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 7 C8 x7 Q! Q! r6 X2 U
; U" F1 h' C) S, U: K
解释的不错 4 g5 r" l# }3 i1 ?8 Q9 D7 z7 p9 g3 | , M& F0 G& F; d5 j- K/ `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。 " n7 b: j/ G7 v8 Z" j7 q+ C e4 v1 q0 ^& L8 V
关键要素6 v( u% ?& }# D0 i. C
1. **基线条件(Base Case)** , L, Q2 e f \% f8 o - 递归终止的条件,防止无限循环& r4 _) ^0 R; A* r& G
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 14 n9 ]! K. q- m. V- L7 |' K
! {4 X8 }! p. O. Y m; M5 _* }5 U2. **递归条件(Recursive Case)**, m# r, Z" n! k
- 将原问题分解为更小的子问题 0 R% _( w Q3 J1 X$ Z0 n$ t - 例如:n! = n × (n-1)! S4 a2 H! n8 f % `9 S7 m9 L% | 经典示例:计算阶乘 : v( e: Z! V0 B7 t# gpython : a+ p# d/ H" \4 Pdef factorial(n):! N) H9 @7 `! d, \
if n == 0: # 基线条件$ b) G; L/ ?% {0 t
return 1' G* r. }. \+ L: c4 |
else: # 递归条件$ b8 j! R U1 w6 o% d& Y$ M
return n * factorial(n-1) ) l8 ?1 B* K, n+ D& V执行过程(以计算 3! 为例): ! T9 U3 F6 i- nfactorial(3)* V7 i V) l2 g8 B [7 {8 Q5 t
3 * factorial(2) & ?4 o0 A- P6 H6 k4 n3 * (2 * factorial(1)) 8 O* @/ K! s% I }3 * (2 * (1 * factorial(0)))& Q0 f' Y9 ]. g3 e. @( ]
3 * (2 * (1 * 1)) = 6 ]2 Q N8 t w, i7 f) z" H2 w 7 ~$ E+ C1 P8 g( E# H$ X0 c# o& x' N 递归思维要点4 {' I5 S3 b+ [0 K
1. **信任递归**:假设子问题已经解决,专注当前层逻辑! H) c$ ?9 I. Z6 a( F! r# m
2. **栈结构**:每次调用都会创建新的栈帧(内存空间) % @3 `$ v7 l$ P! s3. **递推过程**:不断向下分解问题(递)0 R1 o9 }8 z" V
4. **回溯过程**:组合子问题结果返回(归) 3 A0 }6 M8 ` H; g7 z( _. D0 J ; o% j; S9 @& O' W% E9 ` 注意事项' d" c' a( p h9 w
必须要有终止条件8 N! W! i* O9 H
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)3 U. m6 l6 w2 C. V! p- p# l8 F
某些问题用递归更直观(如树遍历),但效率可能不如迭代* O n( u- A! y
尾递归优化可以提升效率(但Python不支持) $ Z/ Q5 }5 a- ~8 o2 U. @4 r& Z ' p, q0 V. L, G7 @* X4 Z# {8 d 递归 vs 迭代 O9 T w4 F& W0 r
| | 递归 | 迭代 |' O# b. h D, }, C9 Q: g
|----------|-----------------------------|------------------|0 \8 h$ d8 F6 k9 q7 H& F0 `
| 实现方式 | 函数自调用 | 循环结构 | * Q8 L' s0 h* K, o9 Q| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | ! t! f8 X% o; U F. b- M3 d| 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 |4 d5 @/ c) o* \: U/ Q* R
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | 9 @1 o0 g& M' p( l G& {' j% {" T * _; u3 p, ?7 f7 X 经典递归应用场景 . Q( |7 o8 e/ Q5 D! c5 L# q2 B1. 文件系统遍历(目录树结构)( ]4 {* B9 Q3 p. D, A
2. 快速排序/归并排序算法; m! {! h" @1 K/ o+ P
3. 汉诺塔问题 ) t% s ]4 D5 p3 n/ `5 Q: a9 Q: N2 H! n4. 二叉树遍历(前序/中序/后序)2 A7 W( o3 n9 k% c' x1 }' {' h
