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 " G* Z( _' H6 Q* K; J r
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解释的不错 8 v$ f/ m1 J e, V( I : `9 |. e2 ?7 o* u. b9 X- Z. O R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。 1 V+ A# q5 l2 C& e& i& `: l; W# z# J' N% q: @5 n$ p
关键要素) ]7 O6 V3 \0 T4 a3 z4 `$ i
1. **基线条件(Base Case)**5 m" y0 a. G+ D4 t! p7 j
- 递归终止的条件,防止无限循环0 m" s' X3 S6 p3 C! b
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 1$ z) j& I1 [5 P3 @0 \1 k" G+ Y. }
5 k+ a% b6 _4 G! p2 K- U% m2. **递归条件(Recursive Case)** 0 g+ L3 ?" a0 `* x! ]8 w - 将原问题分解为更小的子问题 : ~" t% R5 e+ l8 f; Q; G - 例如:n! = n × (n-1)! ( ?; o/ S9 [) n 5 `( k) L) \6 w( g0 _& s# L, v 经典示例:计算阶乘2 _0 _ i* p; T M p
python + _3 s1 W& a. @: S% e7 Sdef factorial(n): 1 ?, u1 N# [$ |. G$ Z2 h$ u5 o if n == 0: # 基线条件7 Q8 v$ {# g4 S* P) y/ _
return 1/ K1 ^( h! p5 G& w: \* n
else: # 递归条件 1 b2 |- Y7 H+ p- l return n * factorial(n-1); D6 d6 R0 o4 q# u9 L6 Z5 ?
执行过程(以计算 3! 为例): 2 F9 @& W1 O# }factorial(3), p1 W1 K' ~: `
3 * factorial(2) ; ?% x% m9 @9 L2 M! a7 F3 * (2 * factorial(1)) " V3 X/ o1 i" m2 L! X. F& [3 * (2 * (1 * factorial(0))) g. f2 T7 O' [7 r4 u# Q4 G& X; p8 X3 * (2 * (1 * 1)) = 6" L H. R; r" T3 h0 j+ C, S
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递归思维要点 % F5 @! L N. G; o1. **信任递归**:假设子问题已经解决,专注当前层逻辑 7 {$ J$ Z O4 a$ t) A2. **栈结构**:每次调用都会创建新的栈帧(内存空间)8 p' T& ]4 s( t
3. **递推过程**:不断向下分解问题(递)2 w& ]7 v; H8 ?9 G- g
4. **回溯过程**:组合子问题结果返回(归)1 {* r$ z" M1 q! }" s
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注意事项 7 X I$ D# m4 u z" p必须要有终止条件* Z( c, Y* O6 b
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)1 w' X* c. T) E* I
某些问题用递归更直观(如树遍历),但效率可能不如迭代 8 I4 e* B( A$ b9 K/ [+ P4 V3 V7 n尾递归优化可以提升效率(但Python不支持): N) b$ i% ~* Q7 [2 e
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递归 vs 迭代 5 c3 n9 }2 p+ G3 {8 h) k3 [- z| | 递归 | 迭代 |. L+ A5 _7 R# M# f! X' {
|----------|-----------------------------|------------------|6 Z- ?3 U5 \. c' S, F& ]
| 实现方式 | 函数自调用 | 循环结构 |; j$ Y9 w: {$ Q2 [4 T& Z/ n
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |0 Z8 J9 L! z: F
| 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 | * t2 T- H5 p5 c2 o) ]1 \3 i& s| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |$ ?* E" s/ e5 v5 c+ @% Q
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经典递归应用场景 . j% [6 ?" ]0 L* t0 m1. 文件系统遍历(目录树结构) $ ]/ M3 H$ L7 B6 g& O+ \2. 快速排序/归并排序算法 1 Z/ B4 F) @% k% v3. 汉诺塔问题 w7 J$ I% N. P+ x
4. 二叉树遍历(前序/中序/后序) ( X& o8 K1 z z! p5. 生成所有可能的组合(回溯算法) 5 c" k! J" p- X8 m. \5 h: q, ]1 B( f M7 P$ H d* z" h, A
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,; |; I5 d0 `/ e' h R) B! B3 C! C
我推理机的核心算法应该是二叉树遍历的变种。 . e4 ^. B0 p# E2 W* D w! [. b% \3 }3 n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: & s4 r# N% _! P- l2 Y( D" I" X6 c% pKey Idea of Recursion 2 w. n% x& i# E; l) {# G& ` 8 R1 D. s5 A7 r6 n! r; k' nA recursive function solves a problem by: : Y+ X$ [. O, }' Z# j: k4 H: w$ Q
