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 2 V' A' O7 o4 B% f8 C) k+ c / F% e. K& n6 a解释的不错 - g* u6 H* b* d! P2 g/ F1 |3 c! Z) i6 V
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。 # q0 |3 P$ @, d! ]$ \/ T& x! X0 T$ ^2 w' Y: r+ @
关键要素6 [- w, w5 A9 ^' j$ g7 Y% u% `/ w
1. **基线条件(Base Case)** 3 y8 G1 \& K7 a9 l) U+ N; s - 递归终止的条件,防止无限循环 1 h' w0 f$ E! M. | - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# [ [5 k. b, y3 _) K; e
# N- D% \' _" O! P3 M+ v. G+ }* [
2. **递归条件(Recursive Case)**, V1 `; c8 [* U4 E1 c( t. Y
- 将原问题分解为更小的子问题 ' n0 [. Z% B6 H" ]# _ - 例如:n! = n × (n-1)! G7 |7 L3 k/ @1 P( ~
$ w5 k, k- A6 U$ x
经典示例:计算阶乘 z8 d* X" U8 Z6 p- Upython& y# _ _1 F9 r# r( S2 M. D! c
def factorial(n): w5 x# k) J7 H; \7 s5 I. \# t
if n == 0: # 基线条件/ e; u; o% J$ {) r
return 1 4 @% P6 Y9 l1 G+ p. E+ t else: # 递归条件5 S i q' T* G' c1 \4 s
return n * factorial(n-1)/ x7 c- B- L. X- `5 s
执行过程(以计算 3! 为例):( M* L! t( ~ T, |: z
factorial(3)9 @8 @8 k9 F8 I. Z
3 * factorial(2) ' ^7 h# E; \8 k) b8 w3 * (2 * factorial(1)) 3 u/ [: }# q8 f6 M3 * (2 * (1 * factorial(0)))0 `: s9 t' f6 H- E2 T9 }0 x
3 * (2 * (1 * 1)) = 69 n% p! j$ q0 g+ t; E% n: k4 F! E4 \
, R [7 V0 d) s
递归思维要点% e ]. N( C- _: q& f v
1. **信任递归**:假设子问题已经解决,专注当前层逻辑% T- s I0 m4 L; t# ~& F5 g& B
2. **栈结构**:每次调用都会创建新的栈帧(内存空间) 2 i- S3 Z1 v$ ~' R# u+ M- ~3. **递推过程**:不断向下分解问题(递); F3 p+ C6 X9 R7 H' l
4. **回溯过程**:组合子问题结果返回(归)7 U/ D _. ~( f( ]% i
; F4 l, M2 X' H7 F6 B! H- F 注意事项 ; G$ M ^. f# G$ Z4 O# n6 d必须要有终止条件 1 g M8 U- M% U" S/ T递归深度过大可能导致栈溢出(Python默认递归深度约1000层)8 l7 W m/ x: x
某些问题用递归更直观(如树遍历),但效率可能不如迭代 + F) Y& W. v/ x: u5 [& @# c尾递归优化可以提升效率(但Python不支持) , S: \5 w% Y h B" j4 A 9 ~" B( U. }6 ~4 }, C 递归 vs 迭代! Z S; w g1 C! H" O
| | 递归 | 迭代 |) C) x+ S9 {. h, b7 n/ B4 Z
|----------|-----------------------------|------------------|% E0 L7 y4 ]& R9 I) } N& B& g
| 实现方式 | 函数自调用 | 循环结构 | & y% n$ v: L% K; Q| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |. a1 d1 ]* P: P1 W3 D
| 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 |4 E" m( n$ `5 z' T8 r
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | . ]' G) n/ p1 `* W ( I, V$ Z8 N/ \& O' t9 S0 x 经典递归应用场景5 z# U0 F8 A% r8 t4 {) }6 d
1. 文件系统遍历(目录树结构) 6 Q5 }# u# |- O; V2. 快速排序/归并排序算法 & R" @! j+ w; f: k4 e3. 汉诺塔问题 3 N3 w; K2 m( m4 x* n& m4. 二叉树遍历(前序/中序/后序) 0 |$ F/ U8 P7 F, @5. 生成所有可能的组合(回溯算法) 4 E2 q3 j/ V( U1 [4 @8 h/ ~8 r7 f* @8 t+ ~
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, 6 p" [" y) L7 c我推理机的核心算法应该是二叉树遍历的变种。! m* I# g, R3 N* C! 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:6 m: H7 P* n% b5 U$ S/ Q1 C1 ?
