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 4 m) ~4 t2 J8 s& |: D4 l" |. E4 `5 P! Q0 i" l3 i( G$ _
解释的不错' F, f! K6 Q, Q# x' P: }
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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。 . @" w5 m# D* o2 F) Q; {& s+ r1 x+ @# @" v R/ X I! e. p
关键要素6 H8 ?- B2 p- F
1. **基线条件(Base Case)** % C7 V! z7 g+ Y) a V - 递归终止的条件,防止无限循环' l/ t3 q" m5 q; h |( W/ I/ F
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 16 U. W+ L/ o" r
& a0 K; x) X! A3 C2. **递归条件(Recursive Case)** 4 k: _7 u' l; \, k9 K - 将原问题分解为更小的子问题 6 Q; H( `% ]1 [ - 例如:n! = n × (n-1)! 7 P8 @+ ]8 A( S; A; d$ ~9 p& L. Z8 M$ f4 a( {( n& s
经典示例:计算阶乘 9 W2 V) N8 h% C8 z3 Tpython0 H8 Y; |5 I$ q: y- J8 y! H
def factorial(n):3 F: @% Q3 A! V0 b) D; K
if n == 0: # 基线条件 ) \5 N; \6 {4 N6 Z) H* V return 1 5 {$ g5 J: U4 ?; p3 ^* Y* F else: # 递归条件 . K$ N: v" s, S: t: o a3 H return n * factorial(n-1)' b6 V' S" e9 Q
执行过程(以计算 3! 为例): / m9 `, q6 _4 D' C; B7 y' Pfactorial(3)' `1 C s ^' d
3 * factorial(2) ) ~, X& Z2 a& `3 * (2 * factorial(1)) % P `9 ~6 w, p; b! a, e7 F3 * (2 * (1 * factorial(0)))% ^& e, a% P. Q! l* H' b
3 * (2 * (1 * 1)) = 6 - z1 F! z" T! [( a ; {; e; e, W" B9 i/ a t8 I 递归思维要点 % l4 I" X) X' j- N% g: I8 ^# [1. **信任递归**:假设子问题已经解决,专注当前层逻辑/ i h* j6 u7 p% s+ }$ J Z
2. **栈结构**:每次调用都会创建新的栈帧(内存空间), \ @' u* }$ b' `* L3 x
3. **递推过程**:不断向下分解问题(递) 1 b4 i" k6 I8 M1 q9 R/ Y9 A4. **回溯过程**:组合子问题结果返回(归) # Z9 P1 |& O; Y3 Q) t7 w8 Q2 V1 M! u# W7 k% q2 P
注意事项1 V$ x% H( i3 v3 { }4 w
必须要有终止条件* o% i' g8 e. B
递归深度过大可能导致栈溢出(Python默认递归深度约1000层) - X/ ^# x" q I$ s' i* x某些问题用递归更直观(如树遍历),但效率可能不如迭代 & ^; _, |0 r) z- X' r. A尾递归优化可以提升效率(但Python不支持) ; S& h; x+ V" ~9 _6 }* }6 W# I0 I 8 t+ u0 i k: I- w( Q8 K, o0 U+ S, U3 x0 ~ 递归 vs 迭代' u& y2 h Y h/ i# V
| | 递归 | 迭代 |+ G+ v. d* C5 l3 e6 _, u' _
|----------|-----------------------------|------------------| 3 n& E# K; n: N; {: R* M& K| 实现方式 | 函数自调用 | 循环结构 | 7 j; t2 L$ h9 r| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |4 U1 k9 x5 ?( a. d, M3 L
| 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 |- Y- L6 b* A2 x. K
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |* Z, g! O' B, t0 f
& `0 A1 S' `9 ?9 o 经典递归应用场景 / p4 C, K0 q, g7 _( P; B' |1. 文件系统遍历(目录树结构) ' o! O8 m$ z, e% C+ k7 k J! l# M0 H2. 快速排序/归并排序算法 , V% q. e* h) E/ ?3. 汉诺塔问题 " m+ r! ]( R/ h$ s4. 二叉树遍历(前序/中序/后序) 1 m" w( o. C1 O H5. 生成所有可能的组合(回溯算法)" [+ M/ f/ d6 W$ p \0 M0 N
