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 1 _4 C# g" i; ~, P+ r t! S
0 k, _; @% W1 h
解释的不错 + l' J! @7 A4 v: O& C9 p 4 |( w6 C2 b8 {: x4 Q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。' q4 h4 o0 s$ S* i9 G$ y
/ m/ w# r: F! U1 | 关键要素! {3 i; o6 u6 o# O2 T3 f; v a
1. **基线条件(Base Case)**" t$ i8 \ P' A' @' ], f- b
- 递归终止的条件,防止无限循环 ; ^1 p% r( K4 Y! u: @5 B - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1# m2 [# h3 F9 f2 h* f
* b5 R9 B, B. q q- G, r6 M: V6 L' k2. **递归条件(Recursive Case)** # F7 V0 b( o- U& }. P - 将原问题分解为更小的子问题 9 e+ ~9 S2 o1 X. f3 E- W - 例如:n! = n × (n-1)!+ p% Y' H3 C3 }# u8 J( E
$ J1 j/ f8 |, W
经典示例:计算阶乘% f9 c' B" B8 b
python 8 V% V# m* L# A( E, A+ i; ^def factorial(n):* T7 z; ? u" c3 G
if n == 0: # 基线条件 % A8 N7 w1 @4 g, q; ^2 n( N: F return 1 # v5 W9 e1 z K3 U4 M else: # 递归条件/ p: z4 Y8 ]$ w" ]
return n * factorial(n-1)3 R' E. M9 S( d# P# E9 f
执行过程(以计算 3! 为例):5 Z) q7 f4 w, d) u
factorial(3)# e$ L7 b& z; _& {" [
3 * factorial(2) 5 s2 O* X2 w. d2 y L0 F6 _8 B$ d3 * (2 * factorial(1)) 8 U4 L; u9 [ l; N& E$ k3 * (2 * (1 * factorial(0))) b0 X* `% z8 m+ y
3 * (2 * (1 * 1)) = 6 % F. b7 f' L7 N8 X7 O) Y; } 4 w1 Z% P# b: v/ q+ K& h 递归思维要点 7 N2 Z O% W7 T( d1. **信任递归**:假设子问题已经解决,专注当前层逻辑6 N8 G$ W" c) a: y4 E
2. **栈结构**:每次调用都会创建新的栈帧(内存空间), L5 E8 _ E0 v; Q4 i8 o
3. **递推过程**:不断向下分解问题(递), W/ G" x' Y1 D& P7 _$ q( B" M& T8 f
4. **回溯过程**:组合子问题结果返回(归)" i, W3 S4 t0 s( c6 u6 H' c
1 J( t& M% ~9 T7 Y) a9 |; F( o
注意事项( R4 y3 v8 g$ R2 z" i
必须要有终止条件 8 l, M8 n! U, V4 s递归深度过大可能导致栈溢出(Python默认递归深度约1000层); z3 n1 V/ I7 A; a! Q
某些问题用递归更直观(如树遍历),但效率可能不如迭代 2 ]0 ~# k* R- R I尾递归优化可以提升效率(但Python不支持) - P/ Q& N8 b4 F( ^; ?: O. d ! S- b, o2 G$ W 递归 vs 迭代9 {& d3 @! T Z5 \$ v" T
| | 递归 | 迭代 |( ]: B6 ~, x0 [: A+ ~
|----------|-----------------------------|------------------| 5 T) c0 E1 _: L! C; [, j7 @7 W| 实现方式 | 函数自调用 | 循环结构 | ( T) \. l1 B3 T) m| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | 5 n% a5 y# H; B+ || 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 |/ d6 b$ ^3 M2 G! u1 w1 v
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |1 @) n$ _1 Q ]2 N" b
0 O" L4 a: q) c& Q
经典递归应用场景 " w, u. E% }3 @. V" h' m1. 文件系统遍历(目录树结构)3 `8 c% m! v2 w( J
