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 . Z- q" Y& B) l7 U2 B& o
( b& \1 U$ D3 ]4 s1 u6 G1 y& v解释的不错 A- q8 Y# a$ N, X3 Q; h4 w \/ T( u! n. d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。 / B* e4 L$ H! [; v3 w! S7 n- `! V; |# V
关键要素0 W& w3 @* g, W2 B' R- l' F
1. **基线条件(Base Case)**3 X9 | [$ ^, f! S/ b9 ]
- 递归终止的条件,防止无限循环 " K8 R, Z+ U% j3 h- k; y - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1 ; C/ Y; Q3 G1 l$ y; n2 E+ I$ z) v1 @& ^$ n1 K7 e7 ^
2. **递归条件(Recursive Case)** . p V2 {5 E" I+ W- w - 将原问题分解为更小的子问题4 S( f* m" U+ d: a7 _7 ]
- 例如:n! = n × (n-1)! 3 \1 t3 t2 {1 L2 [/ l) U- J6 ~$ J5 G! G
经典示例:计算阶乘3 C8 R/ ^9 J1 h$ R
python- e& X. H) L5 j" M; K0 G2 }
def factorial(n):+ P- \! q3 s7 B p& E. d
if n == 0: # 基线条件 ! c4 h1 [& a9 J/ r: w" R return 1 , h' X* i; r5 W9 I! Y, s0 Y* ? else: # 递归条件 - {0 M: z+ a+ ?0 S) j return n * factorial(n-1)( C6 Q4 c" W5 d% b: u2 O
执行过程(以计算 3! 为例): % D& n& J a: E7 v3 }+ ~4 Gfactorial(3)7 N4 S2 D6 W+ Z- n3 @& q
3 * factorial(2) 5 _' R# x$ ]3 V. X$ o- K3 * (2 * factorial(1))9 e& E5 |7 ` Z. {+ E# h8 a
3 * (2 * (1 * factorial(0))) ; P' ]2 i5 A9 B a* E) v# M/ c) g3 * (2 * (1 * 1)) = 6 5 n! h2 E& I! `1 K m % T4 b3 Y" r6 C/ L( ~9 I7 z 递归思维要点 6 L; \0 Z; H c( s1. **信任递归**:假设子问题已经解决,专注当前层逻辑 ; q! R6 [1 Y: ~6 J6 T# t2 Z# A2. **栈结构**:每次调用都会创建新的栈帧(内存空间)! Y3 q0 a* L H
3. **递推过程**:不断向下分解问题(递)% \( O8 L2 @8 m* O1 }
4. **回溯过程**:组合子问题结果返回(归) 7 \, ~* d, C* r3 `" v o; q ( q1 U3 @* r4 r7 {* U5 [2 E) l0 A 注意事项$ H k! a" H& U$ Q
必须要有终止条件 $ Q8 ~- S3 W% F2 a递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( f4 {) N, e" Z6 \- B( \
某些问题用递归更直观(如树遍历),但效率可能不如迭代 : N6 A) F& G5 d& q+ a, y4 ~, p尾递归优化可以提升效率(但Python不支持) # }% {4 P( z6 k5 w" {! {" C+ ]# v( R' o$ [# [0 X7 j
递归 vs 迭代8 O2 [) I S1 H! a* |, S, {
| | 递归 | 迭代 | 5 Z" }! m. x' `|----------|-----------------------------|------------------|# k1 d! x+ C+ I' y/ d
| 实现方式 | 函数自调用 | 循环结构 |" J2 C! [/ ~. u
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | ( U2 W. i8 R) J| 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 C- t: U( W
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 | " Q% {2 w% Z2 E+ E0 r, n |4 A+ R7 D" I e& p# Z) Y* o
经典递归应用场景 ! z. ~1 e" n3 @3 c1. 文件系统遍历(目录树结构)% w& s; K0 @3 E- X1 G) G. b
2. 快速排序/归并排序算法 . K. Z$ V& U" \ `, W( n- f3 t3. 汉诺塔问题% u) Z; l# S$ G( z
4. 二叉树遍历(前序/中序/后序) & a: B/ L A4 J+ M5. 生成所有可能的组合(回溯算法) 6 ~# ~+ G2 D: ]/ ^' F4 ^ 9 `- t a! b% W- S8 N# W1 q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,+ I: U' A# c9 f7 i( b6 _
我推理机的核心算法应该是二叉树遍历的变种。" H' C( ~+ K$ o7 g- x0 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: ; q. m% y- F! ]2 p6 l$ KKey Idea of Recursion/ y$ e9 ]+ D3 H* C- L. w q3 t
