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 d" E1 x8 u( U' s # S. l/ u0 o* W5 R: c解释的不错 2 Y6 J& L: S/ u) q* d8 H" C+ u& 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。% K" a D: L; ?% g% a
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关键要素 . z$ o( m+ N, ^9 N. n5 r+ q1. **基线条件(Base Case)**& t/ ]; W% u& n5 y0 Z; l
- 递归终止的条件,防止无限循环 " D1 D0 X$ l# n1 \5 C6 X - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1* ^- z3 f/ r! K3 l# p3 n7 k5 h
C; U1 p1 [$ ?/ K! r3 Q% k2. **递归条件(Recursive Case)** : l C& `* V+ v4 }/ W" ? - 将原问题分解为更小的子问题 ) d4 r! {( a, J. B# D. g+ _ - 例如:n! = n × (n-1)! ' Z+ i0 t4 z9 E9 `* o: \5 L- m! B( g1 W3 T0 G' R
经典示例:计算阶乘 : Y- Q$ t u% ]python / g5 w& o$ N! odef factorial(n):9 I' l+ |" D3 |) O
if n == 0: # 基线条件; Z+ y6 u8 {7 e9 G: Y7 @8 D+ f
return 1 - z# o3 l% O- b) I1 x. n else: # 递归条件 , ^) g9 f( ?: m, G; M5 ` ^ return n * factorial(n-1)# Z- d3 f; o( R) ?% X: |
执行过程(以计算 3! 为例):2 H5 V; u" i, J- Q& _
factorial(3)/ h3 j8 _# T. E/ z2 C3 T* k' k
3 * factorial(2)- N) z- |$ }# S( c W6 u. d; ?; w3 r
3 * (2 * factorial(1)) ' j9 v5 E! x4 C8 x3 * (2 * (1 * factorial(0))) $ \( M O, x' {3 * (2 * (1 * 1)) = 6 * h, X" q2 ~- k. E / f# \% U3 ?' k _! F 递归思维要点* A- }3 h, K* j" U: C5 Z
1. **信任递归**:假设子问题已经解决,专注当前层逻辑; k+ M0 ?0 f2 r
2. **栈结构**:每次调用都会创建新的栈帧(内存空间) ' j# k7 C+ B; s3. **递推过程**:不断向下分解问题(递) 1 ?0 t, Y& N" {3 s, t+ a4. **回溯过程**:组合子问题结果返回(归) v3 X4 H: Y- ?: K. E. \ M0 a
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注意事项 u. a+ N0 g" V7 r% h- c必须要有终止条件8 p& F2 m, `& F7 E# M( G/ C4 H
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)) D' c& W/ E4 B: k
某些问题用递归更直观(如树遍历),但效率可能不如迭代 8 e7 T& t2 u, ^/ k6 Z" V. b" L尾递归优化可以提升效率(但Python不支持) 8 s: j+ G0 f$ i" d + \/ Z D3 ?+ ^" K5 L 递归 vs 迭代1 K/ W" K6 y; Y0 A H* s3 y
| | 递归 | 迭代 |6 o4 X. C! S* a4 U( R8 E
|----------|-----------------------------|------------------| , V w' ^" [ z2 n7 U: W| 实现方式 | 函数自调用 | 循环结构 | ( g: ]5 Q$ n: Z1 m0 r* T" l| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | 8 L, S: \0 A4 j Y4 U% A L7 \| 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 |* a6 W+ F$ Z' ~% U3 W
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | 2 K: ^+ \ j( ]% p: d6 s. ^; B l j) j6 p
经典递归应用场景6 A" w t1 {& W
1. 文件系统遍历(目录树结构)! s& a6 v; U% ^4 e% Q
2. 快速排序/归并排序算法( c. x4 G! j4 l
3. 汉诺塔问题. ? L, ?; i8 Y( A7 G* z
4. 二叉树遍历(前序/中序/后序) 9 `- e1 ]) E! D% S& q: n) {3 ^5. 生成所有可能的组合(回溯算法) * S$ D, \ k8 c0 B 1 h: t% e) v: k! I7 H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, - S9 r5 X3 R9 ~ L1 V我推理机的核心算法应该是二叉树遍历的变种。 : [4 P- }& y% R3 y; E) f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: / |3 E* v5 X. S6 b; D4 F0 }# ?' cKey Idea of Recursion) l' q3 A4 S2 s1 W- d
$ D' k- V6 R B7 bA recursive function solves a problem by: ) A) t) y; m! V9 W) l9 T9 F1 {- Q+ C9 l3 P
