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 5 Z) C: W0 ~/ P# s& B
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解释的不错& |( R! S! ]% a' l
7 I: V; e; x1 H( X$ m; R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。0 x. z& G: U; T( b' U+ Q2 a
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关键要素 0 d4 o g6 U- [& s: F1 e1 X1. **基线条件(Base Case)**9 g [/ K3 t+ \1 c/ p g
- 递归终止的条件,防止无限循环/ s" \4 M' u' W: u& ~" [0 Y/ {# A0 O
- 例如:计算阶乘时 n == 0 或 n == 1 时返回 1 + }& U: U1 [- s$ {, I* }$ d; F- L; g2 Q2 T0 H
2. **递归条件(Recursive Case)** & P* _0 M8 C* \ g2 Y& U4 G, |" k - 将原问题分解为更小的子问题* P* q; E+ E0 @- A2 ~( x
- 例如:n! = n × (n-1)!- `! R$ i9 M4 n- _9 g, ?
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经典示例:计算阶乘; r4 B5 ?6 t. s& p7 u0 ?0 }5 j" g
python! X/ P1 d; B s9 r0 n" J( s
def factorial(n):3 n- L( V t4 Z& G2 p
if n == 0: # 基线条件, S" s# k9 Z, z7 d6 d
return 1 $ W1 E- W: x7 I) ?, f/ w v else: # 递归条件 & N" ^* r( |7 v" v4 [( X; g return n * factorial(n-1); P4 X& k& z3 ^6 l/ S& Q
执行过程(以计算 3! 为例):0 a& [' k3 k& J2 I3 h! R4 K
factorial(3) 8 o3 f2 F( I& |3 L& R7 ]" i3 * factorial(2) ( v" o M+ p, A2 _- O9 A, S3 * (2 * factorial(1)) 4 L: Q, f+ U- P# ~3 * (2 * (1 * factorial(0)))( y- N5 E1 } y# j
3 * (2 * (1 * 1)) = 6 & U6 ~, m, |. T M& ?1 ] T9 o1 e
递归思维要点) ` _8 l$ a+ X- @/ i
1. **信任递归**:假设子问题已经解决,专注当前层逻辑 $ ?1 P/ w% i! m' _- C: j5 C2. **栈结构**:每次调用都会创建新的栈帧(内存空间)3 b# p7 u7 f0 ?/ B8 h; A+ u5 w! R& s
3. **递推过程**:不断向下分解问题(递) & I, N5 J4 v2 ?( K# @3 Y4. **回溯过程**:组合子问题结果返回(归)' O" Z' b, A$ ^8 W% h( ?
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注意事项 9 }7 l9 n4 v3 U6 ^. y D' o2 v必须要有终止条件 - J* M: `4 K1 r" Y- j递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( f! q; V" w/ D* `: J8 K7 y+ O- Q( V
某些问题用递归更直观(如树遍历),但效率可能不如迭代 2 D# T5 H" m( s& ^+ t* y0 w% `尾递归优化可以提升效率(但Python不支持) 2 H7 E; r2 l* k* y( ]2 G, Z5 q- z* W& G% J8 G ]
递归 vs 迭代 / u8 D; I* ?# }! i/ q+ ~) G| | 递归 | 迭代 | * W% x+ T! R: V+ R% x) o1 V3 t|----------|-----------------------------|------------------| & c* k7 Z! Q: K" Y: l) _| 实现方式 | 函数自调用 | 循环结构 | 4 z: e$ l. z, g' d+ a1 A8 Q| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | 4 e, ?' d8 L; K# v| 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 |; ^, z; m3 o( J ^- s
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |5 k0 c: A: W) l; K" m5 R
+ E' }' S$ @( N: y* N- h7 A 经典递归应用场景 k0 G. [3 J3 c; g' \" u1. 文件系统遍历(目录树结构)+ \4 k0 q* V) N
2. 快速排序/归并排序算法 u1 |5 U2 V: R% c
3. 汉诺塔问题7 z2 x# V. D \# x/ S
4. 二叉树遍历(前序/中序/后序) * J, @7 l! b2 K: r* G2 ~" R5. 生成所有可能的组合(回溯算法); ~0 b+ r, X4 U- o8 J; m& p7 w; Q
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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, 7 ^9 I) \" c+ D' t' @2 u% W我推理机的核心算法应该是二叉树遍历的变种。; ]7 K/ O8 G& k7 v; {% ?$ _3 m1 M
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: * ]* N7 q1 s5 \2 p/ vKey Idea of Recursion3 d& v' J- U) I+ G9 q9 q( W
* K: v- B, Y, sA recursive function solves a problem by: ) T8 A: A& c6 Y 4 y0 L/ m; _* `" h; J6 G Breaking the problem into smaller instances of the same problem. ) v3 n2 [ ]* Z+ l8 C 1 b2 I6 i* o; z Solving the smallest instance directly (base case).1 b' Q! x# }& z4 Q1 M% n
7 L* {$ l& h. {1 T( D7 a Combining the results of smaller instances to solve the larger problem.: g, z7 `8 Q; A7 [; V
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Components of a Recursive Function* b% Q$ D: s2 M& U0 |
" v& D- E6 J2 J5 P3 q0 t Base Case:0 l1 `3 N+ P$ P, F8 u
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This is the simplest, smallest instance of the problem that can be solved directly without further recursion. % g/ [* u' x' A# S & h2 C$ p! g" @ It acts as the stopping condition to prevent infinite recursion. 3 V: `! v" H+ X) d3 x. Z6 b2 I- n# y w
Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% Y# g4 j( p/ o# j \
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Recursive Case:5 q. u. C! E! O. q" J- B
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This is where the function calls itself with a smaller or simpler version of the problem.3 u* E: ?2 M7 [8 e0 P9 P