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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, 7 j" F5 p& p5 @. J& L( s我推理机的核心算法应该是二叉树遍历的变种。+ k2 u! n! V7 r+ F0 U
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: ) B2 m" f2 o6 y! VKey Idea of Recursion% h* l/ f# y0 A. `8 z% }' ~
+ f" |8 \' @5 ~2 UA recursive function solves a problem by: 0 D" L" f( C2 a x8 a0 @% B( R* g# J
Breaking the problem into smaller instances of the same problem. 2 k4 K7 n+ f, E: n( D, [: B: B: w% \4 p2 X) R
Solving the smallest instance directly (base case)., B! W/ h/ ^9 H5 _' }% J" N
" y r+ l2 S& m0 X; f Combining the results of smaller instances to solve the larger problem.; W6 \6 y3 s9 X' E* d$ p
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Components of a Recursive Function # q8 L3 w. D; g* C6 M) N; _! @/ l9 p
Base Case:- ~4 O! g4 V: D1 w2 E' I0 K$ a/ T7 R
' p" o7 q- x, X' c8 o& }' F This is the simplest, smallest instance of the problem that can be solved directly without further recursion. ) I. X F1 x4 n+ T$ z% o3 d; }) F
It acts as the stopping condition to prevent infinite recursion. 1 `7 M# j9 U, V0 \7 |- G% n ) v& L8 c6 |7 K6 C/ ] Example: In calculating the factorial of a number, the base case is factorial(0) = 1. ^7 V! l+ h7 V
) L* L0 @! D; H Recursive Case: 1 e6 \7 j3 z+ l! O) b8 Z" L& [7 J8 E: [+ n
This is where the function calls itself with a smaller or simpler version of the problem. 4 w( h- D+ q' }6 E# P+ w. B s, J2 G0 S. ]- e( a5 x4 G5 M Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 e ~1 s! B! b% _8 p$ N: `
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Example: Factorial Calculation 1 Z8 B' ?# m2 s) h 8 v1 m& J6 P- l- _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:) o8 Y( k( g( l. V# O7 Y: v
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Base case: 0! = 1 * a6 ?2 k* E, F/ U) T 2 ~5 ~4 V4 d" a4 }% F Recursive case: n! = n * (n-1)! ' [: o$ @# H( Y! Y/ h4 D }- `
Here’s how it looks in code (Python):' @# }) ~7 Y% a- W0 A
python 4 ?; m& l! _. ?0 D. X' v7 @% n& L" k$ w r# K
0 I8 T! \7 _5 q, x$ T! Ldef factorial(n): ! i0 L& |- ]3 c7 a8 h # Base case, r& j' Q g" x/ V% K5 y8 k2 O
if n == 0: , S; Z$ @3 z0 V* L1 k' |6 h" R' P return 1" q+ g( e$ Y2 S6 }8 _! A6 l; ?