5. 生成所有可能的组合(回溯算法)+ l1 p' D( \! U
5 \3 W+ t; n4 X; Z2 {; H0 ^' 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, , k# i% M0 {, M3 y2 [7 z7 E我推理机的核心算法应该是二叉树遍历的变种。 5 k* `0 K( c' w) t2 d" D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$ t" J$ t9 ^, {/ [2 M
Key Idea of Recursion 2 n; e* z7 ~( A ! g# p" W' d" L9 @" b$ gA recursive function solves a problem by: $ d/ W* W! z) P. Z, |0 ^; ?" G2 D& H7 `+ g
Breaking the problem into smaller instances of the same problem.% z. Q/ e* L1 ~& b) W4 y3 _: } A5 O; N g
8 _0 x- s( `% [' q
Solving the smallest instance directly (base case)./ F% f. |- k% O2 o7 R' O! r
6 E, r! m9 ]9 z/ m$ Q7 j
Combining the results of smaller instances to solve the larger problem.7 v1 k ]8 s2 q) e! ?) I# n
4 w1 X. h0 p3 A; y+ k
Components of a Recursive Function ) A/ E3 @# i$ G ' J. ?2 s0 Q$ x- T6 x$ ?# S Base Case: 6 p, W3 `- {6 C) n ! Y1 W2 O) N0 d, L( V This is the simplest, smallest instance of the problem that can be solved directly without further recursion.1 f3 ?8 [8 C; F2 `7 n A& m; l( W
4 n0 g. J" T; t
It acts as the stopping condition to prevent infinite recursion.( K# j, v; ^* s7 c/ ^1 N0 h- S
6 P. a& ^3 s6 t# ~$ E+ w! ~( f1 q/ ^
Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% y, ~" l/ q0 ?1 p" p# X% |0 |
B4 Q4 j+ q+ r2 V" o Recursive Case:5 t8 l# e' A& A6 H- o q
8 H6 C/ G5 `+ [1 _, X' _, [) w This is where the function calls itself with a smaller or simpler version of the problem. 3 S2 B6 {* S" q( j. {! E3 n, I) Y2 r$ I8 }
Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).8 Y/ `/ W( F+ A( m1 `; T
1 n4 ^7 x* g( p3 u, Q9 ?
Example: Factorial Calculation2 p+ |6 [4 j5 @
F/ v" c' x( 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: X; ]! O; J4 A5 L" @2 _ % q: W" i8 m' J" [: O$ U Base case: 0! = 1( V) H7 ^7 J4 G5 V6 D+ }
5 N' V. q$ z1 H# t, `% @8 h# z Recursive case: n! = n * (n-1)! 9 c1 f- Q Y2 L. Z, V n, z' ^; u! {' m9 k/ X8 E
Here’s how it looks in code (Python): * H) }. L% C6 U4 i& Z0 {python 5 l( U5 q8 ~& P8 Y! T( G& q* x, |3 S u1 B. X
- ]8 C- ?' q+ L0 g" F! V& w7 y
def factorial(n):- q9 N1 b% d+ X! j4 ]
# Base case' {0 c# b+ S% U6 M5 T- X$ @ l3 h
if n == 0:: V& T# Z& w! [% e
return 1 ' F4 s4 q8 G8 k # Recursive case* ?( V9 k" r" H0 o0 |4 Y
else: . K5 h* u2 @# m0 V2 C8 X* e return n * factorial(n - 1)1 w% N, o+ ` `( ?