Breaking the problem into smaller instances of the same problem.# I0 a2 c1 S8 @2 W% \% O3 c
1 k+ T6 A* l+ ^) y Solving the smallest instance directly (base case).% R. D3 |$ ~# z% M, y( B
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Combining the results of smaller instances to solve the larger problem. $ @7 U+ w% A) W; K# ] : F- K6 c* j9 _Components of a Recursive Function 9 |. T( Q( K0 C1 S* J! q: U3 S1 a$ F9 I( \. Q" {
Base Case: ) { n% t* s# z, s# W0 n" u# p4 v! l% M, _
This is the simplest, smallest instance of the problem that can be solved directly without further recursion. ) V% `: C: [$ `* q. t, {8 T! U# N% k2 x! I4 R* w) v. S% a
It acts as the stopping condition to prevent infinite recursion.) X: d* U% N: D0 x, `; l* N% [
. f' F/ i. W, | Example: In calculating the factorial of a number, the base case is factorial(0) = 1. ! J+ P7 \% v) y, R! `# C% W & ?. J/ J3 l% p& l" L: v Recursive Case:+ t8 ?, ?+ \8 x
0 ?+ E- R! C# H; {) P; U6 q This is where the function calls itself with a smaller or simpler version of the problem. " I) l3 V: L) P' M# ~2 M ; X1 k! s/ f& T9 Z Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% ]6 h3 z, L3 _; ?( _7 X) }
( H( f1 ^" J- YExample: Factorial Calculation- O3 H* E. w: Y" l6 M( \: D) b
9 ~2 u2 I( g, }" _7 r) d9 b- ]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:# u: S$ V% a+ d' @+ ` Y
1 V$ ?7 h. @& a8 g Base case: 0! = 1 0 e" [- ^* ?' z% j2 f" F% \ R# { `$ P: [) A4 D) R
Recursive case: n! = n * (n-1)! - h1 V8 w* Q+ }5 z' G5 V& x 7 ?8 U: X" O) h0 nHere’s how it looks in code (Python):4 ?( p# x) o9 U
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$ k) l) w y* F 6 q- Q' _) \% t. Odef factorial(n):$ O% ~ [! {) A4 W2 P5 I2 c, m3 L3 ?
# Base case7 U/ _4 G2 e3 d9 q; h
if n == 0: $ i$ w3 r1 [7 g8 i: G( @ return 1 x. y! M. `! ]( M6 `6 O
# Recursive case % z( T' u* R" {! P9 v7 y7 @6 }" t else: ; |' ~* b: v% ?% a* \ return n * factorial(n - 1)" A2 B# ?/ g+ T3 K( Y
" I X6 T' x& h! E# Example usage: E9 [. |3 t1 I4 D2 t
print(factorial(5)) # Output: 120 ( I3 A# E7 T3 Y* H ^: Y# E1 q7 l$ L3 \1 i1 a f; l1 B+ M
How Recursion Works9 @0 q7 v- w* Q# l. m1 `. Y
$ N) u6 L9 i# G- @0 j' y) Z The function keeps calling itself with smaller inputs until it reaches the base case. , I& l# J5 |3 \/ X2 Q, Y% G" ? & d* Z* q- b$ V. B Once the base case is reached, the function starts returning values back up the call stack. 7 T$ E n! p3 ^ e; R3 z4 ~4 [1 A/ c" W, H6 Y0 j
These returned values are combined to produce the final result.7 s: C4 @) P: r
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For factorial(5): + J7 a o$ l& U& { ( f: p/ Q. d5 Q6 N6 O; s6 {& m4 g - ]' r$ ?$ f" b" sfactorial(5) = 5 * factorial(4) 1 D* G* A3 i- o; {6 }: h0 ^factorial(4) = 4 * factorial(3)9 Q# s0 R% M+ V