# i+ M, @3 p& c7 e& b8 y+ y Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). ( ^3 F) E, P3 @" D, a9 J5 N1 u% c/ D9 J! i# n. Y
Example: Factorial Calculation 0 n' Y9 n, ?# o# [ s3 s, p9 u3 w8 y1 q: |0 yThe 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:4 L. o! T4 u) W( `
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Base case: 0! = 14 A( O) f: h5 U5 i
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Recursive case: n! = n * (n-1)!2 A8 O; B' f( W/ j; X z% P
$ ?+ ^" k; m# E+ s9 {8 oHere’s how it looks in code (Python): 6 A$ j9 h9 L4 Y7 Ipython / J5 i4 a: h$ i 9 s& I- i: z1 z# D! G! a) P$ z 0 D, U/ {' T* G, M7 h& n0 {def factorial(n): 3 I, `; r5 o5 ^3 }" T # Base case' n; @2 S% B( N; e
if n == 0:: z8 k0 x; ~ D/ J
return 1 4 C4 A7 ^# v# c2 ^. g a # Recursive case. g0 [$ r; d3 ?3 E+ T
else:1 T3 W( H+ f/ w$ _2 Z
return n * factorial(n - 1) I7 G2 x" U7 L% u. @+ e) K6 {# N* F# |1 l, n* `
# Example usage * b9 W5 {7 \0 @8 y2 a& sprint(factorial(5)) # Output: 1203 A6 Z2 [+ S1 D5 i1 F! \
- g' N2 `: o. l7 U. z: eHow Recursion Works 2 o$ l2 W4 I8 w7 S- U! L ! M6 X" E" c( j" N6 r& v7 ?* z' x, M The function keeps calling itself with smaller inputs until it reaches the base case.: z# [1 G9 L+ b/ S j+ d8 c/ _* `4 y% t1 [
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Once the base case is reached, the function starts returning values back up the call stack. % `' C. B9 ~+ k' V' @! o! @/ M" | . N ~2 F+ Q$ v" e/ {* H These returned values are combined to produce the final result.# r* N# H: D6 s0 s
* a. l" D$ |- Y8 l. h& k7 w9 t# JFor factorial(5):! B6 n3 C. Y0 M& r6 N
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factorial(5) = 5 * factorial(4) O1 X0 l0 w" \: T, Ufactorial(4) = 4 * factorial(3) F6 C" N( T7 [" \4 e+ m3 h' r0 yfactorial(3) = 3 * factorial(2) 4 N: @ E3 N3 e4 o h5 sfactorial(2) = 2 * factorial(1)- ?) n1 u! b8 x2 `: z
factorial(1) = 1 * factorial(0)8 c& F% A. ?5 y: Z
factorial(0) = 1 # Base case 1 q/ k* w4 \8 S% F. ^$ [" }# M4 T% P
Then, the results are combined:# V/ P% W. E% R) d- V5 J. c
/ j- b) {. [; G, H o- \ Y 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).9 W5 }% }" p; \' v6 H2 b
6 g- \9 ~9 V3 Z7 K Readability: Recursive code can be more readable and concise compared to iterative solutions.0 r' c; p7 X o1 D1 V5 `
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Disadvantages of Recursion, {3 ], l. f& J
: v2 `7 P; z7 v1 l4 I( G 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., @) F! O; A c$ _+ t' I& J
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Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). 1 i! R3 P3 A T4 K * z3 s: B2 }/ i& t$ D- @When to Use Recursion , x6 @+ l( M$ m0 u. |9 f 1 ?0 M$ R$ U; _8 B1 A0 e! K) ~( j Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). 8 o& a2 c0 E. e: ?# \ v ; Z- W' U5 n o) ~1 {$ y2 n4 O3 M Problems with a clear base case and recursive case. ?: P! `. Q5 b Q: h6 d1 B+ ?1 s1 z, @
Example: Fibonacci Sequence6 _# ]% [6 y3 a8 R
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: 4 o* l3 d- l; n. _6 h , r! @( Z0 t; n9 R6 X8 M. U* P Base case: fib(0) = 0, fib(1) = 1 $ b6 G H* _1 ~- z. F0 v m6 a6 |0 Y0 h. d8 Z Recursive case: fib(n) = fib(n-1) + fib(n-2)) |% L5 }' P6 R6 z( J5 ?
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; f Q2 y% o% Z/ r: u8 S( @ ) G b- U$ @7 ]2 O9 ydef fibonacci(n):2 B+ B. ]8 k* l1 t- s6 L; p
# Base cases# c# ]* a9 m) S# w" }# s
if n == 0:% p# C8 l6 T+ m6 ?, u) y/ z
return 0 3 B0 v/ P1 ]2 L7 x3 Q" W elif n == 1:$ [+ v# `7 T9 M# \- `/ n& b
return 1( @( v$ O5 T9 d0 ]7 K& w
# Recursive case8 T& y8 D5 @, U5 g3 l T ?
else:2 M9 f! ~0 {) Y1 c3 u7 L
return fibonacci(n - 1) + fibonacci(n - 2) # A' X) ]% ^/ T& w& F5 x5 i: a) }) R8 f9 t
# Example usage$ M" x8 Z! b# ~0 |
print(fibonacci(6)) # Output: 8 ! v0 N* S2 L' `. R* `+ y! B9 }. ^/ H; g& h* V/ \
Tail Recursion0 s+ R) U2 u0 K, o
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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).! m" k/ a! D" o
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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。