. W+ T! K; f3 T: w* R z X+ f# Example usage( n. Q# t: e" J0 p
print(factorial(5)) # Output: 120 0 x3 ~6 _1 u d9 i; `7 f3 r & ^6 z, d5 S2 p, KHow Recursion Works ! h9 {. S8 N7 N' w& {4 p' q$ c9 A* J7 C: k3 e+ r. I8 l
The function keeps calling itself with smaller inputs until it reaches the base case. , d5 w+ J* m2 b# V* M* T2 ]! m2 A6 f5 T' T
Once the base case is reached, the function starts returning values back up the call stack.- G9 ] d- q8 g
( i4 J& a% y3 U7 j( i' `4 T% C
These returned values are combined to produce the final result.# h4 ?7 m, z' x c& L2 w5 y- g
. |- ]5 a3 a& w) Y! l! X
For factorial(5):* ?, T3 }# Y% F- v& b/ b+ J% ^
, J0 t+ p) E5 p% F$ A X/ M Y$ o5 C7 i4 v2 C8 x$ A
factorial(5) = 5 * factorial(4)6 C( U' d/ C3 B V; x
factorial(4) = 4 * factorial(3)* p1 d9 a* F$ y) [
factorial(3) = 3 * factorial(2)* }6 [5 o0 D) B: P- t
factorial(2) = 2 * factorial(1)6 b' i8 v! N- i |
factorial(1) = 1 * factorial(0). E+ g) W, d: a
factorial(0) = 1 # Base case ! U9 N" E; l# x$ k- |, U! }) ? / K8 L' f5 _* i/ F7 N! T( O7 T1 XThen, the results are combined:- E! v \1 o5 a1 b1 C! e
# r4 V9 g* b+ Y2 J" t3 ^; AAdvantages of Recursion 2 u0 n! O5 z+ F- v: D6 N. ]7 _ / i& P; }; Z+ m6 N! f, Q9 ~; v 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). : {1 H/ D- G+ V- S& X" {( w : X- T8 {6 \+ k6 e Readability: Recursive code can be more readable and concise compared to iterative solutions. 9 H/ w. Q/ F! T9 D' _7 r% a" y4 y " H: D3 }. {, C$ f2 W/ jDisadvantages of Recursion . P( a3 b) |# ~' c. Q1 A" `; J+ X5 ~5 z( ?# v/ f8 q
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. 9 S% Z M2 Q$ P, o* D, @5 c( ?- U9 S4 r, @5 K$ X9 ^
Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). ( H( W7 v; n5 Z r/ Z4 g+ b5 T0 W/ {! ]1 ]$ U
When to Use Recursion5 U# Y* g7 v4 y t3 c* `7 \8 B5 N% m9 {
# { J) [% H) q& Z0 P/ D2 Z Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 m3 Q3 _) u; Y6 d6 p* X
9 Y& n4 @" S3 Y8 O0 T1 N
Problems with a clear base case and recursive case. " f& s4 B. `/ _! z# H4 e 5 `' O5 U0 W" TExample: Fibonacci Sequence : F H+ \& f0 A4 v" R; G7 V" c, _! \0 h 3 G0 S# `' q, @The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 p" a* q! H* ~) S% C; }
! f Z l2 L, H$ m: X6 }: H Base case: fib(0) = 0, fib(1) = 1' H4 X% T. @- H. \% a" B! _
% ^! W* p+ R f7 c9 N Recursive case: fib(n) = fib(n-1) + fib(n-2) $ S1 w: E% E+ ]7 w' T' v' o. p; O7 C5 A% L! p. ]
python1 r" P/ P' y. {' k0 Q2 T5 I- o
6 }% J( t) `2 b0 O i' o2 m
3 A' L0 l* h1 gdef fibonacci(n): # c! t# [1 L) h T. g # Base cases / _1 h3 T$ _" w; v/ y7 B if n == 0:/ ~5 i1 X0 n/ O% j7 G* b
return 0 # I7 E# p0 k, @8 K+ Z3 W4 g, o2 T# C, J elif n == 1:+ V) {9 W7 q/ @% j2 n" g) Y
return 17 V/ {9 N. O7 u
# Recursive case % A ~1 i) N4 R5 }1 C5 f" i3 R else: 2 e5 E4 O) p8 K3 f3 V8 [) }7 x return fibonacci(n - 1) + fibonacci(n - 2)' J o1 e. k7 B! V
0 Z% q, \, `4 Q1 ~# Example usage) K; P H! n$ l7 j; i* C: {
print(fibonacci(6)) # Output: 8 # t$ q* T8 L1 \/ p% f1 t% \( H0 |0 l* z$ c: L; t
Tail Recursion# K, W) }( U( P0 C& h3 b7 {1 J
( r8 ~/ G) G, ]- Z
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). 6 u% d [# |, \# `2 A: j5 F: @3 j3 Q X/ v8 L
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。