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 6 D- m( _" E! D 7 }* c. |5 `) H5 U4 i1 D$ M6 W解释的不错 - j1 A% `9 ^, G A. d8 ^; _. N+ \2 ]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。 * D* f' P1 b1 A9 x" j . K1 n0 ]+ G0 u- |) Y 关键要素 4 ]8 v# @! s0 \5 m( O' B {) Y1. **基线条件(Base Case)**! B& {0 l$ q d3 ~& T! C
- 递归终止的条件,防止无限循环* s/ u. G: s- |5 H+ ], b8 ?, L
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 1 7 ?* _# x* f" ]; f, K, h9 [ 0 G* ?/ p; v! a/ _; ~# i2. **递归条件(Recursive Case)** - m& e; Y! W/ ~+ u) e% N - 将原问题分解为更小的子问题 $ k% E" M, }* _2 b. t- z7 H. O - 例如:n! = n × (n-1)!4 S# t: ]4 w! @% G' A" g
* x! A' y0 \* Y. f' Y z. V 经典示例:计算阶乘0 Y# X2 }, t9 l) i
python 1 l! C* ^) U/ f+ c7 A& {2 h: ddef factorial(n): \" X N c7 E8 b" ?2 F if n == 0: # 基线条件 V9 ~8 q% a+ R9 \
return 1 ?/ C3 d+ g' V7 a5 p& P% H. m
else: # 递归条件 ' ?2 ^. ~, v7 f' e9 z" X return n * factorial(n-1) 3 ^9 F5 |( R5 x% b2 f* V( K执行过程(以计算 3! 为例):7 b3 I5 E2 o: S' i9 H
factorial(3) + T9 r" ^. M. k7 L3 * factorial(2) ' E4 h l0 u- i" [/ U+ S: w/ A% G/ P2 C3 * (2 * factorial(1)) ' W0 O( G: H% G( v9 w1 k3 * (2 * (1 * factorial(0))) . o2 t1 ~; u' H1 ]( p, D3 * (2 * (1 * 1)) = 6! I8 G. V2 i" K! X9 ^
. M, e6 \6 z9 N7 m' ~! {9 `+ X
递归思维要点 & ]1 v8 M( e8 X7 i1. **信任递归**:假设子问题已经解决,专注当前层逻辑 2 |* \3 w% O9 w5 l, `6 y5 q2. **栈结构**:每次调用都会创建新的栈帧(内存空间)# q: n; t4 r# x6 Z( o
3. **递推过程**:不断向下分解问题(递) 0 ?; ^! D0 |9 p! y$ |7 [4. **回溯过程**:组合子问题结果返回(归)1 y7 }% v/ }5 q1 O
* \/ x: N$ ^; b9 v9 i) v+ i 注意事项. |& w& u! e7 Y& M8 T
必须要有终止条件1 M5 p$ `+ D. W
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)" Q3 c( }4 O3 T2 F3 O
某些问题用递归更直观(如树遍历),但效率可能不如迭代 3 E* w: F# {: a$ Z9 k4 L尾递归优化可以提升效率(但Python不支持); ]2 ^3 K3 m. j; w+ z
" X' p# _& _1 M' `1 A. }3 R4 i
递归 vs 迭代 ' _% _) a! O' M/ d' c E| | 递归 | 迭代 |# | U+ y$ ]6 B3 I
|----------|-----------------------------|------------------| + ]4 n: B8 P) J) Y7 b| 实现方式 | 函数自调用 | 循环结构 |& I# o2 b2 ^- E% [% \
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |. y! H! F; X- O
| 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 |1 E! o1 w, C3 N4 D* S( v# X) B: d/ j
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |, M5 F: y' J, W
/ b: ^3 U z, x# M. ~
经典递归应用场景) i0 S: O, U0 y0 I
1. 文件系统遍历(目录树结构)1 Z4 _+ P! T2 f
2. 快速排序/归并排序算法7 A$ ^+ W5 D* G2 k- [
3. 汉诺塔问题 3 ?$ E4 Y' V) L3 }4. 二叉树遍历(前序/中序/后序) ! I& k% \+ f: _; ~2 E+ G7 j4 t+ L5. 生成所有可能的组合(回溯算法) 1 a+ C) a! Z0 h3 W; |6 g0 `6 k' k' n" I# f8 U* Z, k
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,8 h5 v- T: h5 D. {# l$ p& ~
我推理机的核心算法应该是二叉树遍历的变种。 ) \+ P7 L* m4 m: @0 Q+ 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:' K5 F; p. Y. r7 C/ D% p% z5 m
Key Idea of Recursion. p" S, }) s" d) k2 P: B
7 y! e" c( w6 K, L- C: r