2. 快速排序/归并排序算法 2 s/ n" H; X( H9 G$ a3. 汉诺塔问题 / E* C4 h/ W3 q; L4. 二叉树遍历(前序/中序/后序)0 l# k+ V) B4 h% y; a
5. 生成所有可能的组合(回溯算法)! N, B% v- m$ W4 r
9 u* C, b1 ?2 W3 _
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 P; _+ A7 Q) ~: G" W我推理机的核心算法应该是二叉树遍历的变种。. e* h- A0 r% z4 h6 P, R6 z9 e" h
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:3 Z$ Z `+ F: u( B7 z% a/ a/ K8 v
Key Idea of Recursion 9 U# q( ?5 Y2 U# V1 u) y3 o+ A1 \% s; q# h) C) G
A recursive function solves a problem by: 4 j) t9 s0 H( Y" W4 @9 P. q' [6 h' Z+ D& v ]( F
Breaking the problem into smaller instances of the same problem. - o# z4 E8 d/ w" g. B( b% P6 T- n# o4 D" L5 D+ @
Solving the smallest instance directly (base case).- U' v: P+ q0 `. ~
2 Z* C) ^; Q+ S: D9 l& I) f$ X Combining the results of smaller instances to solve the larger problem.7 M$ R1 T4 S3 [% @9 L1 W6 o
' [, i; t3 k# ?: ?7 t: i% Y1 JComponents of a Recursive Function 7 r, A1 Q; a3 O `2 D* X) m* e7 |# O( Y0 U0 i Base Case:5 N9 s; C% S! U. w; w7 U( a
& m4 e- j' H6 X This is the simplest, smallest instance of the problem that can be solved directly without further recursion.- y r0 B, t+ K
, ]1 v) R' B# k1 x! l U5 X% U) T
It acts as the stopping condition to prevent infinite recursion.7 q Z* C6 Y; O/ b) s( ]
- j! A* f. R5 B6 S0 _
Example: In calculating the factorial of a number, the base case is factorial(0) = 1.0 c1 S# q s5 S, a2 Z4 V5 `# v
2 R' v' E: Y- h& n" C8 [
Recursive Case: & _2 n# D" R. Q R6 Q* Y" m ( Z% N; z% H% u o9 S' {( A This is where the function calls itself with a smaller or simpler version of the problem. ; h" v; d, I$ |; \, Z 2 ^/ |9 v* h( `# u Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). 0 A6 ~. b+ G" Z3 z$ n) { / Q- j; G5 g8 C: L- mExample: Factorial Calculation $ r( g; n- \& k8 [9 T % [0 }" K6 x* @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:9 \0 h$ s% G5 l% P" w& K9 L
# _. t3 H0 s6 r7 q) J Base case: 0! = 1, k1 O7 A+ Y( I( S0 L3 x! H
4 i h4 ~0 K* J Q Recursive case: n! = n * (n-1)! , J' A5 D E/ b2 [. Y3 D, g3 M! |4 t2 M) f T
Here’s how it looks in code (Python): 8 w0 J( q5 v6 s% c- A( j3 q, X Epython! h2 e+ i, n8 A1 m: ]7 i