8 D2 j) U. [# U: @4 bA recursive function solves a problem by: + M3 \. N( P& P $ |; x9 L, M3 U+ c& ]* ] Breaking the problem into smaller instances of the same problem.8 N, D8 [& L x4 y
* U/ G) M8 @3 J Combining the results of smaller instances to solve the larger problem.: [: v' s# G' T! N9 |1 p3 R' T( {
" f$ d4 O8 T5 z( b B; @5 h9 nComponents of a Recursive Function: K _7 R! s7 p) ]7 `0 E
; E$ w8 {+ U, i" @8 F/ D5 @
Base Case:: u/ ?, z; w! Q1 \/ t
% l, e) I8 z0 Y$ v- }5 @
This is the simplest, smallest instance of the problem that can be solved directly without further recursion. 8 V2 | L8 t! {( x" T+ ^" g0 r! F 2 O+ [2 Z! i3 C3 h7 U& X2 r( C7 ^0 Z It acts as the stopping condition to prevent infinite recursion.7 \+ b; a0 Z& q ^3 P
* o' l6 J0 {5 |
Example: In calculating the factorial of a number, the base case is factorial(0) = 1. % D' {) i; k+ ]3 o5 g/ [1 {/ Q 1 {2 o8 m6 h9 E/ P4 g" X Recursive Case: p9 e! {$ Y' g2 B! R d
# \2 @8 o/ Q' x This is where the function calls itself with a smaller or simpler version of the problem. , h. I+ b: b D / F" t& m' G5 x0 [4 s3 S Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). ! y2 C9 ?5 t& W' t8 _; w7 P4 n" s, X * ^0 Y8 @9 Y. zExample: Factorial Calculation" k9 D9 I6 P$ E0 L! E* h
' Z$ B! H3 `1 [6 D4 D
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: ! j7 Y2 ^9 c; _( {" u( D' U, I% I& _: r+ J: t
Base case: 0! = 1) M( p8 e% c: |3 L6 h: ]# L& }$ c
9 u" w. c; O$ P. Y Recursive case: n! = n * (n-1)!* u1 v& W8 @6 e3 N- p
5 v" V. S& y& \: y1 J9 LHere’s how it looks in code (Python):& ]9 A" }) N8 I* P4 M' ~& H
python( e2 x7 g- d1 m6 d2 |, \! t* G _
0 z, O4 k; \9 b. m1 z7 K1 O / m/ T2 R9 ~! pdef factorial(n):7 g5 p3 _# A* L1 g
# Base case3 F' i$ T( A& b
if n == 0: 7 o& v8 X" ^7 [3 S% t return 1* _+ ]3 A. v/ x9 ?* y& d0 f( W/ t
# Recursive case2 g+ d5 U4 l2 I L3 J t
else:0 y% F8 G, R+ p$ I- c% Z+ a
return n * factorial(n - 1)% j# @2 x! |! h: o( G" c
; ?. i" c$ X* d$ `& U. a
# Example usage, U3 r. i& W( m. g
print(factorial(5)) # Output: 120 * G# x% Y; X* e: z8 ?% v' O2 X6 m5 q, a2 A1 e
How Recursion Works $ i, g4 [' `0 f3 U ( b8 j; a2 S2 M$ Y The function keeps calling itself with smaller inputs until it reaches the base case. % j& s4 m* D8 t4 V5 D. A3 E* M : U. G. s' {2 I3 y Once the base case is reached, the function starts returning values back up the call stack. / [! F7 m+ I" W* `/ n, U% m* J% E2 F" |9 _1 S
These returned values are combined to produce the final result.% }# C# w, K/ F8 b2 k
4 K; l/ Y& w$ L% b3 ~7 L9 r* X
For factorial(5):) n% L" Q* Z# C4 c) j
% n# I) L1 F Y
( g9 r) H7 J) G0 K T, Y( _factorial(5) = 5 * factorial(4) 0 F) O. M( w1 ?4 z# bfactorial(4) = 4 * factorial(3) % V% D( r9 ~5 x! l2 ifactorial(3) = 3 * factorial(2): j+ i1 [; | X I) Z