A recursive function solves a problem by:5 _3 h1 q: W0 u# M: a- }, d% T
( x; i! _# Z" X4 u4 v% M7 s' t' O
Breaking the problem into smaller instances of the same problem. / W7 n `8 W* j / [( \7 N9 d# U8 d Solving the smallest instance directly (base case). : r/ }( z8 h* r' {- Z1 w9 [7 d; b1 r! `) \* D% w. R2 l
Combining the results of smaller instances to solve the larger problem.5 T8 P: S" [/ A4 _3 n
7 H; Y& m: U1 o) p/ r: o$ _Components of a Recursive Function$ d* ?5 L1 U, j# y4 ]5 _
0 n4 [, p8 A8 l/ p+ v- I4 ? Base Case:4 z! f- Q6 J b/ j' }
. N. @1 H" |4 Q& i* q$ s! r
This is the simplest, smallest instance of the problem that can be solved directly without further recursion. 4 u1 V* r8 b* S2 K- q! U: A2 V* S! ]) K$ b! T( k# j
It acts as the stopping condition to prevent infinite recursion. # Q, m3 n/ `! ~1 W+ w 1 B% Z. p* L4 _ Example: In calculating the factorial of a number, the base case is factorial(0) = 1.+ k! {0 j" V7 c
: B* u* F$ o& A( e: A } `* O# V* h
Recursive Case: $ t: H" c J7 B5 Q, p) ]( I" V( g f
This is where the function calls itself with a smaller or simpler version of the problem. - B |/ O) |' q! B) z& D a. ~! j7 R2 M D3 B! k/ { Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). - e" G% D; n$ g4 A K+ n- }2 a9 H7 {6 l9 I' h
Example: Factorial Calculation) n3 e5 {5 `9 C8 P' B3 S
" L7 t1 h/ |" B* ~4 }0 FThe 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+ \; Y; c! q/ K& @8 j
3 v6 {) d, A$ ?# [* q* m
Base case: 0! = 1% n5 q/ e2 p* O' O9 K3 k+ ?4 y
+ \( W. }, M k) O) [3 }
Recursive case: n! = n * (n-1)! s4 u) ?7 I% g n1 Q( A. s
6 P( w; `# E! O8 ~' j
Here’s how it looks in code (Python): 9 Q' H1 L& t, o1 cpython g) p' F$ Y0 b k0 M1 N5 w2 d6 B: a& g& r$ i
& d' I2 y: T, idef factorial(n): 9 V0 g) I3 [0 R: c4 _5 [/ k# { # Base case $ J- h8 f/ g8 ~) x6 |8 M3 }2 z& V5 y5 ? if n == 0: . M d0 T- P" l% [1 s9 Q return 1 t: l+ k; x; {" L
# Recursive case * T5 M H+ P7 @9 T) ` else:+ r8 y& z# ~, |8 Y }
return n * factorial(n - 1)' q9 v7 Y0 E& `4 g4 `
4 H: W& D5 V. O* f: X8 O# Example usage- u2 s9 w' N' Y. t
print(factorial(5)) # Output: 120 ' \* o, B) d, p$ Q/ c! K; L# ~2 Y5 T9 _& K5 \; O* v
How Recursion Works) m, c* h' o; W
. l/ W/ x% A& v
The function keeps calling itself with smaller inputs until it reaches the base case.$ _; D- D/ J+ _: F- ^5 i
6 p& B' q0 y0 ^: s/ M
Once the base case is reached, the function starts returning values back up the call stack.# p& d0 `$ a( x( h: {7 M
" x& F7 F& e% A2 k, J( | These returned values are combined to produce the final result. 8 c! e# W5 e x& U2 x3 |/ Z( d D 1 q E4 v4 v ^7 Q9 O2 cFor factorial(5):- A# h& ?. a& c$ N
! n: o( `& ~3 A
& U+ G% d+ P9 j: z) I4 k
factorial(5) = 5 * factorial(4) * t/ @# d! }9 E9 Tfactorial(4) = 4 * factorial(3) I1 G/ L6 a O f0 |7 y
