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 % B# J8 E9 a+ F( a" R& A
6 K3 Q* \! C; [/ {; l
解释的不错/ S+ Q" M( M% X
4 z' H. e. `' H) g( |% A
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。 * r4 D! @- @4 n5 d: Q D2 t& C) o7 ^1 @
关键要素 G8 S, j/ q0 ^ i2 T6 |
1. **基线条件(Base Case)** 4 p$ j7 I7 }; m8 Z: ] - 递归终止的条件,防止无限循环% g0 ^" J& N$ B c. `. F
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 1/ Q* z8 v9 ^+ w- C. [7 _. I* P
- A& P+ K- r# H4 A/ o2. **递归条件(Recursive Case)**" g8 q7 k2 }; @3 S- Z
- 将原问题分解为更小的子问题; Z# p( {; l7 c2 T' M! Y
- 例如:n! = n × (n-1)!! D5 }; f1 t& @6 `! x, f
% U. X+ r4 n7 a$ ^7 d! o/ E' f8 { 经典示例:计算阶乘% _7 w- d% C# N: ]
python! w$ U/ d) D. w: c5 t* E( E" P9 P
def factorial(n):4 c3 Q7 `' k8 ]& ` Y+ f
if n == 0: # 基线条件 ( _6 K5 f7 k) o3 r8 } return 1 5 [$ Q/ {, n; _' V: | else: # 递归条件 / \$ D# L1 S) w8 ^! g3 l return n * factorial(n-1) . F1 E* o9 r( o2 k执行过程(以计算 3! 为例):4 W r) L( }1 A9 x$ y
factorial(3) # k9 U; f, o, M# g: {. c3 * factorial(2)* K B$ Y, P$ M
3 * (2 * factorial(1))9 L4 Y# |/ {% V, F# M; S
3 * (2 * (1 * factorial(0))) 5 {# S9 {( u4 w% N7 N, x3 * (2 * (1 * 1)) = 6 # W9 ]3 n* R5 @, q' A5 U2 O / H N, ^1 K1 @ W9 g4 R5 Q 递归思维要点 + T. u/ @/ V. g3 d' @! L1 G9 j1. **信任递归**:假设子问题已经解决,专注当前层逻辑 % g, ?2 U" G! T; A2. **栈结构**:每次调用都会创建新的栈帧(内存空间) # Q1 y) W- T0 W4 C' e: N3 J a1 l3. **递推过程**:不断向下分解问题(递). ^* b/ Q& e, o$ p7 I; ?
4. **回溯过程**:组合子问题结果返回(归) / |/ P2 Q4 |9 B6 P 2 y+ W& q, B# U& m 注意事项 , q% G$ T! R/ w9 x% C. G必须要有终止条件1 I# x3 w7 Y% i/ r+ E* Z
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)# z) z& R* A0 q0 S% d" w" d4 V
某些问题用递归更直观(如树遍历),但效率可能不如迭代4 N6 N& {9 |: v1 ?
尾递归优化可以提升效率(但Python不支持)& O+ u6 d8 f# F% x" Z L
J3 {& A4 K7 l7 C9 O6 L
递归 vs 迭代 7 X6 ]. g, v% s) L| | 递归 | 迭代 | $ B2 P7 R3 s& q& x0 q( c+ e|----------|-----------------------------|------------------| 2 E* g1 x$ L% A+ m. A8 p| 实现方式 | 函数自调用 | 循环结构 | 3 K `% g) [! v( L; F! r| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |) }9 ^0 t' y# g, f" H) k b& o$ m+ Y" N
| 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 |5 J) t, r& i# d. A! `2 ?
