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 / V+ k1 y2 V. |! W/ `# l & K, Z. x# i0 O; i. a9 ~% H解释的不错+ @8 g' ?' @( |' ]6 m7 y
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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。 : r5 P$ k6 R" _- k' d& o+ i7 ?6 g5 R
关键要素 4 d+ E9 t; b o0 J# h1. **基线条件(Base Case)**$ S- R0 W2 n7 p- h/ k
- 递归终止的条件,防止无限循环3 F. b2 b6 s* a2 T( @- I
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 10 `- u5 ~5 v7 c0 i) S$ h4 ]; w
$ i9 _, K v% c- w) z3 a# U# W2. **递归条件(Recursive Case)**5 I( g# c4 l; V2 D# _7 U' y
- 将原问题分解为更小的子问题 4 Z9 P' ~# W5 r# O1 q9 m( t - 例如:n! = n × (n-1)! # q; e9 s+ Q, W# r* \/ }' U; l9 k* u, z" f
经典示例:计算阶乘0 A# }4 g! a5 ^. K: _0 y
python! ^( C; t* p9 B4 A
def factorial(n):' F5 I8 Q8 F3 O) _0 j
if n == 0: # 基线条件* ~: U7 X* m, Y+ z5 j
return 1 ; ]2 s) B; v9 p1 l" M1 Q" T2 n else: # 递归条件 . p6 C- z. W/ m2 y3 c return n * factorial(n-1)2 o! M0 V1 x8 q
执行过程(以计算 3! 为例): # d5 z! a0 I" e" [& \. q( p" mfactorial(3): ]9 J: B) w! w) u* g: j! D
3 * factorial(2)% H% w9 d$ W6 b! M
3 * (2 * factorial(1))$ i4 X* F0 I7 D# b# [- Z/ i9 W
3 * (2 * (1 * factorial(0)))7 R! E5 W* B2 p4 a+ e! a" P8 V0 f
3 * (2 * (1 * 1)) = 6 6 d( s( V& }6 Q/ p0 z. i9 K ! l& p1 S" F) _7 `' ^7 a 递归思维要点 9 r3 m) Z2 w2 Y; [1. **信任递归**:假设子问题已经解决,专注当前层逻辑! ~' u7 U; q ]4 g0 y, X/ v
2. **栈结构**:每次调用都会创建新的栈帧(内存空间) 3 B/ o! K: m0 c3 P( {* u2 ]3. **递推过程**:不断向下分解问题(递) # c* J0 i3 j. b. b4. **回溯过程**:组合子问题结果返回(归)1 F+ }3 g- v6 X% C5 s
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注意事项 * y" U d6 P) N( [2 X0 A6 A必须要有终止条件* d' }; I6 M" Z+ R
递归深度过大可能导致栈溢出(Python默认递归深度约1000层) ! [4 B8 i% ~ s# M" L0 B某些问题用递归更直观(如树遍历),但效率可能不如迭代& p2 ^5 P8 j0 d5 m8 v: t
尾递归优化可以提升效率(但Python不支持) 2 Q. V% g4 R$ w, X9 B7 S1 F" t3 F 9 j- i" t4 v% x: U 递归 vs 迭代& @& k" }: C/ S2 Y3 k, Z3 j
| | 递归 | 迭代 |$ N+ Z. J9 J; K' Q$ K" T: Q
|----------|-----------------------------|------------------|1 H! Y1 ` i9 j: I- i/ i
| 实现方式 | 函数自调用 | 循环结构 |% O- p* X: h7 e3 @( D+ R# y
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |$ h/ T- v3 ]' E
| 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 |6 g/ {/ k/ ^* ]9 L. o
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | 3 {. Y: E" r: j/ _0 X6 T/ m * H7 x, k3 Y/ U* [3 g5 _ 经典递归应用场景0 p! U* ^) ^; t+ z
1. 文件系统遍历(目录树结构)' I ]/ ]! m- \6 Y
2. 快速排序/归并排序算法$ _" Y0 F9 \: z& _
3. 汉诺塔问题2 ]. @! r/ c3 I3 ?' B& [
4. 二叉树遍历(前序/中序/后序)! t' U+ X, N+ p) M. W K5 Q
5. 生成所有可能的组合(回溯算法) 8 F- h2 M+ L2 I/ i+ W/ w5 p; i; l' G1 K: B2 y* Z- Y
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, ' D$ U# x8 J* q" R y我推理机的核心算法应该是二叉树遍历的变种。, E! ~) v i% B
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:. a: `) t7 l0 m' X/ I* z0 R
Key Idea of Recursion `7 W% Y" @) I+ {
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A recursive function solves a problem by: " M# \! l: Z$ x" q* [5 H( k/ x2 L! x0 }: s' z5 V2 Q0 `- X
