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 0 I4 X2 h- {; W1 l E2 A
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解释的不错1 z( ~1 j6 F `0 U% B6 u
0 g9 R3 C% Q; e4 @& h: j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。 ( l0 Y- B) a d$ P6 U; n$ H# ^; d/ R) P5 T) n% J& G
关键要素 ) x6 I: m6 x1 i! x" j; S9 {1. **基线条件(Base Case)**, x7 e0 K9 k6 K+ B6 z8 ^( O. Y/ E R
- 递归终止的条件,防止无限循环9 M) Q" U5 l7 }4 W! w
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 1- H; F7 K; h) F) c! d5 t* b$ u0 x
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2. **递归条件(Recursive Case)** 5 d( W. c& @7 e: d) ~0 o - 将原问题分解为更小的子问题* q/ F& E' ?. H3 y7 m4 A6 y6 E
- 例如:n! = n × (n-1)!" o. q" n8 {: A# R' [3 \0 s* z' ]6 L
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经典示例:计算阶乘4 s) V" c" k7 Y
python 9 b" j, }" n( M5 U }8 _/ h$ Tdef factorial(n):; {7 q6 u/ X8 ~% Z' f. x
if n == 0: # 基线条件 * F, W/ q2 b' b i5 g# K return 1, x5 ?* G- M f% \2 Z# Y
else: # 递归条件) U$ P# `1 e. C' C' n: M
return n * factorial(n-1) : y/ C/ E7 [, ^1 g j% @6 w7 t执行过程(以计算 3! 为例):! p7 S1 v' f& b; ]* r
factorial(3) * m2 G# H6 G- r y$ a3 * factorial(2)9 v' \* x: V6 @( `) u
3 * (2 * factorial(1))+ Q; Q8 ~9 I$ ]9 x: G: h
3 * (2 * (1 * factorial(0))) 8 |+ g, l7 y0 H# m3 * (2 * (1 * 1)) = 6 * L5 x1 W% S- h9 s9 C+ l2 c* `. I- Y$ ?
递归思维要点 " V' p9 J& W. n1. **信任递归**:假设子问题已经解决,专注当前层逻辑 # `! ]" G. ] `' ]2. **栈结构**:每次调用都会创建新的栈帧(内存空间) 6 ^6 `9 q% t% w8 W, P3. **递推过程**:不断向下分解问题(递) & @: M# o; l* K# T' x4. **回溯过程**:组合子问题结果返回(归) f) U: @. i' g$ O0 Z* `
3 O" ~" Q# r% `. \ 注意事项$ n" k, ?! U5 j6 @9 r. y
必须要有终止条件 , ` h7 N8 b; L& P递归深度过大可能导致栈溢出(Python默认递归深度约1000层) 1 `* H3 B. u& l% S某些问题用递归更直观(如树遍历),但效率可能不如迭代 * O8 T' v _2 v4 i6 E4 a尾递归优化可以提升效率(但Python不支持)( O2 i7 e) C% L9 u$ n
) M+ M' O! P _# E+ r 递归 vs 迭代( q; C/ w% I2 `- q1 m R1 ?