Key Idea of Recursion - u3 E! [. ^( v& n+ \" `3 |5 Z' ?9 Y' O* ?. J4 l8 R
A recursive function solves a problem by: 5 j& p! g$ ?7 n" n% ?( R" l% v" [4 }& _; t/ [9 Q
Breaking the problem into smaller instances of the same problem.' t3 }2 i! f6 Z: r% ?4 G$ b0 ^ W
! L* s4 l" z h8 X( l
Solving the smallest instance directly (base case). ! b8 e+ W+ e8 E4 ~) h ! T- G% K: A5 ]: C Combining the results of smaller instances to solve the larger problem.0 S" S5 m0 W( E; X, t* n% K
4 z P7 R$ I$ mComponents of a Recursive Function0 q! M6 l1 e+ {0 Z
, E( J8 x8 @. U z6 \1 u# a! }
Base Case: * t) \- w- P& e9 r* F ! J! s. ?9 j; W! N! b This is the simplest, smallest instance of the problem that can be solved directly without further recursion.% R9 r& d% }, |6 f" w' R
; y1 O; z7 h: h It acts as the stopping condition to prevent infinite recursion. " W/ }$ E. I$ t- |& h5 H/ q! w3 M; |/ w
Example: In calculating the factorial of a number, the base case is factorial(0) = 1. . r$ y4 t- k% u; w: r5 w' J+ o7 W ( Z* B7 e& ]/ O, M Recursive Case: 5 q3 v4 `( G a6 ]5 s! i0 O4 i E7 S+ q* j- g
This is where the function calls itself with a smaller or simpler version of the problem.# _. Y$ ]. _7 I. U! d
# S+ O H$ J' X+ K5 h/ b' `
Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 m9 _, M! @/ {9 m- |: K8 N8 m8 |. z
+ E0 G4 R. i5 K0 h( |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: 5 c, M$ A- p- z( T6 [: R7 F! |: n; X: i; X
Base case: 0! = 1 0 @- v4 R6 ]; f) O9 c) e$ I3 f8 Y+ h9 g. M; T
Recursive case: n! = n * (n-1)!: @# m& Y5 u5 t* Y
5 e! F9 |4 K K7 w. W- q8 jHere’s how it looks in code (Python): # p; l0 ~( y7 I2 O/ ]python ) w$ w1 u1 E, C5 ^/ v 7 V* i7 s2 o# v6 G6 ?, @) y3 g9 c* `4 B
def factorial(n):$ ]! Y5 D: a0 f' N: z$ [( m
# Base case . ]! A2 D) G5 v+ } if n == 0:& k& o B4 h3 W: D$ o8 Z6 M
return 1 * H5 ]" v* G7 Z6 l o% \! w # Recursive case + ? [( _9 F8 v/ ?1 U0 u" J else: l2 T: o: \6 z, j0 g$ b8 z
return n * factorial(n - 1) 4 c O$ [5 T) k6 J- t3 H# w 2 [2 Z% }$ `" T( V7 D# J# Example usage / F+ n2 u* ^6 f% Jprint(factorial(5)) # Output: 120# F; d s. y+ z9 m9 l$ a# z
& s8 f4 n B n* W$ G/ WHow Recursion Works& B% t- e3 b' a! |$ F6 u
! L5 M$ i& a9 E5 R0 M The function keeps calling itself with smaller inputs until it reaches the base case. 8 K; i' \3 A7 Y' |+ J 6 z/ d9 h+ w$ t6 C/ _; l Once the base case is reached, the function starts returning values back up the call stack. : i8 r2 T# K9 b2 f% } 3 l2 k, N7 J3 c. h0 o, ] S These returned values are combined to produce the final result. 9 ^/ W5 `1 J* @# H3 ?& k% A+ S) [5 ?: {, _) k
For factorial(5):" |' X# m. R- \% }0 h, ~