factorial(3) = 3 * factorial(2) $ C3 O. ?- p, p5 ]9 r+ jfactorial(2) = 2 * factorial(1)- L. L- f5 C( f8 I
factorial(1) = 1 * factorial(0) w+ W9 K$ x6 J% p
factorial(0) = 1 # Base case. D/ s5 n8 U. q4 |, ^8 ]
# n( u0 o- B4 i! E' j$ ~1 }5 z6 X" Q0 JThen, the results are combined: . ~2 E8 f7 @+ @6 c) b O, A2 b+ Y% P% }1 Q, s
2 u* |5 S4 Q C/ u$ U ?* rfactorial(1) = 1 * 1 = 1- B. m4 |3 _0 R+ v9 i
factorial(2) = 2 * 1 = 29 v2 i3 u6 A" s' t) R
factorial(3) = 3 * 2 = 6 + i6 I3 Q6 }8 F2 ]factorial(4) = 4 * 6 = 241 s3 i3 G( s! o
factorial(5) = 5 * 24 = 120/ e. K4 o$ O5 k2 w: W. J
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Advantages of Recursion , V5 b7 f% _$ {( P6 G ) W6 h9 I1 E9 ]$ N- C/ c. [ 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).$ x C* L1 T) l7 y0 l, c1 |
y6 K" P3 r3 s Readability: Recursive code can be more readable and concise compared to iterative solutions. 0 {- c, F, X6 E' v' @+ H8 D5 v* W! h; X& O# A9 v2 M& R. ]5 O
Disadvantages of Recursion $ Y. f' Q( x4 U* n+ Y- A% B1 r% K8 f
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. + O* z* y( P- G t8 V+ w ( T' B. M- ~* G; r# S/ w Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). ( [2 O$ d. ?% b . Q. Q. \+ v! S; @( [2 WWhen to Use Recursion 1 g- R: ^4 ^; |9 Z F/ a" o; ?$ t
Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).# a- S5 |7 l6 F7 o( p$ G1 ^
1 n L6 }0 P* k& _/ t* \$ ] Problems with a clear base case and recursive case. 0 v2 Z/ x0 Z! |6 L9 f% O o+ I5 S/ M: |0 c' |# S+ BExample: Fibonacci Sequence4 N+ e* z. m) [. A9 h( ~
1 d7 a7 Z- [# C, {The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:! k2 P' I6 A6 D0 f' p& ^- {& i8 [
* T' V+ H* B2 r% L+ E Base case: fib(0) = 0, fib(1) = 1) A4 U* ? y) ^3 T q4 H
" H3 \: O+ ?3 O! d Recursive case: fib(n) = fib(n-1) + fib(n-2), y) s9 f) a) H3 Y6 [
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python : H4 l+ \/ R% J. p5 u ! m3 P7 B J8 i) @) z6 { ) m$ X& l' Q8 `7 v" ydef fibonacci(n): % Y/ f& b# Z" @1 ^ # Base cases 9 K" O( f4 H2 F/ \* r if n == 0:! m2 w% \+ F4 J! X$ a c
return 0 ) P% _! [8 i$ Y p1 ~& G+ l elif n == 1:* t& R/ r j5 W, \& x
return 1; ?2 O& |( S& _" B
# Recursive case) u- |6 {3 L8 M2 N
else: & c w+ g: ^ P0 U; H' G return fibonacci(n - 1) + fibonacci(n - 2) * H! A* @: z& o) G& D$ z' }4 H" d, x1 } |# v) b k5 r( s$ [8 h
# Example usage : r. u/ q0 J3 a& W$ K# k: V4 X$ Lprint(fibonacci(6)) # Output: 8 : ?1 c$ i p; Z3 O" @- A8 @ W* ^, x1 a4 o* A0 ^) D/ N, z
Tail Recursion # K+ G- b: R* i: o6 N' F; I. A" Y9 K6 V3 X* E( v! M1 _' g
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). 2 E* B6 C6 t6 w' j; `/ O + s. |5 q3 `: Y4 n' j" t9 N# hIn 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。