| | 递归 | 迭代 |0 z3 t, _( k- \+ C
|----------|-----------------------------|------------------| ( |! D ^( Z- P/ p/ y| 实现方式 | 函数自调用 | 循环结构 | ) l3 k5 S: B3 ?- ^, G' I9 Z| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |9 U$ d4 o# W5 E- j, G
| 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 | # r' D' n, W) W7 d| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | 7 @, F8 ]( ~; c- |( N9 M) O3 b 3 S. J* `/ T4 O+ y" \4 ]- @! G! c 经典递归应用场景! |9 \7 r& H% d! K' B
1. 文件系统遍历(目录树结构) 3 f' p. f' i3 k! z! g4 g9 ]2. 快速排序/归并排序算法 3 t& Y; r( N: K% i) A* D3. 汉诺塔问题 $ @# S: F3 z8 ?5 V! Z5 B+ X4. 二叉树遍历(前序/中序/后序) 1 i- F% X+ Y" P) I5. 生成所有可能的组合(回溯算法) ; V+ a" @+ L6 g- N, h/ H, w) b5 b% D+ I- M. C
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, - b, G1 \" `- s9 L! n$ w我推理机的核心算法应该是二叉树遍历的变种。 / u# z" u( D3 j# a, \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:% J+ l* ^8 @+ R6 T
Key Idea of Recursion $ G% H$ t( T5 o$ w6 J6 L+ t ( f* d0 h l$ v1 h# Q, O4 rA recursive function solves a problem by: . d* L4 i, A2 K" F8 ~ P8 q. p) w. Q4 }0 z
Breaking the problem into smaller instances of the same problem. 0 T. e( b; ^+ S R+ L2 e 4 U) s* S, _; ~3 y7 ?8 X Solving the smallest instance directly (base case). : s+ @0 p( H' s. R, I- s Q1 C8 e( o) d8 q- J
Combining the results of smaller instances to solve the larger problem. 7 ?* _5 n* \1 E) }: E* h# w " z4 v) k8 H) }+ E% qComponents of a Recursive Function9 t0 y! f' q( |) [
, x* Q9 F1 j' b2 U. M; Z2 w( x Base Case:! H& N. q, Q6 Q; j: w/ `
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This is the simplest, smallest instance of the problem that can be solved directly without further recursion. . W* i% n) q, ^8 l; @ 4 {1 @1 w% [0 s4 E6 T" ]5 J It acts as the stopping condition to prevent infinite recursion.: n/ L+ y! j5 L2 n0 f ~7 c/ A
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Example: In calculating the factorial of a number, the base case is factorial(0) = 1.$ A8 }* T9 w# Z7 D( f
) b' E- G1 v. T9 @( s* o Recursive Case: 9 d: K% K- b/ j5 J8 P7 o3 j% ?/ u H5 c1 ?: ~
This is where the function calls itself with a smaller or simpler version of the problem.$ O# G3 t6 N4 N$ r% M
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Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). ; E) u% |- L+ l: T* o9 m! x8 v6 m' h- Z: |# T) E8 F" m
Example: Factorial Calculation% _# ^ j, E0 @0 J/ H* S Y {
6 h4 q3 u0 P* C+ \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: , g. o# R; @ p: N& t; a6 n3 Q' J% _
Base case: 0! = 1 5 k- u" v, u2 [+ f& H: D3 w/ Q. Q# |! ]* P0 |: d
Recursive case: n! = n * (n-1)! ) E, Q( ]5 }; _1 o: K4 f w( x% A2 Z) |8 n5 m# j8 wHere’s how it looks in code (Python): { c* l/ L# ^9 [$ ^6 L! rpython . P+ ~( ^7 k( j" Z6 N. T* X; [; o# N0 k# a( x' K
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def factorial(n):4 ~/ ?, i7 C! H7 Y$ D
# Base case ' [& c( H$ w( ^ if n == 0: % K* o2 r* s# |! B5 k return 14 f8 p6 @$ S; g5 {( I( i8 Z
# Recursive case 0 C3 { p' V4 X3 \6 ?1 ^% { else: " e8 h0 _6 n5 q1 G return n * factorial(n - 1) z H; y, g9 K1 c( [: \
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# Example usage 4 ~! a! P& T* U0 fprint(factorial(5)) # Output: 120 7 [4 r1 q; D4 {: v$ @ / t+ Q: [0 k! nHow Recursion Works 8 B: W# E9 |# L7 {6 F. v @- t2 W3 a( E' ]8 V- p