* d" C0 v$ o( m( _) ^9 m( M. m5 r* ? ! U& Q& D% K9 b% e1 w2 Q% Cfactorial(5) = 5 * factorial(4) : X M8 A) I7 O5 {3 W9 gfactorial(4) = 4 * factorial(3)# U7 i! M( {$ R! m' @$ r- W
factorial(3) = 3 * factorial(2): M: w1 _+ ~. T
factorial(2) = 2 * factorial(1) : }( k- J, s- y2 r; I+ z6 nfactorial(1) = 1 * factorial(0)2 }% }6 }3 ]/ D( S+ \ [- x
factorial(0) = 1 # Base case4 Y# i' {+ x8 w+ \
0 o( o1 Y% i7 J) g* T) ~ {
Then, the results are combined:0 R9 {, F' ^5 z3 q/ H
. `5 B ^1 {& h }% B
9 p% V% J+ j! b
factorial(1) = 1 * 1 = 13 _2 w; S& j! a- a7 ^. ~2 ^' H* x
factorial(2) = 2 * 1 = 2" L: J8 Y" l* O. C, \6 r9 A
factorial(3) = 3 * 2 = 6' m7 C$ a# E6 B2 W
factorial(4) = 4 * 6 = 245 B2 H A2 d, V5 r7 u
factorial(5) = 5 * 24 = 120 & z3 ^8 z- t3 M) \: V ! p! k& b Z( JAdvantages of Recursion ! N- x2 D/ T8 Y" g4 N1 { $ v" ~/ ?9 s: J: K! J 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).( ?7 D2 L/ @- Z; ~
4 G1 e* J Y% N2 p
Readability: Recursive code can be more readable and concise compared to iterative solutions.4 @$ S4 G6 l+ x/ k* N) q
; |3 c5 l: x" e, u" u5 U! z/ ADisadvantages of Recursion' ]) B& z3 ~5 l3 g# o2 Z
7 z1 s3 g+ I1 n8 }! Y+ B2 e |( G9 A 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. 7 X& \; H- I% g o2 ] ! l, a1 Y/ S; Q$ t Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# Y( t" Q2 T. l4 [' u+ o- `
2 p: h6 @- e+ g( \
When to Use Recursion 1 `0 j6 Q+ N. y) j% x; L: i5 p- Z . p. @0 z- f E# ?. Q5 D6 L Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).* U+ D) j+ w8 S) N& j
: q' U% ]: |0 W
Problems with a clear base case and recursive case. ( a/ }4 @- Y) I6 L$ q' m/ }9 v S8 F( `/ c. H0 n5 G
Example: Fibonacci Sequence , S( Y1 a) F! {/ j# V5 n8 F " f/ V" ?5 m$ r# d, u2 BThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:' Z ^' c, ]) Q# }& v$ u9 z- B6 K
1 _2 g0 h# ^: h, p9 ^% o) _
Base case: fib(0) = 0, fib(1) = 17 ?3 T1 f, h2 _
- ]! q" K: F# _ e- D
Recursive case: fib(n) = fib(n-1) + fib(n-2) ! X3 m' G7 y5 G# D, P0 ~" S; l9 x" ?0 e: ~$ U- k2 g
python * Q" h9 u9 c0 g X+ y, ?5 J1 p( O& Q + X) z- o; m+ e1 Y) [def fibonacci(n): 5 E3 [' a) X- e/ |# m # Base cases 7 S( x1 n) N9 g9 [3 s) G# [ if n == 0:8 j+ {/ I% `3 W8 Z
return 0, L# t q1 _+ R5 b0 r2 H: q2 B. o
elif n == 1:) B1 ^3 i7 A1 ^, G& z
return 1 8 {& B0 v6 O* L! h # Recursive case ' b0 C i' [( ]! Q else:( G! K3 c& t& d% R1 q3 u( {
return fibonacci(n - 1) + fibonacci(n - 2)+ K# \/ U `$ r4 H* ~, F, M
2 e- I* [2 e1 l; u+ ~$ S. s/ D# h
# Example usage 0 a7 ]+ n) s" N4 h( i2 M2 R% lprint(fibonacci(6)) # Output: 82 @8 D2 P9 {9 Z
- ^7 U7 \. m5 X, K
Tail Recursion 0 ?3 s/ w. L) Q" z! l, X0 y W 4 |6 Z M; E, D, STail 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). 8 a& P8 W8 L0 P, x9 Y' u m0 ]; Y. D( ?1 a: u
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。