 D( [0 U% u# A0 F+ ?! o; y. u! b; ?, j9 b3 u0 y, \
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: + `3 B: y: U8 w2 ~' {: ^0 d* b# k) I) b8 j9 N" e5 F. o: O
Base case: 0! = 1 . X! k# `" y. N7 p& e X# k( E9 t) ]) ]# M/ e- p% |' i( n
Recursive case: n! = n * (n-1)!4 Z- ~( u( N* J1 z# S9 s/ {
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Here’s how it looks in code (Python):& t; s+ p/ c3 q. v' q9 E! N" L; ~
python 7 Z; O% b Z$ |* M9 o9 @ j7 [: o) C5 U; h
Q: P2 n# m y$ D, w% vdef factorial(n): 5 R$ x& \0 S; N7 l+ |% ` # Base case : ^/ w1 ~5 D7 X5 }+ } if n == 0:7 l' c4 g2 [+ K/ V) a
return 1( J' G0 M) O# k9 y
# Recursive case . C' ^+ T8 ` x; Q8 T- v else: s; K) c; C: r/ ?
return n * factorial(n - 1)4 J6 X4 c2 O7 o4 g7 I) U
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# Example usage; K0 a6 r+ d) o) h, _0 J R
print(factorial(5)) # Output: 120 8 M. L: e" V7 A+ Q/ F' F: A Z( ~5 M+ o& d) ^ I) c* `
How Recursion Works! ^3 ]$ _) u& S9 ?
+ j% Z- l! B0 |# c8 l The function keeps calling itself with smaller inputs until it reaches the base case. 5 \* f! {( a& g9 A- G" @8 j3 r 8 o6 s2 o4 z, F3 a$ O- U3 p Once the base case is reached, the function starts returning values back up the call stack.* y" f+ b' d/ t/ i" y+ B+ b3 X* p
& L8 ` L; Z' z$ c y d These returned values are combined to produce the final result. 1 z6 j) K. N. D4 j ( S8 W% w+ {1 P5 k- X# n: eFor factorial(5): , }: N; C0 y9 n: P9 J4 ?3 u5 S2 v; C5 R7 T. B! y1 V% S
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factorial(5) = 5 * factorial(4)9 ^# u( X" K5 c8 s3 D
factorial(4) = 4 * factorial(3) 3 v0 C) d* K( N# g: J3 P, \1 Kfactorial(3) = 3 * factorial(2)( W$ N2 N5 } B: w+ u* K- ?- _3 ~
factorial(2) = 2 * factorial(1)! T, z3 H, j5 h( k1 R/ L/ }
factorial(1) = 1 * factorial(0)! y2 m5 ^" N$ I8 G2 K
factorial(0) = 1 # Base case $ V* b ?+ Y- F* u: l" a& Q" e" y. l
Then, the results are combined:! Q7 Q4 [" y# `! Y' Z, r, q- \6 y
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factorial(1) = 1 * 1 = 1 5 {5 B9 r$ H8 }6 N9 D7 I# N" f8 zfactorial(2) = 2 * 1 = 2 $ S7 k1 P5 }0 h) w+ lfactorial(3) = 3 * 2 = 62 b v n& o% P5 ?( \- M
factorial(4) = 4 * 6 = 24% y- l: I! X% o' E: y) W- @3 {
factorial(5) = 5 * 24 = 1207 j0 K1 w' P( w1 y# H! y
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Advantages of Recursion2 t1 G8 F F# s6 t4 z9 F* C
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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).( K' D; [4 H( u$ U
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Readability: Recursive code can be more readable and concise compared to iterative solutions.0 _: ^% @9 ~" B1 z4 l9 v
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Disadvantages of Recursion 4 i' \% ~0 \5 W" _" _ ^0 n, T
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. / ?% q. d9 V) J1 G$ A4 c 1 q4 N# _( j5 e1 w* b2 D+ x$ \ Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). * {1 I5 ]7 n9 y8 X9 ^$ W+ L/ \. G5 `. m# b+ S
When to Use Recursion7 d# a' u8 P1 ?/ J) c# [6 i
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Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). ' X5 l- J: M- ]( t; B3 O" S 2 N+ ~. ~0 i% a# m Problems with a clear base case and recursive case.% L9 m5 w7 |) h7 }& r
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Example: Fibonacci Sequence9 R# e' r( t; K
; d' w. ~" ]" LThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:* G. Z! p9 W- }+ F+ F( i) ]
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Base case: fib(0) = 0, fib(1) = 10 {3 O; |# s- O8 r; Z
& z% z7 W$ o Y8 G* f1 { Recursive case: fib(n) = fib(n-1) + fib(n-2)4 Q: {. z. D2 j! D+ m9 F7 U
" V) m x1 Z e/ k . u* R2 M7 J4 r B# A9 Hdef fibonacci(n):3 B3 Z# n' z; {; g9 y
# Base cases ! I( K: u, B& S8 [8 t" I if n == 0:( H% R5 v$ }. B3 E! k
return 0 % d' Q- o; j& B# Z elif n == 1:# h7 S0 ?) l) t( v+ C
return 1& v* V" c0 H5 C; K( u/ q
# Recursive case 1 [/ I5 c) m/ J& j# a: T else: ; B3 x _; m+ l9 y return fibonacci(n - 1) + fibonacci(n - 2)( f# Y, h7 B; s, M8 s: ]) |: T
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# Example usage ( P6 _1 k7 n ]* Y7 wprint(fibonacci(6)) # Output: 8 0 ~7 P4 N! R6 @7 ] ? 1 M& |8 V4 m% ^/ C0 Y; W) U/ D/ YTail Recursion2 q. y. ?: T5 E7 m9 z q# y
2 [% @5 T/ _ |( z+ Q! e8 M, Y; HTail 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). # r3 B* R+ C/ X+ E H- x5 [ ( g8 |3 s- b$ Y) H+ S$ A" j* yIn 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。