$ C: E. L4 O. Y. ?# a6 b
7 I, l+ y3 u& H+ k pdef factorial(n): : H( w9 Q+ W' m) T' y" z # Base case- | ?/ P! k8 |
if n == 0: 5 p3 [2 e1 X- ]8 N/ U& n return 1$ s. f( Z$ ~. Q! t% @2 M
# Recursive case " B; ?* q- c/ o( g; j else:+ @; @7 K8 B6 ?3 ?' q8 x- ]0 {0 T2 K
return n * factorial(n - 1) , w6 J/ o8 v) `$ `3 G& J1 I' ]( D l1 f
# Example usage z. n0 e6 {# k! p7 K# ?
print(factorial(5)) # Output: 120 c7 _( b, n+ L% P; B# K5 i" ]- |1 F, q/ p" n
How Recursion Works: `1 a C+ A0 X+ S. v% N2 T8 V. h
% E. T' u% g# S. c+ b; F
The function keeps calling itself with smaller inputs until it reaches the base case. 2 s3 K( n2 a% y3 D 3 z0 n% x. D/ j$ o" `% D4 a5 ~& G( m Once the base case is reached, the function starts returning values back up the call stack. % c7 B) l4 \% n% n8 x+ _6 D' ~, M9 h& E* S+ D6 ]' t T8 X
These returned values are combined to produce the final result. + Y ~, Z" Z& |, m( c! E" B $ ^0 f! K& G4 X4 r; T: WFor factorial(5):7 s( P1 K4 H. S' U
% p: b! e* t& _6 {) {3 w7 E; S6 A& j7 w! L# W
factorial(5) = 5 * factorial(4) : Q! r7 Z: `( c) G0 vfactorial(4) = 4 * factorial(3)! h" {( V! h# {6 M5 u
factorial(3) = 3 * factorial(2) + i/ P3 L) [: b2 D4 s) {0 Lfactorial(2) = 2 * factorial(1) B! n; F" m: s: j) L/ v1 ffactorial(1) = 1 * factorial(0). R+ X" A4 A' C, x5 w Y
factorial(0) = 1 # Base case0 g3 ~7 H$ `: g7 W ^
1 b: Z) V0 V" D0 v5 ^, v
Then, the results are combined:0 R5 \! m) ?7 z \" B+ e2 j0 T0 i
4 x- o* S6 u) q) H- Z; G7 k# F
+ h4 o- a( p, ifactorial(1) = 1 * 1 = 1 9 f9 ]5 G) n% j- |9 ufactorial(2) = 2 * 1 = 2) c5 R9 x% v! }5 e4 k2 L5 ]
factorial(3) = 3 * 2 = 6 ( }9 [8 A/ o4 Z" N- _) cfactorial(4) = 4 * 6 = 24 ' p q- O7 ]7 n4 T$ @factorial(5) = 5 * 24 = 120 ) g% I9 B- D6 _- e% y 0 |9 m2 L; J; q8 a1 NAdvantages of Recursion, d* Y2 a. B. K0 r0 w5 f
1 _$ D' B: @' K$ z0 B
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). / u3 k. _1 B$ f4 ~) @( y9 o& `) N" \. K7 t! ~- W4 f% Q
Readability: Recursive code can be more readable and concise compared to iterative solutions. 2 [6 o. }7 ^( g- Z8 J* f) v* X" R' p' n# \
Disadvantages of Recursion 2 x9 U* m* H& w" G2 A' R) B0 ~- e* ]4 K2 |( [7 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. % s8 k' t3 D0 l: ]! d- Y7 b* |7 i0 Z* Q7 }1 C+ t0 D0 X) ]1 R- e4 R
Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). 4 l+ W+ A% D/ b ! E% O& S! a& p/ p& N" [: j' UWhen to Use Recursion* K% n; s6 i: z
8 h# e, J5 S. }
Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). 2 B. w/ b* n" S L7 v0 G; i2 y. Y/ P
Problems with a clear base case and recursive case.4 d" }" ^% ~6 s4 L3 V
: z' j3 T6 m$ e: l; T
Example: Fibonacci Sequence. U3 f9 K3 e5 b/ V6 \
~+ h* C) H5 i5 Q" m E/ gThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: . b- z' Y* D* d, y: Q/ S L7 Q, ?# W! Z+ y4 A
Base case: fib(0) = 0, fib(1) = 1 : h6 O& K d6 O7 Y! X3 b 4 W0 `) F% E" f) A Recursive case: fib(n) = fib(n-1) + fib(n-2) 4 u+ L- ?4 V# m* p* l" \2 [, _+ ?4 G9 L; C, A5 X H5 B
python % j& W5 j& b6 b( A 7 ~$ f2 W8 q7 l; R & s; [) D1 c, F$ P0 b1 rdef fibonacci(n):+ N/ l4 u+ W; p! M7 n# K+ X/ r+ h# t7 N
# Base cases* w6 p* q# j: h
if n == 0:' L; i5 D* e( d4 x/ P- ^3 \% x* I) P% O4 H
return 01 K( A4 ?& b* w6 w0 a8 a4 r
elif n == 1: & V6 x9 L' F8 F3 v return 12 l+ x4 j5 _+ d/ X. r
# Recursive case * S6 M3 i* A- Q; m else: , }6 R5 n0 f$ T9 P) o return fibonacci(n - 1) + fibonacci(n - 2) # ^2 m1 d$ q9 N! \- K4 _, B& G$ {: I. {. F1 @4 y+ l
# Example usage# w% A n& X( `% F6 M+ w
print(fibonacci(6)) # Output: 8 , y' N* T" Q( s$ f4 D# _! y5 z( j7 g+ U% s
Tail Recursion. e. g2 w* w8 {
0 R% R! s( K, m4 P/ ITail 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).; L9 H8 x! i0 ?2 W# Y% f
# ]8 g9 x% M& A) P2 v
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。