Breaking the problem into smaller instances of the same problem. 8 ~1 h$ |9 y5 M0 @# `% v! E$ b + Z7 ?7 {- Y6 E/ [$ W. x5 K4 ^ Solving the smallest instance directly (base case).1 \" n7 g2 {' A# r' x/ P8 W
9 @0 W& s% Q6 U" T6 h- q Combining the results of smaller instances to solve the larger problem.) \; y; @( E% V: s, {
* K5 T7 E; }- A$ n/ R- i GComponents of a Recursive Function ) N$ V0 M1 p9 w, u / z" N& E- B# _4 w7 J Base Case: ! t# o' y' T z2 M6 g6 y % k/ ~) j; X3 h Z6 b9 A This is the simplest, smallest instance of the problem that can be solved directly without further recursion. : ` u+ p# @9 Q [; d* m( l0 [6 m& H6 Z4 i
It acts as the stopping condition to prevent infinite recursion.1 l3 b; i1 \, E" C: R
( G7 W1 O# K7 J& ^$ w6 k1 \ Example: In calculating the factorial of a number, the base case is factorial(0) = 1. , I$ b, z+ L2 g6 z1 G/ e% Q # M: u& P: U: `4 H" T0 L3 { Recursive Case:& l V- p% e6 T H, K. @. o
: q) y9 L+ F# L0 F$ {& i( N+ m This is where the function calls itself with a smaller or simpler version of the problem. 9 W. S* K' h9 j$ ?& H5 i! u' s 9 Y7 G) B- k, z6 ?4 h: V) b Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). ; C5 S% p6 n# Z$ }4 Z- l! F$ n( G+ p
Example: Factorial Calculation ' \& X* D# {8 P- \ 4 w# Q( y! z6 b* Y6 t, YThe 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: 7 s9 H! Q) X) }0 {, R . E, J) a7 o* E Base case: 0! = 1 , f2 O: D3 Q! I$ O" @ . {1 n! o/ p3 d1 g' w w Recursive case: n! = n * (n-1)!6 L1 w$ A$ u! s; \
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Here’s how it looks in code (Python):% k4 B9 L# x/ `+ ], }7 b
python ( m- u @- L- R g. r; A4 q( O# t: @' J
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def factorial(n):, l& }) Q h3 G7 N+ x( J- r2 |' ] y5 t
# Base case + r ]4 Q7 q6 ^7 ] u, F0 v" y if n == 0:! r* E( \: ~" D5 y- b& E+ U
return 17 Z% C/ p5 b4 a4 O5 Q* y
# Recursive case 5 x" ]! Z5 V% a+ x6 P else:8 W9 H; p5 |$ n2 { w3 ~: e/ d& r7 p
return n * factorial(n - 1) 2 { s5 D% R2 y* }8 f9 ~5 G# G& C 1 R2 t) l* H! \: |6 Q# Example usage: K. P: O) b) d4 r& d! l7 i$ ~* ~
print(factorial(5)) # Output: 120 3 g) g2 w( o: j6 Z/ I. O6 B' U5 F+ C* r
How Recursion Works " p$ e; Q S( M+ u( @ L" I0 t" c5 I: b
The function keeps calling itself with smaller inputs until it reaches the base case. + J3 F6 S% ~+ f* w' x/ I% D- f1 l* ?; Y3 U# R6 O8 _, X) J" c5 Z
Once the base case is reached, the function starts returning values back up the call stack. ; N) W7 U: K3 o) L: D. L 8 ^. C! B" ?1 u/ X/ S0 w# d9 m These returned values are combined to produce the final result. & I4 V2 @6 ]' m6 y9 O6 [. v0 y: y8 W
For factorial(5):% o; x9 K0 l( u& P
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factorial(5) = 5 * factorial(4); P) _4 P3 F/ y1 e
factorial(4) = 4 * factorial(3); {! R2 Y$ d0 d, ?