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | ) S( J2 |7 u" D0 r+ X% ?& D) B 7 P0 m2 r. d# b P$ L. l1 [ 经典递归应用场景- N$ y4 [4 U/ v }0 d0 a
1. 文件系统遍历(目录树结构) 8 g+ c2 {; s7 G2 z& L1 } I2. 快速排序/归并排序算法 # }. p* f V2 F0 h3 o5 X. m3. 汉诺塔问题 1 q) P4 ]( h; { A) w4. 二叉树遍历(前序/中序/后序) # u. X& ~7 m5 x, F5. 生成所有可能的组合(回溯算法) - r2 _4 k0 O4 F* p 0 D" J9 C3 E6 e+ K6 ~% F- R 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" Q; F) ]: M我推理机的核心算法应该是二叉树遍历的变种。3 D- U( `- j' u6 @) A* 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: ) O1 A% w: @0 @5 q5 G/ B/ QKey Idea of Recursion $ e) z8 v6 w* j! y% C- C6 ^ 5 {# V6 s9 u; G7 x: DA recursive function solves a problem by: E; {7 V: w$ i/ A6 D
1 C7 J3 z/ E* L6 m; _0 u Breaking the problem into smaller instances of the same problem. N0 l# ~$ ?% u! s2 C) s : w6 Z! c" F1 e: b' a$ Y# ] Solving the smallest instance directly (base case). * [) F7 D$ k5 E3 {( N6 y' E- p. z c8 Y* }3 c0 E
Combining the results of smaller instances to solve the larger problem. % |& K" X& A* ~0 D o5 m% n 6 n- T: |& r) zComponents of a Recursive Function* R0 @( y% |9 U7 P
+ I5 l$ W! }7 N3 g% G+ O0 a
Base Case: ) z. v" e# s: R5 a' n2 t& A 1 }# g2 V6 N6 }$ v& i) J, b This is the simplest, smallest instance of the problem that can be solved directly without further recursion. ; ^& a/ M# |$ @# E: H3 T2 a& N" E8 u* \! ]- w4 o2 }
It acts as the stopping condition to prevent infinite recursion. , a9 M3 h2 y) r3 ~ 8 r0 u5 B3 X6 ?. z/ z, h Example: In calculating the factorial of a number, the base case is factorial(0) = 1.2 v8 f: w' F& _0 T {2 y& B
0 X( h; h" s: Z3 m) } Recursive Case: 2 t$ z0 G- l3 ]4 v5 @ e- |# t! z( L$ c$ s7 c
This is where the function calls itself with a smaller or simpler version of the problem. 8 i& [5 J/ W1 F3 v% g' P6 P. P' e5 B v6 f. v6 @
Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).6 Q% }6 L1 @" {+ S: L: {' b( {
0 _( t/ i# M9 ]1 [! aThe 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:+ d. a F- v: M; r
; e- }. n" p0 L M
Base case: 0! = 1/ {. y/ z% u$ ~2 V6 |1 I
9 O, P& p% d$ ]. Z Recursive case: n! = n * (n-1)! / F) L& X$ z8 j# r5 c7 | 5 _) `5 c' ?$ ^; G+ NHere’s how it looks in code (Python): . ` b l& z4 }: d# f o4 I5 B- Lpython ( A% i9 U$ N5 k8 x3 v& Y! K8 @) l1 b: f0 a0 w8 l
a: R5 }- q; sdef factorial(n):% E- K: \4 |3 _7 }
# Base case & z m* z' F- o* p$ d if n == 0:( }$ @4 g; M" G0 J
return 1 1 J* U$ W/ W- b # Recursive case* U. R4 \4 d* P. Q! C
else: / V, R$ f* Y% V( q A; G return n * factorial(n - 1)1 P8 a, a" U) D' }" `- ~
( E3 b5 `, p& a+ A" I# Example usage " ^1 t S; d! R2 a ~0 pprint(factorial(5)) # Output: 120# J5 y! h. p- h# I5 ^( g; @
8 ?: k" z- @7 M! `+ OHow Recursion Works: X8 i$ R* i$ Z0 [& f
+ F& P% @/ ^; y' i# ]
The function keeps calling itself with smaller inputs until it reaches the base case. 5 Y7 I" B8 y2 r/ E) t, x; B. q- {# L0 H1 U0 j; ~
Once the base case is reached, the function starts returning values back up the call stack. 2 f: {' D0 F5 ~. J2 A: `( b 4 }, V L& q* j" y These returned values are combined to produce the final result.1 ]" v% s' s K7 g* h+ F {