Breaking the problem into smaller instances of the same problem. 0 ]- E6 W. v* S. a, |* e$ B/ I. R# ]1 C* C: R$ T
Solving the smallest instance directly (base case).) L+ k( D- U: B1 i) n1 U
& |2 H! C6 W9 t Combining the results of smaller instances to solve the larger problem. 8 W& U2 R4 b; y9 ]0 R9 e( s: M0 w1 t) ~" A2 q6 i( D
Components of a Recursive Function 6 Y3 a; ^& Y0 T: g$ m9 r" W) f _7 m9 E: x4 ?- h3 E/ m2 |+ U; I Base Case: ) l+ y1 A/ W3 L+ u. Q: { T8 o6 B2 Q0 n9 e) w" z8 Z/ T% y( `+ N* T
This is the simplest, smallest instance of the problem that can be solved directly without further recursion./ @8 J- l' `, K
1 ^7 I% L4 [1 N% a$ i6 k! P It acts as the stopping condition to prevent infinite recursion. 4 s$ {+ G% k/ j9 _. A. e( U% K$ @ # z* N$ ^$ i; B* X& M Example: In calculating the factorial of a number, the base case is factorial(0) = 1. $ l/ k6 t& Z" a4 F3 B5 V - q! G1 f6 k2 Y5 Q$ e# D Recursive Case: ( } e. d$ x2 D3 N: v+ P" C- D- c( a% `% [* ~2 l3 D8 p
This is where the function calls itself with a smaller or simpler version of the problem." K4 U- B3 P9 u/ K( ~, q' M
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Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). / f: S( u6 S) Z# r- F$ n3 ~, a) O# T0 C0 l; p, Z
Example: Factorial Calculation7 I1 w4 A6 q3 m. U1 ?2 R1 P
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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: * ^ J+ m; A# Q6 O3 c* [, B : G; ~2 d* C: z/ [# o7 Q9 H; B Base case: 0! = 1 6 X1 e; f' v9 e- i% F: s% k% F0 s' ]2 S4 o, ]6 w, G5 A' [
Recursive case: n! = n * (n-1)! 5 p t& `; y; p2 f" s1 B+ N4 V8 K6 ~ - S8 ]. ~; g* F4 x: ]$ ZHere’s how it looks in code (Python): 2 A: d# _" \" K1 ~python! s: x; h9 ?5 l4 c0 I' G
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def factorial(n): 1 |; G% F Y! G( E- v # Base case c. B% J4 l4 v( ]" t6 W
if n == 0: - g0 e0 L4 O! T& D return 1 & k @1 F% m: C* E # Recursive case2 U* t, k. a( {" W, v
else:+ M. P$ Y# `/ @2 \% L& @& e
return n * factorial(n - 1), }- n! w( P$ {: Q, A$ d
# P6 ?/ P1 y. `& R+ F5 F. B9 i# Example usage 7 u+ W# S* C1 b% i* W! d: K3 |5 A7 [! V% zprint(factorial(5)) # Output: 120 ' l" h) e0 Y3 W6 _# | 8 X: b( i) g4 `, J$ cHow Recursion Works 5 u, U* U# J' Q' o. \* y' _3 G2 j( ?7 @& y* D
The function keeps calling itself with smaller inputs until it reaches the base case. 5 b# t3 S% Q# g( \+ L$ ]$ y ' z {, B! B5 |% _4 }( [6 A Once the base case is reached, the function starts returning values back up the call stack. ! ?2 A' [; B8 e2 D+ `# B' h0 \' E% a. s3 A2 Z$ K# g
These returned values are combined to produce the final result. : U. M" J" y6 e8 i5 S ^: Y/ w r. x$ O4 e
For factorial(5):* I$ |# k \; V3 z6 U5 \, J4 c
( \2 K$ o8 x8 e! W* I " S! Q* x& J+ l! h7 k \factorial(5) = 5 * factorial(4) 9 |# W* T, {6 jfactorial(4) = 4 * factorial(3)' G+ u5 H# z; d