factorial(2) = 2 * factorial(1)- @: q2 j5 P, C8 @+ ^1 v
factorial(1) = 1 * factorial(0). n [) j0 o: E% {; r. B3 `
factorial(0) = 1 # Base case % N7 f' w* u4 E' J, l" ?) D% l/ H1 f; ` 3 S" k- K+ z( Z& @Then, the results are combined: 0 f+ \7 o3 f5 f( s% e6 J 5 G7 f% a5 @5 I. c) a e% W9 a0 }3 a4 ]* u0 V6 l0 S2 C) q
factorial(1) = 1 * 1 = 12 S9 y5 w4 V" p; T; p
factorial(2) = 2 * 1 = 2 5 ] |6 G$ M; afactorial(3) = 3 * 2 = 60 N9 H7 \& o( g. t f. x
factorial(4) = 4 * 6 = 24" u0 O8 J& B6 W6 h( T- [! \5 F
factorial(5) = 5 * 24 = 120: f1 `9 S6 G5 D# F
/ n0 @/ a( V, L
Advantages of Recursion5 ~( W9 o: m) F- ?! n
- x; ?. }, P0 \. U& N9 ~ 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).5 {9 s, Z0 x6 x' N# c/ Q
. |6 _% t/ z, H4 @' u4 d, ? w Readability: Recursive code can be more readable and concise compared to iterative solutions.3 c o' }! G: c6 k2 \. j
' J4 l' e0 |4 i5 {4 \
Disadvantages of Recursion! V/ R% i& H4 J" E5 G' A: E. m
, N9 F: g2 u1 i) |0 O! a
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.+ Z) A. h- R/ v4 s
$ f8 P& `; c$ K% ]1 I+ L5 ^ Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).7 T& v2 J* r$ n/ }6 q. D6 U3 D4 n5 \+ p
) G2 ]* V3 h9 a: k
When to Use Recursion" a. L6 G+ q0 x1 S6 T! l- V
/ `4 N& N/ g$ j: G/ Y) y
Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). 1 V) G% j; H) ^ * v& D6 h( ^7 k! k Problems with a clear base case and recursive case. ( u. \( W9 o- K0 ?5 P. s+ w( I1 p6 ~9 a" y* ~
Example: Fibonacci Sequence6 y+ D+ }: T7 n7 s
1 x n, @. l9 C% qThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: : M/ G. F2 H5 u" j3 P: f; | $ H G; K5 P3 Y' E# G Base case: fib(0) = 0, fib(1) = 1 2 t6 H# P! }& q+ e2 h4 A' f2 G4 y, W( e0 v/ S
Recursive case: fib(n) = fib(n-1) + fib(n-2)- D' h9 W' }# X
3 d: D& j8 N: F* apython, m! n7 h$ B- k" C/ I* W/ B3 z
2 [" C! X4 r; a% g- v: [3 z% g3 x
2 Z8 m4 m/ b! X- e( m) A
def fibonacci(n): : j, }- P$ ~% A3 e. [9 S9 l # Base cases 4 r) o {, J8 c! m$ j" [" l if n == 0:6 ?0 k& Q4 l' k4 T S- B
return 0 # D, F1 p; Q+ d1 E0 ~: t7 ]2 u* A8 j elif n == 1:; u, o# S, P: E) Z. D
return 1 1 t1 \0 u% h! d4 R # Recursive case* \2 K) B. R8 s% G- @1 C& u
else: # ]- \% t; L$ t& Z1 Z return fibonacci(n - 1) + fibonacci(n - 2) 6 e4 Y2 h+ t& h6 X7 f " \. F( }: I* o0 [# Example usage 3 ^# h5 H& _( V- {print(fibonacci(6)) # Output: 8% N( W3 }" m; V+ o
# _6 n) L2 b" P! r
Tail Recursion- N7 o& h5 g( u: K% _
/ ? g, A5 e3 x, X. n0 {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)./ O: g6 Q3 R" [) |3 J5 T
: c( G5 z& Q$ }# a: ^0 j8 o
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。