# Recursive case & l5 S& B/ N7 U4 u else: : J+ g7 B8 |& Y! |8 F; _ return n * factorial(n - 1) : g+ Y" N$ z; i. z; d) N' p6 F: F: K% q% E Y0 Y
# Example usage / q$ z( ?; h% ^+ dprint(factorial(5)) # Output: 120 v% ]2 k7 y& f9 {2 L3 t: B
& H! e9 g, M: \' K; SHow Recursion Works2 ]. F8 @. A+ E
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The function keeps calling itself with smaller inputs until it reaches the base case.9 r8 ?. V) W9 G e. h9 {2 k
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Once the base case is reached, the function starts returning values back up the call stack.8 P; c& Y: f1 {$ e- e4 H5 B4 d4 U
9 W3 ~3 d8 D- |) f These returned values are combined to produce the final result. ' S8 K* e! I. `+ Y1 m ' v7 l3 l' b6 YFor factorial(5):# q; @$ A" H- n3 L
% o" ]5 ^# [/ A& H' uThen, the results are combined: @8 B( k6 \& J# M' w- k3 k6 F
( N8 ^2 z+ a5 ~4 e D/ I. q9 g. f' E5 z& b: Xfactorial(1) = 1 * 1 = 1/ q$ x- n2 h# g3 F
factorial(2) = 2 * 1 = 2 7 V4 @# {' r- E" z6 Tfactorial(3) = 3 * 2 = 6 / [; Y$ m$ x! Q/ D4 P) z/ Y3 C) Dfactorial(4) = 4 * 6 = 24% s |" N. s0 x, a6 _( n
factorial(5) = 5 * 24 = 1208 s; Z0 [- |8 A' S
1 Y) g( |) r( k3 \Advantages of Recursion 3 Z$ ^1 t6 `) J7 I # s3 }3 ~9 h( v' [1 `1 @( H: M% g. \ 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). / h* r( ^& ]7 H5 L/ N+ K0 u! Y1 b # b% ^# G/ Q. k Readability: Recursive code can be more readable and concise compared to iterative solutions.! J) F9 F0 K. _4 o! q
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Disadvantages of Recursion 3 M/ }8 z5 }1 a$ ?* M9 h" g( p8 m8 h3 s& O3 x7 I* V
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. 3 ]8 S4 W! {* d# y7 U7 Q+ A' W8 Y9 S# U1 y$ W
Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).' _! {+ @; U, P# l2 ^: Y9 H8 V/ \ n! L
. c5 b. Q1 }8 |When to Use Recursion% h" [- g6 F6 f k
$ y$ Y# g8 c. v E6 _# d Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 g& j) s* Z+ i. X% _3 {
" E- i8 v. C8 g2 V7 q3 R Problems with a clear base case and recursive case.5 Z. g( R6 D& ^
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Example: Fibonacci Sequence 8 L* M8 t/ O/ {% i8 t, o8 t$ n! W$ Z- z n, ]& }$ j' o
The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: $ C3 ?4 V; u1 U4 r/ G# b0 F$ Y( C , j3 s; l3 N r u& e7 \ Base case: fib(0) = 0, fib(1) = 1 0 {0 {/ F M4 ?* u8 D e: d! `* x, M
Recursive case: fib(n) = fib(n-1) + fib(n-2). b7 ]1 S3 ]6 ]& ?7 b+ w" a
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python ) o% l; Z( A( @3 Z" e0 N 0 W5 _ g7 C8 N6 H; X* O 8 ^* D9 B# N5 K- r! a; vdef fibonacci(n):: S' ?3 |5 T( c6 `2 B! |; p
# Base cases j$ l( C3 ]" ?1 b) d2 K; k. c2 | if n == 0:" J! Z# G. x8 q+ h
return 0 6 O" F; R4 T3 {$ O elif n == 1: 4 g( H/ j' l6 ] return 1 $ H1 M; e- R9 T0 ` # Recursive case R9 x% f! I" b4 K c9 n* ` else: d# g* \7 k B6 c8 z return fibonacci(n - 1) + fibonacci(n - 2) ; F: S. W) T |& ^0 A4 E& I* ^5 G- ~" }% g, F$ ?/ U
# Example usage 4 i( s2 x6 g; C) |4 U, O$ Lprint(fibonacci(6)) # Output: 8 : y, x S i# K | " ]7 w7 X6 o. C$ W" P3 @Tail Recursion 8 T9 L/ ^2 [7 W+ G1 S8 v( E- j& L' J9 ^4 M% ^' l3 }7 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). 3 ^6 v D- ?: X* R , _# s o6 P0 R; ~+ N8 NIn 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。