factorial(3) = 3 * factorial(2) 1 u5 v# N7 |8 T3 M* l, F- lfactorial(2) = 2 * factorial(1) % i+ |" l, e3 M8 [0 Y8 zfactorial(1) = 1 * factorial(0) 7 V6 K) i0 p7 z& A) J& p& Yfactorial(0) = 1 # Base case0 Y2 g# M! H# ^0 M9 M
" ]3 g% I/ A$ K3 S- ?Then, the results are combined: 9 e- T+ k0 g0 `9 k. }+ d, S( E! b2 R/ D L0 e+ L' M
) C) m8 Y, c- ?' E* H4 E) }! M+ \factorial(1) = 1 * 1 = 1( H% S! \: u6 `7 V
factorial(2) = 2 * 1 = 2 5 f( Q6 N3 Z6 [5 I3 U( ]factorial(3) = 3 * 2 = 6 " Z& s/ Z+ J/ [. U: A* U% Mfactorial(4) = 4 * 6 = 24 6 H/ u0 G: E. r! i# h+ @0 `' a# ufactorial(5) = 5 * 24 = 120 ! N! _6 L+ S" ~( U* _2 L" X; l+ r f" g: h
Advantages of Recursion3 e& M* l. P6 a9 j7 P q, q6 P% k% u
1 _$ I# F$ V' F9 Z: x
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).# T( x5 Y3 d) a9 _5 n
6 y$ {7 ~7 j* ?* j& Y Readability: Recursive code can be more readable and concise compared to iterative solutions.5 v- n# J) ]2 v# K0 H4 Y. _- {+ t
6 W9 ?5 i9 X# g& d
Disadvantages of Recursion8 v( e: W) r: D& i+ w
# s8 i2 s3 Y$ N/ F2 }6 b: s" @ 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.& L, E; R- l% a3 }# i6 L0 @1 g1 Y
# F( |% y: ` L8 f" ^9 \8 C; w( V: A; ~ Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).1 g+ f" o" ^7 A, J; V
, v0 l% |( m% R. v4 pWhen to Use Recursion 3 B( y" D. ?# P$ r, V6 }- P7 G4 g% m9 P g3 r/ H6 h
Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).1 @3 A! ~+ k0 K ~/ L, A
. k1 Y! I) L- t- n9 T8 w* b Problems with a clear base case and recursive case.: e( \$ y! j+ i$ C0 g
8 K- W: X6 J# W9 W% O
Example: Fibonacci Sequence & |. \" O. z( Q8 m2 @, H- t( q+ L6 }& ~! }3 u3 `
The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: 7 o5 d# T( O, d7 }; N + J) i: {& F' w+ q Base case: fib(0) = 0, fib(1) = 1, }0 q1 q' F7 @8 N: c: i; V1 o4 E; S z
1 h/ {4 }3 E+ D' W3 z" e2 t. P Recursive case: fib(n) = fib(n-1) + fib(n-2)+ v2 }2 O/ \) V
( a7 C. Z5 [* M: e7 _3 [* cpython ' Y: H B) \0 t1 v" u9 s: b4 y. W% l w) v
. o; e3 I+ y+ |2 h& f! f
def fibonacci(n): $ V4 [, k% Z4 \9 S # Base cases/ l9 M$ {. e, O$ o
if n == 0: % V! L3 [+ k8 h3 W ^, A return 0/ E* q' K% f$ @1 z
elif n == 1: 5 _. l5 l- R6 o: ^ return 1 ! g" I% }' h7 B! e. u& G0 V \1 N # Recursive case # U% L3 w2 G0 Z. { else: 2 @( e$ f4 B/ j0 Y: k8 n7 r# \ return fibonacci(n - 1) + fibonacci(n - 2) 3 d' ~* O6 e4 w" I. ?. v$ G8 u: }4 Z7 r. _
# Example usage& l# y1 I7 J* M1 P9 M3 b" d1 Y1 u/ D7 N
print(fibonacci(6)) # Output: 8 ! Y5 P# R# \9 ?) ~/ m * H" Y6 s5 Q$ B8 n9 RTail Recursion ( A" K3 c! B) y! ~9 O. i: |" k2 g1 ~/ E% l
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 @* x \$ R9 ^/ { E; z' s% v
$ w- c3 j7 u3 E1 { D6 z5 IIn 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。