The function keeps calling itself with smaller inputs until it reaches the base case. 3 \6 m' T$ s+ Q3 V# c/ h, W2 F6 J2 ^0 R3 Z9 w6 E ?% S5 j, ^
Once the base case is reached, the function starts returning values back up the call stack. 3 L5 F. L! W' W# J! H # J/ o- @* `+ q1 d9 D These returned values are combined to produce the final result. ( {: M" L. T- V3 j* B3 C8 i4 f6 l+ | " K8 x1 I. n0 M7 jFor factorial(5): 9 Z0 R- J; R, ^& Q4 `/ n7 ` ! _+ e# [- |+ u2 T8 O0 N& e$ ^+ S+ w
factorial(5) = 5 * factorial(4)2 k2 W, E. m# [! Z2 j! I) [; y
factorial(4) = 4 * factorial(3) , U3 p% J$ }% i3 b" Tfactorial(3) = 3 * factorial(2) & m. L( E/ P* X S. hfactorial(2) = 2 * factorial(1) $ C9 U8 z) r0 Zfactorial(1) = 1 * factorial(0) ; o& W, Y c/ n+ j$ a$ T* Bfactorial(0) = 1 # Base case & n# s% N3 g& P# s; |9 F) ?! W. B
Then, the results are combined: ; B1 [) E8 L6 w8 R4 c: V7 }: K& G7 `
2 d) l; _" i" }9 V' j, m$ Ffactorial(1) = 1 * 1 = 10 H. Q; Q; z1 x! y3 F( \! Y
factorial(2) = 2 * 1 = 2- F6 c7 u H6 U6 m
factorial(3) = 3 * 2 = 6 : D }" O( B4 y2 D( O1 rfactorial(4) = 4 * 6 = 24 $ P& g G7 y6 G; sfactorial(5) = 5 * 24 = 120 - K3 N* {" F9 X+ P* J9 z8 ^- Z % l+ _) E& d* X# Z7 `/ ^Advantages of Recursion3 e, \3 G) ~* \7 i" s
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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). / g7 m/ v* A- ^5 T7 K# Q( e6 s' p8 t- c5 s# ?. I! `- i3 N
Readability: Recursive code can be more readable and concise compared to iterative solutions. ; L9 N, O% }( T . U" a% o2 l" M3 Y( f+ MDisadvantages of Recursion4 f% ~8 G2 R5 T/ ]+ y6 i8 f' d
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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. % j5 ^* y# T, G& n# |7 r $ u; I5 O4 J' }! Z: W2 V5 ?$ I1 X9 | Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). 8 @8 W7 n0 s6 @" @ }8 _( Q/ B' _# u/ \3 }' u
When to Use Recursion9 j, P" ~9 f6 Z! h1 B2 |0 ~
5 h) u7 f! F# Z* {" L/ K- T, M Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)." x' n% o& T( o6 o6 F: P
: k. r3 \. ^8 W Problems with a clear base case and recursive case.& Y0 `+ E/ ~- W7 T3 A/ x4 D
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Example: Fibonacci Sequence # C8 _7 ^$ d. H" j5 t: a0 c9 ?, ?) j 7 H" j/ y/ P# U7 Z O! zThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:6 S3 ^8 M8 T- ], a% ?# X
5 z* Y6 R E- i/ T' E Base case: fib(0) = 0, fib(1) = 1 " W- e3 F: S3 M" k$ f # V/ E7 O0 f6 K6 z0 |2 H Recursive case: fib(n) = fib(n-1) + fib(n-2) * s* q2 G" N- g8 z. @& m3 F4 x; F9 p( p6 x
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2 I+ [4 Q5 `! P / |" S8 _- g" R' U+ ` @0 X7 u. Udef fibonacci(n): ' h" D8 u( J6 q6 @- I # Base cases4 n" y, X/ ^5 l& _' a7 Z
if n == 0: " H P& v+ [4 G8 h' d: W$ | return 0 * i3 z7 s/ J. s. M& e$ h# v elif n == 1: 9 j; R0 |0 O u% i2 {7 F2 d return 1 8 c# q8 n4 @ ~+ O) O6 t # Recursive case ! u- J. W. ~- }+ A0 L else: 9 _" V+ a% j5 r* V( w9 Q return fibonacci(n - 1) + fibonacci(n - 2) . U! S. i; B1 F3 I# D8 ^- w3 W6 D3 _5 j; G! q2 K
# Example usage 1 t# E, G4 J6 Z' ]0 u# z* G+ Jprint(fibonacci(6)) # Output: 87 s5 N' ` @8 V) @! Z- r
6 C0 z V; F' @" v# LTail 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). " F* l0 y4 C ]/ f2 J: R4 W! U$ v0 i% T* |& g
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。