) {3 d; i8 O( g, N6 v0 i
For factorial(5): ! n+ k/ Z0 P9 q7 C7 k* R5 @4 a9 f, d
0 A' t! r5 [" ?/ m3 |( i. yfactorial(5) = 5 * factorial(4) * p' y& Q" r2 O8 k/ Lfactorial(4) = 4 * factorial(3) 9 B: f9 j* K* h$ Y9 Y+ V9 bfactorial(3) = 3 * factorial(2) / m" s0 {, X$ Efactorial(2) = 2 * factorial(1) % L9 t% _" H* b) m. ~4 f2 \2 b$ `factorial(1) = 1 * factorial(0) ; u- k, ~. x! _# i3 Ffactorial(0) = 1 # Base case 2 o% o+ ^, S- j5 w* Y! K n1 e6 J6 r# i o: g
Then, the results are combined: 0 N# m* ?/ y$ |3 K) u1 X7 X& X4 N; J/ K& C
' \5 r) b5 \( x4 i0 K, m: e0 @
factorial(1) = 1 * 1 = 1$ O$ N. R2 ~. R
factorial(2) = 2 * 1 = 2 7 }. B+ s1 `/ L) K$ X) sfactorial(3) = 3 * 2 = 6 5 N- F2 r7 H6 V4 x: h) P" F" c3 tfactorial(4) = 4 * 6 = 24( R* ~7 j. d5 ~, w3 \( u' ~, s
factorial(5) = 5 * 24 = 120 % C/ G5 F/ n& u, m/ I4 T5 ^" z" P) K. K1 `' E# L. O5 n. c
Advantages of Recursion / M# y+ S# f3 X/ {9 K, h & W/ i1 x7 I. D0 M, j+ F3 q" Q Simplicity: Recursive solutions are often more intuitive and easier to write for problems that have a natural recursive structure (e.g., tree traversals, divide-and-conquer algorithms).' S7 n# }4 R3 ^) n
5 I$ T7 e; Z) r7 K. g Readability: Recursive code can be more readable and concise compared to iterative solutions.0 g5 S7 G5 D8 M. @
9 |+ F7 E% f8 X2 fDisadvantages of Recursion ' u( T+ G2 I& v" G2 \# u0 ?# |0 ]& r
Performance Overhead: Each recursive call adds a new layer to the call stack, which can lead to high memory usage and potential stack overflow for deep recursion. " j9 y$ c0 u# }9 o0 b# ` _; P# Y/ [8 [5 e9 W* I$ Q! B
Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).& B$ o5 x! z8 c5 L {. a( _: X
) ]7 ?1 {3 g8 X. I8 U
When to Use Recursion2 F" H0 d- \) n1 Y
* x) |2 J& ?. h* K x# x) d
Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). * k \) H: b) k$ ]% }3 O3 W" V; Z; | p4 K# [
Problems with a clear base case and recursive case. " [& b c" s# V* U 4 T# m8 J- L. M5 D, ^( tExample: Fibonacci Sequence$ v0 O) d. Y2 b" U8 _; x8 [: c. m
2 r) ~% e4 v; Q/ h0 ZThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: ) R6 f* J, a y+ w" S% y! s. c " J; _, ^2 L- C- X Base case: fib(0) = 0, fib(1) = 1( ^5 C _& Q: F# F% g
: x+ ?( X( a0 s0 W. c
Recursive case: fib(n) = fib(n-1) + fib(n-2)- C; H% s3 s8 Q( V
_0 S6 s2 T5 d, Epython x7 Y, ^5 |& {+ T6 r- a' m" C9 {% y3 [ m7 Q% D' l
. k8 J, I B+ X o
def fibonacci(n): , ?& E) E3 W7 W" Y* y7 j, S; a% Q1 } # Base cases% A1 e" Q. S+ d' ]) d0 X9 o: i
if n == 0:6 {' F- e0 m$ B' ?7 x
return 0 M. J8 B" q. H' g. ?6 I; e elif n == 1: 9 [7 q1 G6 Z/ U" V8 Q9 M return 1, c$ q* G; T6 _: [7 ]
# Recursive case : }0 K/ {. |7 g8 q else:) @+ q) k/ U) Y6 q
return fibonacci(n - 1) + fibonacci(n - 2) 4 W9 f; X# d8 T, v5 Y4 i 0 R1 N( u2 Y3 x+ j% c# Example usage% q2 j; @1 u! k. @& \
print(fibonacci(6)) # Output: 8 , m# d) {( x, s' K9 V# J7 L0 Y7 ~; U4 W. W! w- ]
Tail Recursion 9 e$ A& M8 R* Z8 S2 k 7 l u3 `+ u$ ?$ {3 f5 ` c! d" bTail 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 Z- V- E. W2 ^5 Q2 A' j# b
, V' K- N2 Q: {3 }, @
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。