factorial(3) = 3 * factorial(2)9 B. r$ i; d1 `/ x( ]* y0 u
factorial(2) = 2 * factorial(1)$ Z! ]; v7 s9 B
factorial(1) = 1 * factorial(0)7 Z0 C7 [; _) e' C: U+ m. a
factorial(0) = 1 # Base case& w0 x9 U& Y6 n% h& c' z
5 f$ `8 Z# N7 @1 EThen, the results are combined:/ h: Y. \; L# q+ u
$ L |% K. ], r: d$ t x, j 0 o, b: X: X+ Bfactorial(1) = 1 * 1 = 12 x. g0 m: D7 B4 ^$ z; c% N
factorial(2) = 2 * 1 = 2 7 E# r6 R; i$ j Wfactorial(3) = 3 * 2 = 6" X. k1 u0 {( N& e+ A% ^
factorial(4) = 4 * 6 = 24/ {9 W, g& A7 g0 l; t7 p
factorial(5) = 5 * 24 = 1209 ]' p6 X6 t& Y) _
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Advantages of Recursion 2 q% j' Z5 s$ R8 n: v# M8 i. Z+ w- d- {# m. 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 W; b8 U( ]9 K2 h4 {8 N
' i/ ]( w4 k L6 G# m Readability: Recursive code can be more readable and concise compared to iterative solutions.8 b' @' c1 `4 {, ?) p3 ]) I7 @
) N$ o. G2 U4 A$ g2 C5 ~& i& k$ nDisadvantages of Recursion2 I! Z8 z! I1 @! }' }
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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. 5 c) s% Q4 K8 J6 {, H, l2 a " p$ P" t5 z1 m2 o; _ Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 T* ]# m) g' L! [$ e
- [# N( n N" c# M7 q) f! k6 GWhen to Use Recursion 7 m- h# Z0 N# v+ t+ A$ N% Z* G9 d" o! U2 S3 c
Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort)./ O7 r6 h* k( o( l; Y! c
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Problems with a clear base case and recursive case. : S" q& W4 C) v T+ W+ t% A( T! f 5 @$ U! f6 _' G2 d( T) o5 XExample: Fibonacci Sequence: A$ h6 ]; `: `9 f D
* J& f* G) V5 }/ MThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: & _" V& b1 ]- A! S! O 4 _/ }0 J2 U f' n" h$ [ Base case: fib(0) = 0, fib(1) = 1. G/ k; H6 ?1 x% \ u! k; B
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Recursive case: fib(n) = fib(n-1) + fib(n-2): C+ ]" ~$ _8 u) q. u/ [
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def fibonacci(n): 2 M: n7 I' o% z. c6 b6 z # Base cases' w2 q% m% B! g+ S9 G
if n == 0:) z3 |. q8 Q9 S: s& r3 l
return 0 7 j" _" ]% @7 h elif n == 1:* _( L- R2 D! {3 {0 H3 z3 _" ?# d$ O" f
return 1 2 p- u d" F p0 _ # Recursive case2 |4 I6 c" l) Z
else:. K% C9 H C4 @. j' A" s- ?$ B3 E
return fibonacci(n - 1) + fibonacci(n - 2)& r0 f# d' L8 U- _9 d
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# Example usage & ], q% S" ^) g6 B$ a, F Kprint(fibonacci(6)) # Output: 84 [. r8 b/ a: B" z$ ?4 m
4 T- f0 i4 @) C; M: e4 c" b8 o! pTail Recursion7 l: k' H# @, R# G7 x& G0 K: U
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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).0 o# u) i0 _1 m2 M5 n
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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。