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 $ ~$ [2 i% _; T7 _3 n/ a2 P9 b
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解释的不错 $ J J6 b- }& ~3 ?; v. E % V8 p# \" q$ f/ N& U9 j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。 , H; ?4 U; e- Y" }6 ~) u9 N; A' s' l
关键要素 1 l/ I1 T* q$ X" @1. **基线条件(Base Case)** * x+ p1 k; U ]# A# y0 j - 递归终止的条件,防止无限循环 $ ^" b; k/ C0 ^! k% ^2 d - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1 % `7 C$ \/ A$ f$ y7 ~) J ! D* r$ V$ B' i. Q; ]" G \9 S2. **递归条件(Recursive Case)**5 ?4 q% _( X. Q4 v
- 将原问题分解为更小的子问题 + i# @& H8 M4 \9 }; V - 例如:n! = n × (n-1)!& @5 W6 u5 L, s1 r* @* q& e1 C
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经典示例:计算阶乘% ~5 M p9 L" c5 u" X: V
python% V: i3 {. u3 d
def factorial(n):; k+ ?; W" O) V" U) O0 e* r Z. u
if n == 0: # 基线条件 # d4 S# L* `* m4 u5 U1 H return 1 ( f6 F, a* I) T5 d9 A0 i else: # 递归条件 + P) k2 s, H9 F/ i' o return n * factorial(n-1)& l7 k3 C" K: w- ?7 W/ R
执行过程(以计算 3! 为例): 1 p2 f G/ R5 w6 I6 ?factorial(3)0 l' E' O$ z1 Y" P
3 * factorial(2) ; e8 c7 b" N% J) W# d' G" n9 j3 * (2 * factorial(1)); ]; e2 V7 e0 B8 Y; L
3 * (2 * (1 * factorial(0))) 4 K5 @4 e7 N( J) z( o* _3 * (2 * (1 * 1)) = 6 & V3 r f( u; b. y * E8 O% t; Q! }" j% ? 递归思维要点 - y) r/ [5 z4 n0 }6 |1. **信任递归**:假设子问题已经解决,专注当前层逻辑# m: W J6 b7 v) i* q
2. **栈结构**:每次调用都会创建新的栈帧(内存空间) I2 ?" O: _7 F$ Y' f1 e3. **递推过程**:不断向下分解问题(递) 8 M" a, g2 x3 V( i' A0 v4. **回溯过程**:组合子问题结果返回(归) k! v- g1 h7 z1 b
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注意事项 5 L. N2 O4 A8 m2 E( R* j" _必须要有终止条件8 A5 y$ ~4 }/ J8 s+ ` u- o$ F
递归深度过大可能导致栈溢出(Python默认递归深度约1000层)! `6 F7 T M2 W d1 X* e, o
某些问题用递归更直观(如树遍历),但效率可能不如迭代 ( H4 o" R) P; w6 ^/ f尾递归优化可以提升效率(但Python不支持) ) b5 A8 t7 D4 m* Q! h0 g4 ]3 J 0 W9 n( e' a. U \+ Q k 递归 vs 迭代 # [2 e5 l8 R% s c5 {) j- \| | 递归 | 迭代 |9 e; [0 a1 G2 H. _9 V% \. ]; p
|----------|-----------------------------|------------------|9 x/ W. Z* ]. {/ z- x8 U5 B3 k1 e
| 实现方式 | 函数自调用 | 循环结构 | ! p U; H/ O6 J| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 |8 |8 M& r0 H; c7 z
| 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 | ! L8 L7 i3 K# d8 [7 r9 }$ Z| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |/ N6 G7 F1 k& P
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经典递归应用场景+ N; z; B% X: {& z5 H
1. 文件系统遍历(目录树结构)' f; h- {% \7 B1 f- X( V+ d1 C
2. 快速排序/归并排序算法 " H, n& d) ?% L9 M: M, L3. 汉诺塔问题 ; r8 I1 N, b5 f) K. A8 {+ G" [2 p4. 二叉树遍历(前序/中序/后序)6 I. ^0 E u2 u6 I. _# ~
5. 生成所有可能的组合(回溯算法)* O' P4 o- l4 q: O1 o# ]
/ S. K0 g( w# _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, 1 K% G5 e; a+ L5 G. ?+ u# W我推理机的核心算法应该是二叉树遍历的变种。 % H+ |1 w9 h' q' i3 @7 e3 T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:6 {/ i" Y' \$ j4 y, L
Key Idea of Recursion0 y0 T5 H) Z# D% a# w
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A recursive function solves a problem by: $ V' {: P6 G2 c& o4 B: f: q$ z% q8 L& n2 ?1 {8 G. G) f* Z$ M
Breaking the problem into smaller instances of the same problem." }& D8 N* |2 b# E0 `) G
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Solving the smallest instance directly (base case). 4 T2 _2 ~" }6 _) }, I+ I5 r( X |3 t; t+ e- J9 h* e9 [: P2 O
Combining the results of smaller instances to solve the larger problem.1 l; @5 [! V- s# F8 c% ~
7 n# L% x K' F4 f) dComponents of a Recursive Function , a% F9 k) Q* g7 W6 Z$ k1 E0 \; q) p. g4 f
Base Case:9 {8 s8 q2 I, G! k, T7 K) j
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This is the simplest, smallest instance of the problem that can be solved directly without further recursion. ) u' A# ], l; @0 u) I/ R 7 U2 U3 F( {6 @% s! e4 o It acts as the stopping condition to prevent infinite recursion.1 p9 T' v4 `$ x0 U/ } C t% E
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Example: In calculating the factorial of a number, the base case is factorial(0) = 1.% N' A7 M( n$ ^: h
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Recursive Case:: K* K: e. ^" m1 S" C7 S
' }4 R- \# T) `% }) j This is where the function calls itself with a smaller or simpler version of the problem. ' n7 z1 X1 ~) F4 x+ n' ?+ Y$ I- U1 M) h
Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1). - J" F B$ Q7 d. }2 D) ~ 7 C7 R! s, [0 zExample: Factorial Calculation |; x' r- N$ Z0 Z' o: P8 P: j5 [ 5 e3 I0 z o/ S7 T6 vThe 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 {7 D7 l" U! S! ~: [1 |
+ b! B6 w7 i) q) h' f Base case: 0! = 1 : R; e7 B& U) ~& a& s, {, I2 b ! ~4 E4 W' e: o! a Recursive case: n! = n * (n-1)!. @' r% m3 t$ e! M% z+ W! T
. L. L, d* d. iHere’s how it looks in code (Python): 1 v a" b( T" v& d3 Fpython ) G5 t/ n% b6 u3 s+ @, }; B4 n* X 9 j0 Z- W9 u( d 7 t+ o: h# d' Y' F+ E' k& idef factorial(n):& _2 b8 K' n9 \8 R1 p+ ?