factorial(3) = 3 * factorial(2)( n _, p( I0 e& K
factorial(2) = 2 * factorial(1) ( l2 z/ Q, G, t g% F# ufactorial(1) = 1 * factorial(0)5 y! Q+ w$ S O, A6 j! r
factorial(0) = 1 # Base case 6 u, }" s% c) }/ F% U! }3 {, W+ k" N j
Then, the results are combined: , S- c. p( d+ p8 @* J4 O4 p7 g, P8 D! H5 m
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factorial(1) = 1 * 1 = 1 8 Y3 A6 I6 w9 z# I. D4 \* }- rfactorial(2) = 2 * 1 = 20 E: z4 M1 ]! U- h! p( j" Q( u8 }
factorial(3) = 3 * 2 = 6 , U5 A1 K- l. x9 P( u6 b2 L: `factorial(4) = 4 * 6 = 24 8 i9 V0 f- z+ `( s: {factorial(5) = 5 * 24 = 120 " G& D* i" Z8 M9 `' J o8 T/ D% Q# I
Advantages of Recursion% U6 P, N1 s3 X* _2 _
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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).3 ]0 ~' M0 B) z- g5 N
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Readability: Recursive code can be more readable and concise compared to iterative solutions.5 m( o7 S" N$ \4 t# a8 u
4 {$ Z$ S* h3 ODisadvantages of Recursion; l& _' b5 D4 e) ~7 O( o) e+ R
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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. % |$ d. @& x5 F% d ; `+ T) W- F8 w0 ` Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).& j: E* q5 w! ~! P
, A7 I5 o$ C2 r: l1 W& wWhen to Use Recursion$ |( `1 Q6 i- M- d
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Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 ~; N# Z% [4 K2 `( B3 n9 \ U5 z
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Problems with a clear base case and recursive case. ! C8 \1 u l+ e v 1 n4 \/ ], @6 ^4 C7 N$ `) I; n1 E- GExample: Fibonacci Sequence4 v3 O1 Z6 z$ T: v+ x( T3 V/ m
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: 4 h" ~' N6 Y$ S% V1 \3 z , k9 Y" b k2 e8 Z! G; Z8 P; L Base case: fib(0) = 0, fib(1) = 1, q5 u9 r" |) N- }( Y7 y# G
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Recursive case: fib(n) = fib(n-1) + fib(n-2)& o2 q: f. s4 b% Y7 t& M% @
0 ]% z' b# w f+ c6 vpython ) O5 C8 t% F$ a) @ ( e; B4 a! j) v& Y+ \7 H7 X5 g* J0 N- y4 g- G& S
def fibonacci(n):4 z: c- L+ x0 ~4 }5 |( U7 x
# Base cases + i6 O0 l) P# i% Q8 y3 l if n == 0:6 V9 R! |- J( G" i* N7 Z w' {+ ]0 G
return 0 # q1 l8 j9 G1 W( \. |, z; g elif n == 1: ( o* R7 G% S! C: f, P) l6 J3 v& f return 1+ g. H' n2 x9 q
# Recursive case8 U0 r* {4 S3 u) o
else: . n. W! @) a3 ?( X) X return fibonacci(n - 1) + fibonacci(n - 2)& n# g1 J! K, i d( M
! g$ f7 K1 b! C8 i; e4 m. y/ E5 Z# Example usage- R6 F) @2 x2 @# i) n# I e" G c+ Y( a
print(fibonacci(6)) # Output: 8 # t5 e; {( N8 i2 Q8 A$ f 1 W" n- A! |: L! n" T# h" MTail Recursion- x' j8 ^( Z. [2 ]1 Z* y* z$ z
" I2 _% {4 M: y# o0 s$ nTail 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).$ Z* l6 U% Q. h4 z' S% G, ]
5 z: ?6 K% u" |2 xIn 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。