# Base case " D# } y, Q; ^* t if n == 0: * V2 ?* x6 K" P" o return 1! a' Z2 \& g U, s' ]
# Recursive case- ^# @9 P2 _3 a! l: M- u/ M$ s
else: 7 L& M# |6 ^" r$ O( g( y return n * factorial(n - 1) ( z: |2 R$ ]+ j. [' l: H* Y/ {9 e1 y7 o9 h3 C, r' V- v! c' X6 q
# Example usage9 H% N* S( M6 O2 [2 p d, U
print(factorial(5)) # Output: 120 , n+ h7 m+ W: U$ Z% {. T + B: T, b3 o; z4 }7 i- MHow Recursion Works5 t: ?/ \. i3 g2 u S( }' H
6 i8 ?9 G1 `$ z$ c( B0 c The function keeps calling itself with smaller inputs until it reaches the base case. ' ~, k8 B# T: \2 _/ E3 R 6 \; I& \$ r+ S, X Once the base case is reached, the function starts returning values back up the call stack.' Z6 j' w6 m3 \7 g/ z3 |: U
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These returned values are combined to produce the final result.' v; v: ^: }4 E V
5 x% u [* V$ v2 KThen, the results are combined:# }( Q3 C8 q/ M Z5 f
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factorial(1) = 1 * 1 = 1 4 U2 L+ ]% d- h% R* Q3 Ufactorial(2) = 2 * 1 = 2. |8 U6 Z4 G+ o" F: `# ]1 M
factorial(3) = 3 * 2 = 6 7 R4 v+ u6 s$ L' c$ S, B# k' cfactorial(4) = 4 * 6 = 24 % H9 I; }8 |/ b2 j2 H* Yfactorial(5) = 5 * 24 = 120 : `8 \; Q; w* |) ?: c7 Z" u9 q( x3 ?
Advantages of Recursion % i# j) E6 U7 R" a" A. d0 n0 U; s+ i. D$ S. `: S4 |' C
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). 2 ]; w9 x" Y: b4 V) ?! I0 N: T7 X7 Z! x* M
Readability: Recursive code can be more readable and concise compared to iterative solutions.* |2 x9 b2 y' \( C
4 Q, A+ \# m/ Z* `! U+ JDisadvantages of Recursion " S0 @4 T( L, }" Z+ p9 w. H6 s
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., T% s0 Q# \ N
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Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).4 R* k# ]2 D. C( x
( c* k- M1 C8 V5 hWhen to Use Recursion 0 c; K' T( }( k9 s8 Y , Q' g4 P" s2 E- e1 u Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). 0 ~) d: K: Q, r. p% `) R& {- ^! L' j0 @% @! `3 m
Problems with a clear base case and recursive case.* _0 l7 ]% A# i+ h. v
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Example: Fibonacci Sequence " M2 k* T1 }) A3 Z 3 S4 X/ O9 K4 i/ Q" o' [! qThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:, H* N- h3 _8 M" L9 S2 J
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Base case: fib(0) = 0, fib(1) = 1 & j+ ]1 N. V) Y# K5 ~8 v. ~1 [/ ~# F# d+ C8 c8 B
Recursive case: fib(n) = fib(n-1) + fib(n-2)! E" j4 H \+ r
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python 4 w4 m+ O# I, {( l; z/ B1 M% } p: X( V
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def fibonacci(n): 5 i/ W) s( s/ ^, k3 G! ` # Base cases ; ?9 f4 R% o. {; b Y/ I0 c! I if n == 0: ) `9 b& l: d2 o2 [6 J' j return 0( s" n+ K6 I @
elif n == 1: * u$ ~. s1 R! O. L0 X x return 1 $ I( q- }9 v7 n9 O O1 M" O. C # Recursive case 2 a" b6 F1 e& W$ s0 @+ s, T else:' U" t" Y! N8 `, @+ e4 B9 m2 ?% N9 X: h
return fibonacci(n - 1) + fibonacci(n - 2)8 i" E+ D" P) Z' R1 F
3 h" t$ S: M+ H5 l# Example usage 4 y9 o+ X W* F" |! ^7 ]0 Sprint(fibonacci(6)) # Output: 8; ]. }! G2 f6 ?% G
1 \* }: @& b' J0 @; ?) eTail Recursion5 m: _% @. F7 b: c" X/ m/ o. \" A
$ L: @! u6 [2 uTail 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 J& H2 S+ [: |' M6 E, m& j, g+ x" z
1 J4 L3 {6 Z s* cIn 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。